KITP - 15

Quantum Phase Transitions

Participants
Blackboard Seminar
Tuesday Discussion
Main Seminar
Thursday Discussion

Week 15.
April 18th - 22nd, 2005

Bloggers: Piers Coleman, Andrey Chubukov.

The rattler that put Qimiao Si in a locally (quantum?) critical state.

joerg's 40th...

Joerg's birthday party

Over the weekend the workshop had a rattle snake drama. Qimiao Si almost stepped on a rattle snake during a Sunday hike. Fortunately, providence was on his side - and he lived to tell the tale! On Monday, Andrey Chubukov gave a wonderfully honest blackboard seminar about the application of the spin fermion model to electron doped superconductors. The pro-Mott contingency in the audience gave him a hard time, but we all learnt a lot and went away with much food for thought. Monday was the anniversary of Einstein's death in 1955, but we also learnt that Joerg Schmallian was born 10 years later, so the event was celebrated...

Frank Steglich was our experimentalist of the week, but since he could not arrive until Tuesday afternoon, we used the Tuesday seminar time for a group discussion. On Wednesday Frank reviewed the problem of quantum criticality in heavy electron systems, hypothesizing that local quantum criticality is bad for superconductivity. Our Thursday discussion session .....

Participants present. Click on participant to read questions that they have posed

Chubukov, Andrey
Coleman, Piers
Haas, Stephan
Ho, Andrew
Khodel, Victor
Krotkov, Pavel
Le Hur, Karyn
Posazhennikova, Anna
Ramazashvili, Revaz

Schmalian, Joerg
Si, Qimiao
Stewart, Greg
Steglich, Frank
Vekhter, Ilya
Weng, Zheng-Yu
Yakovenko,Victor
Zhu, Lijun

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Blackboard Discussion. 10am Monday, 18th April.

