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

Dan Sheehy: Superfluidity in "magnetized" fermionic atomic gases

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"

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

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.

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"

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

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"

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

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"

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

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 ?"

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"

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

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

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

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

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

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"

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

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"

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

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

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"

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

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.