Aharon Kapitulnik (Stanford U): What can Kerr effect measurements teach us about high-Tc superconductors (especially the pseudogap)

Aharon started by saying that he had changed the title.

He began with an introduction about the puzzle of the pseudogap in High Tc. It is he said some kind of state in which we see a change of state - seen in measurements. Initial knight shift and T1 measurements suggested a spin gap appearing at T* >> Tc. Later, ARPES showed a pseudogap - a gap in the DOS that survives above Tc, but without a coherence peak. This was also found in STM measurement - Aharon showed nice data from his own group - noting the presence of a non-dispersing charge modulation that persists up to some temperature, called T*. (He will use T* as a general term for a crossover temperature where something happens in the measurement...)

Two types of T* discussed by theorists:

(1) representing a crossover into a state with preformed pairs and a d-wave gap.

(2) marks a true transition into a phase with broken symmetry which ends at a QCP, such as loop currents in the d-density wave (Charavarty et al Q > 0)  or Chandra Varma's theory (Q=0).

The T* phase may have various different configurations - (a) T* line merges with Tc line at large doping (preformed pairs) (b) T* crosses into the dome (c) where it ends at the dome. Various experiments invented to look for signatures of the various scenarios.

Key questions posed in review by Norman, Pines and Kallin (Advances in Physics, 54, 715 (2005).

In particular:

• Is there LRO associated with T*?

• Is charge ordering central to the  pseudogap

YBCO and Bisco as examples - Pseudogap is found to be above the peak of the SC dome.  On the other hand, for 221, looks like T* line appears to suggest a convergence of T* to the top of the dome.

Aharon wants to concentrate next on expts that do suggest T* is a real phase transition, by looking at a particular Bragg diffraction peak.  YBACuO and HgBa2CuO. Onset of magnetic scattering in Bragg peak on a T* line that seem to follow one-ahother, extrapolating to x=0.2.   On the other hand, a similar type of exploration on Lr2-xSrxCuO4, eg for x= 0.13, T* is about 450K from optical, but neutron saw development of short range order at 120K.  Seemed not to be a unique quantity. 

Question from Andriy - since they claimed to see real phase transitions - did anyone try to study specific heat around these points. 

A - yes people tried, but no feature seen in S-Heat - its very difficult to see electronic parts to the s-heat. i know of any good high resolution sheat that shows these features.   There is perhaps some unpublished data from Greg Boebinger's group, but I can't talk about it.  I'll come back to it data.

Anyway - in LSCO, there is a big discrepancy.   

In LaBCO, near 1/8th seen CO followed by Spin order - a feature at around 50-40K, butfrom susceptibility, the T* is around 400-500K.  Big discrepancy between the phase transition and T* from susceptibility /knight shift.

Observations:

• Some material systems, different probes indicate different T*
• Most probes extract T* from some change in the systems behavior. Generally crossover
• Only expts to indicate true broken symmetry below T* are neturon scattering, also Kerr and possibly Nernst.
• In seceral systems, BLCO, charge ordering seems to be the most proninent phenomenon below T*. 

Insight from Kerr measurements

Magneto optics have advantages probes bulk, highest quatlity samples and can probe SC and normal states. In addition, polar Kerr effect using the loopless Sagnac Interferometer provide high sensitivity.

Kerr effect measures the difference of nR and nL.  Different to Faraday, which is transmission, Kerr effect is in reflection.  Kerr effect depends on the off-diagonal component of the conductivity, which when non-zero indicates time reversal symmetry breaking. 

Theta_K = - Im ((nL-nR)/(nL.nr-1)]

really measuring the imaginary part of sigma _xy. 

Sagnac interferometer - use fiber optics, and 1/4 wave plates to produce circularly polarized light that passes through the sample.  Very high sensitivity for non-reciprocal effects. (TRSB).  Insensitive to losses.  Sang Cheong asked about intrinsic depolarization from the fiber. Apparently not an issue. 

Results on YBCO  Method really a descendent of an earlier machine used to test for anyonic superconductivity.  Method motivated by current loop or scenario with staggered AFM, expected to get a Kerr effect in the case of orthorhombic  distortion with AFM. 

YBCO, 1/8th doping, Tc = 65K, A signal produced by trapped flux in SC, but small signal above Tc from 65 to 155K. Cooling in zero field eliminates the vortex effect, see a signal that disappears at a higher than Tc.  

All the data together, get a Kerr effect line at Ts(x). Kerr effect, consistent with muSR and elastic neutron scattering . 

Optimally doped YBCuO, Ts ~ 105K, Tc=89K, 

Summary of single xtal/thin filme results. Conclusion - some kind of charge ordering at 35-60K

near optimal doping (below Tc). 

Recent Nernst claims a breaking of rotational symmetry (Nature 463, 519 (2010). )  Nernst effects and Kerr and neutron broadly agree.  Aharon mentions that the Nernst signal can be reinterpreted as a break-down of time reversal symmetry.  