Dr. Andrey Chubukov University of Maryland	Spin-Fermion Model Near a QCP: Non Fermi Liquid and Pairing
	Andrey introduced his recent work with Krotkov. The objective of the work was to apply the spin-fermion model to the case of electron-doped cuprate superconductors. using Eliashberg theory to understand the nature of the superconductivity which surrounds the magnetic quantum critical point. The talk turned into a quite fascinating discussion on the assumptions that go into the spin fermion model. There were many energetic dissenters, and some of the questions and remarks that arose are given below.
	Andrey began with a summary of the phase diagram in the hole, and then the electron-doped cuprate superconductors. Unlike the hole-doped case, in the electron doped superconductors, there is almost no sign of a pseudogap. The superconducting phase is nestled right up to the antiferromagnetic quantum critical point, having a finite T_c at the quantum critical point. Can this be explained within a spin-fermion approach?
Andrey began by stating the assumptions of the spin fermion approach. Suppose one starts with the Hubbard model, he said. In the spin-fermion approach, one assumes the most important low-lying collective degree of freedom to interact with the electrons is a soft magnetic modes, with a large susceptibility around Q= . This motivates the the spin fermion model. where the spin degree of freedom S_q is a Gaussian field with no intrinsic dynamics. This prompted a barage of questions and dissent! Karyn Le Hur asked: Q(Karyn le Hur) Andrey - close to half filling we know that the particle-particle channel competes with the particle hole channel, so why don't you include both, and what is the justification for only considering the particle-hole spin channel? A.(Andrey) We will avoid a nested Fermi surface that gives rise to the situation you describe. Andrey admitted that this was a weak point, and had to admit that this was effectively a kind of phenomenology. He agreed that there were in fact, a number of issues that the spin fermion approach sweeps under the rug, and to assuage his detractors, wrote them up on the left hand panel of the blackboard. (see below).
"Under the rug" issues in spin fermion model. (1) For an instability, one requies a coupling constant g~W, the bandwidth, yet for the derivation of the model requires that g<<<W is at weak coupling. (2) That, at least for hole doping, the unstable Q vector is not Q=. (3) That one can ignore all other collective channels - particle-particle, charge density are irrelevant.	The next assumption of the spin fermion approach is that the bare spin susceptibility of the spin boson has an Ornstein Zernicke form, and that the susceptibility is peaked around Q₀= . This then led to the second issue that is swept under the rug. Andrey admitted that in hole doped superconductors, the best efforts to calculate or measure this susceptibility show that it is not peaked at the commensurate Q vector, and is at best, a very flat function around Q₀. However! He said that in electron doped superconductors, chi is indeed peaked at . This prompted a remark from Qimiao Si Remark: (Qimiao Si) Andrey - just because there is no difference between RPA and Heisenberg at half filling, this doesn't mean RPA is good at finite doping. Surely there is a fundamental difference between Heisenberg superexchange and this contact interaction. (Bloggers aside - so when is it reasonable to treat magnetism as a non-interacting boson? Various methods - the spin fermion approach, and also the Extended Dynamical Mean Field theory that Dr Si has pioneered, make this assumption.)
The important point about this band-structure, is that the velocities of the Fermi surface at the magnetic Brillouin zone are equal and opposite. This does not occur with hole doping, where the velocities are not antiparallel, and the self energy is proportional to . However, here the self-energy acquires a marginal Fermi liquid form over intermediate energy scales.	Andrey sketched the band structure and the q-dependence of the susceptibility for the electron doped case. The next question came from Ilya Vekhter - Q: (Ilya Vekhter) How is the asymmetry between hole doping on Oxygen sites and electron doping on Copper sites built into your theory. A: Good question..... Q: (Qimiao Si) It seems very unfair to me that you use a different Fermi surface as a reason to delineate between electron and hole doped cuprates. A: Andrey drew a funny diagram that I didn't understand.
	Andrey then summarized the kind of self energy that is obtained from the spin fermion model. He showed us the real and imaginary parts. At low energies, the self-energy is Fermi liquid like, but above the spin fluctuation energy produces a marginal Fermi liquid self energy which extends up to a scale . Above this scale the scattering rate depends on frequency to the power 3/4. Remark: (Joerg Schmallian) didn't Altshuler, Ioffe and Millis have vertex corrections to the self energy that contained logs?
At this point Andrey turned to the treatment of superconductivity within an Eliashberg/spin fermion model. He wrote down an equation for the vertex part in the d-wave pairing channel, which took account of the unusual frequency dependences of the self-energy. When these equations are solved, he claims, the transition temperature of the superconductor in the paramagnetic phase peaks at the magnetic quantum critical point. Andrey told us that the characteristic scale of Tc is determined by his scale , i.e the superconductivity is not really aware of the quantum critical magnetism. He then reported on an interesting consequence of this treatment. He claimed that the d-wave order parameter has an interesting dependence on the direction of momentum. In addition to the node along the diagonals, their theory also predicts that the gap will vanish along the x and y axis, with a dependence on angle that goes as . Andrey claims that something reminiscent of this is indeed seen in Arpes around a 12% doping. It seemed interesting.....	Andrey's conclusions

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Discussion session 12.30pm
Tuesday 19th April
Open questions on quantum phase transitions

Frank Steglich's talk was scheduled for Wednesday, so we used our usual lunch-time Tuesday slot to come up with
the most essential issues in the field of quantum criticality, which we still need to address. A similar session was held at the beginning of the workshop, and it was thought it might be interesting to contrast the issues that arose. Here are the questions that arose as part of the discussion.

1. Do we have the right starting mean-field theory for the description of the emergence of antiferromagnetism in 3D heavy fermion/Kondo lattice systems? (Piers Coleman)

Piers argued that our understanding of classical criticality is built upon an expansion around Landau or Weiss mean-field theory, that in this case at least, the mean-field theory uniquely defines the interacting infra-red modes that dominate the universal long-wavelenght physics. He argued that the failure of Hertz Moriya theory is not a failure of the general approach, but an indication that perhaps we need to look for a new class of mean-field theory, one that might furnish new kinds of infra-red modes (such as fermionic zero-modes) that co-exist with the zero modes of the order parameter.

Leon Balents remarked that perhaps the pretext of the question was wrong, and that one could imagine a wholley different approach, such as that of Sentil, Balents, Sachdev, Vishwanath and Fisher, where the critical behavior is not associated with fluctuations of the order parameter, but topological fluctuations in the background of the order parameter.

