Novel Aspects of Superconductivity

Condluding Discussion: What did we learn, and where do we go from here?

2007-09-01T13:15:00.001-06:00

Dan Sheehy: Superfluidity in "magnetized" fermionic atomic gases

2007-08-31T14:39:00.000-06:00

Dan Sheehy presented in his talk work on cold atomic gases done in collaboration with L. Radzihovsky (PRL '06, Ann. Phys. '07, PRB '07).

Dan started by reviewing recent results on the BEC-BCS crossover in fermionic gases. Typical experiments are done with Li6 or K40 at temperature of the order 10-100nK (cold) and densities of the order 10^10-10^13/cm^3 (dilute). Two different hyperfine Zeeman states
("spin up" and "spin down") are trapped in a harmonic trap and their interaction is tuned using a Feshbach resonance. In this way strong attractive interactions can be achieved.

QUESTION: Is it only possible to trap two different species?
ANSWER: In principle it's possible to trap more different species and one would expect interesting physics as a result.

QUESTION: How does one measure temperature?
ANSWER: Temperatures aren't very well known experimentally. In the non-interacting case the spatial profile of the atomic cloud can be fit to a Fermi function and thus one can extract the temperature. In the presence of interactions this is, however, not possible.

The attraction between fermions can be considered point-like:

H_int = g(B) \int d^3r psi_up^dagger(r) psi_down^dagger(r) psi_down(r) psi_up(r),

where g(B) is tunable by an external B field. The scattering length varies as

a_S ~ - 1/(B-B_0).

The divergence at B=B_0 signals the appearance of a bound state. The change in sign of a_S does not mean that the interactions switch from attractive to repulsive. At B > B_0 the attraction between fermions is weak (BCS regime). At B <> n_down. He presented a phase diagram as a function of the inverse scattering length and the polarization

P = (N_up - N_down)/(N_up + N_down).

By contrast to conventional condensed matter experiments where one fixes the magnetic field, in cold atomic gases the number of spin up and spin down particles can be fixed, i.e., one works at fixed P.

In the BCS limit it is known that a Zeeman field kills superconductivity if h = h_c = Delta_0 / sqrt 2 (Clogston limit). The transition is first order. Therefore, at fixed P one finds phase
separation. The BCS state exists only for P = 0. At finite P, three different phases occur:
1. phase separation at small P
2. FFLO state in a small region of intermediate P and far enough away from the resonance
3. normal Pauli paramagnet at large P

Experimentally phase separation is indeed observed by the Rice & MIT groups: in the middle of the trap a condensate forms while the excess up spins accumulate towards the edges of the trap.

QUESTION: Is the normal region at the edge of the trap fully polarized?
ANSWER: No, it is only partially polarized according to the conditions for chemical equilibrium.

Far enough on the BEC side, instead of phase separation a magnetic superfluid is predicted consisting of molecules (singlets) mixed with the excess spin up fermions. This is to be compared to He3 - He4 mixtures. No experiments in the deep BEC limit testing this exist so far.

COMMENT: For non-s-wave pairing a quantum critical point exists between the BEC and the BCS regime.

QUESTION: What about the formation of quartets predicted by Nozieres et al?
ANSWER: There is no experimental evidence for that.

QUESTION: How can one measure the FFLO phase?
ANSWER: It has not been observed, yet, but it should be visible in the density profile.

QUESTION: In He3-He4 mixtures the solubility limit depends on the nature of interactions. Is this true here as well?
ANSWER: Within mean field theory the transition to the polarized superfluid depends on the interaction strength. So far there are no results beyond mean field available.

M. Eschrig: "The pairing state near superconductor/half metal interfaces"

2007-08-31T09:39:00.000-06:00

In this talk, Matthias Eschrig discussed the modification of the pairing
state of a superconductor (SC) due to proximity with a magnetic material.

He began by reviewing the case of a ferromagnet (F) to SC junction, studied
by Buzdin in 1982. Due to the Fermi energy mismatch in the F, characterized
by the parameter J, one expects a split Fermi surface. This furthermore implies
pairing at a nonzero wavevector q, i.e., oscillations in the pairing phi(z)
as a function of the position z in the F region:

phi(z) ~ Exp[-z/xi1]*Exp[i z/xi2]

Where xi1 and xi2 are parameters that can be determined theoretically. xi1
decreases with increasing T, while xi2 increases with increasing T.

Matthias discussed two experiments verifying this picture, each with
a SC-F-SC function. The first (by Kontos et al) showed a transtion between 0 and Pi phases
of the junction (signalled by a vanishing of the critical Josephson current Ic)
with increasing width df of the F region and the second (by Ryazonov
et al) showed a Pi-to-0 transition with increasing T

At this point Dirk Morr pointed out that one can have a zero of Ic without
a transition between Pi and 0 states. But the location of the transition
agreed with theory.

Turning to the case of an interface between a SC and a half metal Ferromagnet,
the main topic, naively one expects no proximity effect. However, recent experiments
by Keizer et al, Nature 2006, on Josephson junctions with NbTiN SC
linked by Cr02 half metal, showed a large-distance Josephson effect.
An initial clue was that the Josephson effect was observed to be
very sensitive to surface properties. The experimentalists
observed hysteresis of the Fraunhofer diffraction pattern. After
subtracting the hysteresis, the pattern was shifted by Pi from
the usual case.

Matthias's work on this problem was published in 2003 in PRL and in 2006 on
cond-mat, and is based on the notion that the important physics occurs
at the interface.

The first effect to consider is spin mixing. Thus, one expects different
phase shifts of spin-up and spin-down fermions scattering at such an
interface, characterized by an angle theta. By itself, this leads to
singlet (S) - triplet (T) mixing, the magnitude of which is proportional
to Sin theta.

To understand the experiments, however, additional scattering properties
must be included. The additional properties included were surface scattering
at the interfaces that were assumed to have a local interface magnetization
m. The two relevant interface magnetizations, m1 and m2, can be labelled by
their angles alpha_i with respect to the magnetization M of the FM regime
and also by the angle between them. The resulting critical Josephson current
is sensitive to the angles alpha, and theta, while the shift in the
Fraunhofer diffraction pattern depends on the difference between the
local interface magnetizations m1 and m2. Future work will focus
on determining the precise physical mechanism behind the interface
magnetizations m1 and m2.

Dieter Belitz: Skyrmion Flux Lattices

2007-08-30T18:46:00.001-06:00

Dieter Belitz presented a theory for the skyrmion flux lattice in
triplet (p-wave) superconductors.

Dieter started out by noting that in singlet (s-wave)
superconductors, the superconducting order parameter possesses an
SO(2) symmetry, in which case the topological excitations are given
by (conventional) vortices. The energy per length of the vortex is
E_vortex=Phi^2 * ln(R)/\lambda^2, where R=lambda/xi, lambda
is the Kondo penetration depth, xi is the superconducting coherence
length, and phi is the flux quantum. In an applied magnetic field,
the vortices form an Abrikosov flux lattice with one flux quantum
per vortex.

Dieter then pointed out that in a triplet superconductor, the spin
sector forms an SO(3) subgroup, which allows two different types of
topological excitations: vortices and skyrmions.

Dirk Morr asked whether Dieter considers a particular spin state, as
represented by the d-vector in a triplet superconductor, and Dieter
replied that he consider the non-unitary spin state described by
d=(1,i,0), representing |up,up> - pairing. Diete then drew a picture
of a skyrmion, in which the spin part of the superconducting order
parameter rotates from |up,up> to |down,down> as one moves radially
outward from the center of the skyrmion. Diete noted that there is
no singularity at the center of the skyrmion, in contrast to a
vortex. Dieter showed that the energy per unit length of the
skyrmion is E_s= Phi^2 /\lambda^2 which is smaller than the vortex
energy E_vortex for R>>1 (Dieter noted in passing that this result
was obtained in a purely classical theory). The skyrmion lattice
contains two flux quanta per skyrmion.

Dieter then described a perturbative result (in 1/R) for the energy
of a skyrmion as a function of the skyrmion radius, which is given
by E(R)= Phi^2 /\lambda^2 *(1 + 1/R - ln(R)/R^2 - 1/R^2 + ...) (R
is given in units of lambda). Dieter noted that this result agrees
very well with a numerical solution of the problem by Rosenstein.
The resulting skyrmion potential is then given by V ~ 1/R, in
contrast to the vortex potential that is given by V ~ exp(-R)/R.

This long-range interaction leads to some distinct differences betweenvortex
flux lattices and skyrmion flux lattices. In particular, they have qualitatively
different melting curves. Dieter sketched a phase diagram for a vortex flux
lattice, which always melts if one gets sufficiently close to the lower critical
field H_c1, and one for a skyrmion flux lattice, which never melts close to H_c1.

Thilo Kopp: h/e periodicity in loops of nodal superconductors.

2007-08-30T16:51:00.000-06:00

Thilo discussed the periodicity of the ground-state energy and
the supercurrent as a function of the magnetic flux threading
a superconducting ring. He presented a joint work by F. Loder,
A. Kampf, J. Mannhart, C.W. Schneider, Yu.S. Barash and himself.
He first reviewed what is historically
known: flux quantization, periodicity of the ground-state energy,
and of the supercurrent in units of phi=h/2e. He recalled for us
that the states corresponding to q times phi, where q is an even
integer (London states) are related by a gauge transformation.
However, there is no such a relation between states corresponding
to q even and odd. The degeneracy between q even and odd is lifted
in s-wave superconductors when the diameter of the ring is smaller
than the coherence length of the system, since in this case, the
discrete nature of the electronic states becomes relevant showing
in general differences between half-integer and integer flux
quanta. The aim of the work by Thilo and collaborators was to look
for a mesoscopic superconducting system where h/e periodicities
become observable.

The theoretical work consisted in the numerical solution of the
Bogoliubov-de Gennes equations for a BCS-Hamiltonian with a Peierls
phase factor corresponding to the coupling to a vector potential
for a magnetic field threading a 100x100 lattice through a 30x30 hole.
The superconducting order parameter was chosen to be a d-wave one.
The idea is that while the main contribution to the supercurrents
comes from the states closest to the Fermi energy (E_F=0), most of the
condensation energy comes from the lobes. In such a way a d-wave
superconductor is protected from reaching the critical value of
the superfluid velocity by the Doppler shift, in contrast to s-wave
superconductors.

Due to the nodal character of the order parameter, discrete states
very close to E=0 are present. Thilo discussed first the evolution
of the eigenenergies as a function of flux close to q=0. As the
magnetic flux is increased, supercurrents are present and the discrete
states of the finite system shift accordingly (e.g. the states closest
to zero increase their energy). However, an abrupt change takes place
very close to h/4e (where the parabolas in the infinite case cross).
From there on, one enters the regime with q=1. Both the ground-state
energy and the supercurrent show an h/e periodicity, and the change
from states with increasing q takes place at odd integer multiples
of h/4e. At such points, the condensate reconstructs.

The finite size effects discussed by Thilo vanish as 1/R, where R
is the radius of the ring. Claudio Castellani asked whether an estimate
can be given for the sizes required to see the effect. Thilo said
this should be the case of rings in the micrometer range where a
percent effect should be still observable. Zlatko Tesanovic pointed
out that in an s-wave superconductor presumably a length scale should
exist, where the effect essentially vanishes.

Ilya Vekhter: "Probing anisotropic superconductivity with magnetic field"

2007-08-30T16:48:00.000-06:00

Ilya presented his work done with Anton Vorontsov on extracting the nodal structure of an order parameter by applying a magnetic field in varying directions and predicting what the zero-bias density of states averaged over the vortex unit cell looks like as a function of field direction. Starting with an experimental overview, in particular specific heat and heat conductivity measurements were singled out for testing nodal quasiparticles. Based on work by Volovik '93, the local density was calculated in the Doppler-shift approximation and averaged over the vortex unit cell. Ilya pointed out that although many experiments contain information about the presence or absence of nodes, not so many tell where in the Brillouin zone the nodes are situated. In order to tackle this problem, Ilya presented studies where a magnetic field is rotated and the averaged zero-bias DOS is monitored.

Question by Dirk Morr: is the field in plane? Answer by Ilya: we do not assume any Josephson vortex structures to be present, all vortices are usual Abrikosov vortices.

Ilya then draw a picture with an oscillating behavior of the averaged N(w=0) as a function of field angle, with minima where the field is parallel to the node. Ilya then pointed out contradicting experimental findings for CeCoIn_5, with nodes either consistent with a d_x^2-y^2 or a d_xy order parameter. Ilya suggested that this can be explained by a more careful study in the high-field region. Solving Eilenberger-Larkin-Ovchinnikov equations employing the Brandt-Pesch-Teward approximation again the averaged zero-bias DOS was calculated, this time also covering the high field region. Ilya summarized the results of these calculations in a H-T phase diagram, with a low-T low-field region showing minima in N(w=0) as tested by C/T when the field points in nodal direction, another not so interesting region near T_c, and an 'inverted' region in the rest of the phase diagram where a maximum occurs in C/T as function of field direction when the field points in nodal direction.

Q: Hartmut Monien asked if subdominant order parameters in the vortex core regions would change the results. Ilya answered that he does not think that this happenes in the materials that were studied.

Q: Ben Powell was interested in the role of pancake vortices, however this was beyond the model Ilya was considering.

Alejandro Muramatsu: Massive CP1 theory for doped antiferromagnets

2007-08-30T15:11:00.000-06:00

In his talk, Alejandro discussed field-theory oriented approach to weakly doped antiferromagnets.
He began his talk by briefly reviewing the algebra for Hubbard operators, along the lines first discussed by Wiegmann back in 1988. The algebra for Hubbard operators contains both commutations and anticommutations, and to reproduce it one needs to express Hubbard operators in terms of fermionic and bosonic fields, subjects to three local constraints (one of them is a constraint on the length of the bosonic field). Using this representation, Alejandro re-expressed t-J Hamiltonian in terms of these two fields. He then considered the limit of small fermion (hole) density, integrated out fermions, and used CP^1 representation for the bosonic field in terms of z-spinons (z and {\bar z}. He then arrived at the CP^1 action for the z-fields in the form

S = \int d\tau d^2 x \frac{1}{g_\mu} [\partial_\mu {\bar z} \partial_\mu { z} + \gamma_\mu ({\bar z \partial_\mu z)^2]

where g and \gamma are expressed in terms of the parameters of the t-J model. The quartic term may be decoupled using the gauge field.

Alejandro argued that at zero doing, \gamma_\mu =1, in which case the gauge field is massless, spinons are confined, and the system has a critical point (at some g), which belongs to O(3) universality class. At a finite doping, \gamma_mu is smaller than one, and the gauge field acquires a mass. In this situation, spin configuration becomes incommensurate, spinons are deconfined. In the limit \gamma =0, the system has another critical point (at some other g), which belongs to O(4) universality class. He presented the full phase diagram and discussed RG flow.

In the discussion after the talk, Kim and Castellani both asked questions about fermionic damping.
Muramatsu answered that the Landau damping is not present in his z=1 theory.

Hae-Young Kee: "Electronic Nematic Fluid and Metamagnetic transition in Sr3Ru2O7"

2007-08-30T10:01:00.001-06:00

Hae-Young discussed a theory of electronic nematic order (also known as Pommeranchuk distortions), which, she argued rather convincingly, explains the metamagnetic transitions in the bilayer strontium ruthenate (the title compound).

Hae-Young began by pointing out that the monolayer ruthenate is isostructural to La2CuO4. She then moved on to a discussion of the experimental results of Andy Mackenzie’s group on this material. When a field is applied along the c-axis at low temperature the resistivity is more or less constant until a critical field, Hc1, where it rather rapidly doubles in size. As the field is increases above Hc1 the resistivity traces out a dome before return to its low initial value for H>Hc2. Peaks in the imaginary part of the susceptibility and jumps in the magnetisation are observed at both Hc1 and Hc2. From this data Hae-Young sketched a H-T phase diagram with first order transitions at Hc1 and Hc2 and a dome of second order transitions connecting them.

Throughout this discussion John Mydosh wanted to know why no specific heat experiments had been performed. John pointed out that this is important to rule out the possibility of a spin glass. Hae-Young stressed that this is an itinerant system and so one should not expect a spin glass. These two then debated the gradient of the first order line with Hae-Young saying that it was vertical and John arguing that it concave.

Hae-Young then wrote down a Hamiltonian with hopping terms, as well as Hubbard U and V and a correlated hopping term, tc, (which is where most of the physics appears to arise from). Andrey Chubukov asked about the relative magnitude of these terms and Hae-Young said that the size of U was not very important, but that tc>V. She then wrote down her order parameter, which describes a d-wave Fermi surface distortion. Hae-Young spared us the details of her calculations and simply sketched the dependence of the order parameter on the chemical potential. This looked remarkably like the experimentally derived phase diagram she had sketched earlier, except for the fact that the x-axis was chemical potential rather than field. However, Hae-Young quickly pointed out that the Zeeman term of the field acts exactly like a spin dependant chemical potential and so the down spins may undergo a nematic transition if they hit a van Hove singularity while the up spins are simply spectators.

Hae-Young then argued that the formation of domains is responsible for the anomalies seen in the resistivity. (The two domains correspond to elongating the Fermi surface in either the x or y directions.) Then she moved on to discuss the experimental observation that tilting the field removes the resistance anomaly. She argued that this is because of the bilayer structure, which allows for circulating interlayer currents, which pick out one domain over the other. Claudio Castelani then asked whether one could use this hysteresis effect as a test of the theory. Hae-Young thought that one could but that the experiment had not been performed.

In question time John asked whether de Haas-van Alphen experiments had been performed as these could look directly for the Fermi surface distortion. Hae-Young said that they had and that although nice quantum oscillations could be seen below Hc1 and above Hc2 nothing could be seen in the intermediate region. She interpreted this as evidence for domains.

Andrey asked if there is direct evidence for the van Hove singularity that is required for her thesis. Hae-Young said that Takagi et al. had seen evidence of this in STM experiments.

Your humble blogger asked what was known about the effect of disorder on Hae-Young’s nematic phase, as the experimental anomalies appear to be strongly suppressed by disorder. Hae-Young replied that although there are not any definitive calculations, arguments have been supplied by Kivelson, Fradkin and others that suggest that the nematic phase is suppressed by disorder.

Claudio Castellani: Superconductivity near a multiband Mott transition

2007-08-29T23:11:00.001-06:00

Claudio Castellani introduced a DMFT toy model, motivated by the
fullerenes in order to describe the emergence of superconductivity
near a multiband Mott transition.