Beyond the thin-film YBCO  LBCO - 1/8th doping - see large signal developing at 54K, max at 41K where there is spin ordering, and reduced at 19K.  54 - charge, 41- spin, 19K - sc.  Canfield says - doesn't this mean that the charge ordered state has a FM moment?  Aharon says - no - lets assume that I mean, that it could be, but in the most general case, suppose I have a piece of magnetic material that is not magnetic, coupled to the charge ordered system, OK - I can couple it in several ways - I can couple it in several says - an impurity phase - some kind of squushing - all the way to they are linked together. 

Canfield said - I did not say it was causal - Aharon ultimately agreed. FM is something else - and may have been so in the earlier data.

Laura Greene asks - what kind of length scale do you need for these results?  

Aharon answers - can only measure a global FM moment

The blogger is left very confused about how charge ordering can induce FM, unless close to a FM instability or canted AFM - or some other thing that has a bit of FM order. 

Another material - found Kerr effect turns on at the charge ordering temperature.  (bisco 2201 at optimally doped). 

Finally if I go back to Bourges  LSCO - here T* 450, but transition at 120K -  

Summary - 

• The Kerr effect seems to be an indirect probe of charge ordering - some coupling

• Charge ordering is a pronounced effect in all hi Tc sc.

• Charge, followed by spin ordering is the last effect before sc

• There are several magnetic effects, some may occur through phase transitions above Tc.

• This motivates, he said, studying the gneeral problem of charge order coupled to other order parameters. 

Question from Andriy Nevidomskyy in Lanthanum Barium.  Dont' you see a cusp in Kerr at the point it goes sc? 

A: Yes - of course I do.

Q - what is the reason for the cusp?

Q:  its vortices. 

Q - from Sang Cheong.  I'd like to clarify the connection between charge ordering and Ferromagnetism. wouldn't it be fun to look at other charge ordered systems?

A - yes

Q Blogger - don't you need to have quadratic couplings of the charge order to the FM order, that is close to criticality.

A- yes it would need to be close to criticality.

There followed a discussion about whether one needs quadratic couplings between the charge order and
finally tuned ferromagnetism. Another suggestion was made by ZT followed up by Sang Cheong, and AK seemed to be in broad agreement.

Zlatko Tesanovic:- "this issue of close to FM criticality" - everything is - if there are spins around- charge ordering will produce a FM order.  But Sc - this a singlet state - and this diminishes such a response. 

Aharon K. Might not be quadratic coupling but - because of the smallness of the effect, a piece of stuff is squshed, and this might make it FM. Its a kind of indicator of charge ordering - a possibility that I can not rule out. 

Cheung - if you buy the best quality - best you can get is 5 -9s - you know how much impurity you have - if the background is metallic you don't see it - but its very possible that it responds to charge order.

Hideo Takagi (Tokyo): New superconducting transition metal pnictides and quasi-particle interference in Fe(Se,Te) superconductor

Hideo has started by an outline which divided the talk on two parts:

I) Exploration of the new SC : basically 122 systems without Fe [ATM_2P_2 and TM being Rh, Pd, Ir] and

II) quasiparticle interference in the superconducting FeSeTe using STM/STS spectroscopy: evidence for the s+- symmetry of the order parameter

Then he moves to the part I. The first materials to look are BaRh2P2 and CaPd2P2 and BaIr2P2 and SrIr2As2 (remark by Canfield that these are most expensive chemicals!), all of them show low-Tc superconductivity. Piers: How do you know that these systems are unrelated to Fe-based compounds, Answer: there is an indication from LDA: the Fermi surface (FS) [he then has shown for comparison the FSs of BaIr2P2 and BaFe2As2 - indeed they are different]

Next he continues with the crystal structure of Ca(La,Sr)Pt_3P, newly synthesized systems. These are either I4mm (polar) or P4/nmm (non-polar) crystal structures. Tc is about 1.5 K, 6.6, and 8.5 K, respectively.

Looking at the transport (resistivity) and thermodynamics (specific heat) the estimated Sommerfeld coeff. gamma \approx 12.7 mJ/mol K^2, and the ratio of \Delta C /\gamma T_c \sim 2.5. The latter number may indicate strongly coupled SC (remarks: or disorder -Andrey, or multiband effect - Raphael).

Again new systems, different possible pnictide materias TMP and TMAs, where TM again stands for one of the 3d, 4d, or even 5d (W) elements. Examples are RuAs and RuP (all in MnP structure): 3D network of distorted MnP6 octahedra. Transport for RuP - Metal-Insulator transition at 250 origin is not clear but definitely not nesting as FS has no sign of it, RuAs - obvious anomaly at 270 K but what kind of the order is not clear. Both (RuAs and RuP) can be doped by Rh (instead of Ru). In RuRhAs series the anomaly ("the order") is suppressed at 25% Rh doping. No SC yet observed down to 1.5K. Here Takagi finished with an optimistic remark that there is still hope to find it at lower temperatures!

In RuP doped by Rh a suppression of the order occurs at 50% and no sign of SC as well is found. Cheong raised an issue of the critical doping, Kapitulnik noticed saturation at low T, some discussion on the chemistry aspects [size of the crystals and so on], blogger has not followed all the details.

