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.