Claudio began his talk by briefly reviewing the effects of strong
electronic correlations on superconductivity. He pointed out that
there are two opposing effects. First, correlations reduce he
spectral weight of the electronic states (with quasi particle weight
Z<1) and hence reduce the electronic band width from W to Z*W. This
in turn leads to an increase in the density of states from rho to
rho/Z, which should give rise to an increase in Tc. On the other
hand, the attractive pairing potential, V, is reduced to V * Z^2,
which implies a reduction in Tc. Hence, in order to increase Tc
through strong correlations, it is necessary to increase the density
of states without a reduction in the pairing potential.

Claudio then introduced a 2-band Hubbard model with interorbital J,
whose ground state possesses 2 electrons per site (Claudio also
mentioned its relation to the 2 orbital Kondo-model). Since this J
gives rise to the formation of on-site singlets, this model allows
one to study superconductivity in an RVB environment. However, the
symmetry of the resulting superconducting state is s-wave, and hence
this is not a model for the cuprate superconductors, but is likely
more applicable to the fullerenes. Claudio studied this model by
using dynamical mean-field theory (DMFT).

Claudio identified a quantum critical point of the model which
separates a Fermi-liquid regime from a pseudo-gap phase. In the
vicinity of the QCP, the superconducting Tc is enhanced, and the
physical behavior of the system is determined by two energy scales,
that of the pseudogap and that of the SC gap. For large J, Claudio
found the uusual Migdal-Eliashberg type of reduction of Tc, while
for small J, the superconducting Tc is enhanced by the Coulomb U.
Claudio pointed out that there is a Drude weight gain in the SC
state, but that in the metallic state, the Drude weight goes to zero
before the QCP is reached. Claudio identified the pseudogap state as
an unstable metallic phase.

Last, Claudio raised the question of whether superconductivity is an
instability of the pseudogap, or whether these two have no relations
at all. He stated that the processes leading ot the pseudo gap are
not competing with pairing but with coherence, in analogy with
non-pairbreaking impurities in conventional superconductors.

Thilo Kopp asked whether other interactions (besides the
interorbital J) are included in this model, which Claudio confirmed.

Catherine Pepin asked how one can make any statements about the
Kondo physics of this model within single-site DMFT. Claudio
answered that this is possible since the model is a 2 orbital Kondo
model, in which the QCP separates the Kondo-screened phase from the
unscreened phase.

Dirk Morr: "Impurities, collective modes, and magnetic droplets in the cuprate superconductors"

2007-08-29T19:14:00.000-06:00

Dirk Morr noted that there has been much interest recently in analyzing the spectroscopic signatures of collective modes in superconductors. He pointed out that if a collective mode is close to critical, pinning it on impurities creates a local droplet of locally ordered state, and cited Alloul/Bobroff NMR experiments in Ni doped high-Tc as an example. Dirk therefore suggested that STM spectra obtained near impurity sites depend on the type of the collective mode, and have the potential to distinguish between different types of short-range order.

He considered two examples of such droplets: small-q charge density wave order, and the spin-density wave order at Q=(pi.pi). The interaction with the conduction electrons is local, and the question Dirk asked whether a spin droplet looks different from a charge droplet when viewed from an STM tip. He showed the results of both T-matrix and Bogoliubov-de Gennes calculations for the spectra on the impurity site. An important point is that the local value of the spin/charge polarization that enters the calculations of the density of states is proportional to the static part of the spin/charge susceptibility, and therefore one may read off spatial dependence of the susceptibility.

The spectra for the two cases are different. The charge droplet is essentially an extended potential impurity, the potential varies slowly, and therefore in this case there exists a resonance state inside a gap. In contrast, for the AFM SDW droplet the oscillatory spin polarization (on the scale of the lattice spacing) reduces the scattering, and leads to overall suppression of the density of states with no sign of the local resonant state on the impurity site. Dirk emphasized that these qualitative different effects of charge and spin droplets on the local density of states allow one to identify the nature of collective modes via STM experiments.

Dirk also pointed out that for the spin droplet spin-polarized tunneling will give unequivocally distinct results from the charge droplet, and discussed with Hartmut Monien what the time scale for such measurements may be.

Benjamin J. Powell: RVB Theory of Organic Superconductors

2007-08-29T18:24:00.001-06:00

Ben Powell gave a short survey of work that has been recently published by him and Ross H. McKenzie (PRL 94, 047004 (2005), PRL 98, 027005 (2007) and J. Phys.: Condens. Matter 18 R827 (2006)). A paper on similar grounds has been published by J.Y. Gan, Yan Chen, Z.B. Su, F.C. Zhang (PRL 94, 067005 (2005)). The work concerns the kappa- and beta phases of (BEDT-TTF)_2X, a layered organic superconductor. Here, X is an anion complex such as Cu[N(CN)_2]Cl. Ben discussed the pressure-temperature phase diagram which reveals an insulator -> metal transition for increasing pressure (the pressure is supposed to be related to t/U in a Hubbard-like model, see below). At low temperature, a transition from an antiferromagnetic insulator (AFI) to a superconducting state (SC) is seen. At higher temperature, the AFI is replaced by a paramagnetic insulator and the SC takes a transition to a Fermi liquid. At even higher temperature, a "bad metal" is observed. A pseudo gap phase (PG) seems to be present above the SC at the low pressure side.

John Mydosh: does the PG end at the "bad metal". --- Ben: Yes.

Ben then presented a sketch of log(T_c) versus log(lambda) with slope -3 which is still a puzzling observation (BJP and R.H: McKenzie, J. Phys.: Condens. Matter 16, L367 (2004)).

The organic molecules are placed on a unit cell whereby two of the molecules (a pair) takes a lattice site. The transfer energy within each pair is the largest energy scale so that in the kappa and beta phases, they present a dimer. Then there is hopping t between adjacent dimers and a hopping parameter t' between one pair of diagonal dimers (the second diagonal hopping is much smalller and may be neglected. In the kappa phase t'/t ≤ 1 and in the beta phase t'/t > 1. The band structure may be described by a half-filled tight-binding model, with each site representing a dimer. For his modelling, Ben proposes a Hubbard-Heisenberg model with parameter space (t,t',J,J',U). A RVB-evaluation with a projected BCS ground state is set up.

Hartmut Monien: Doping dependence? --- Ben: The system stays basically undoped. The calculation is therefore at half-filling.

The evaluation of the Z-factor (which is 8(1-2d)d, where d is the average number of doubly occupied sites) displays a first order Mott transition with increasing U/t. With increasing t'/t, the transition is shifted towards larger U/t. The calculated phase diagram (U/t versus t'/t) is then as follows: For large values of U/t a Mott insulator is established. The peak in the fluctuations switches from wave vector (pi,pi) at small t'/t to some incommensurate value (q,q) close to t'/t=1 to (pi/2, pi/2) at larger t'/t. For sufficiently small U/t various superconducting phases are identified. For small t'/t <> 1 a d_xy -s phase is found. They are representations of the c_2v symmetry. For t'/t equal to 1 the symmetry group is c_6v and a d+id phase is expected. Numerical work and GL expansion suggests that it is stabilized in a finite interval. Finally, Ben expects that, with a lowering of the symmetry, a splitting of the SC transition may be observed for t'/t close to 1. First a d-wave state is stabilized and then, with lower temperature the d+id state. As this parameter regime is realized for the organic systems with X=[Cu(CN)_3], Ben proposes to search for such a phase transition scenario, for example, measure the state with broken time reversal symmetry with muSR.

John Mydosh commented: The anomalous Nernst effect is observed in the SC at sufficiently low pressure but is lost when you move to higher pressure (that is, also to different compounds).

Claudio Castellani asked: Is there a coupling to the lattice observed with the phase transitions?
Ben: A jump in the lattice constants has not been clearly observed. However in Raman frequency shifts of the modes have been seen.

Claudio Castellani: Comparison with the cuprates? Ben: you would have to introduce a new axis t/U in the phase diagram, pointing vertically from the undoped system.

Brad Marston: "Do gapless spin liquids exist in 2D ?"

2007-08-29T17:19:00.001-06:00

Brad Marston started by saying that the topic of his talk is a controversial one. He said he will present some works done in collaboration with Ookie Ma and Arun Paramekanti.

The first thing he mentioned was the Matt Hasting's theorem for two dimensional spin systems, namely a generalization of the Lieb-Shulz-Matthis theorem in 1D. The theorem applies to the systems with half-odd-integer spin per unit cell and it says the ground state of the system can only be one of two types.

1) When there is an excitation gap about the ground state, the ground state has to be degenerate. The examples are dimerized states and Z2 spin liquids.

2) If there is no gap, then the ground state is a U(1) spin liquid.

Thus the non-generate ground state with an excitation gap is not possible.

Brad mentioned there are some interesting materials with half-odd-integer spins per unit cell, where possible signatures of spin liquid phases may have been discovered.

There examples are as follows.

1) Cs2 Cu Cl4 (Radu Coldea's neutron scattering experiments)

Cu spin-1/2 moments reside on an anisotropic triangular lattice. The lattice can be regarded as a square lattice with an additional strong diagonal bond only in one diagonal direction or a coupled spin-chain system with the chains along the diagonal direction. The exchange coupling J along the diagonal or chains is three times stronger than the inter-chain J'. DM-interaction also exists and it is about J/10; this leads to highly anisotropic response to external magnetic field.

At this point, Andrey Chubukov asked how people know about the magnitudes of J and J'. Brad said that one can go to the ferromagnetic state by applying external magnetic field, look at the
magnon spectra, and get the exchange couplings. John Mydosh asked whether this problem is related to the BEC of spin excitations. Bard replied that it may be so in the high field limit.

Brad then mentioned that the ground state below T < 1K has a magnetic long range order. At intermediate temperatures, neutron scattering sees continuum of excitations that could be due to deconfined spinons of a spin liquid phase (other interpretation may also be possible).

2) \kappa-(ET)2 Cu2 (CN)3 (Kanoda's experiment)

Effective spin-1/2 moments reside on almost isotropic triangular lattice. Spin-liquid-like behavior is seen in the insulating phase near the Metal-Insulator transition. Charge fluctuation may be important and lead to substantial ring-exchange type contributions. No long range order is seen down to 20mK (NMR). There are anomalies in thermodynamic quantities around 5K.

3) Zn Cu3 (OH)6 Cl2 (Herbertsmithties, Young Lee's experiments and many others)

Cu spin-1/2 moments reside on the isotropic Kagome lattice. No order is seen down to 50mK (NMR, \muSR). Power law in T is seen in the specific heat.

Then Bard switched the gear and started the discussion on possible instabilities of gapless spin liquid phases. This is mainly because there have been some proposals advocating that Herbertsmithites may be an example of 2D gapless spin liquid phases. He wants to see whether this really works and there is any alternative ground state or not.

He listed the following instabilities or other possible ground states.

1) Long range spin order
2) Various forms of dimerization
3) More exotic candidates; spin-nematic, time-reversal-symmetry breaking, charge-conjugate symmetry in the case of the SU(N) magnets.
4) Spin liquid with an extended Fermi surface.
There may be spin-triplet (V. Galitski + Y. B. Kim) or Amperian pairing (S. lee+P. A. Lee).

Brad talked about an interesting theoretical paper by Y. Ran, M. Hermele, X. G. Wen, and P. A. Lee. This is a variational calculation of various candidate ground states. The paper claimed that the ground state seems to be a gapless spin liquid where spinons have a Dirac spectrum. It can be obtained by putting the (fictitious) \pi-flux in each hexagon, to be seen by the spinons. Then they performed the projection (remove double-occupancy) on the mean-field wavefunction.

\Psi_{variational} = P_G \Psi_{MF}

Their variational energy is remarkably good in the sense that it is quite close to the result of
the exact diagonalization, namely

E_{\pi flux state} = - 0.42866(2) J

E_{ED} = - 0.43 J

But Brad believes that the true ground may not be a spin liquid and Y. Ran et al's calculation may have some problems. First he mentioned his old work with C. Zeng (1991), where they realized that the ground state of the Heisenberg model in the 1/N expansion is some kind of dimerized states. Basically the system wants to maximize the number of hexagons with (non-touching) three dimers. One can show that at least 18 site unit cell is needed to achieve this. Thus one ends up with the dimerization patterns with the unit cell of some multiples of 18 sites.

Then he mentioned a recent work by Singh and Huse, where dimer expansion is used to obtain the dimerized state with the unit cell of 36 sites. The energy of this state is

E_{dimer} = - 0.432 J

In order to see how good Y. Ran et al's \pi-flux state is, Brad examined the effect of small number of dimerized bonds on the \pi-flux state. With 5% dimerization (calculation done on the 12 X 32 X 3 = 432 sites), he gets

E_{5% dimer} = - 0. 42860(3) J

which is not terribly different from Y. Ran et al's result. This implies that energy land scape is quite flat ! Brad also mentioned a problem with Y. Ran et al's calculation, namely they imposed the anti-periodic boundary condition (by inserting a fictitious flux) in one of two directions. This breaks the rotational symmetry. Apparently this has a big effect; in fact this is the way that the degeneracy was broken. Brad said there exists 10% dimerization in their numerical calculation because of this choice of boundary condition.

Brad stressed that according to Singh this is the first time example where dimer expansion
converges so nicely; this may be a strong evidence that the ground state is indeed a dimerized state with a huge unit cell.

Brad then mentioned a work by White and Singh (Physical Review Letters 2000). They looked at a "kagome-strip" or coupled chains where there are "crossed" inter-chain couplings. The ground state turns out to be dimerized and the gap is very small (gap = 0.01 J). Brad looked at this problem and found the detailed ordering pattern of the dimerization (which turns out to be rather complicated).

I asked whether these results are in contradiction with Y. Ran et al's argument based on the PSG analysis, where they claimed all physically relevant perturbations about the \pi-flux phase seem to be irrelevant and the spin liquid phase is stable. Brad replied that perhaps the gauge field fluctuation is so strong that the PSG of the mean field states are not that useful.

Igor Herbut: "Coulomb interactions, ripples, and minimal conductivity of graphene"

2007-08-29T15:25:00.001-06:00

Herbut started the presentation by stressing that, in his view, there are two problems in graphene physics which might be worth the while of a theoretical physicist interested in correlated systems. First, QHE, where he argued the interaction effects are essential and, second, the problem of minimum conductivity, where the interactions might be a part of the ultimate solution. He then proceeded to define an experimental puzzle: the measured minimum conductivity of graphene sheets tuned to Dirac point is about 300% larger than what one would compute for the clean or weakly disordered system.

Herbut then set up the theoretical background. The symmetry of the honeycomb lattice of graphene sheets guarantees two Dirac points under rather general conditions. Near these points the electron-hole spectrum has an appearance of a relativistic massless Dirac fermion and the gate voltage can be tuned so that the Fermi surface passes right through them. When this is the case, we can view the problem as a nice example of a fermionic quantum criticality. The universality of the familiar result, \sigma_0 = \pi/2 e^2/h, derived by Fradkin and others, is a manifestation of such criticality.

Next, Herbut introduced interaction. This is just the unscreened 1/r Coulomb interaction, which, one can show using the technology of quantum critical phenomena, turns marginally irrelevant at low energies. He demonstrated this by computing the correction to \sigma_0 arising from such interaction. Indeed, the correction due to the Coulomb interaction was found to fall of logarithmically, as one moves to low frequencies. Importantly, however, this correction was positive – the conductivity at some low but finite frequency was enhanced relative to \sigma_0. This set the stage for an intriguing piece of physics: in a typical experiment, the frequency scaling of conductivity will generically be cut off by temperature, disorder or some other effect. It could be that the observed access conductivity is actually due to such a phenomenon.

The specific example worked out by Herbut, Juricic and Vafek is due to a disorder effect arising from rippling of graphene bonds. When graphene sheet is fixed onto a substrate, such ripples act as a gauge field frozen into a particular configuration – the effect arises through the modulation of hopping integrals on bonds. Such “magnetic field” disorder is precisely marginal and it acts on the interaction to arrest the logarithmic decline of its contribution to conductivity. The result is a line of fixed points along which the minimum conductivity takes on a non-universal value, set by the rippling disorder, but always larger than \sigma_0.
This picture supplies a rather attractive explanation for the available experiments. The details of their work can be found in http://www.arxiv.org/abs/0707.4171.

Several comments and questions were lobbed at Herbut by clearly animated audience members. Chubukov inquired about the work of Efetov and Aleiner and its relation to this presentation. Herbut answered that they were considering a “non-critical” case, where the gate voltage moves the chemical potential away from Dirac nodes and thus the system acquires a small Fermi surface. Several audience members, including Morr, Vekhter and Eschrig, wanted to know more about the ordinary potential disorder, resulting in a random variation of a chemical potential. Herbut replied by pointing out that, for the current experiments, he felt his picture of the rippling disorder was the most appropriate.

John Mydosh: Nernst Effect in NdBa2(Cu_1-y Ni_y)3 O_7-d

2007-08-28T16:51:00.000-06:00

In a very inspiring talk John Mydosh presented us recent experimental results about the Nernst effect in the compound NdBa_2(Cu_(1-y)Ni_y)_3 O_{7-x}. This work was published recently in Phys. Rev. B 76, 020512(R) (2007). John started by saying that in those compounds it’s very difficult to get bulk sample, which is what his student had. The system has the same anisotropy as YBCO
and is at optimally doped for x=0. John had drawn a phase diagram for this compound, which looks very similar to the one of YBCO, with a maximum Tc of 85 K, an AF phase at low doping and a ``pseudo-gap’’ phase in between. The interest of doping with Ni is that it reduces Tc without changing the amount of oxygen doping.

Then John embarked on a very nice pedagogical review on Nernst and Seebeck effects in metals. He recalled that, for one-band metals, the Sondheimer cancellation is the cause that Nersnst effect is very small, almost undetectable. This is not true, however for semi-metals, like Bi, which have some energy gaps. Ilya Vekhter asked whether this property was true for all semi-metals. John answered that generically, if you have partial gaps in the Fermi surface the Nernst effect will be big. Andrey Chubukov asked what `` big’’ meant within these units? the answer was that already a few micro Volts per Kelvin can be considered as ``big’’.
At this point, John recalled a review of the compounds for which anomalous Nernst effect has been observed with putative explanation. I reproduce it below:

CDW-NbSe_2: counterflow of electrons and holes
PrFe_4 P_12: quadrupolar ordering
URu_2Si_2: ``hidden order’’ transition
(B_12)-(BEDT-TTF)_2 : SC near a Mott transition
CeCoIn_5: co-existence of SC and AF
MgB_2 : ?

Then John recalled that for a superconductor, a non zero Nernst effect is expected, du e to the presence of the vortices. The motion of vortices generates some phase slip and creates a perpendicular response to a temperature gradient (E_y = - e_y \grad T, with e_y the Nernst coefficient). In a superconductor, one can even see the pinning force of the vortices, which reduces very strongly the Nernst signal at very low fields. This vanishing of e_y at low fields can be considered as test mark that we are in a superconducting phase.