Ok some theory consideration for the next set of systems: the idea is to look on the non-centrosymmetric crystals with 4d or 5d elements. In such a case (like in a well-known example CePt_3Si) there is a hope to find some exotic SC with a mixture of s- and p-wave symmetries. This is allowed due to lack of inversion symmetry. Overall Hideo reported 9 new superconductors with POSSIBLY non-centrosymmetric ATMSi_3 structure. Examples discussed are BaPdSi_3 and BaPtSi3 with an emphasis on the specific heat data. Unfortunately, looks very much like standard BCS superconductors. Another two examples mentioned are Rh2Ga9 (1.9K)and Ir2Ga9 (2.2 K).

NOW the second topic (II) - iron-based superconductor Fe(Se,Te) and QPI in the magnetic field. The idea starts from a earlier work in cuprates by Hanaguri et al. in 2009 and is based on the assumption that in a d-wave superconductor the constant energy maps of the conductance should be in direct correspondence to the LDOS of system at the same energies. Namely, the impurities, resposnible for the QPI, induce the scattering between the banana-like shapes of the LDOS which appear in a d-wave symmetry of the superconducting gap (He means here cos kx- cos ky function of the gap). Then the lobes of the banana will have a large density of states and the scattering between them dominates the QPI maps at all energies lower than Delta_0 [maximum of the gap]. Overall 9 wavevectors can be identified.

 In addition, remember that these are Bogolyubov quasiparticles - the coherence factors originating from the u-v transformations are involved. As a result, QPI induced by magnetic and non-magnetic impurities looks quite different. Now if on top of this, the SC order parameter has some specific symmetry, the QPI pattern due to magnetic and also non-magnetic impurities will look quite unique which allows to identify the order by means of FT-STM. The most efficient way is also to apply the magnetic field which generates vortices which then act as magnetic scatters. By substracting the QPI maps made with and without magnetic field you can identify at which momenta the scattering is suppressed or enhanced which tells you about the symmetry of the sc gap.

In cuprates one finds two types of the scattering wave vectors either with suppressed or with enhanced intensity in the magnetic field. Looking at the constant energy maps of the LDOS for d-wave superconductor one observes the effect of coherence factors and then confirm the coskx-cosky function to be the symmetry of the SC gap in cuprates,

Now Takagi shows the result for the iron-based superconductors. He gives short introduction to s+- symmetry of the SC order and theoretical arguments based on the spin fluctuation theories why it may occur in iron-based superconductors. Basically as most of us know for the FS given in pnictides with pockets at Gamma and (pi,0) and (0,pi) points of the BZ the gap is constant at each of the pocket but with opposite phase, i.e. opposite signs at the pockets separated by the (0,pi) or (pi,0) momenta.

In the QPI one expects to see several wave vectors: (pi,0) and (0,pi) which change sign of the SC order parameter as well as diagonal scattering and intraband low-q scattering where no change of sign occurs. Now the experimental data comes: Fe(SeTe), Tc~14.5. We look at the conductance maps and the first thing one finds is that the sc gap is only ~2meV [Piers has asked about why it is particularly small, the answer blogger did not understand] which closes (from the coherence peak ) at around 11K. Another important remark is that there are some extra features (dip-hump) at around 4meV at both positive (+4meV) and negative (-4meV) voltages but with an asymmetric shape. Some speculations are made that these are most likely the many-body effects.

Then Takagi shows the conductance maps which should refer to the QPI. The wave vectors are clearly identified (first at zero magnetic field), they roughly can be associated with wavevectors of the scattering between the FSs in ferropnictides (here actually in FeSeTe). By applying the magnetic field one notices that at those q which have opposite sign for the s+- the intensity is indeed enhanced while at all others [diagonal and due to intraband scattering] it is suppressed. The results are consistent with the s+- wave symmetry.

Summary: magnetic field dependence of the QPI is consistent with s+- symmetry but not consistent with any other symmetries (d, p). issues for the future direct observation of impurity states in STM, antisymmetric shape of the dI/dV curves. Few minutes Takagi spent on he discussion that it is important to understand why s+- is not that sensitive to disorder, the blogger, however, thinks that this is not an issue, as in the s+- only the interband (large Q, thus small) scattering is bad for superconductivity while intraband (low q, thus strong) scattering which should be the largest does not produce a significant effect on superconductivity.

Zlatko: 1) as compared to cuprates the scattering wave vectors are commensurate in FPs
which would be indistinguishable with regard to the Bragg peaks. Do you have a different doping level
Reply: in 11 compounds there is a magic doping whether SC is bulk, so the answer is no
Blogger: did you observe vortices Reply: Now here we do not know where they are located as compared to cuprates, thus we did not know where we could see them
Andrey remarks on the interband versus intraband impurity scattering which blogger already mentioned above.
Lu was wondering whether the materials shown in the first part are indeed non-centrosymmetric, the answer was that despite the expectations the were not. The second Lu's question was about the origin of the transition in RuAs which Takagi in the reply attributed to the orbital order though without going into the details of it.
Piers and few others [blogger was disrupted by some discussion on the non-related topic which happened next to him, thus missed the names] were curious on what is the particular effect of the magnetic field in iron-based superconductors is [as there were no signs of vortices]. The overall opinion at the end was that even in the case there are no well defined signs of vortices the magnetic field still enhances the scattering induced by magnetic impurities, therefore the argumentations holds.

