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.

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.

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.

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"

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

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

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

1. If the measurement is made on hole pockets, then why is the Hall constant negative (corresponding to electrons)?
2. Are we sure that 60T - or even higher 80T measurements are really in the "normal state".
3. 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.
4. 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.

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''

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

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, ¹H NMR and static susceptibility measurements have been performed in an organic Mott insulator with a nearly isotropic triangular lattice, κ-(BEDT-TTF)₂Cu₂(CN)₃, which is a model system of frustrated quantum spins. The static susceptibility is described by the spin S=1/2 antiferromagnetic triangular-lattice Heisenberg model with the exchange constant J∼250  K. Regardless of the large magnetic interactions, the ¹H NMR spectra show no indication of long-range magnetic ordering down to 32 mK, which is 4 orders of magnitude smaller than J. These results suggest that a quantum spin liquid state is realized in the close proximity of the superconducting state appearing under pressure.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

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:

1. Can we really be sure that there is such a diverse set of pairing mechanisms? If so - what is the unifying thread between them?
   
2. 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?
3. 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

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

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

(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.