John reviewed the results obtained by Ong for LASCO and Nd_(2-x)Ce_xCuO_4 (electron doped). He emphasized the striking difference in the Nernst signal between the eletron and hole doped compounds, with a much stronger Nernst effect in the hole doped case. He stressed that the common view by the Ong group is that the Nernst effect is due to some pre-formed pairs and that the theory of Anderson seems to work well.

Back to his own data on NdBa_2(Cu_(1-y)Ni_y)_3 O_{7-x}, John compares three dopings O_7, O_{6.9}and O_{6.8}. One can see on the curves, that generically the Nernst signal is present in the normal phase at optimal doping, in the under-doped regime it grows to then vanish for the very under-doped case. By then playing with the doping in Ni (which semsibly reduces T_c) John compared the variation of the Nernts signal wth T_c for the three O-dopings mentioned above. For the O_7 case, the temperature below which the Nernst effect is present- let’s call it T_N- follow T_c (namely T_N= T_c + 20 K). But when one under -dopes the behavior of T_N changes and becomes independent of T_c: it’s more or less stationary. It is important to note that T_N doesn’t follow the pseudo gap temperature T^* as well, which is shown to increase with Ni doping.

John concludes that the behavior of T_N remains mysterious, but there is indication that this temperature is very sensitive to the presence of impurities. He cites the work of Alloul and Albenque using the van de Graaf in Orsay, where due to irradiation of the sample one produces intrinsic disorder. This hand-made disorder creates a Nernst signal.

Discussion followed.

Zlatko Tesanovic asked what was the difference with the work of Ong, and in particular whether there is a vortex liquid phase in this compound. John answered that the beauty of his experiment is that the Ni doping enables to reduce T_c without changing other parameters ( in particular the oxygen doping). This is due to the fact that Ni has spin 1.

Claudio Castellani commented that the Nernst effect should in general depend on the superfluid density. John answered that it is the case except if there are inhomogeneous regions, which could well be the case here.

Zlatko Tesanovic: d-wave duality and its reflections in cuprates

2007-08-28T16:27:00.000-06:00

In an inspired talk, Zlatko Tesanovic described his latest work on the on-going research program that attempts to understand the physics of cuprates as strongly fluctuating d-wave superconductors. The talk was mostly conceptual, and those interested in fine technical points are advised to look at his preprint arXiv:0705.3836.

Zlatko began by dividing all (singlet) superconductors into two general classes: the weakly correlated, BCS-Eliashberg type, and the strongly correlated, to which presumably all of the cuprates belong. It is for the second class, which inevitably suffer from strong phase fluctuations, that the notion of duality becomes useful. Duality was first discussed in a simpler problem of the negative, strong-U Hubbard model, which exhibits local singlet pairs. If one freezes the amplitude of the gap, the remaining theory for the fluctuating phase degrees of freedom may be cast in the dual language, in terms of the "disorder parameter" that signals the proliferation of infinitely large vortex loops. In this formulation, the dual condensate represent the non-superconducting phase. The theory has therefore two phases: superconducting, in which the original order parameter is finite while the dual vanishes, and the non-superconducting, in this case a charge-density-wave, in which the reverse is true. Zlatko mentioned an example of compound (BaKPb)BiO3 in which the observed strongly diamagnetic CDW phase may possibly be an example of a such "phase incoherent" superconductor. (For an introduction to the standard "Peskin-Dasgupta-Halperin" duality, see my book, "A modern approach to critical phenomena", Ch. 7.)

Turning to the d-wave superconductors, Zlatko observed that unlike in the s-wave case, pairs here are necessarily non-local objects which live on bonds. The phase of the superconducting order parameter is therefore a bond variable, which leads to richer physics. In the continuum limit, the bond-phases get approximated by the site-phases, at which point some information about the phase configuration, namely the relative (fluctuating) phase between two bonds emanating from the same site is lost. To retain the complete set of configurations of the bond-phases Zlatko wrote it as a sum of the "center of mass" and the "relative phase". The quantum disordering of the former then leads to the old QED3 theory of the cuprates, and the concomitant pseudogap phase which, essentially being just the disorder d-wave superconductor naturally exhibits a large Nernst effect and the surviving nodal quasiparticles. (Although a small gap at the nodes, which would signal an incommensurate SDW order is possible as well.) Disordering of the relative phase, on the other hand, proliferates the monopole configurations in the emergent U(1) gauge field of the QED3, and erases the last memory of the d-wave superconductor. Zlatko identifies this final state with the transition into the commensurate Neel antiferromagnet near half filling.

In the question period several people raised the issue of what should all this mean on the electron-doped side (Eschrig, Castellani), to which the answer was that the electron-doped superconducting state appears to be more of the BCS, non-fluctuating variety, and the transitions therefore more mean-fieldish, or first-order (for the dSC-AF transition). Muramatsu wanted to know what kind of topological singularities are actually present in the theory: vortex loops and monopole-antimonopole configurations. Castellani went back to the negative-U example, and if I heard correctly, guessed that a finite doping in the dual theory would appear as a finite magnetic field, which is correct. Abrahams asked about the difference between the SDW that arises as the chiral instability of the QED3 and the Neel antiferromagnet at half-filling. Zlatko's answer was that the latter obviously does not show a large diamagnetism, while the former does, and that there is presumably a quantum phase transition between the two. Kee wondered where would the place for the standard Fermi liquid be in the whole story. The answer was that Fermi liquid is actually outside the present theory, which assumes a finite amplitude of the gap; setting the amplitude to zero would restore the Fermi liquid. Monien asked about the status of the experiment at low dopings and temperatures, to which Zlatko replied that there is a large Nernst signal there as well, so the ground state itself should be a disordered d-wave superconductor below the critical doping. Finally, Pepin could not see the difference between the present theory and the gauge theories of several other prominent workers in the field. Zlatko, after admitting he was sad to hear this, explained that the crucial difference is that those fatal monopole configuration that more often than not undermine the usefulness of the (compact) gauge theories in condensed matter here are kept in check by the BdG quasiparticles and the associated Higgs mechanism. And on this uplifting note the discussion was ended.

Catherine Pépin: Kondo Breakdown as a Selective Mott Transition in the Anderson Lattice

2007-08-28T14:07:00.000-06:00

Blog from talk presented in week 4.

In her talk Catherine Pépin presented results published recently in PRL (I. Paul, C.
Pépin, and M. Norman, PRL 98, 026402 (2007); C. Pépin, PRL 98, 206401, (2007)). The work was devoted to study of a Mott transition of the f electrons in the Anderson lattice. The model of the Anderson lattice offers a way to relate the Kondo breakdown (vanshing of the effective hyrbidization between the f and c bands) to a Mott transition of the f electrons. The suggested idea has analogies in description of the cuprates superconductors because both in the Anderson lattice and in the Hubbard model, there is a competition between the Coulomb and kinetic energies. When the Coulomb energy is stronger this can lead to a localization of the f-electrons. A spin liquid is needed to stabilize the localized phase, but both in the Anderson and the Hubbard model a spin liquid is believed to be apearing when one approaches the insulating state, at least in a slave boson treatment, used in the presented work. Around the QCP associated with the Mott transition, one observes flucutations of the hybridization. Using a fermionic representation for the localized spins the deconfined quantum critical point was studied within this model.
The main idea presented in the talk was that an unusual behavior in thermodynamics and transport might be due to critical fluctuations of a nonmagnetic order parameter associated with the vanishing energy scale T_{K}, where T_{K} is an efective Kondo temperature. To be more precise, the Kondo effect breaks down because the effective hybridization is renormalized to zero.
In the Anderson lattice this occurs exactly when the f electrons localize. This contrasts previous approaches based on critical contributions of paramagnons.
Expressing the spin varibles in terms of fermions one comes to a fermion model with quartic fermion interactions that is further studied using a mean field approximation for the slave bosons. As the next step, flucutations around this mean mean field were also taken into account, in order to describe the thermodynamics around the QCP.
Assuming that the mean field solution does not depend on coordinates it was shown that, above a very smal temperature scale, the critical fluctuations associated with the vanishing hybridization have dynamical exponent z=3, giving rise to a resistivity that has a TlogT behavior. At the same time, it was found that the specific heat coefficient diverges logarithmically in temperature, which is in agreement with results of observation in a number of heavy fermion metals.

Dirk Morr asked:

Is the temperature dependence of the resistence linear?

Answer:
Above a certain temperature, the temperature dependence linear but it becomes quadratic below it.

Andrey Chubukov asked:
Why is the the crossover temperature so low?

Answer:
It depends on the interplay between the Fermi momenta of the two bands.

Ilya Vekhter asked:
Why was it that z=3 in the intermediate temperature regime?

Answer:
It is a q=0 transition. When the critical modes are damped by the
continum of the electrons, one gets z=3 (Landau damping).

Viktor Galitskii asked:
What kind of the spin liquid is needed?

Answer:
In the U(1) gauge theory one gets naturally a massless spin liquid.

Hartmut Monien asked to explain why te resistivity is quasi-linear in T?

Answer:
In this model, the f-electrons llocalize and form a reservoir. Hence,
although one has a q=0 transition (hybridization fluctuations ) one does
not need impurities to break translational invariance: the f electrons are
on the lattice, which leads to umklapps. Then, the one impurity life time
is the same as the transport lifetime-- quite a unique feature.

Enrico Rossi: Neutron resonance in electron-doped cuprates

2007-08-24T15:45:00.000-06:00

In his talk, Enrico Rossi discussed the work with J-P Ismer,
Ilya Eremin and Dirk Morr on the neutron resonance in electron-doped cuprates. His main idea is that the resonance is a spin exciton, shifted to a higher frequency by a finite fermionic damping rate. Enrico started his presentation with a brief review of the excitonic scenario for the resonance. He then argued that, in distinction to hole-doped cuprates, where the resonance is well below 2 \Delta_[max}, the resonance in electron-doped PrCeCuO and NdCeCuO is observed at 11 meV, which might be larger than 2\Delta (the measured gap
maximum is less than 5 meV). as determined by ARPES experiments Enrico presented RPA-type calculations of the resonance, which include a finite broadening of the fermionic linewidth. He argued that due to a finite broadening, the resonance shifts to a higher frequency, which may exceed 2\Ddelta_{max}. Enrico then argues that in the presence of a magnetic field, the resonance is split into three distinct peaks. Due to the smaller magnitude of the gap, and a resonance frequency which is much smaller in the electron-doped cuprates than in the hole-doped ones, the experimental resolution in INS experiments is sufficiently good to resolve a splitting of the resonance in field of about 8 T, a splitting which is of the order of 1 meV.

Finally, Enrico argued that in those electon-doped cuprates, in which superconductivity exists with antiferromagnetism and T_c>T_N, the resonance shifts down to lower frequencies as T_N is approached, and reaches zero frequency at T_N.

The discussion after the talk focused on the intensity and the linewidth of the peak.
Enrico was asked whether a large width of the peak may prevent the development of three sub-peaks in a field. Enrico replied that even if a large quasi-particle damping prevents the resolution of the three peaks, the resonance will become highly asymmetric.

Maxim Vavilov: Quantum Disorder in Andreev Billiards

2007-08-24T11:25:00.001-06:00

Maxim Vavilov discussed the effects of quantum disorder in Andreev
Billiards. These billiards consist of a small grain of normal state
material that is brought into contact with a superconducting
reservoir. These systems are realized, for example, by connecting a
quantum dot to a superconducting leads

Maxim first discussed the various energy scales that are relevant
for this problem. The largest energy scale is set by the (isotropic)
superconducting gap, Delta_sc, which implies perfect Andreev
reflection at the interface between the normal and superconducting
systems. The next smaller energy scale is set by the Thouless energy
E_T=hbar/tau_f where tau_f=L/v_F is the flight time of the
electrons, and L is the size of the normal state grain. Another
energy scale is set by E_g=hbar/tau_d where tau_d=tau_f*L/b is the
dwell time of the electrons, and b is the length of the interface
between the normal and superconducting systems. The last energy
scale is set by the mean level spacing, delta_I, of the normal state
system. The relative order of energy scales for the system that
Maxim studied is given by

Delta_sc >> E_T >> E_g >> delta_I

The objective of Maxim's work was to study the properties of the
electrons in the normal state grain, which are reflected in the
averaged density of states (DOS). Of particular interest is the
question of whether Andreev scattering off the interface leads to a
suppression of the normal state DOS at low energies. Maxim then
proceeded to outline a calculation using Random matrix theory (RMT) (see "Induced superconductivity distinguishes chaotic from integrable billiards", J. A. Melsen, P. W. Brouwer, K. M. Frahm, C. W. J. Beenakker Europhys. Lett. 35 (1996) 7) and a Gaussian Orthogonal Ensemble, which can be exactly solved in the
limit hbar/(tau_f * delta_I) -> 00. In this case, the DOS opens up a
hard gap at low energies up to an energy scale set by E_g, and
increases as DOS ~ sqrt(w - E_g) for energies w>E_g. At this point
Daniel Sheehy asked whether this result is achieved by averaging
over ensembles. Maxim answered that in the case he considered,
averaging over ensembles is equal to averaging over many energy
levels. Hence the RMT result should be valid for the average DOS of a
single normal grain.

Maxim then proceeded to outline a different calculation based on the
Eilenberger equations developed with Anatoly Larkin ("Quantum Disorder and Quantum Chaos in Andreev Billiards", M.G. Vavilov, A.I. Larkin, Phys. Rev. B 67, 115335 (2003)). This approach corresponds to the semiclassical approximation only if impurity scattering is not taken
into account. Without disorder, this approach yields an averaged DOS
in the normal grain that is suppressed at low energies (below E_g),
but does not show a hard gap, in contrast to the results of the
random matrix theory. Finally, Maxim considered the effects of
disorder, as realized by a distribution of short range impurities.
In the limit of strong disorder, when the scattering time is
comparable with the dwell time, the Eilenberger approach recovers
the RMT result, and a hard gap opens in the DOS up to a frequency of
E_g. However, even in the case of weak disorder, a gap opens in the
DOS.

Daniel Sheehy asked whether the Andreev reflection at the interface
is perfect. Maxim answered that this is the case as long as the
superconducting gap is the largest energy scale in the problem, and
in particular, as long as Delta_sc >> E_T.

Andrey Chubukov asked whether this averaged DOS can be measured
experimentally. Maxim pointed out that in general, it can be
measured by studying quantum dots connected to superconducting
leads. However, the main experimental problem seems to be the
interface between the superconducting and normal state materials.
Finally, Maxim remarked that while his theory was developed for
two-dimensional grains, the effect might be more easily observable
in three dimensional systems.

A. Auerbach: "Quantum Tunneling of vortices in underdoped cuprates: theory and experiment"

2007-08-23T17:14:00.000-06:00

Assa Auerbach (Technion, Israel) told us about a new experiment performed by the group of G. Koren also at the Technion (cond-mat/0707.284) where variable range hopping (VRH) of vortices was observed in a special YBCO film. The film was 1m long and 14 um wide wire arranged in the form of a meander. This allowed the experimentalist to perform magneto-resistance (MR) measurements at low currents
(1uA) and low fields (up to 6T) where most of the contribution to the MR comes from single vortex tunneling.
The major finding in the experiment is a VRH type temperature dependence of the MR at low T, namely,

1) MR~exp[-(T_0/T)^1/3] .

Assa argued that VRH is not expected in a conventional BCS type superconductor, with large coherence length.
However, for underdoped cuprate superconductors, where the ratio of carrier density n_s to pinning site density n_pin can be low, Auerbach, Arovas Ghosh (PRB 74 2006) have predicted Eq. 1 based on a interacting boson model, and have calculated that

T0~(n_s/n_pin)^2(1/n_lay)*dV,

where n_lay is the layer density and dV is the fluctuations of the pinning potential. Reasonable values of the parameters agree with the measured T_0.

In the context of tunneling vortices, Assa also described recent studies of vortices in a model of half filled, hard core, lattice bosons (Lindner et. el. cond-mat/0701571).
He reported numerical estimates which found the vortex mass to be quite low (~3 times the boson mass), and hence the critical melting density of the vortex lattice was estimated to be of order 10^{-3} vortices per lattice site. This implies that quantum vortex liquid phases could be achieved by relatively weak rotations (in an optical lattice) or magnetic field (in e.g. cuprates), much lower than Hc_2.

Victor Galitski: Mesoscopic disorder fluctuations in a d-wave superconductor

2007-08-23T17:09:00.000-06:00

Thursday, Aug 23th

Victor Galitski started our Patio Discussion by returning to the recent STM experiments by Ali Yazdani showing an inhomogeneous spatial gap distribution above the superconducting transition temperature Tc in the cuprates. Taking these as motivation for his today's presentation, he first pointed out that the important features seen by Ali Yazdani are that the experimental gapmaps are static and reproducible when varying temperature. In particular this means no phase separation takes place.

Victor went on by stressing the break-down of Anderson's theorem in d-wave superconductors in the presence of disorder potentials, leading to a dependence of Tc on the disorder. As the density of impurities is random, there are fluctuations in real space. These are according to Victor associated with a local Tc larger than the Tc for a corresponding homogeneous state. A picture of paddles of superconductivity within a normal background emerges, where each of the paddles have their private Tc. Victor now continued with an overview over what is known from s-wave superconductors, in which case Tc does not depend on disorder in leading order in accordance to Anderson's theorem. In this case fluctuations are not important.
Victor proceeded by reminding us that in s-wave superconductors with magnetic impurities there is an Abrikosov-Gorkov formula
ln (Tc0/Tc) = \Psi(1/2 + \Gamma/[2\pi Tc]) - \Psi(1/2)
that determines the actual Tc in terms of the critical temperature for a system without disorder, Tc0. The crucial parameter in this formula is the pair breaking parameter Gamma. In a magnetic field and in the diffusive limit it is proportional to D*H, where D is the diffusion constant and H the magnetic field. This leads to the well known Hc2(T) curve. Victor draw our attention to the fact that for s-wave superconductors this theoretical curve is smooth at low temperatures, whereas experimentally often an upturn of the Hc2-curve is observed. A possible explanation would then be that Tc depends on disorder via the diffusion constant D, and thus Hc2(0)~n_imp. Dan Sheehy asked the question what happens for n_imp=0, and Victor stressed that he restricts his discussions to the dirty limit, so that Tc0 \tau <<>

Next Victor draw a picture of superconducting islands connected by the Josephson effect and mentioned the works about Josephson networks by Spivak/Zhou PRL '95 and by Larkin/Galitski PRL 2002. At this point a specific model in terms of a Ginzburg-Landau action followed, in which spatial fluctuations of the order parameter where taken into account.