Silke Paschen - Recent developments in heavy fermion quantum criticality

Introduction

Silke begins with a general theoretical introduction to quantum criticality, as it is simpler than the experimental picture. She shows a generic phase diagram where some sort of ordered phase is destroyed by quantum fluctuations as some parameter (magnetic field, pressure, chemical composition) is tuned. Above the quantum critical point (QCP) where the ordering temperature goes to zero there is a `quantum critical fan' encompassing quantum fluctuations, while far from the QCP, the fluctuations above the ordering temperature are classical. Silke notes that experimentally it is hard to find any classical critical behavior in real materials as the QC fan always dominates.

Question (Andrei Chubukov) Wasn't this phase diagram written for localized spin system without any conduction electrons?

Answer: Yes - written for insulating spin system, but this is more general, for any continuously suppressed order parameter.

Silke notes that quantum criticality is `everywhere' - high TC, heavy fermions, metamagnetism, Sr3Ru2O7, FeAs, Ising chain... and its study is a very big problem in solid state physics. However, in materials like the cuprates, the temperature and field scales needed are inaccessibly high. The energy scales in heavy fermion compounds are much lower, making this study much easier.

She introduces the heavy fermion experimental picture with the canonical image of the critical resistivity exponent in YbRh2Si2. In these materials, unlike the cuprates, we really know that some order parameter (the antiferromagnetic order) is really going to zero.

Next Silke introduces the theoretical picture of heavy fermion quantum criticality using the Doniach phase diagram, where by tuning the hybridization, the relative strengths of the Kondo and RKKY interaction scales tune between antiferromagnetism to heavy Fermi liquid physics at larger hybridizations, with a QCP inbetween.

There are two competing theoretical pictures of heavy fermion criticality: spin density wave (SDW) versus Kondo destruction, and we had some vigorous discussion about the meaning of these two pictures.

Standard spin density wave scenario:

The magnetic moments are screened throughout the phase diagram to form heavy electrons. The magnetism is a SDW of the heavy quasiparticles and the criticality is described by theories discussed by Hertz, Millis, Moriya, Continentino, Lonzarich... Here there is no omega/T scaling, the Fermi surface evolves smoothly (in that there are the same type of heavy quasiparticles on both sides - as was discussed, there will still be folding of the FS due to the magnetic order).

Kondo destruction scenario:

This picture is motivated by experimental results. Here local moments are essential to the QCP. While there is a single ion Kondo temperature, TK indicating the onset of Kondo screening at high temperatures, there is also a second Kondo scale TK*, which goes to zero at the QCP. The antiferromagnetism is then that of local moments, and there should be a change in the Fermi surface volume as the local moments are included in the Fermi surface in the heavy Fermi liquid, but not in the AFM. Main point: two energy scales go to zero at the same QCP.

Question (Paul Canfield): What is the origin of bifurcation of the Kondo temperature?

Answer: There is some Kondo screening, but not complete - the AFM moments are small.

Q: simple indication for why two Tks? (Andrei Chubukov): Specifically, what do the two scales mean?

Answer (Piers): Upper one is the single ion scale - above that see Curie behavior, departure from it before below (in the susceptibility), the resistivity turns over at these scales... The lower scale is empirically based, it is a coherent Fermi liquid scale. A theoretical example is given by the two impurity Kondo model, when tuned to a QCP it shows two channel physics, and there is an emergent Fermi scale growing quadratically with distance from QCP while the single ion Tk is constant and large.

Question (Andrei): How is this different from Fermi liquid scale scale on the left hand side? eg - how are TK* and the Fermi liquid temperature different?

Answer: There are two distinct energy scales.

Question (Andrei): either we have local moments or not?

A: We do have local moments, but small. Local moments in/out of FS whereas in SDW order local moments always in the FS.

Additional important point: this scenario contains omega/T scaling in chi(omega).

Experimental discussion:

Now Silke moves on to discussing the experimental observation of this jump in the Fermi surface - first showing the de Haas van Alphen data in CeRhIn5, where the Fermi surface orbits are discontinuous in pressure, where the Neel temperature is suppressed to zero. The second order nature is indicated by the divergence of the effective masse. However, this technique fails for YbRh2Si2 as fields tune it away from quantum criticality. Here, the Hall effect can be used the measure the Fermi surface jump - R_H shows a crossover at finite temperatures that extrapolates to a sharp jump at low temperatures, indicating a sharp change in carrier density. This answers the question about T* vs T_FL - Hall coefficient jump is distinct from T_FL.

Question (Andrei Chubukov): Suppose we take conventional SDW, large FS split into hole and electron pockets - as in cuprates, also have jump in FS, as the FS folds over at a SDW transition, so get different Fermi surfaces.

A: As the FS evolves continuously, the size of jump should be infinitesimal at QCP, as the OP is infinitesimally small, so the hall coefficient jump should extrapolate to zero at QCP. No B dependence.

