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.

## Friday, August 24, 2007

### Enrico Rossi: Neutron resonance in electron-doped cuprates

Posted by Andrey Chubukov at 3:45 PM 4 comments

### 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.

Posted by Dirk Morr at 11:25 AM 1 comments

## Thursday, August 23, 2007

### 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.

Posted by Amit Keren at 5:14 PM 4 comments

### 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 <<>

Posted by Matthias Eschrig at 5:09 PM 5 comments

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

(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.

Posted by Brad Marston at 4:19 PM 11 comments

### 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.

Posted by John Mydosh at 1:44 PM 7 comments

## Wednesday, August 22, 2007

### 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.

Posted by Assa at 9:40 PM 38 comments

### 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.

Posted by Artem Abanov at 6:34 PM 5 comments

### 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.

Posted by Ilya Vekhter at 6:28 PM 6 comments