Victor mentioned in passing that in cuprates in principle Tc0 depends on doping, such that Tc is determined by an interplay between the intrinsic x-dependence of Tc0 and the induced one by the spatial disorder. This leads to a superconducting dome resembling very roughly that of the cuprates.

The spatial randomness of the gaps introduces via the eigenvalue equation

(1/v) \int C(r,r') \Delta(r') = (Tc/Tc0) \Delta(r)

also a random Tc. The random operator C(r,r') is the Cooperon. The statistics of C(r,r') can be expressed diagrammatically, and leads to a distribution of Tc's as function of coherence length, mean free path and (Tc-Tc0)/Tc0. Victor finished his talk with developing a picture of underdoped cuprates in terms of superconducting islands separated by normal regions, however with a fluctuation gap. This also implies a reduced local density of states in the normal regions.

In the discussion part, Phil Anderson commented that all this does not seem to be related to high-Tc cuprates, but to d-wave BCS superconductors. The nature of the phase transition in cuprates is that of an x-y model, where Tc~\rho_s, not ~\Delta. Thus, fluctutating gaps are not related to fluctuation Tc's. Victor basically agreed and mentioned that he studied a BCS model, not an x-y model. Andrei Chubukov commented that Tc in the calculations should be related to the pseudogap temperature T*.

Dirk Morr asked how the distribution of local Tc's is related to the global Tc. Victor answered that the distribution of Tc's is related to disorder, but that there were no direct relation to a global Tc. Claudio Castellani commented at this point that he thinks Tc as a local quantity is only a technical parameter of the BCS model, any real Tc has to be global. Victor disagreed in the sense that if the puddles are in size larger that a coherence volume, it makes sense to talk about a local Tc for each puddle.

Yong-Baek Kim: "Heisenberg Antiferromagnet on the Hyper-Kagome Lattice: Application to Na4Ir3O8"

2007-08-23T16:19:00.001-06:00

Yong-Baek Kim discussed very interesting recent experiments on a new three dimensional antiferromagnetic compound [Okamoto et al., "Spin liquid state in S=1/2 hyper-kagome antiferromagnet Na4Ir3O8"] and recent theoretical work by his group directed at understanding the magnetic behavior of the material [Hopkinson, Isakov, Kee, and Kim, "Classical antiferromagnet on a hyperkagome lattice," PRL 99, 037201 (2007) and "Topological spin liquid on the hyper-kagome lattice Na4Ir3O8"].

Stoichiometry shows that the Ir ions are in a 4+ valence state and the five 5d electrons form a spin-1/2 state in the t2g level. The 3D lattice is like the better known pyrochlore lattice except for the fact that the Ir ions occupy only 3 of the 4 sites of each tetrahedron; the (spinless) Na ions occupy the 4th site. The Ir ions on the decimated pyrochlore lattice form a network of corner-sharing triangles that has been dubbed a "hyperkagome" lattice. As the unit cell contains 12 spins, the material doesn't have an odd number of spin-1/2's per unit cell (in contrast to several 2D candidate spin liquids that may support gapless spin excitations). Nevertheless, magnetic susceptibility measurements find a large Curie-Weiss temperature of -650K and no sign of ordering down to 2K. Due to the large nuclear charge of Iridium (Z=77) there could be a sizable spin-orbit interaction, but as the lattice has inversion symmetry, Dzyaloshinsky-Moriya interactions are apparently forbidden.

Kim's group first investigated the behavior of a classical Heisenberg antiferromagnet on the hyperkagome lattice. Assuming nearest-neighbor exchange, the Hamiltonian can be rewritten as a sum over all the triangles of:

(J/2) * (S_triangle)^2

where S_triangle is the sum of the three spins on each triangle. At zero temperature the ground state is specified simply by setting S_triangle = 0 on each triangle; as there are many choices of the spins that satisfy the constraint, there is a macroscopic degeneracy. Here Andrey Chubukov asked if the constraint could be implemented independently on each triangle, and Yong-Baek clarified that the triangles are not independent, but nevertheless the ground state does have macroscopic degeneracy.

The question then arises as to whether or not there is an order--by-disorder transition induced by classical thermal fluctuations at non-zero temperatures; Hopkinson et al. addressed this by classical Monte Carlo calculations and found a transition from a "cooperative paramagnet" at high temperatures to a spin-nematic phase at low temperatures (below 0.3 to 1.5 K for an exchange constant of about 300K). The rather unusual spin-nematic order parameter is given in terms of cross-products of pairs of spin operators on the triangles. Correlations are found numerically to become long-ranged in the ordered phase, and the change in the entropy is consistent with the formation of spin-nematic order.

Analysis of the corresponding quantum Heisenberg antiferromagnet by Read and Sachdev's bosonic Sp(N) method finds two other phases: An ordered co-planar state and a Z2 spin liquid with massive deconfined spinon excitations. Thus quantum fluctuations have a markedly different effect than classical thermal fluctuations. I noted that spin-nematic order would be difficult to find within the Sp(N) method, as the order parameter doesn't generalize to Sp(N) in a natural way.

John Mydosh suggested experimental investigation by neutron scattering would be interesting as it could detect a spin-ordered phase, and Yong-Baek agreed but pointed out that the common isotope of Iridium strongly absorbs neutrons. Amit Keran suggested that NMR measurements on the sodium atoms would be the next logical step, especially as lower temperatures can be reached.

Phil Anderson: Nernst Effect in the Cuprates

2007-08-23T13:44:00.000-06:00

Phil Anderson presented an experimental/theoretical talk entitled "Theory of the Nernst Effect in the Cuprates: Is not Black Hole Physics". Here Phil gave an overview of the anomalous behavior of the Nernst effect above Tc in the so-called vortex liquid state. First and foremost was his phase diagram (temperature versus hole doping) which contained an additional phase line, concave in form, above the usual superconducting Tc - dome. This new phase represents the vortex liquid where vortices form due to the charge pairing and a diamagnetic response is found. The pseudogap appears above these domes with its monotonic decreasing structure from the antiferromagnetic region to an intersection with the supoerconducting dome.In a superconductor the Nernst effect tracks the vortex motion due to a temperature gradient and from the second Josephson equation a transverse voltage develops as a function of the perpendicular applied magnetic field, i.e., a phase slip voltage. In a type 2 superconductor a large Nernst signal results below Tc up Bc2. Now for the generic high Tc superconductors the Nernst signal remains far above Tc and according to the physical model it is proportional to the vortex velocity. Phil's theory enables one to relate the Nernst and Ettingshauser coefficients to the order parameter of the vortex fluid phase, i.e., the energy gap of preformed pairs which now appears as a distribution of gap sizes. Since phase coherence is broken the material in not in a conventional superconducting state.Phil showed that the distribution of energy gaps in the vortex liquid phase is related to the Nernst coefficient minus the field derivative of the Nernst signal. Thus one can now determine the distribution of gap sizes and the probability distribution of order parameters. Note that the pseudogap, as usually determined from NMR, ARPES, optical conductivity, etc., is distinct from the vortex liquid phase. And for certain materials there seems to be no correlation between the pseudogap temperature and the onset of the Nernst signal. One needs further experimental studies to map out the the vortex liquid phase boundary and to fully establish its properties in a variety of high Tc materials.

Amit Keren: Magnetic "Isotope Effect" in Cuprates

2007-08-22T21:40:00.000-06:00

Amit Keren (Technion, Israel) told us about an accumulation of 10 years of
research by his group, of a family of YBCO-like high Tc cuprates called
CLBLCO, = (Ca_x La_{1-x})(Ba_{1.75-x}La_{0.25+x})Cu_3O_y.

While most systematic studies
of cuprates involve changing just one doping parameter, such as the oxygen
concentration y, CLBLCO presents a unique opportunity to continuously vary TWO
parameters, x (family index) and y (oxygen concentration), without
significantly disturbing the structure or varying the disorder in the CuO_2
planes. In fact, the primary effect of changing x on the CuO plane is to
slightly vary the copper-oxygen buckling angle which is known to change the
magnetic superexchange constant.

Amit showed his group's mu-SR data for the superconducting transition
temperature Tc(x,y), the spin freezing temperature Tg(x,y) at intermediate
doping, and Neel temperature T_N(x,y) at low doping. Moreover, the 2D AFM
exchange J(x) was extracted from the Neel temperature (by fitting the
T-dependent staggered magnetization to estimate the interlayer exchange).

At first, the data seems scattered on the (T,y) phase diagram. Amit chose to
collapse the data by rescaling all transition temperatures by T_c^{max}(x), and
also rescaling the y axis by an "effective doping" Delta p = K(x) (y-y_max),
which collapsed all the Tc(Delta p) "domes" onto one universal curve.
Collapsing the Tc domes is hardly surprising. However the same axes rescaling
completely collapses the -magnetic- freezing transitions onto one curve as
well!

The conclusion is that T_c^{max}(x) \propto J(x).

Apparently, the data collapse indicates that a single energy scale determines
both antiferromagnetic and SC ordering temperatures!

This blogger feels that this finding, although simple, is far from obvious. It
puts a serious constraint on theoretical mechanisms of cuprate
superconductivity: One would naively expect more than just J, to determine T_c (say e.g. some
additional kinetic or interaction energy might be important). These scales have
no apparent reason to stay proportional to each other as the two material parameters x, and y,
are independently varied.

Amit also showed uniform susceptibility data, which was used to define
the "pseudogap temperature" T*(x,y). While this energy scale did not precisely
collapse by rescaling the axes, it seemed to follow for some reason the 3D Neel
temperature T_N, which depends on magnetic interactions both in and out of the CuO planes.

Frank Marsiglio "Issues Concerning the Optical Sum Rule Anomaly below Tc in the Cuprates"

2007-08-22T18:34:00.001-06:00

TOC
1. Conventional Theory
2. Experiment
3. Phenomenological Explanation
4. Issues

Conventional Theory

Kubo sum rule
Integral of the real part of conductivity over frequency is
constant -- temperature independent.

Single band sum rule (theoretical construction)
Integral of the real part of conductivity coming from a single band
over frequency (denoted by W(T)) measures the average second
derivative of energy over momentum.

In conventional cases it is proportional to the minus average energy.

In Fermi Gas as temperature increases the distribution function smears and
particles get transferred to higher energy, so W(T) goes down.

If one now decreases the temperature the Superconducting transition occurs,
the distribution function gets smeared, kinetic energy increases and W(T)
goes down.

So W(T) has a maximum at T_c.

Experiment

Experiment shows the decrease W(T) as one decreases temperature through T_c
in overdoped materials, but in optimally doped and underdoped materials
it goes up.

It means that in optimally and underdoped materials the kinetic energy
decreases in superconducting state. That gives us
"kinetic energy driven superconductors".

Phenomenological Explanation

Norman & Pepin (2002) showed that interactions decrease W(T).
Microwave experiments show that there is a collapse of the scattering
rate (scattering rate is due to interactions) below T_c

Taking together those two statements mean that above T_c W(T) is suppressed
by the interactions while below T_c interactions are suppressed and W(T)
goes over to the one of the noniteracting case -- increases.

Issues

There is no issue of the low energy cutoff as Kuzmenko et al explained that
although they cannot measure conductivity at low frequencies accurately
enough, they can measure the contribution to the sum rule.

The main issue is the upper cutoff.
In order to measure the sum rule for the single band one has to introduce
an upper frequency cutoff which is below the frequency of the interband transitions.

Imagine that we have a simple Drude behaviour of the conductivity.
The Drude peak sharpens up as one lowers temperature. If one then checks the
sum rule up to some upper cutoff in frequency one finds that it is more
weight below this frequency. So although the total weight is conserved
the total weight below a frequency cutoff is temperature dependent.

So the normal state ~T^2 behavior can be explained by a mundane upper cutoff
effect. We are currently investigating whether the anomalous rise of W(T)
below T_c can be attributed to a mundane cutoff effect as well.

Questions

Assa: What should the high frequency cutoff be in order to recover full temperature independent sum rule?

F.M. Large, depends what "full" means.

Chubukov: comment, the increase or decrease of W(T) due to cutoff
depends on the valueof \Delta\tau

Pepin: Has anyone investigated the influence of the van Hove singularity
on W(T)?

F.M. Theoretically, last year in a PRB paper we showed that the change below
T_c can be anomalous, using just a BCS approach. As far as I know no one
has measured this same quantity in High T_c samples that are doped beyond the van Hove singularity.

Konstantin Efetov: Transport in Graphene

2007-08-22T18:28:00.000-06:00

Konstantin Efetov started by drawing parallels between high-T_cs (which
dominated earlier discussion) and graphene. He emphasized that both are 2D
systems which have been lauded as materials of the future for energy and
nanoelecronics respectively. Konstantin reminded the audience that grapehene
has 2D honeycomb crystal lattice, and therefore contains two sublattices. The
Brillouin Zone has two valleys with linear, Dirac-like, energy dispersion, so
that the effective hamiltonian for pure graphene is a 4x4 block-diagonal
matrix. Gating the substrate with graphene film on top allows changing the
filling fraction easily.

Konstantin then described the effects of impurity scattering on transport in a
system with such an energy spectrum following his recent work with Igor
Aleiner [PRL 97, 236801 (2006)]. The work was motivated by experimental claims
of delocalized low energy states and universal metallic resistivity in
graphene (which, it seems, is no longer universal), and theoretical analyses
of weak localization corrections.

Efetov and Aleiner considered general purely potential impurity scattering
(spin-orbit interaction is graphene is weak), which replaces zeros in the
block-diagonal Hamiltonian with finite values. Symmetries of the problem
(time-reversal, translation, etc.) dictate that there are 5 independent
parameters that characterize disorder. Konstantin pointed out that the
self-consistent Born approximation does not work for Dirac spectrum as there
are many logarithmic corrections that need to be resummed using the RG
methods. The main conclusion is that all 5 impurity constants grow under RG flow.

The work considers a finite filling fraction and proceeds by looking at the
free energy functional using the
supersymmetry approach. The main conclusions are that, if one neglects the
scattering between bands, the system maps onto a symplectic ensemble,
resulting in antilocalization: increase in conductivity upon lowering the
temperature. However, upon lowering the temperature, intervalley scattering
becomes important, and one finds an orthogonal ensemble for which all states
are known to be localized.

The prediction is for a non-monotonous behavior of the conductivity with
temperature. As T is decreased, first the conductivity is reduced in accord
with the log corrections. At lower T antilocalization kicks in and the
conductivity increases when the temperature is lowered. At yet lower T, the
intervalley scattering takes over, and the conductivity drops to zero as the
states become localized. Konstantin concluded by saying that there is no
chance for minimal metallic conductivity due to generic disorder.

Assa Auerbach asked whether this behavior had been found numerically since
this is a non-interacting theory that lends itself easily to modeling.
Konstantin replied that exploring the phase space of 5 parameters is hard, and
that the localization length is expected to be large.

Claudio Castellani and Andrey Chubukov both asked what happens when graphene
is tuned very close to the Dirac point, i.e. filling fraction is small. The
system is right away in the strong disorder limit, but Konstantin believes the
conductivity still goes to zero at T=0 since the states are almost localized
already.

John Mydosh asked what are the potential impurities, and apparently these are
mostly charged impurities on the substrate.

Victor Galitskii asked what determines the crossover scale between
antilocalization and localization. There is no unique answer, this is related
to how different components of the impurity scattering vary with T.

Catherine Pepin asked whether interaction correction have been considered and
what they do. There seems to be some work done on that, but lunch truck was
about to arrive, and we stopped.

2007-08-17T10:09:00.000-06:00

Thursday, August 16, 2006
Andy Millis gave a talk titled "Are the cuprates really Mott insulators? Comparison of optics and DMFT"

Andy started by stating that this is a quantitative question which may have conceptual implications and went on to say "In 1987 Phil Anderson proposed that the high Tc materials were Mott insulators. I want to suggest maybe that is not quite true."

Andy reviewed the optical conductivity data for the cuprates in both the antiferromagnetic state and the doped state. In the AFM state the conductivity turns on around 1.8eV (for LCO) with a peak and a dip before increasing at higher frequencies. The peak structure near 1.8eV contains about half of the oscillator weight, and with doping this weight is shifted to a Drude peak at low frequency.

Chandra Varma argues that it is wrong to talk about electrons in a CuO2 antibonding band, and Andy responds that he will assume this is fine for scales below 2eV.

Andy poses the questions: How fast does the Drude peak fill in with doping and what does this tell us about correlations? He then turns to results from single-site DMFT to study this problem. This ignores spatial correlations but allows a description of the Mott insulator. The 1-site DMFT phase diagram for the half-filled Hubbard model shows a critical value of U, Uc2~1.5W, above which one has a Mott insulator at zero temperature, and a first order line extending to lower U at finite temperature. Comparing the spectral weight transfer below 1eV, Andy concludes that U~0.8Uc2 for LCO. In other words, the cuprates are not Mott insulators, but, rather, the gap at zero doping is due to the formation of a spin density wave.

Andy summarizes by saying that within the uncertainties of this model, the insulating behavior is due to magnetic order and the doping properties are not due solely to strong correlations, but also to magnetic correlations.

In the questions, there were objections to the use of 1-site DMFT and the absence of the physics associated with the energy scale J. Andy responds that cluster DMFT with J correlations gives support to his analysis.

Zlatko Tesanovic, continuing about SdH in superconductors.

2007-08-16T19:49:00.001-06:00

Thursday Aug 16th

Our Thursday morning Patio Discussion began with the continuation of Zlatko Tesanovic's
presentation about electrons in the vortex state of superconductors. Since yesterday, he
has written a set of notes on this topic, which are linked here (Zlatko's notes.)

Zlatko said he'd finished yesterday, but would take questions. Piers Coleman (the blogger here) asked him if he could give his opinion about the semi-classical approach that Grisha Volovik has introduced for describing the motion of quasiparticles in a vortex lattice.

In a moment of gest, Zlatko said that the main difference between his theory and Zlatko's theory, was that "He is wrong, we are right"! Actually, later, Zlatko made it quite clear that he has the highest opinion of Grisha Volovik. Here's the main point of the discussion. Volivik's idea, he said, is that in a uniform superflow of superfluid velocity v_s, the Green's function becomes

G(k,\omega) --> G(k, omega - k v_s)

where the change in omega is the famous Doppler shift. Zlatko said, this is good if there is a length scale to compare with, but in a d-wave superconductor, the nodes mean that the system is critical, so at absolute zero, the relevant length scale is infinite. In point of fact, once the temperature is finite, provided the thermal length is small enough there is a cross-over to the regime where the semi classical regime is OK.

Zlatko then gave a quick resume of the previous day's discussion.