If you have finite local moment, then whole moment goes into the FS. But Hall coefficient jump does get smaller with smaller temperautres, but it extrapolates to finite size at T = 0. All samples have such a finite jump. Meaning - the FS is rather complex, but the finite jump is robust even to different |R_H| due to impurities changing mobility ratios for the electron and hole bands.

Silke shows other results measuring the energy scale in thermal expansion, magnetization, which tells us its a thermodynamic energy scale. Now Silke moves on to more recent experiments (just put online one week ago). The full width half maximum (FWHM) of the crossover in the hall coefficient scales linearly with temperature, with a perfectly sharp jump at T = 0). Implies E/T scaling.

Question (Piers): Is it an intuitive link or more concrete?

A: The crossover reflects the relaxation time of electrons, as you go from one FS to the other - at finite temperatures should see relaxation from one FS to the other FS.

Piers: So the idea is that the electron scattering rate growing linearly in T implies this FWHM ~ T. (Andy Schofield): But the scattering is a single particle property, while the Hall effect is rather complicated.

Silke next shows some new thermopower data, plotting S/T vs H which shows a pronounced feature at the T* line, and gets much sharper as T decreases. There are also drastic changes going into the FL -S/T becomes constant. Gives same phase diagram as previous measurements. On the magnetic side of the phase diagram, the thermopower changes sign with temperature, but beyond the QCP, it doesn't change sign.

Separation of scales:

Silke now introduces the experiments doping Ir and Co onto the Rh site, which correspond to positive and negative pressure, respectively. Doing this can separate the points where T* and TN go to zero. Co doping moves T* inside the magnetic dome, while Ir doping separates the two points, leaving a broad region of NFL behavior.

Question (Paul Canfield): For the Co doping, how does the Neel transition above and below T* line differ? Is it the same transition?

A: Local moment order vs SDW order. There is also an additional magnetic phase transition, T_L (T_L <>

Question (Zlatko Tesanovic): Is the lower line T_L is really a transition? or a crossover? Does it have to be a phase transition? Increasing size in FS is, in principle not tied to any magnetic order.

A: It is a magnetic phase transition, seen in thermodynamics (specific heat anomaly).

Silke shows data on YbRh2Si2 under pressure. Interestingly, T* is robust to changes in pressure, but TN increases with increasing pressure. Next, she shows data doping Ge on the Si site, which resembles the Ir doping data, as there is a finite region where both T_N and T* are zero. Resistivity is linear for all this finite range of fields

Question (Rafael Fernandes): At Bc1(where TN -> 0) there is no change in the FS?

A: There doesn't seem to be, this change is measured T*, which goes to zero at Bc2, Fermi surface jump means local moments or not, does not mean that they have to order - large or small FS not tied to magnetic order) Also, not that pressure can be inhomogeneous, possible hinders ordering.

Rafael: For the QCP at Bc2 - we don't know how the symmetry is changing, don't know if it's first or second order.

A: Yes, that is for us (experimentalists) to find out.

Towards a global phase diagram:

Silke sketches a general zero temperature phase diagram for magnetic Kondo materials - the Doniach axis provides the y-axis, while frustration (or decreasing dimensionality) provides a second axis (which is also true for pure spin compounds) . How do these two QCPs join up? And also the FS must jump somewhere, so add another line. Silke's diagram, on the right, contains a quantum tetracritical point, where these two lines cross, while Qimiao Si has a similar diagram, but the two critical lines coexist for some range. Both phase diagrams contain four phases: Spin liquid metal, AFM local moments, AFM SDW, and heavy Fermi liquid. Various heavy fermion compounds can be speculatively placed on this diagram - for example - pure YbRh2Si2 would be tuned right through the QCP.

Tuning the dimensionality - how to change the value on the vertical axis, and also theories of local criticality relied on two dimensionality. How does changing dimensionality change the critical behavior?

• CeIn3/LaIn3 superlattices really turn the 3D materials into 2D. TN is suppressed, as expected. See initial signs of NFL...
• Silke's group is studing a 3D system (cubic): Ce3Pd20Si6. It contains two different phase transitions, from two different Ce sites: AF Quadrupolar order on one and then AF magnetic on other site (which collapses at the point where T* goes to zero - although within the quadrupolar ordered phase). She sees a jump in resistivity, with again a linear full width half maximum - phenomenologically similar to YbRh2Si2.

Questions:

• T* lines in other systems? Scaling?
• What are the excitations in NFL/SL metals?
• How can we understand unconventional QCP in 3D systems, is it protected by the ordered phase?

Question (Satoru Nakatsuji): What does the specific heat in NFL range of YbRhSi2 do? (Piers) And entropy?

A: Can't really tell. No phase transition, obviously.

Question (Andrei Chubukov): At T = 0, before reach QCP in HFL phase - if transition is second order, would you expect sharp jump? or continuous hall jump? How can you tell if it's second or first order QCP?

A: Spins either in FS or not. SDW volume changes discontinously, but physical measurements are continous.

Question (Andy Schofield): At a simple SDW transition, transport coefficients change discontinuously in a finite magnetic field [AJS adds reference JJ Fenton, AJ Schofield, Phys. Rev. Lett. 95, 247201 (2005)]

Paul Canfield: For doped materials, FS change is not tied to the transition, so long range magnetic order a red herring.