Chandra Varma asked whether Zlatko could provide a simple criterion for the characteristic field or energy scale where the quantum nature of quasiparticle-vortex interactions becomes important. (Beyond Volovik's semi-classical theory). Zlatko did not directly answer Chandra's question, but made an alusion to work by Doug Bonn, who found a break-down of the Uemura scaling behavior rho~T_c in highly underdoped YBCO. In Bonn's work, the superfluid stiffness in 2D cuprate systems scales like T_c^2. I did not understand the link between this remark, and Chandra Varma's question.

Dan Dessau, U. Colorado: Laser ARPES studies of the cuprates.

2007-08-16T16:23:00.001-06:00

Delayed Blog from Tuesday Experimental Discussions
Laser ARPES studies of the cuprates.

Dan Dessau, U. Colorado

In his well-prepared and carefully worded talk, Dan told us about his
new laser ARPES data for Bi2212. He started by comparing the normal state
data for the spectral function along nodal direction,
obtained with 6eV laser and with higher energy synchrotron light sources.
The quasiparticle peak is much more narrow in the laser ARPES data.
Dan extracted the fermionic self-energy from his data. T0 a blogger, it looks
quadratic in frequency at small frequencies (as in a Fermi liquid). At higer frequencies, it crosses over to near-linear behavior.
He next presented his data on the temperature dependence of the Fermi velocity
along nodal direction. His temperature dependence is linear in T
and is quite strong -- velocity at 300K is about 35% larger than the
value extrapolated to T=0. Finally, Dan discussed isotope effect.
He argued that he and his collaborators didn't find a large isotope
effect on the high energy incoherent states. This result is in disagreement with earlier
measurements by the group at Berkeley. Dan, however, found a 3meV isotope
effect on the position of the kink in the dispersion below T_c. He argue that this indicated that that the kink may be due to interaction with phonons.

In the discussion following the talk, A. Millis asked why the absolute magnitude of his Fermi velocity is larger than in measurements by other groups. Dan replied that this may be due to calibration, and that one should focus on relative variation of $v_F$ with $T$ rather than on absolute values. C. Varma
pointed that in the electronic scenario for the kink in a $d-$wave superconductor, the kink appears at $2\Delta$, and
asked whether Dan measured the isotope effect on the gap value. Dan answered that he didn't.

Joerg Schmalian: Superconductivity in a Shastry-Sutherland model.

2007-08-16T16:09:00.000-06:00

Joerg discussed a special case of a doped Mott insulator, namely
doping the Shastry-Sutherland model, that is closely realized
experimentally in SrCu_2(BO_3)_2. The Shastry-Sutherland model
consists of an antiferromagnet on a square lattice with exchange
coupling J and frustrating couplings along alternating diagonals
with strength J'. For J'/J > 2 the ground-state was shown by Shastry and Sutherland to be given by a product state of singlets on the alternating diagonals. Later numerical results showed that this is
so for J'/J larger that approx. 1.44. In this regime, the ground-state is exactly described by a RVB wavefunction, i.e. a projected BCS wavefunction, making this case an appealing one for describing a doped Mott-insulator within the RVB frame.

The doped cases include single-particle hopping t along the bonds
of the square lattice and t' along the alternating diagonals. Joerg
and his collaborators considered both electron and hole doping,
that in the frame of a t-J model correspond to a change of sign in
t' after a particle-hole transformation for the electron doped case.

The electron doped case had strong singlet correlations along the
alternating diagonals, i.e. strong spin correlations for sites i and j
on the same diagonal but decaying fast for i and j further away.
In spite of the fact that the RVB wavefunction was based on a BCS
one, no off-diagonal long range order (ODLRO) could be detected after projection, such that although local pairing was present,
ODLRO could not be preserved. Also, a finite Drude weight was
found. A question (Andy Millis) arose about Pauli susceptibility.

In contrast, the hole doped case showed d-wave superconductivity
after 3% doping. The amplitude of the order parameter was around
10% of the BCS value. Coexisting with such a state, plaquette
ordering was obtained on the plaquettes containing the alternating
diagonals. In such a state Joerg and collaborators found that
ODLRO is strongly enhanced in comparison to a homogeneous state.

Yong-Baek Kim remarked that he had slightly different results
using different methods (bond operators in contrast to variational
Monte Carlo). Chandra Varma asked if there is any property that
differs from BCS after projection. Zlatko Tesanovic remarked that
at half-filling, after full projection, gauge invariance is restored.
Finally, it was asked whether Dzyaloshinski-Moriya interaction
present in the actual materials may change the situation.
Although they were not considered explicitely by the authors, the
expectation was that they would mostly change the triplon
dispersion but not the findings discussed by Joerg.

Predrag Nikolic (Harvard): "The role of vortices in unconventional properties of d-wave superconductors"

2007-08-16T15:30:00.001-06:00

Nikolic presented his work with Sachdev dealing with quantum nature of vortices and their interactions with fermionic degrees of freedom. He was seeking to understand several STM experiments on vortex cores, their immediate neighborhood as well as the observed 4x4 checkerboard modulation in underdoped cuprates. He was also interested in understanding the shape of the “Nernst dome” on the underdoped side of the phase diagram.

The basic calculation is as follows: take a single vortex. Let it fluctuate in quantum time from its equilibrium position and compute its generic quantum action. Focus on the quasiparticle contributions to various terms. This is very sensible since vortex cores in cuprates are very small and basically gapped so we do not expect much contribution from the conventional Bardeen-Stephen effect. Instead, the major effect must come from nodal quasiparticles which are present even for H = 0.

Nikolic makes FT transformation to reduce the effective Hamiltonian for nodal quasiparticles to a Dirac problem with two gauge fields describing the influence of vortex on quasiparticle motion. He then computes the response of this Hamiltonian to small quantum fluctuations in vortex position. The results are the following: first, there is no Ohmic dissipation. The reason is that the nodal fermion density of states vanishes at the Fermi level. This can also be understood by a general scaling argument using the fact that massless Dirac fermions represent a nice quantum critical system (with z = 1). Second, they compute the vortex mass and find it to be very low, of the order of only a few electron masses. These two results confirm the speaker’s expectation that vortices in underdoped cuprates are highly quantum objects and that they can be used as such to construct useful effective theories of cuprates.

Using the above results, Nikolic, Sachdev and collaborators argued that the STM structure within vortex cores (the minigaps observed at ~ 7 meV or so) can be understood as arising from quantum fluctuations in vortex core position within a vortex lattice. Furthermore, strong quantum vortex-antivortex fluctuations in zero field can be used to construct the so-called dual theory of fluctuating superconductors. This theory takes the form of a dual Hofstadter-Abrikosov problem with vortex-antivortex (bosonic) “particles” moving on a lattice and in a dual magnetic field fixed by the density of original electrons. The condensation of such dual bosons naturally leads to suppression of superconductivity in favor of a charge density wave of Cooper pairs. Such tendency is the strongest near x=1/8 where pairs of holes (electrons) can arrange themselves neatly into 4x4 checkerboards.

Several questions were raised by members of the audience. Millis inquired about dissipation and how much of it was observed in actual experiments. Norman pointed out that the STM minigaps in vortex cores were found to scale with Delta (Fischer et al). Nikolic responded that their scaling was somewhat different. Auerbach stressed the importance of the “hydrodynamic” contribution to the mass (as opposed to the quasiparticle one) and a lively discussion ensued which was brutally interrupted by a lunch truck (see Comment by Sachdev on how the hydrodynamic mass is dealt with in their approach).

Much of the physics Nikolic spoke about can be found at www.arxiv.org/abs/cond-mat/0511298 and in references therein.

Chandra Varma: Derivation of Quantum Critical Fluctuation spectra for Orbital Currents

2007-08-16T15:25:00.000-06:00

Chandra started by drawing a phase diagram of high T_c cuprates with the quantum critical point inside the superconducting dorm and finite temperature transition line terminating at that critical point. He said he will describe the formulation of the microscopic theory that gives this phase diagram and explains the physics almost everywhere in the phase diagram.

He briefly mentioned the marginal Fermi liquid phenomenology that says many things can be explained if one assumes a q-independent dynamical response function,

\chi (\omega, q) = \chi_0 tanh (\omega/2T) for \omega < \omega_c, where \omega_c is some cutoff. In the putative quantum critical region, he said that

"Lots of things have been predicted, and no alternative has been found".

The question is "how does this spectrum arise?." Chandra began with the description of the ordered phase associated with the quantum critical point. He said 1) the cuprates are charge-transfer insulators and large U is present both at the Cu and O site. 2) The interaction between Cu and O is responsible for new phases. He wrote down the interaction term that involves the repulsive interaction between the density at the Cu site and that at the O site, namely V \sum_i n^{d}_i (n^p_{i+x} + other three neighboring O sites). He said this interaction can be rewritten in terms of quadratic forms of the currents, -V/4 \sum_i (J^2_{ix} + J^2_{iy}) + ... Here J_{ix} and J_{iy} describe the currents through the O and Cu sites in the horizontal and vertical directions. He said the mean field theory gives a local minimum that does not break translational symmetry. This solution is also characterized by finite expectation values of J_{ix} and J_{iy}.

He calls these currents "coherent" parts or the currents that order. He draw an example of such a (translationally symmetric) ordered current pattern. He said that there is now evidence for such a current-carrying state in the pseudogap region of the phase diagram.

He said that he is now going to present the derivation of the marginal Fermi liquid spectra starting from this picture of the current-carrying state and the corresponding quantum critical point. He pointed out that there are four possible current-carrying states with the broken time reversal symmetry and they are characterized by four possible directions of "staggered" magnetization within the unit cell. He then claimed that the effective model describing these four states are the so-called Ashkin-Teller model or two-coupled Ising models. This model has two kinds of terms; J_2 describing the spin-spin interaction for each Ising spin degrees of freedom \sigma and \tau, and J_4 that involves energy-density-energy-density interaction of two kinds of Ising spins, namely

H_{AT} = J_2 (\sigma_i \sigma_j + \tau_i \tau_j) + J_4 (\sigma_i \tau_i \sigma_j \tau_j).

He said that in some range of J_2/J_4, basically a Gaussian theory is valid. Here the model is supposed to be equivalent to an XY model with a four-fold anisotropy;

H = \sum_{ij} \kappa (J_2,J_4) cos(\theta_i-\theta_j) + h \sum_i cos(4 \theta_i).

Here the XY degree of freedom correspons to the direction of the "staggered" magnetization within the unit cell. Thus the ordered phase of this model is supposed to correspond to the current-carrying state mentioned above. He claimed that the four-fold anisotropy is irrelevant in the fluctuating regime (disordered state) while it is relevant in the ordered phase. He then said that, according to the analysis of the Ashkin-Teller model, the specific heat is completely smooth across the finite temperature ordering transition (to the current-carrying state or the Ising-symmetry-broken phase); thus it is expected that there will be no anomaly in the specific heat across the transition.

He went on to describe the effective model in the quantum fluctuation regime. He added a simple dynamic term for \theta and the Ohmic dissipation term proportional to |\omega| (which is supposed to arise after integrating out underlying fermions). He calls the strength of this Ohimic dissipation term, \alpha. He said this model has a quantum phase transition at \alpha_c = 4\pi and described the finite temperature phase diagram where the finite temperature transition line terminates at the critical point at zero temperature. Then he said this model allows the "exact" computation of the correlation functions. He said this is achieved by some clever trick that leads to the separation of two degrees of freedom that depend only
on space and time, respectively. As a result, the total partition function can be written as the product of two parts; each one involves only either the space or time fluctuations. The consequence, he said, is that the "staggered" magnetization, M (now it became an XY degree of freedom in the disordered regime) within the unit acquires a peculiar form of the correlator that is completely local in space;

\delta (r-r') 1/(\tau - \tau').

The Fourier transform of this correlator gives the marginal Fermi liquid spectrum in the frequency-momentum space.

He then turned to the question of superconductivity. He said the Ising degrees of freedom would couple to the underlying fermions; this coupling has the form of the current-current
interaction where the "coherent" part of the current (or the collective part) couples to the fermion current. He said the effective four-fermion interaction arising from this current-current interaction is strongly momentum dependent and gives rise to an attractive interaction in the d-wave channel.

Several questions were asked after his 15-20 mins presentation.

Andrey Chubukov asked how the strongly-momentum dependent effective interaction can give rise to the momentum-independent self-energy expected in the marginal Fermi liquid. Chandra said if one works with the circular Fermi surface and the q-independent correlator, the self-energy turns out to be momentum independent.

Catherine Pepin asked whether there is anything one should worry about the transport coefficient because after all this is a q=0 fluctuations. Chandra said there is no vertex correction.

Piers Coleman asked how and why the partition function can be written as the product of two contributions that only depend on space or time. Chandra started with a Villan form of his effective model and said that a clever choice of two orthogonal degrees of freedom (integer fields in the Villan action) leads to this construction.

Philip Anderson said there must be some peculiar response to the magnetic field; the "staggered" magnetization would become asymmetric within the unit cell and it will lead to some kind of distortion. Chandra said such a piezo-magnetic effect does not arise in his state because of some symmetry reasons.

Mike Norman asked whether there is any consequence from some kind of chirality (in the current) fluctuations. He mentined the case of MnSi where some kind of chirality effect has been discussed. Chandra said this may be a different issue.

Patio Discussion on Shubnikov de Haas oscilations in the cuprates

2007-08-16T14:55:00.000-06:00

On Wednesday morning, the group met for a very animated and very exciting Patio discussion about the implications of the Shubnikov de Haas oscillations recently observed by the Taillefer group (Doiron-Leyraud et al.) at high fields in YBa2Cu3O6.5 and more recently in the double layer YBa2Cu4O8 compound (Bangura et al arXiv:0707.4601). There is a lot of excitement about these measurements, which may be linked to the mysterious Fermi arcs seen in underdoped cuprates using ARPES spectroscopy.

The discussion was hosted by Andre Marie Tremblay.

Andre summarized the key observations. In underdoped YBCO with a nominal hole doping of

p=0.1

The effective mass of the carriers, obtained by fitting the temperature dependence of the oscillations (See c) is

m* ~ 2-3 m_e

Measurements were made at above 50T on YBCO. Paradoxically, even though the SdA oscillations suggest small hole pockets. Andre Marie discussed how band theory can not account for these small hole pockets.

Here are some of the key issues that came up in the discussion

If the measurement is made on hole pockets, then why is the Hall constant negative (corresponding to electrons)?
Are we sure that 60T - or even higher 80T measurements are really in the "normal state".
The huge size of the pseudogap, the observations of the Nernst effect all suggest that the flux flow regime of the underdoped cuprates may extend far further than these fields.
Do the oscillations represent conventional Schrodinger Landau level oscillations (possibly damped by pair fluctuations) - or could this be some kind of Landau level quantization of quasiparticles - even Boguilubov quasiparticles around a nodal point?

Five speakers then gave brief presentations.

Subir Sachdev discussed the effects of holes moving in an antiferromagnetic, or possibly a quantum critical spin background. In a antiferromagnet, the unit cell is doubled, and work carried out long ago by Schraiman and Siggia, supported by numerous subsequent work leads to the prediction of two hole pockets, so now

n = 0.075/2

and the hole density of 0.075 is closer to the nominal p=0.1 - there is a smaller discrepancy with Luttinger's theorem.

Subir asked: is it possible to get hole pockets without broken symmetry. He argues (see previous blog) that if there is topological order, with gauge excitations, one can have holon pockets - which are spinless - invisible to ARPES but which still give dHva and SdH oscillations.

The audience cruelly asked Subir if this scenario predicts a negative Hall constant. Subir admitted that it probably would not.

Subir also discussed the possibility that a superconducting-insulator transition might be able to give a negative Hall constant, but there was not enough time to pursue this point.

Michael Norman gave a brief review of earlier attempts to carry out quantum oscillation measurements on cuprates at high fields. He pointed out that three other measurements gave
oscillations in broad agreement with the Taillefer measurements - if less reliably.

He also mentioned Zlatko Tesanovic's work on dHvA in the mixed state - the main point here is that the average gap around a quasiparticle orbit is zero in a d-wave superconductor. Providing that the gap is smaller than the cyclotron frequency, one can have Landau Levels.

Such oscillations have been seen in conventional superconductors too - such as V3Si and NbSe3.

Norman also discussed the Hall constant, which changes sign as a function of doping in the cuprates in the flux phase. Larkin et al made a theory of this, correlating it with dT_c/dmu,
but the sign was wrong. He reported that whereas old measurements had found negative Hall constants over a narrow range of the phase diagram - Taillefer now finds it extends over a large region at high fields, also in the 248 material.

Mike raised the following questions

Is there a conflict with Photo-emission?
Are the pockets electrons or holes? He noted that this can be determined by looking at the relative phase of the rho_xx and rho_xy quantum oscillations, or M and rho_xy - and work is now underway in this direction?

Mike discussed whether the observed pockets might somehow be associated with the "banana-shaped" surfaces of constant quasiparticle energy that are known to exist around the nodes of the d-wave superconductor. He pointed out that since the ratio of v_Delta/v_F ~ 20, these long, thin regions would have to extend out to the zone edge to get the right areas.

He ended by asking

Is it a field induced effect?
Is it a field induced state?
Is it oscillations around a nodal particle-hole ordered state like a d density wave?

Phil Anderson then gave a brief presentation. Phil pointed out that we really don't know Hc2 for these systems. He argued that we know that T^*, the pseudo-gap temperature scale sets the scale of the gap anti-nodes - this is a pairing energy at energies of order J/2 he said, which is several 100 meV. This is much larger than the observed fields, so he said, crossing Hc2 is simply out of the question, even at 80T.

Anderson then went on to talk about the Fermi arcs. Here's his argument, as close as I could capture, verbatim

"Near the nodes, there's still a gap (at these fields) and its fairly hefty. The Fermi arcs form by the electrons buming against the gap in the anti-nodes, giving rise to Andreev scattering. In Andreev scattering a pair of electrons go on, a hole goes back in the opposite direction. The arcs form because electrons are bumping against the gap. Its Andreev reflection. "

Phil mentioned an old paper of his, which he said that Marcel Franz and Zlatko Tesanovic (in the audience) had effectively destroyed- but he felt it is still relevant. He said that when an electron turns into a hole through Andreev scattering - it leads to a kind of Zwitterbewegung. He drew a picture of electrons scattering back into holes inside a square well.

Phil ended by saying that he worries that all the current proposals don't take into account
Andreev scattering, and Andreev scattering is not normal scattering. The advantage of such a scenario, he said, is that the sign of the Hall effect would not be important.

As you can imagine a lot of discussion followed. Here's a brief summary

Chandra Varma:" We would like to know Hc2"

Phil Anderson: "Its some kind of Vortex matter state."