Andrei Chubukov: wants the finite jump in Kondo destruction scenario to be proved rigorously.

Markus Garst (Köln) -- Multiscale quantum criticality: Nematic instability in metals

References for what Markus will talk about:
M. Zacharias, P. Wölfle and M. Garst, PRB 80, 165116 (2009)
M. Garst and A. Chubukov, PRB 81, 235105 (2010)


Markus began by apologizing to the experimentalists for giving a (possibly) technical theoretical talk, but there was no mass exit from the lecture theatre so all should be OK.

Introduction: Quantum Phase Transitions (QPT) and multiple scales

A 2nd order QPT is an instability in the ground state at T=0, as a function of some control parameter we will call r (e.g. magnetic field, pressure, doping, etc...) While this transition happens strictly only at T=0, it has a strong effect on finite-T properties due to an abundance of low-energy fluctuations. So on phase diagram, should have T too - region in phase space where properties controlled by Quantum Critical Point (QCP)

Some technical things:
--Correlation length exponent \nu : \xi \sim |r|^{-\nu} (\xi = correlation length, r=control parameter)
--dynamical exponent z : spectrum of critical fluctuations goes as \omega \sim k^z which gives a vanishing characteristic energy scale \epsilon\sim\xi^-z
--enhanced dimensionality: correlation volume in space: \xi^d and time \xi^z gives you an effective dimensionality d_eff = d + z

This set of exponents will give you a scaling ansatz of the critical free energy
F(r,T) = b^{-(d+z)} F ( r b^{1/\nu}, T b^z)
for some arbitrary scaling variable b.
Such scaling behavior are widely used to interpret a wide range of experiments where you don't know exactly the microscopic details of what happens, but the scaling may still work.

Quantum-to-classical crossover: we have the correlation volume in time \xi^z, but this is limited by the temperature \xi^z < t="0),">0). However, the flow is such that for T>0, a crossover temperature (\xi_T) may be defined where the flow leaves the T=0 path to divert to the classical critical point.
(see e.g. Nelson 1975, Millis, 1993)

As a brief summary: the relation between the thermal length \xi_T and correlation length, \xi gives us a crossover in the phase diagram.

In many systems, there is a coexistence of low-energy fluctuations.
For example, near a magnetic instability, have critical magnetic fluctuations as well as ballistic electrons - so can find sometimes two dynamical exponents z_1 and z_2.

With two dynamical exponents, this raises a lot of points:
--coexistence and interacting fluctuations -> entanglement!
--identification of proper critical degrees of freedom?
--two dynamical exponents-> breakdown of scaling and power laws?
--crossovers in phase diagram
--fluctuation driven first order transitions
Markus tells us that these are the `big questions' in this topic, many of which are unanswered. The remainder of this talk will be about a simple specific case.

Nematic Instability of Fermi liquid

Pomeranchuk instability: instability of FS towards development of a quadropole moment
Order parameter is a tensor object, similar to nematics. The traceless part of the strain tensor gives us the shear modes of the Fermi surface. In d=2, there are two shear modes.

We will look at a simple model Hamiltonian introduced by Oganesyan, Kivelson and Fradkin (2001). They developed an effective bosonic model which acts as the Ginzburg-Landau theory in the usual way by Hubbard-Stratonovich and integrating out fermions. More complicated than regular GL due to tensor order parameter, but otherwise standard. We note that in d=2, no cubic term is allowed, which means a 2nd order transition is possible.

OKF analyzed this model, finding a Pomeranchuk instability, with a criterion basically identical to the Stoner criterion as a function of some control parameter r depending on the density of states and the coupling constant (a control parameter Piers referred to as pretentious, but this story can wait for a rainy day...)

Markus then says the most interesting point about the instability is the dynamics - which he will now discuss.

One can have polarization of the excitations both longitudinal and transverse to the quadrupolar momentum q tensor - but because of different phase spaces for exciting particle hole pairs, one finds
--longitudinal polarization, Landau damping z=3 dynamics
--transverse polarization, ballistic z=2 dynamics.
This is the multi-scale property that was introduced early.

Two energy scales, two modes. Which mode is more important?

Naive answer: longitudinal z=3 mode has larger phase space: \Omega_n \sim q^z, so dominates the specific heat. This led OKF to claim that the z=2 mode plays no role in the critical theory.

However in reality, things are much more interesting. Transverse d=2 mode has smaller effective dimension d+z=4 - so generates logarithmic singularities in loop corrections, and it is the interplay of both modes that determines the critical properties.


We now become a bit more technical and look at the structure of the theory at T=0.
The transverse z=2 fluctuations allows us to write a logarithmic RG flow, giving mass renormalization. This flow is marginally irrelevant, but introduces a logarithmic scale dependence of the correlation length
\xi^{-2}(\epsilon) \sim r / [log(1/\epsilon)]^{4/9} for \epsilon>\xi^{-2} (z=2 energy scale)
This exponent 4/9 differs from Ising and XY universality, and is characteristic for Pomeranchuk.