Chandra Varma: "These are just murky words that don't amount to anything very much."

Zlatko Tesanovic: "There is very strong sign of some kind of vortex matter state"

Chandra Varma: "Zlatko has solved this problem at large field ,he should talk by himself."

David Pines: "I'm quite taken with Phils notion that one has to understand how the quasiparticle goes (Andreev reflects) around the Fermi surface. The gaps are so big that they are not going to be affected by the magnetic fields."

Next up came Assa Auerbach. Assa presented a low energy Hamiltonian called the plaquet fermion model, which he argued has features that can account for the observed phenomena. The plaquet fermion model is a Hamiltonian containing mobile 2e bosons and mobile fermions that scatter via Andreev scattering. The dispersion and existence of these objects he argued, can be determined from finite size diagonalization of the Hubbard model on small plaquets.

The blogger is not sure he understood the full gist of this theory, but he said that the f-fermions in his theory have fermion arcs, and when the bosons condense, this produces a standard BCS dispersion

E(k) ~ Sqrt[(epsilon(k)-\mu)^2 + (d_k b)^2]

where epsilon(k) is the dispersion of the electrons moving on a small pocket.

Chandra Varma argued that this is not consistent with ARPES, where one has never seen two peaks in the spectrum. Mike Norman agreed with him.

Muramtsu argued that the finite plaquet diagonalizations were, in effect, holes moving in an antiferromagnetic background - in other words - he felt that Auerbach's pockets were really hole pockets in an AFM. Assa Auerbach disagreed strongly.

After this, by popular request, Zlatko Tesanovic stood up to discuss his ideas, developed with Marcel Franz, on the theory of quantum oscillations in superconductors. Zlatko has since posted a set of notes on these ideas, which you can obtain here. Zlatko began by remarking that the main issue divided into whether the underlying order was

particle-particle (pairing or pair fluctuations)

particle-hole (density waves, circulating currents..)

In this latter category would like graphene, d-density waves, Chandra Varma's theory of circulating currents (see blog by Y. B. Kim).

Conventional normal state de Haas van Alphen, he said, is caused by the Landau quantization of the Fermi sea. According to conventional wisdom, there could be no dHvA oscillations in a superconductor, because the gap mean there were literally, no states that could undergo any sort of Landau Quantization. ("No density of states, no oscillation").

However, both experiment, and detailed theory, paint a different picture. If one lowers the field at low temperatures, he said there were three regions

I - Above Hc2 - normal dHva oscillations

II- Below Hc2, where Delta is smaller than the cyclotron frequency - dHvA oscillations with the same frequency, but damped by the pairing

II - Region III, where Delta is larger than the cyclotron frequency -here - dHvA dies.

He made a remark that this was an example of KTN^2 - a kind of topological transition.
I did not understand.

Zlatko then turned to the situation in d-wave superconductors. He referred to the ideas of Gorkov and Schrieffer, and later Anderson - who proposed that in a d-wave superconductor, some sort of Landau quantization would occur around the nodes of the 2D d-wave sc.

Unfortunately, this idea turns out to be wrong at the lowest energy scales, because the phase of the superconductor has to be taken into account, and when this is done so- the gauge field associated with it cancels the effect of the field for those quasiparticles that wind around the vortices in the mixed state. There are he said, unfortunately, no Dirac Landau levels
in a d-wave superconductor.

He sketched the energy levels of the superconductor, showing how the Schrodinger Landau levels ultimately die as one reduces the energy down towards the node. Here's the point he said. In a Dirac Landau Level, the Hamiltonian looks like

However, in a field, the field is replaced by p-> p - e A, so that this becomes ( a minimal gauge coupling)

The change of sign in the coefficient of A for particles and holes is significant. This is different to the situation in graphene, where the vector potential couples to both diagonal elements with the same sign.

Zlatko's discussion was interrupted by lunch, and resumed on Thursday morning.

Zlatko's online notes on electrons in the mixed state.

Weds 15th Aug. Implications of the Shubnikov de Haas Oscillations in the under-doped cuprates.

2007-08-15T16:55:00.003-06:00

Wednesday 15th Aug. 11.30am - 12.30 am
(Patio).

Discussion, led by Andre-Marie Tremblay

What are the implications of the Shubnikov de Haas Oscillations in the under-doped cuprates?

Subir Sachdev, Michael Norman, P. W. Anderson, Assa Auerbach and Zlatko Tesanovic

Greg Stewart : ``Short review of non centro symmetric superconducting compounds/recent results on CePt3Si''

2007-08-15T15:38:00.000-06:00

Greg Stewart presented an experimental study of the non centro-symmetric superconductor CePt3Si. The main issue here is to understand the pairing state when the superconductor is lacking an inversion center. Parity is no longer a symmetry of the Cooper pairs, so that the pairing state is expected to be a mixture of even parity singlets and odd parity triplet pairs.

Superconductivity in CePt3Si develops inside an antiferromagnetic phase with Tc=0.75 K and TN= 2.2K. The extrapolation to zero temperature of the specific heat coefficient inside the supercondcuting phase seems to suggests the presence of nodes in the gap. The large enough value of the specific heat coeficient in the normal state (C/T ~ 160 mJ.mol.K^(-2)) shows that the compound has some heavy Fermi liquid character. Then Greg showed a cleaner compound by the group of Takeushi, which shows a bigger normal state specific heat coefficient (\gamma_n ~ 335 mJ mol K^(-2) ) as well as a bigger residual specific heat coefficient in the superconducting phase. The overall decrease of the thermodynamic coefficients with disorder reamains mysterious. Greg then compared CePt3Si with various othernon centro-symmetric superconductors recently sudied, such as Li2 Pt3B. It appears that in some compounds the specific heat coefficient has large values but in some others it has small values,!
so that not all of them can be considered as heavy fermions. Moreover Greg says that some of the compounds have residual specific heat coefficient in the superconducting phase while some others don?t, so that there is no systematic evidence for nodes.

The discussion became lively about what one can expect for such sueprconductors. It was asked by Chandra varma whether one could find at least one of these superconductors which is not at the same time antiferromagnet. Greg's answer was that there is not any. Then the discussion went on the fact that one theoretically expects triplet superconductivity, at least for part of the pairing. To show this experimentally is very difficult, Greg said.

Kazushi Kanoda, Spin liquid behavior in organic charge transfer salts

2007-08-15T14:40:00.000-06:00

Kazushi presented his results for spin liquid behavior observed in a member of the two dimensional charge transfer salt family κ-(ET)₂X with counter ion X=Cu₂(CN)₃.

ET-molecules form a two dimensional pattern where two molecules form dimers and dimers form an anisotropic triangular lattice. The counter ions take one electron out of a dimer, leading to a half filled band. Kazushi stressed that in the organics, different ground states can be stabilized via changing the counter ions and tuning the pressure.

In distinction to the ambient pressure Mott insulator κ-(ET)2X with X=Cu2[N(CN)2]Cl,which undergoes antiferromagnetic long range order below 27K, the triangular lattice of the X=Cu2(CN)3 compound is almost isotropic. The key observation from NMR, µ-SR and susceptibility measurements is the absence of magnetic long range order down to 20mK (much smaller than the estimated size of the exchange interaction J~250K).

In the discussion it was pointed out that the Heisenberg model on a triangular lattice has an ordered ground state, but that a disordered ground state can be stabilized by ring exchange terms that are expected to be important close to a Mott transition. Indeed the material undergoes a pressure induced Mott transition from insulator to metal at p=0.3-0.4GPa.

Next to the existence of a Mott insulator without broken symmetry, the most spectacular observation of Kanoda and collaborators is the fact that the low T behavior of the system seems gapless. The heat capacity vanishes linearly with T and the susceptibility is finite. This is consistent with the observation that the entropy of the spin liquid is larger that the entropy of the pressure induced superconductor as deduced from the Clausius-Clapeiron relation.

While there seem to be no long range order at low T, ¹³C-NMR, thermal conductivity and heat capacity measurements show indications for the onset of some inhomogeneous state below T=5K. Also, the spin lattice relaxation rate decreases with T^3/2 below T=5K (it decreases as T^1/2 at higher T). A broadening of the ¹³C-NMR lines and stretched exponential relaxation is the key argument supporting a magnetic field induced inhomogeneous state.

Open problems in this context are:

-is the gapless nature of the ground state related to the anomaly at 5K and the onset on inhomogeneities?

-what is the nature of the superconducting state? Is it different from other organics? (In this context Kazushi mentioned that the Knight shift in the superconducting state does not seem to vanish)

-Why is the charge response of this system rather different from other organics?

Buttom line: This material seems the first gapless spin-liquid in a quasi two dimensional Mott insulator. Upon pressure it becomes a superconductor. So far no theory was able to account for the rich behavior of this system. It is clearly a very sharply defined outstanding challenge!

Relevant papers:

Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, ¹ Phys. Rev. Lett. 91, 107001 (2003)

Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, and G. Saito, Phys. Rev. Lett. 95, 177001 (2005)

Frank Steglich: Superconductivity of heavy electrons: New Insights into the Enigma

2007-08-14T16:32:00.000-06:00

Frank Steglich (MPI CPfS, Dresden) Pictures to be added.

Frank Steglich discussed some new insights into CeCu2Si2. CeCu2Si2 is the first heavy fermion superconductor to be discovered. Thirty years later, this material continues to provide new clues into the nature of the glue involved in heavy electron superconductivity. In the last 5 years or so, it has become apparant that the phase diagrams of heavy electron materials are frequently controlled by a quantum critical point - and in this respect, CeCu2Si2 is no different.

The ground state of CeCu2Si2 is highly sensitive to the method of preparation, and there are two limiting types of sample - "A" and "S". The former are antiferromagnetic, whereas S samples are superconducting. The S-type samples have a slight excess of Cu.

"A" type samples have the property that the application of a pressure drives T_N down to zero, where they pass through a quantum critical point arpimd 0.7GPa. In the vicinity of the QCP, the A type material becomes an S-type material, and develops heavy electron superconductivity.

In the vicinity of this QCP, A-type CeCu2Si2 appears to behave as a classic quantum spin density wave, or "Millis-Hertz" quantum critical point. In particular,

resistivity rho(T) ~ T^1.5
Cv/T = gamma(T) = constant - b T^ 0.5

(See Gegenwart et al, PRL, 1501, 1998, Sparn et al, Rev. High Press Sci Technol. 7, 431 (1998)). These features are hall marks of a quantum critical point associated with the formation of a spin density wave.
Frank showed us recent data that has confirmed this hypothesis - showing a clear incommensurate magnetic peak in the elastic neutron scattering around

Q=(0.226,0.226, 1.467).

(See Stockert et al, PRL 92, 136401 (2004) ). Frank asked the question - do these soft magnetic fluctuations provide the glue, at the QCP, that drives the superconductivity?

He then showed us some hot new data taken by Oliver Stockert and collaborators at the MPICPFS in which a "resonance" has been observed in the inelastic neutron scattering
results, in the superconducting phase. As one cools below Tc, this resonance develops at the incommensurate Q vector of the antiferromagnet, and the energy of the resonance grows as the temperature is reduced, maxing out at 0.22meV. This energy is closely related to the gap of this superconductor, and indicates that 2Delta/ k T_C ~ 4.3, around the right value for a weak coupling sc.

Frank Steglich argued that these results support the idea that the incommensurate magnetic fluctuations provide the glue that drives the pairing in S-type CeCu2Si2. In this respect,
this material is quite close to UPd2Al3, which develops superconductivity around 2K. This material has an acoustic crystal field excitation around Q=(0,0,1/2) which can be directly related to an anomaly in the conductance. In UPd2Al3, a Macmillan-Rowell-Eliashberg analysis was able to relate the neutron data with dI/dV, providing support for the idea that here too, the superconducting glue is provided by a lightly damped, acoustic excitation.
(See Sato et al, Nature 410, 340 (2001). )

However! This is not the whole story, for it turns out that one can apply chemical pressure to CeCu2Si2, by alloying with Germanium, to form CeCu_2(Si_1-x Ge_x)_2. When one does so, one finds that a second "island of superconductivity" develops. at an effective pressure beyond the quantum critical point. (See Yuan et al., Science 202, 2104 (2003) , Yuan et al, PRL 96, 047008 (2006) ). This second island of superconductivity is believed to be connected with a valence instability. Various groups have suggested that here, the soft excitation responsible for the glue, is a soft valence fluctuation.

These results lead Frank Steglich to conclude that more than one pairing mechanism
is operating in heavy electron systems - one has signs that pairing is produced by

valence fluctuations
antiferromagnetic fluctuations
ferromagnetic fluctuations
quadrupole fluctuations (skudderudite compound)

Frank ended with a note of caution - remarking that there also appear to be heavy electron systems such as the CeCoIn5 and YbRhIn5 that do not fit so naturally into a critical spin, or valence fluctuation picture. These "hard" quantum critical systems remain an open challenge, he remarked.

Remarks and questions from the blogger:

Can we really be sure that there is such a diverse set of pairing mechanisms? If so - what is the unifying thread between them?
What has happened to the Kondo effect in systems like CeCu2Si2 - how can a density wave form without this effecting the screening of the ions - in which case - can we really be so sure that a density wave scenario works in this case?
If CeCu2Si2 has a magnetic and a mixed valence pairing mechanism - then why is it that the superconducting domes for these two mechanisms merge together? It would seem quite unlikely that two completely different pairing mechanisms would produce share a common superconducting ground state.

Week 3 Experimental Talks and Discussion

2007-08-14T12:59:00.000-06:00

Tuesday, 14th August.

10.30am Flug Auditorium.

Kazushi Kanoda (U. Tokyo)

"Quasi-two dimensional organic conductors" (nominal title)

Dan Dessau (UColorado)

"New insights into the cuprates from Laser ARPES" (nominal title)

Frank Steglich (MPI CPfS, Dresden)

"Superconductivity of heavy electrons: New insight into the enigma"

Greg Stewart (UF)

"Short Review of non-Centrosymmetric Superconducting Compounds/Recent Results on CePt3Si"

M. Norman, 8/9, "What is a Fermi Arc ?" P. Hirschfeld

2007-08-13T10:06:00.001-06:00

Delayed Blog Posting of M.R. Norman talk 8/9/07 “What are Fermi Arcs?”

Definition of Fermi arcs observed in underdoped cuprates, particularly Bi-2212: spectral function in pseudogap state near (pi,0) with k on Fermi surface is pulled back from Fermi level after normalizing by Fermi function. Thus max of A(k,w) is at finite w<0, style=""> On the other hand, as one goes towards the node, this pullback disappears, and maximum is now centered at w=0, indicating gapless normal metal like excitations, and the dispersion through the Fermi surface is observed.

Define the point in k-space on the Fermi surface where the gapped character disappears and the gapless character begins as the endpoint of a Fermi “arc”. The distance in k-space along FS of this point from the node is called the length of the arc, and it has been measured by Kanigel et al Nat. Phys. 2006 to depend linearly on the temperature (note this is not a trivial effect of thermal smearing since f(w) has been divided out).

One can’t go all the way to zero, since SC interferes, but the implication is that the pseudogap state at T=0, if one could get there, is a nodal liquid. Mike quoted Taillefer’s result showing that the limiting T->0 thermal conductivity in the SC state is continuous across the underdoped Tc->0 transition as indirect support for this point of view.

One alternate point of view which has been discussed is question of hole pockets instead of arcs, discounted until recently since the “back side” of these pockets has never been seen. These should be weak because of matrix element effects, however, and apparently there is a report of their observation by Valla et al. Mike is skeptical, as this data has been discussed for some time but not published. Questions about relation with quantum oscillations observed by Taillefer postponed because this deserves session to itself.

As one goes through Tc, arc length is observed to collapse, i.e. length vs T goes like some power greater than one. Since we know nodal lifetime is collapsing (to some extent even in BSCCO) in SC state, could this simply be reflecting d-wave + lifetime effect?

Gap extracted by usual methods is flat in underdoped samples – 2-gap effect?

Question by blogger: Loram, Tallon pointed out recently (cond-mat that arclike physics could be obtained with large T-dependent scattering rate in pseudogap phase modelled with constant d-wave like pseudogap. Then as true gap gets small close to nodes, scattering smears spectral function into single peak centered at w=0, looks like arc. Millis points out this idea is not new. Norman responds by saying that models of this type may describe spectral function at Fermi surface, but they will fail elsewhere in Brilllouin zone.

Final point by blogger: there is an inconsistency somewhere in the comparison of the underdoped thermal conductivity story and the nodal liquid picture, since the value of the thermal conductivity, interpreted in the universal transport scenario, implies an extremely large gap slope, in contradiction to ARPES and Raman determinations of same quantity. “Universal” interpretation of this value may not apply.

Boris Shklovskii: Superfluid-insulator transition with strong disorder

2007-08-12T15:57:00.000-06:00

(Delayed Blog posting from Week 2, Thursday 9th Aug).

Boris Shklovskii spoke about the critical density of particles, n, needed to enter
a superfluid state from a localized insulator. He considered the phase diagram
as a function of n and the Cooper pair size xi. For small xi, this becomes a boson
localization problem, while for large xi, we instead have to consider the metal-insulator transition of fermions (it is assumed that once the fermions become metallic, the BCS instability will convert the metal into a superconductor). Boris argued that bosons localize more easily than fermions: for bosons one has to compare the variation in the effective potential energy with the kinetic energy (of order 1/L^2, where L is the size of a wavefunction) of a single state, while for fermions the potential energy has to be compared with the Fermi energy. Consequently the critical density for small xi was parametrically larger than the critical density for large xi - this was one of the central results.

A crucial ingredient in the argument was the computation of the effective potential energy experienced by a single boson or fermion. This was determined in a classical (Hartree) manner, by adding the interaction energy to the random potential. In particular, Boris considered a model of a strongly compensated semiconductor, with many impurities of both positive and negative charges. Then a self-consistent non-linear screening argument was used, similar to that described in the classic book by Efros+Shklovskii.

Doug Scalapino, UCSB: Is There Pairing Glue in the Hubbard Model?

2007-08-09T17:23:00.000-06:00

Doug's presentation was his response to the recent article in Science magazine by Phil Anderson questioning the validity of a bosonic glue for the high temperature cuprate superconductors (Science 317, 1705 (2007)). Phil's contention is that the pairing is coming from the superexchange energy, J, which is in essence instantaneous in nature. As a consequence, he contends that a proper theory must be quite different from the strong coupling theories developed in the 1960s in regards to electron-phonon mediated pairing. The implication is that J is the "elephant", and that any dynamics causing bumps and wiggles in ARPES, tunneling, and optics spectra is the "mouse".

What Doug and his colleagues like Mark Jarrell have done is to use a dynamical cluster approximation to calculate the singlet pairing vertex in the Hubbard model for values of the Hubbard U ranging from 4t to 12t, where 8t is the bandwidth of the electronic states. This vertex, Gamma, can be thought of as a sum of an irreducible part, Lambda, plus induced interaction terms due to repeated scattering in the particle-hole channel. The latter can be divided into an S=0 and S=1 part. As the temperature is lowered, Gamma develops a momentum structure with a peak at q=(pi,pi), despite the fact that Lambda is structureless (the latter defines an effective U, denoted as U-bar). This behavior of Gamma is mirrored in the S=1 part of the induced interaction (the S=0 part is depressed around q=(pi,pi) instead), indicating that it is this term where the real action lies. They then write down a gap equation, and find out that the dominant eigenvalue displays the d-wave cos(kx)-cos(ky) behavior.

Now what about the dynamics? The pairing self-energy decays as a function of Matsubara frequency out to an energy scale of t, and this is mirrored by the frequency dependence of the dynamic spin susceptibility, chi. Equating the two, he finds that Gamma is equal to (3/2) (U-bar)^2 chi(q,omega), as expected for a spin fluctuation picture for the pairing. Therefore, Doug concludes that at the level of the Hubbard model, one can indeed think in terms of a pairing glue.

Next week, we will find out what Phil has to say about all of this.

Several questions were raised after Doug's talk.

Jorge Hirsch - What are the consequences? Doug - There will be structure in the ARPES and infrared conductivity that can be related to the frequency dependence of the normal and pairing self-energies.

Chandra Varma - Is this approach equivalent to RPA? Doug - Yes, except that the dynamic spin susceptibility is quantitatively different from what RPA gives.

Gabi Kotliar - What about analytic continuation, and how is this work related to my own? Doug - We have a lower Hubbard band, coherent structure near the Fermi energy, and an upper Hubbard band, as in your work.

Peter Hirschfeld - How does U-bar vary with doping? Doug - We don't know yet.

Chandra Varma - Does your pair vertex, etc., scale with the antiferromagnetic correlation length? Doug - We don't know yet.

Peter Hirschfeld: extracting new kinds of information from recent STM data

2007-08-09T16:13:00.000-06:00

Peter Hirschfeld

"Extracting new kinds of information from recent STM data"

Peter remarked that he was going to give an extremely rare type of STM talk - one without color pictures, presented on a blackboard. Two topics were presented :

The extraction of life-times from STM data
Modulation of the gap in response to the BSSCO supermodulation.

The first topic concerns a new method, developed by Aldridge et al. They have found, empirically, that one can fit the local density of states
with two variables, a local scatering rate, and a local gap. The fit form looks something like

Emprically, these two variables are correlated with the gap determined from the coherence peaks in N(r,E), i.e. Delta_1(r) is found to be basically the same as the former type of gap maps.
Now however, one can pull out a scattering rate as well. All of these variables follow the approximately 30A (five unit cell) correlation length seen in previous gap maps.

Hirschfeld argued that this indicates that the local DOS is therefore sensitive to disorder effects that must be on length scales that are shorter than 30A. The local Green function is determined by

and he argues that r-r1 and r-r2 must be smaller than about 30A for consistency.

One can also extract an effective scattering rate

What is the meaning of this?

Chandra Varma asked whether this is sensible, because the scattering is low-angle scattering?

Hirschfeld pointed out that the scattering rate extracted this way is about 10 times smaller than anisotropic scattering rates extracted from ARPES measurements. From STM

Gamma(max) = 10meV (p=0.08) - underdoped
Gamma(max) = 2 meV (p=0.02) - optimally doped

Comparable figures from Arpes are 100 and 40 meV respectively. The origins for this mismatch may be

Broadening effects of the inhomogeneity in oxygen content
Resolution of the ARPES

Hirschfeld then turned to the second topic. In Bisco, there is a 26A, or 4.8 unit cell modulation of the structure. Slezak et al have been able to map out the phase of the modulation, defining contours of constant phase across the maps, and from this, they are able to correlate the local gap with the phase of the modulation. It is found that the gap drops by 10% as the phase goes from zero to pi. In Boguilubov-de Gennes phenomenology, this corresponds to a reduction of the coupling constant by about 30% (?). Peter mentioned similar types of conclusion in a t-J model by Zhang and Rice.

Of course, perhaps if once one correlate this modulation of the gap with the structure, one can gain insight into the pairing mechanism of high Tc. Is this possible?

Hirschfeld pointed out that the supermodulation corresponds to a "tipping" of the octahedra, and that in the region where the gap is largest, the apical oxygens are furthest apart. This may cause the t' - next nearest neighbor hopping to be larger, giving rise to a large Delta.

Philip Phillips, UIUC: " Exact Integration of the High Energy Scale in Doped Mott Insulators"

2007-08-09T13:59:00.000-06:00

Philip points out a problem with the naive procedures used to integrate out high energy degrees of freeedom.
As an example, in eliminating the upper Hubbard band in order to
derive the low energy spectra of the large-U Hubbard model,one might miss the enhanced spectral weight for the addition spectrum of holes in the lower Hubbard band. (Comment: a fact which is missed in simple Hartree Fock theory.)

In his talk, Philips sets out to preserve the "2X" sum rule (i.e. the holes' spectral weight is -twice- their doping concentration, as hole doping pulls states down from the upper Hubbard band).

Philips' method of choice is an introduction of addtional fields, a fermion field which counts the high energy doublons (sites with two electrons), and a constraint field -phi- which projects the enlarged Hilbert space back to the original electrons states.
The result is a formally quadratic Lagrangian, with matrix (and space-time dependent) coupling parameters which is a starting point for a saddle point expansion. Its saddle point includes the t-J model terms and is argued to be a better description of the Hubbard model's low energy spectral weight. Predictions were made about a second dispersing peak which may be observed in the ARPES data.
Time limitations have restricted questions to a minimum, but Patrick Lee commented that
although the effective Lagrangian is formally correct, its fluctuations are large in the large U/t limit.

I argued that the traditional renormalization procedures (Brillouin-Wigner perturbative expansion, Real space Contractor Renormalization (CORE)) are somehat simpler and that the t-J model correctly captures the low energy spectrum. However in a later discussion with Philip and Patrick, Philip's approach was understood as an attempt to simplify the calculation of
quasiparticle renormalization and the intermediate energy scale excitations.

Thursday 9th Aug, Patio Discussion

2007-08-09T13:06:00.000-06:00

Thursday 9th Aug. 10.30am-1.00am Patio

Short talks and Discussion: provisional schedule

Philip Phillips, UIUC

"Exact Integration of the High Energy Scale in Doped Mott Insulators"

Boris Shklovskii, UMN

``A simple model of superconductor-insulator transition in Coulomb disorder"

Douglas Scalapino UCSB

``Is there pairing glue in the Hubbard model?"

Mike Norman ANL

"What is the Fermi arc?"

Peter Hirschfeld UF

"Extracting new kinds of information from recent STM data"

Ribhu Kaul: Lattice Deconfined Quantum Criticality

2007-08-08T08:20:00.001-06:00

In a beautiful and inspiring talk, Ribhu Kaul described his research on DQC ( a pun on QCD) - deconfined quantum criticality. He began with a review of the various ideas about spin liquids - mentioning two ideas -

the algebraic spin liquid that occurs in RVB, involving Dirac fermions in a gauge field.
DQC - the unusual fixed point involving deconfined spinons that is conjectured to lie between the valence bond solid and the Neel state in certain two dimensional antiferromagnets.

The physics of the DQC is thought to be described by the non-compact CP1 gauge theory - in which the fields are spinors interacting with a U(1) gauge field. The critical physics of this model is distinct from that of the O(3) sigma model.

The big question however - is how should one disorder the Heisenberg model? Frustration
is difficult to treat using numerical methods. There is a sign problem for Monte Carlo approaches and direct diagonalization can not reach lattices with more than about 40 spins. He described the ring exchange approach of Anders Sandvik, the so-called "J-Q" model, in which a nearest neigbor Heisenberg model has an additional term of the form

Q(S_iS_j - 1/4)(S_k.S_l-1/4)

where the spins are arranged around the plaquet. This model can be treated using an overcomplete RVB basis, and there are no sign problems.

Kaul described his new work with Roger Melko, to be found at http://arxiv.org/abs/0707.2961
where, by using an S_z basis, they have been able to show that the model has a kind of Marshall sign property, where all off-diagonal matrix elements in the Hamiltonian are negative. They can treat this model using Monte Carlo methods at finite temperatures.

They are able to see many interesting things in their simulation. They can measure the spin spin correlation function, and find that it has an anomalous dimension

eta = 0.35,

to be compared with eta - 0.038 for the O(3) non-linear sigma model.

One of the questions raised by Coleman, was whether these techniques can see the two correlation lengths expected in the deconfined quantum criticality scenario? Kaul reminded us that in this scenario, there are two correlation lengths = a "short" one that describes the size of the critical magnetic region, surrounded by a larger one, defining the length scale on which the spinons are deconfined. Beyond this length scale, one either develops valence bond, or neel order. The upper length scale is currently too large to be measured independently, but they certainly see the regime with power-law valence bond order.

Ribhu Kaul raised two key questions

Can one find another lattice model that displays the same anomalous dimensions?
Can one confirm this anomalous dimension by a lattice simulation of the CP(1) model?

Tesanovic mentioned that work on the non-compact CP(2) model seems to have ended with a first order phase transition. When questioned by Sachdev, Tesanovic cited this work - of Prokofiev (et al?) - but it seems that this may not apply anyway, becuase it was an x-y model with U(1) rather than SU(2) symmetry.

This blogger came away from this great talk with a very optimistic sense that genuine progress is bein made on the topic of deconfined criticality in spin systems. My questions: When will we have evidence that the same physics can occur with both spins and charges? Can we find a controlled expansion, eg a 1/N expansion, epsilon expansion, in which the observed anomalous exponent can be obtained approximately?

Subir Sachdev: "Theory of the Nernst Effect near the superfluid-insulator transition"

2007-08-07T14:52:00.001-06:00

Sachdev presented general approach to transport in quantum critical systems based on the (broadly speaking) Ginzburg-Landau-type field theories. An example of a 2D bosonic superfluid-insulator transition at integer fillings, with its 2+1 “relativistic” symmetry, was worked out in some detail. Elegant connections and parallels were made to various CFTs of highly supersymmetric field-theory models where the universal numbers, critical exponents and critical scaling functions entering quantum transport frequently can all be computed explicitly.

Sachdev stressed that the well known misfortunes of condensed matter physics, with its paltry supply of symmetry, limit our ability to compute quantum critical transport to the hydrodynamic (as opposed to collisionless) regime. Still, he showed how general hydrodynamic arguments and conservation laws in 2D can be effectively used to infer various transport coefficients from the knowledge of only one and how that particular one, say electrical conductivity, can be computed from the 2+1 relativistic quantum critical field theory. He then discussed the effects of perturbations taking one away from the ideal relativistic (particle-hole) symmetry, like the chemical potential, as well as finite magnetic field and impurity disorder. He formulated the connection between his theory and the phenomenology of cuprates and discussed how the calculated Nernst coefficient appears to fit the experimentally observed trends. He also briefly discussed how the theory can accommodate Dirac-type fermions which do not demand a finite Fermi surface. For those interested in the overall philosophy and inner workings of Sachdev’s theory the best resource is http://www.arxiv.org/abs/0706.3215 . Those interested in the Nernst effect in cuprates might also enjoy www.princeton.edu/~npo/VortexNernst/Nernst.html .

The presentation was punctuated and followed by a spirited debate: Patrick Lee noted that Sachdev’s Drude-like expression for conductivity implied significant temperature dependence and stated that this does not seem to be the case for the experimentally observed Drude part of optical conductivity in cuprates. Balatsky and Varma both inquired about the values of various parameters and their connections to measurable physical parameters of cuprates. In response, Sachdev noted the importance of the “speed of light” in the theory as a dimensionful parameter that plays a crucial role in phenomenology. Coleman wanted to know more about supersymmetric theories and to what extent was the loss of supersymmetry injurious to our ability to calculate everything analytically. Sachdev explained how loss of supersymmetry makes it difficult and often impossible to compute general \omega/T scaling functions; one ends up limited to the hydrodynamic regime where the conservation laws can be utilized to evaluate transport coefficients. Scalapino asked about pair correlations.

Cigdem Capan: Superconductivity and Quantum Criticality in CeIrIn_5

2007-08-07T10:35:00.000-06:00

Cigdem Capan began with a reprise of the key physics of Kondo lattice symmetries, describing how the physics of heavy electron physics is driven by a competition between the screening of the local moments and the RKKY interaction between them.

She raised three general key questions:

How does the single ion Kondo effect relate to the Kondo lattice effect?
Is the phase diagram organized by the singular Quantum Critical Points?
Does the presence of a QCP favor superconductivity?

Capan introduced the 115 heavy electron materials, already discussed in blogs last week. She introduced the Pagliuso phase diagram that links the Co, Rh and Iridium versions of these tetragonal systems. She mentioned CeCoIn5, for which a variety of measurements indicate dx^2-y^2 pairing. Her main focus is on CeIrIn5. This is a HF superconductor, thought to have a line of gap nodes, but the symmetry is currently unknown.

This was followed by a brief summary of the physics of CeRhIn5 - for which I will refer you to the previous talk by Joe Thompson.

The talk then proceeded to discuss CeCoIn5, where there is a field tuned QCP at the upper critical field, yet to be identified. Here, as one lowers the field towards Hc2=5T,

The linear specific heat grows and appears to diverge at Hc2 (Bianchi et al)
The A coefficient of the resistivity (rho(T) = rho(0) + AT^2), A diverges as one approaches Hc2 (Paglione et al)

It turns out that the QCP is not pinned to the top of the sc phase diagram. First, the position of the QCP is suppressed much faster than Hc2 under pressure. One can carry out an analysis of the Hall coefficient RH(T) which has an interesting field dependence. Cigdem claimed that the Hall data can be collapsed onto a single scaling curve, by scaling the field with respect to the field at which the Hall constant is a minimum. This procedure suggests that the QCP appears to be located at Hc =4.1T.

There is also recent work with tin (Sn) and Cadmium(Cd) doping. Tin doping fails to separate Recent Cd doping by Pham et al that indicates a link with antiferromagnetism. At the "top" of the Superconducting phase diagram, there is an additional phase - which may have links with antiferromagnetism and the FFLO incommensurate superconductor.

Returning to CeIrIn5 - here there is indication that some characteristic temperature scale in the specific heat drops with magnetic field, suggesting a QCP at 27T. The specific heat at high temperatures shows a transition, which appears to extrapolate to this same point. The talk focusses on this putative phase diagram. Here, more careful measurements suggest that at
the system narrowly misses a QCP at 27T, but makes a close "flyby", leading to a rapid evolution in the magnetization, or "meta-magnetic transition".

Sachdev suggested that this transition might be a very weak antiferromagnetic phase transition.

Capan showed dHvA data and argued that there is no significant change in the alpha orbits in the passage past the "metamagnetic transition". There is no significant change in the effective mass m* through the transition.

Varma pointed out that the observed masses were far too small to account for the huge linear specific heat (1000mJ/mol/K2) in this system. The amplitudes are however, anomalously depressed near the MMT - Capan would like to understand the origin of this anomally.

Key questions raised by Cigdem Capan:

Is there a simple phenomenology to understand the scaling of the Hall effect in CeCoIn5?
How do we undertand the electron hole assymetry in CeConIn5 (sn vs cd doping)?
What is the damping mechanism for the dHva oscilations at the MMT transition in CeIrIn5?
What is the origin of the upturn in the resitivity near the MMT transition in CeIrIn5?
Why is there no QCP in CeIrIn5?

There is an interesting dichotomy between the larger Fermi surface of CeIrIn5, and the much higher mass of the system. Are the f-electrons more localized - as suggested by the large mass - or are they more delocalized - as suggested by the large Fermi surface. Is the Co or the Ir closer to the antiferromagnetic instability?

Maxim Dzero: Symplectic spins and Pu 115

2007-08-07T09:23:00.000-06:00

Time reversal and the symplectic spin of the electron: application to Pu 115 superconductors

Collaborators: R. Flint, P. Coleman

(1) The discovery of the talk

- Superconductivity in the fluctuating valence compounds Pu 115 may arise from two-body interference between two-Kondo screening channels.

Results and contact with experiment:

- The superconducting critical temperature reaches its maximum when the energy levels of excited valence configurations are almost degenerate. This is the case of PuGaIn5.
- One can probably explain the fact that Curie Weiss behavior in these compounds ends at the critical superconducting transition temperature.
- It is predicted that the symmetry of the order parameter is determined by the product of the Wannier factors in the interfering conduction channels. For example, kz2(kx2-ky2).

Assumptions

- That it is the virtual valence fluctuations of the magnetic Pu configuration that create two conduction channels of different symmetry.
- That the two-channel Kondo lattice model is an appropriate description.
- That the mean-field theory corresponding to the large N limit of the symplectic Sp(N) representation of the SU(2) spins is an accurate description.

(2) Questions it raises

Subir: Superconductivity does not occur in this formalism in the single-impurity limit. What symmetry garantees that the V2 and D2 coefficients in the Hubbard-Stratonovich transformation are identical?
Answer: Particle-hole symmetry.

Does the Sp(N) large N solution correspond to the SU(N) large N solution?
Answer: Yes at mean-field but not for the fluctuations.

Why should we consider that the Sp(N) representation is better?
Answer: Because it gets rid of the "dipole" degrees of freedom of the SU(N) representation that do not transform like spin under time-reversal and charge conjugation symmetry. Sp(N) preserves that fundamental property of the physical spins.

Can the antiferromagnetic phase be described in this formalism?
Answer: One probably needs to use bosons.

(3) Questions left open

How big are the fluctuations, at least at the gaussian level?

Should we expect that all two-channel Kondo systems should have superconducting ground states?

Reference

Experimental Talks and Discussion: Week 2.

2007-08-07T07:36:00.001-06:00

Tuesday 7th July. 10.30am-1.00pm, Flug Auditorium

Experimental Talks, Short Theory presentations and Discussion

Cigdem Capan, LSU. (50 mins)

``Superconductivity and quantum criticality in CeCoIn5 and CeIrIn5"

Chandra Varma, UCR (10 mins)

``Brief update on experimental evidence for a Phase transition entering the pseudo-gap region of the cuprates".