What about theory at finite temperature?
multiple z -> multiple thermal lengths (\xi_T \sim T^{-1/z} )

For each thermal length, there is a quantum to classical crossover - so can have coexistence of quantum and classical fluctuations because of the multi-scale dynamics. (This is Markus' answer to Piers' question at the beginning).

This overlap regime in fact controls the thermodynamics in a wide range of the phase diagram.

Lets look at this again, in terms of RG: on the RG phase diagram, one mode wants to push the system away from the primary quantum critical point, while the other competes with it pushing in the other direction; and it is this wide range of scales that is important for thermodynamics in a wide range of the phase diagram.

Now look at the temperature boost of the RG flow -> some technical calculation, but the answer ultimately is that there is a universal correlation length at criticality (r=0):
\xi^{-2} = cT
which is universal in that it is generated, but independent of, the bare quartic coupling, u. This feature is entirely due to the competition of these different z=2 and z=3 modes.
In other words: multi-scale dynamics leads to a new kind of universality.

These multiple scales are also present in the phase diagram:
two thermal lengths -> two crossover lines in the phase diagram.
The sensitivity of something on the crossover depends strongly on specific thermodynamic quantity that you are looking at. Means that there is no scaling in terms of a single dynamical exponent; things are just a bit more complicated.

Markus then briefly shows us the results for a few thermodynamic quantities (specific heat, etc..) , which all have `funny' logs in them.


Electron spectral function

One-loop self-energy from z=3 mode (see Oganesyan Kivelson and Fradkin, 2001; Metzner, Rohe and Andergassen 2003, etc...) gives singular correction.
Also look at one-loop self-energy from the transverse z=2 mode; which gives interesting contribution to off-mass shell part, and singular correction to Z.

Sum up more logs, RG, ... find that the combination \Gamma Z = 1 is invariant. Hence the polarizations are unaffected by the electrons, and the critical dynamics are preserved.

Question (Piers): Is this a Ward identity? Answer: Not strictly speaking (but unfortunately I missed why - the slides are getting mode and more technical and difficult to blog...).

Hence the electron propagator has three important parts:
-- non-Fermi liquid frequency dependence at z=3 energy scale
--anomalous dimension at z=1 energy scale (z=1 is electrons)
--interesting correlation length dependence at z=2 energy scale.

This is still no the full story though, as there are further logarithms, including some that appear only in 3rd loop order (see Mross, McGreevy, Liu and Senthil, 2010)

Summary

--Nematic quantum criticality in metals: multiple energy scales
--extended quantum-to -classical crossover where quantum and classical fluctuations coexist and interact, and may lead to new forms of universality.
--In thermodynamics, all energy scales can appear, depending on quantity in question

Questions:

Q) Does this theory obey sum rules?
A) It should, no good reason why it shouldn't although must take into account all correct crossover scales to make them work. Chubukov extended this by commenting that this is a low energy calculation, high energy modes will adjust to make sum rules work

Q (Nevidomsky) In many cases, longitudinal modes don't couple to things, is that also true here?
A) (mostly given by Chubukov) - longitudinal and transverse for this quadropolar order are a different notation to what we are used to - should be careful trying to draw analogies

Q) In which (real) materials might you expect this to occur?
A) Crystal lattice may make big differences in this theory e.g. z=2 mode may become gapped (did I hear that correctly?), so at the moment, this work is without reference to real materials.

Q) (missed)

Q (Schofield) In the Stoner criterion, the Pomeranchuk instability condensed around q=0, rather like a ferro-magnet. If the condensation was about finite q (like SDW), would the structure of the theory change?
A) Yes, dramatically. No z=3 mode, lots of other stuff

Comment (Nersesyan) There are also cases where this multi-scale criticality can arise, even without different z's (e.g. in spin-charge separation in Luttinger-Liquid)
Answer: Absolutely true, although these sorts of single z cases are somewhat simpler, as you know how to rescale momenta, etc... Lots of new stuff when more than one z present.

Satoru NAKATSUJI: Quantum criticality and spin liquid in Kondo Lattices (YbAlB4, Pr2Ir2O7)

Satoru begins by presenting the classic Doniach type phase diagram (of course in a its more "modern" form) showing a quantum critical point between the magnetism and Fermi liquid. The question is whether the magnetic order can be suppressed using geometric frustration for the f-electron system. This could reveal a quantum spin-liquid. Here the case to be presented is Pr2Ir2O7. The alternative is to use the itinerant electrons to weaken the order (perhaps by mediating a feromagnetic interaction) and this is the case in YbAlB4 where a chiral spin-liquid is stabilized.

Time reversal symmetry is fundamental in physics and can be broken say with magnetic dipole order. It can also be broken without dipolar order via a chiral spin state (where there is a net S.S x S around a plaquette) as proposed in the curpates in the distant past. It is difficult to detect this order, but the Hall effect may provide a signal. The anomalous Hall effect (AHE) in a ferromagnet would be an example (with a contribution proportional to the magnetization). However a chiral state would also give a contribution to the Hall effect because the Berry phase accumulates as electrons move in the presence of chirality. Here you get a Hall effect without M or B.