Subir Sachdev, Harvard (30 mins)

``Theory of the Nernst Effect near the superfluid-insulator transition"

Ribhu Kaul, Harvard (30 mins)
``Lattice deconfined quantum criticality: The search for exotic physics in spin models"

Maxim Dzero, Rutgers/Columbia (30 mins)

``Time reversal and the symplectic spin of the electron: application to Pu 115 superconductors"

Nick Curro: Droplets of Magnetism in Cadmium Doped CeCoIn5

2007-08-06T08:14:00.000-06:00

(Delayed Blog posting from Week 1, Thursday, 2nd Aug).

Nick Curro described the results of a new series of NMR measurements that his group (see Urbano et al.) has carried out on Cadmium doped CeCoIn5, Ce(Co_1-x Cd_x)In5 . CeCoIn5 is a heavy fermion superconductor. One of the recent excitements has been the discovery by Pham et al, that the addition of Cadmium induces antiferromagnetism within the superconductor. There is a lot of similarity between the phase diagram of the Cd doped CeCoIn5 and the phase diagram of CeRhIn5. (See Joe Thompson blog, below). Cd is like "negative pressure" in the CeRhIn5 phase diagram. Thus the Cadmium helps to experimentally unify the physics of the 115 materials.

Nick described CeCoIn5 as a Kondo lattice material, in which mobile electrons move through a lattice of localized moments, interacting with the moments via an antiferromagnetic super-exchange J. He showed the Doniach phase diagram, and compared it with the phase diagram of this Cadmium doped material. Cadmium doping is loosely equivalent to "hole doping", and by reducing the size of the conduction sea, the system is driven to the left on the Doniach diagram. (See figure).

NMR measurements are carried out on the Indium sites of this 115 material - there are two indium sites - one of high symmetry, lying in the Ce plane (I) and another of low symmetry, out-of-plane. When NMR is carried out, the NMR line at the I site splits into three peaks, corresponding to three different environments (A, B, C). The A environment is the predominant "bulk" environment, and the 1/T1 signal shows the higher Neel temperature and the lower superconducting temperature. Two interesting features here:

The normal state above TN does not display a Korringa relaxation rate, but a slower T^1/4 temperature dependence.
There is no observed effect on the T^1/4 region, due to Cadmium doping.

But the SC Tc is Cadmium dependent, and at high dopings, completely disappears.

The B and C sites are exposed to progressively higher local Weiss fields, and indicate a non-uniform antiferromagnetic environment. Curro says these results show that the Cd is inducing antiferromagnetic droplets, and the C and B sites may correspond to the nearest and next-nearest neighbors, respectively.

In conclusion, the effect of the Cadmium seems two fold:

It changes the uniform bulk environment by reducing the hole density and uniformlysuppressing the superconducting Tc
It induces droplets of antiferromagnetism which percolate to produce long-range order.

Joe Thompson: "Proximity of Superconductivity and Magnetism in the 115s"

2007-08-03T13:14:00.000-06:00

Joe described his work with T. Park and other collaborators on CeRhIn5,
a member of the 115-family of heavy fermion materials that are layered
derivatives of the cubic CeIn3. The attached figure shows the
magnetic/superconducting phase diagram as a function of pressure,
magnetic field, and temperature.

At a sufficiently high magnetic field (above 9T), increasing pressure
induces a quantum phase transition (at P1) from an antiferromagnetic metal
phase to a non-magnetic metal phase. There is some indication that
the transition is second order. Yet, dHvA measurements of Onuki's group
find a jump in the Fermi surface, with effective mass showing a tendency
of divergence at P1. In the low-pressure AF phase, the f-electrons are
localized since the measured Fermi surface is similar to that seen
in the f-less reference material LaRhIn5. In the higher-pressure non-magnetic
phase, on the other hand, the same f-electrons are itinerant because
the measured Fermi surface is similar to that of the bandstructure
calculations in which these f-electrons are assumed mobile.

Joe went on to describe the pressure-induced transitions inside the
superconducting part of the phase diagram. For finite fields (which
are smaller than 9T), there is evidence for the second-order nature
of the transition from a co-existing AF+SC phase (which appears to
be homogeneous) to a pure SC phase. At H=0, the residual specific
heat coefficient was found to undergo a rapid decrease as the pressure
is increased through a threshold value (P2, smaller than P1),
as did the Fermi velocity fitted from the finite-T Hc2.

Joe suggested that the f-electrons remain localized in the SC+AF
phase. He did so based on the aforementioned dHvA observation
of localized f-electrons in the high field AF state, along with
the observation that the ordered moment at temperatures just above
Tc of the AF+SC phase is large.

There was discussion about how strong an evidence the above entail
for the localized nature of the f-electrons inside the AF+SC phase.
There was also discussion on the extent to which the change of \gamma
and v_F across P2 should be associated with the transition in magnetism,
or is instead a reflection of a distinction in superconductivity between
the co-existing AF+SC phase and the pure SC one.

Joe went on to describe the effect of Cd-doping in Co-115 and Ir-115.
A co-existing AF+SC region occurs in the Cd-doped Co-115, but is absent
in the Cd-doped Ir-115.

The main questions that Joe raised are:

* What is the nature of the 4f electrons when they participate
simultaneously in magnetism and superconductivity?

* If we accept the notion that the f-electrons are localized while
participating in the superconductivity, what implications does it
have for the non-Fermi liquid behavior in the normal state?

* In particular, does it suggest some form of Kondo breakdown?

Herb Mook: Search for Magnetism in YBCO Superconductors

2007-08-02T23:33:00.000-06:00

H. Mook, ORNL

“Search for Magnetism in YBCO Superconductors”

H. Mook described his work with P. Bourges and their collaborators at the Laboratoire Léon Brillouin (LLB), Saclay, France. The doping corresponded to O(6.6). The Oxygen chain ordering was striking and corresponded to the Ortho-II state (for details, see the reference below). The large crystal weighing 25g was previously studied for the detection of d-density wave: Mook et al. Phys. Rev. B 66, 144513 (2002). Mook reiterated that the previous experiment did exhibit evidence of d-density wave although the signal was weak. The present experiment was designed to detect circulating currents proposed by C. M. Varma in the same sample.

What makes this effort puzzling is that in an unpublished work done at NIST no evidence for circulating currents was found, while the same sample examined in LLB showed strong evidence of circulating currents in agreement with the experiment of B. Fauqué et al. Phys. Rev. Lett. 96, 197001 (2006). To quote Mook, “If you can see it in France, why can’t you see it here?” The joke apart, there may be some serious reasons for this. The beam at LLB was better, with a higher flux and therefore higher intensity. More importantly, the flipping ratio, crucially important for polarized neutron scattering, was much larger, 70 as opposed to 23.3 in the NIST experiment. According to Mook these factors contributed significantly to the success of the measurement in LLB.

Indeed, the rise of the spin flip intensity at (101) at 200K was very sharp and continued through without the slightest hint of the superconducting transition. However, no such scattering was found at (200). The size of the moment is not yet known but the scattering cross section is 1 mb, probably corresponding to a few hundredths of a Bohr magneton. The real difficulty is that the observed direction of the moments is at an angle of 45 degrees, not along the c-direction.

P. A. Lee suggested that it would be useful to repeat the previous measurements at
(1/2,1/2) at LLB. I completely agree.

The bigger issue involves the elusive signatures of magnetic order in the pseudogap phase of high temperature superconductors. I find it particularly puzzling that the same sample shows two signatures of orbital magnetic order: one that breaks translational symmetry of the lattice and the other that does not. However, it is heartening that competing groups are now collaborating to unravel some of the mysteries of the pseudogap. This is real progress.

Laura Greene: Andreev reflection in Ce 115's

2007-08-02T14:18:00.000-06:00

Laura presented point-contact spectroscopy results on CeCoIn5/Au contact,
namely the data on normalized tunneling conductance for the wide range of
temperatures. Data shows an asymmetry which appears below certain temperature,T*, most probably associated with a formation of a heavy Fermi liquid. As CeCoIn5 becomes superconducting, the small (~13%) Andreev signal is
observed. Laura presented fits to the data based on the the Blonder-Tinkham-Klapwijk (BTK) model. To account for an asymmetry, the "Kondo-like" resonance peak is included in the density of states. The Andreev reflection data for various tunneling directions convincingly shows the d-wave symmetry of the order parameter. To account for an amplitude of Andreev signal it is assumed that there are two contributions coming from the light gapless Fermi surface and heavy Fermi surface which is gapped. Questions from the audience (C. Capan, A. Balatsky, A. Millis, Q. Si) were mostly concerned with the origin of an asymmetry in tunneling conductance and whether one might think of this asymmetry as a result of the formation of a heavy quasiparticles. Laura's BIG QUESTIONS are:

1. Why does BTK model works so well?

2. What would be theoretical justification for "two-fluid" picture of gapless
and gapped Fermi surfaces?

3. Does the peak in the density of states exist?

Jim Allen: Quasi-1D "Purple Bronze" Evidence for a new state of matter.

2007-08-02T10:10:00.000-06:00

J. W. Allen, U. Michigan

"Evidence, arguments and challenges for showing a new quantum state of
matter in the normal phase of quasi-1D Li_0.9Mo_6O_17"

Jim Allen gave an extensive review of the highly anisotropic physics of Blue Bronze. This material is a quasi-1D conductor that develops superconductivity at 1.9 K. This system has a resistance anisotropy

rho|| : rho _|_ (1): rho _|_(2) = 1:10:25

Down to about 26K, rho|| follows a T^0.4 variation and rho _|_ follows a T^1.15 variation.
Below this temperature, both resistivities show an upturn. The upturn was initially thought to be the result of a spin or charge density wave, but susceptibilility, optics, photo-emission and X-ray diffraction seem to rule this out.

Jim described that ARPES appears to suggest this system is more likely to be a Luttinger Liquid, and in the spectra, they can see a holon peak and a spinon edge. The photo-emission spectra show a power-law A(E) ~ (E-E_F)^alpha, where alpha is temperature dependent .

This temperature dependent exponent can, it seems be understood in terms of a two band Luttinger Liquid, undergoing a cross-over from a two band fixed point to a one-band fixed point.

There are several questions raised by this system

Can the large t_perp (of order 100K) be reconciled with the quasi-one dimensional behavior seen at lower temperatures?
Band theory predicts two quasi-one dimensional bands, yet only one is observed in ARPES (figure above). Is this because a gap is starting to develop in the second band - and if so - can this be modelled theoretically?
Does the superconductivity restore the 3 dimensionality to the electron fluid?

Experimental Talks and Discussion.

2007-08-02T07:14:00.000-06:00

Today's experimental talks will be individually blogged by five members of the audience. Please feel free to comment on any of the postings.

J. W. Allen, U. Michigan

"Evidence, arguments and challenges for showing a new quantum state of
matter in the normal phase of quasi-1D Li_0.9Mo_6O_17"

J. Thompson, LANL

"Proximity of Superconductivity and Magnetism in the 115s"

L. Greene, UIUC

"Andreev reflection in heavy-fermion superconductors and order parameter symmetry
in CeCoIn5"

H. Mook, ORNL

"Search for magnetic order in the YBCO superconductors"

N. Curro, LANL

"Antiferromagnetic droplets in the 115 superconductors"

Open Discussion and Short Presentations

2007-07-31T15:13:00.001-06:00

Tuesday, 31st July.

Open Discussion and short presentations.
10.30am-1.pm Bethe Meeting Place.

The group met to discuss the structure of the workshop. Five speakers
gave their perspective on some of the open questions in this field.

1. Andre Marie Tremblay, Sherbrooke.

"Antiferromagnetism vs d-wave superconductivity: Insights
from the organics"

2. Gabi Kotliar, Rutgers.

"Superconductivity near the Mott transition."

3. Qimiao Si, Rice.

"Quantum Criticality and Superconductivity in Heavy Fermions"

4. Doug Scalapino, UCSB.

"Some issues motivated by the cuprate problem".

5. Subir Sachdev, Harvard.

" Fractionalization on the route from Neel order to d-wave superconductivity"

Summary of Discussions

1. Andrey Marie Tremblay: "Antiferromagnetism vs d-wave superconductivity: Insights
from the organics"

Andrey emphasized that an ultimate test of our understanding of unconventional superconductivity, is to see how successful the theory is when applied to a diverse set of componds. The Organic superconductors display many aspects in common with the 115
and the high temperature superconductors - proximity to antiferromagnetism, frustration and Mott physics.

Andrey discussed the K- (ET)2 X layered organics. The physics of these systems is believed to be described by a 2D Hubbard model on a triangular lattice, with hopping t and t'. You can think of them like the cuprates, but with only one t' cross-link per square plaquet. Unlike the cuprates, by changing the anion X that separates the layers, you can tune t'/t

U ~ 400 meV
t ~ 30 meV
t'/t - (0.6 - 1.1 variation).

Andrey presented the results of a cluster dynamical mean field theory calculation on this model. Without pairing it has a 1st order Mott Transition line. For larger U/t, the system develops antiferromagnetism, or spin liquid. For smaller U/t, it enters a d-wave sc phase and then a metal.

Questions: At large frustration, does the d-wave sc become unstable to new symmetries - e.g p-wave symmetry?

Gabi Kotliar questioned the similarity with the cuprates. Here, the superfluid stiffness is a maximum near the Mott transition, whereas it goes to zero at the Mott transition in the cuprates. Others questioned whether this is due to the difference between doping and U/t tuning.

Also - here the temperature dependence of the superfluid stiffness has an anomalous T^3/2 variation.

2. Gabi Kotliar, Rutgers.

"Superconductivity near the Mott transition."

Gabi discussed the phase diagram of the t-J model of high temperature superconductors, contrasting the predictions of slave boson theory, with that of cluster dynamical field theory
(CDMFT). CDMFT predicts significant anisotropies in k-space that are absent from a slave boson theory. In particular

The rate at which the quasiparticle renormalization constant Z goes to zero in the approach to the Mott transition, is much faster in the antinodal regions. (Measured in the superconducting phase).
The v_Delta - the component of the qp velocity coming from momentum dependence of the gap, decreases with the doping (linearly? ), whereas v_F remains doping independent.

This last point has some important consequences. In particular, the T coefficient of the
superfluid spin stiffness, "a" in

rho = rho_0 - a T

becomes doping independent, rather than proportional to doping squared, as in RVB theory.
A similar feature is seen in the omega coefficient of chi''(omega) in Raman spectroscopy, which is predicted to be doping independent.

Gabi's questions:

Are these features observed experimentally?
What is the origin of this doping dependence?
Can we understand the solutions to the CDMFT in a simple language, perhaps analytically?

3. Qimiao Si, Rice.

"Quantum Criticality and Superconductivity in Heavy Fermions"

Qimiao started his presentation with the remark

" Most of the really interesting questions about heavy fermion superconductivity have not been deeply explored.

What are these questions? Qimiao began his talk with a summary of the key properties of heavy electron quantum critical points, taking as examples, CeCu_6-x Au_x (doping tuned), CePd_2Si_2 (pressure tuned) and YbRh_2Si_2 (field tuned).

He later turned to discuss the case of the 115 materials, showing a phase diagram that was quite similar to that presented by Andre-Marie Tremblay. 115 materials have the chemical formula

CeX In_5

where X= Co, Rh, Ir. They are layered heavy electron systems that display antiferromagnetism and superconductivity. Application of pressure to CeRhIn5 leads to a transition from antiferromagnetism, to a region of co-existent superconductivity, then into a purely superconducting phase. However, if you apply a magnetic field to remove the sc, there is a single QCP between the antiferromagnet and paramagnet, where the Fermi surface volume appears to "jump". This jump is associated with the delocalization of f-electrons.

Qimiao asked:

Can superconductivity co-exist with a state that appears to undergo "fermi surface fluctuations" ?
How should one characterize superconductivity that forms a dome above a second order QCP, particularly one where the Fermi surface jumps?

4. Doug Scalapino, UCSB.

"Some issues motivated by the cuprate problem".

Doug Scalapino asked the question:

What is the (mother) phase that underlies superconductivity?

As an example of the importance of this question, he discussed two different models of how superconductivity emerges from an antiferromagnetic Mott insulator:

Antiferromagnetically mediated pairing, in which the "mother state" of the superconductor is
a nearly antiferromagnetic metal. In this scenario, the omega dependence of the gap function
Delta(k,omega) should reflect the underlying spectral function of the spin fluctuations.

RVB model of superconductivity, in which the "mother state" of the superconductor is a spin liquid - a Mott insulator without Neel order. In this case, one might expect a disconnect between chi(q,omega) and the frequency dependence of the gap function.

But beyond this, Doug listed all sorts of possible "mother states" of high temperature superconductors:

Stripes
d-symmetry CDWs
d-symmetry SDWs
2-leg ladders
Orbitally ordered states

"Now you may not agree of the starting point, or the mother state, but the important point is that there are a lot of different materials, and room enough for everyone!" (paraphrased).

Doug also mentioned the challenge of the s-wave superconductor Barium Potassium Bismuthate (Ba_1-xKx BiO_3) where T_c ~ 40K. He asked whether the underlying mechanism here is solely phonons, or does charge disproportionation

2Ba(4+) <----> Ba (3+) + Ba(5+)

play an important role?

There was a lot of discussion.

Andy Millis raised the issue of the Taillefer group measurements, that reveal a small Fermi surface in the high field state of underdoped high temperature superconductors. He asked - should we regard this as a "mother state" of the high temperature superconductor, or is it just a competing state?

Thomas Vojta felt that if the transition between competing states was second order, then they would still influence one another.

Shankar asked whether we should consider Grandmother states?!

Sudip Chakravarty pointed out that in simple systems like Pb, there was no ambiguity about the mother state - it was a Ferm liquid with e-phonon interactions, but already once one gets to V3Si, the underlying state might already be a CDW.

There was a philosophical discussion about why it was, when physicists manage to link
Arpes data with spin fluctuation data, it is no widely accepted..... Doug said something about Physicists being quite artful at fitting selected data....

5. Subir Sachdev, Harvard.

" Fractionalization on the route from Neel order to d-wave superconductivity"

Subir posed his questions at the beginning of his talk. They are:

Is the low doping limit of cuprate superconductors a BCS state (+ some other unconventional order), or is it exotic or "fractionalized"?
If there is an exotic state at low doping - can this state be obtained by doping a Neel state (and not a non-existent spin liquid?)

Subir then discussed some work he has recently done with Senthil, Levin and others,
in which they discuss the effect of doping the deconfined quantum critical point that is thought
to separate a Neel state from a valence bond solid. The key point about this, is that the quasiparticles of the state that emerges are "fractional", in the sense that they carry a non-trivial gauge charge. These particles, they believe, are possibly the origin of the pockets seen in the Taillefer experiment. Subir also described how, when they pair, they form a conformally invariant fluid, in which the superfluid stiffness has a T-dependence that is independent of doping - rather like the results of the Kotliar work

rho_s(x,T) = constant - R T

where R is universal.