Pr2Ir2O7: (S. Nakatsuji, PRL 96, 2006) has Pr3+ 4f2 localized Ising moments in a pyrochlore lattice of corner sharing tetrahedra. It is highly frustrated with no order down to 0.3K much less than the Heisenberg scale of 2K. The Ir4+ 5d5 provide the conduction electrons. Piers asks about the crystal fields and in response to Satoru's comment, Paul Canfield: asks about the point symmetry and worries about the declared doublet groundstate since this is non Kramers. Satoru parks the question about the evidence for a magnetic doublet. The pyrochlore lattice has the ice rules physics (2 moments in, 2 moments out on each tetrahedra along the local 111; direction of a tetrahedra) with the residual entropy classically. Evidence that this is the case in the Pr2Ir2O7 is the magnetic anisotropy which is consistent and neutrons. There is also a metamagnetic transition only for B fields along 111 which is when the system switches to a 3in 1 out state. S-W Cheong: says why is there no magnetization plateaux then? The numbers seems to match expectations. The claim is that there are ferromagnetic correlations with a Jff of 1.4K as seen as a peak in the specific heat. Chi3 has a steep negative increase and saturates to a large negative value. Andriy: why negative? The explanation comes in the form of the M(H) curve being convex not concave (which seems like a restatement of the facts to me). Normally you would expect spin freezing at around 1.4K but this does not happen.

Below 1.5K there is an enhancement and hysteresis in rho_xy between zero field cold and field cooled. Yet muSR shows no freezing of the moments down to 20mK. Field is along [111] and current along [110]. There is also a remnant Hall effect when B is returned to zero, though no evidence in the magnetization of a net moment. So this points to a spontaneous breaking of time reversal symmetry (TRS) at 1.5K. S-W Cheong as a question and demands to see the next slide(!). He means previous of course to much amusement. Is it really a zero field state on domain related state? Answer: domains are being alligned which need to be trained by the field. Paul asks about a specific heat signature. Answer: there is a peak but no jump.
There is anisotropy in this hysteresis effect: largest with B//[111] and smallest with B//[100]. So the 3-in, 1-out state may be stabalized not only at B>B_c but also may have some overlap with the groundstate.

Zero field quantum criticality in beta YbAlB4

We now turn to the second material which may show a spin-liquid groundstate. In beta YbAlB4 the Yb ions are organized into a honeycomb lattice which is slightly stretched and the B ions form a lattice of 5 and 7 fold coordinated rings. Here the ordered magnetism is suppressed apparently by the enhanced hybridization of the local moments with the conduction electron sea. Hard X-ray XPS reveals that this is ain intermediate valence compound with Yb averaging 2.75 rather than either 3+ or 2+.

Application of pressure enhances the hybrization as seen in the resistivities. The evidence for quantum criticality comes from the magnetic Gruniesen parameter which diverges but with two distinct power-laws. T^{-1.5} for 0.4K < t="0.">300). The critical field is quite anisotropic and seems to be paramagnetically limited. SdH oscillations are seen '(mean free path of order 1 micron) anbd show a 3D Fermi surface, and a mass of 30m. It is a combination of 2D cylinders and 1D sheets. A surprising thing is that X-ray PES shows that Yb3+ and Yb2+ coexists (roughly 2.75) . This mixed valence would normally yield a Pauli paramgnetic suscpetibility because of the screening/itinerancy. However in this material you actually see shows 1/T. Paul says, no it does not look like 1/T. Coleman says: is there one or two regions of Curie-Weiss. Answer: there are two. Resistivity exponent (rho ~ T^2) colour maps suggest a quantum critical point at B=0. Moreover it is a rare example of M diverging as a quantum critical point is approached. With interesting power laws. Rafael Fernandes asks is there something in the specific heat? Answer: yes but wait.... The interesting thing is that there is scaling of the magnetization (dM/dT) at B< f="B^\alpha" b="0". Oh no... it looks like my attempt to put in the scaling form as deleted the rest of my blog....help. Is there anyway of recovering it? Here is my attempt to recall the other later aspects of the blog and the questions.

Yuri Grin and Satoru Nakatsuji discusss alpha and beta YAlB4

There was a nice experimental contrast with the alpha phase of this material which differs by having a chequerboard arrangement of the distortions in the honeycomb lattice. It seems to be a Fermi liquid. So the claim is that this (ie the beta material) system is right on the border (within 1 gauss) of a quantum critical point. This seems an unlikely fine tuning so Satoru offered an argument that this is part of a quantum critical phase where magnetism has become a spin liquid. The mechanism was argued to be ferromagnetic interactions effectively frustrating the system.
Question time (apologies to those I have not remembered)
Amy Briffa and Rafael Fernandes: both asked about the evidence for the first material being chiral as opposed to a ferromagnet. The fact that there is no hysteresis in the magnetization but only in the Hall precludes ferromagnetism. There is no microscopic signature yet of the chirality - just the Hall anomaly.
Juri and Thamizhavel: both asked about the growth conditions of the alpha and beta phases. They can be discriminated by colour and morphology from the flux. However, why such similar materials emerge in pure form from a flux did not seem to be understood.