2. Do the phases of the two sides of the QCP uniquely specify the
nature of the QCP? (Qumiao Si)

This was a follow up question to question 1.

3. Do we have to take Mott physics into account near AFM transition in
one-band systems? (Andrey Chubukov)

At this meeting, there has been a fervent and continual discussion about the degree to which Mott physics is important to the quantum critical behavior near an antiferromagnetic transition. Andrey would agree that Mott physics seems to be vital to the AFM QCP in two band-systems, such as the heavy electron systems, but would argue that the situation is unclear for one band systems (where he favors a spin fermion approach).

4. Is Hertz-Millis description internally inconsistent by construction? (Joerg Schmalian)

This question is motivated by the observation that the Moriya-Hertz-Millis description of quantum criticality, indeed, the entire spin-density wave
and Stoner approach, relies on taking an essentially weak-coupling approach, and then pushing it to a point where the strength of the interaction is comparable with the band-width. Joerg would argue that this is fundamentally inconsistent. However, no one can actually point to a concrete
internal inconsistency of the approach, and many felt that it was more likely that there are two types of QCP, those in the intinerant category
described by the Moriya-Hert-Millis framework, and others governed by a more localized description.

5. Do QPT in continuous systems (without lattice), such as 2D electron gas and 2D He3 have something in common with lattice systems/Kondo systems? (Victor Yakovenko)

Victor Yakovenko reminded us of a wide class of quantum critical points where the lattice, and the presence of localized moments, does not play any role. He mentioned the "solidification transition" in 2D He-3, where measurements by Saunders et al, reveal a transition strongly reminiscent of a Mott-Hubbard transition. He also cited the 2D metal insulator transition as another possible example. Victor argued that we should examine these continuous cases as we discuss the problems of quantum phase transitions.

6. Why these are so few examples of insulating QCP? (Qumiao Si)

Various people questioned the tenet of this question, arguing that the list is short, but growing fast.....

7. Should we think about NFL phases rather than QCP separating Fermi liquids,
and is there a possibility to experimentally prove that there indeed exist NFL phases?
(Joerg Schmalian and Andrey Chubukov)

This is a question that repeatedly surfaced at the meeting. The case of MnSi has really brought back to center field, the possibility that we should consider non-Fermi liquid phases as a serious posisbility. If this is the case, then MnSi should have a second higher pressure QCP in which it
transforms back to a conventional Fermi liquid.

During the discussion about the last question, Qumiao Si argued that the
Gruneisen parameter may be an indicator of such a phase - suggesting that this quantity might be measured in the putative NFL phase of MnSi at high pressures.

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Seminar, 12.30 Wednesday 12th April

Dr. Frank Steglich MPI CPFS, Dresden	MPI CPFS Quantum Criticality in Heavy Fermion Materials[Slides][Aud][Cam]
	On Wednesday, Frank Steglish, the head of Max Plank Institute for the Chemical Physics of Solids, in Dresden, presented the talk on the experimental studies of quantum criticality in heavy fermion materials. To come to our workshop, Frank crossed more than the half of the globe (he arrived from China). Besides giving a talk, Frank spent all his time at KITP discussing non-stop his experiments with other participants and patiently listening to their, in many ways conflicting, theoretical ideas. In his talk, Frank presented the comparative analysis of Ce-based materials and YbRh₂Si₂ , lightly doped with Ge.
	He started his talk by showing us the data for CePd₂Si₂ and CeCu₂Si₂. The phase diagram for CePd₂Si₂ is rather ``conventional''. The system is antiferromagnetic at low T at a normal pressure. When pressure increases, the Neel temperature goes down and vanishes at P_c ~ 28 kbar. The antiferromagnetic QCP at this pressure is surrounded by a dome of superconductivity. Superconductivity is very sensitive to small potential scatterers, what implies that the pairing likely has non-s-wave symmetry (d-wave is the primary candidate). Above the superconducting dome, the resistivity at P_c scales as T^1.2 i.e., is almost linear.

	Frank then presented the data and phase diagrams for CeCu₂Si₂ and CeCu₂(Si_1-xGe_x)₂ for x=0.1 and 0.25. He first focused on the phase diagram. The phase diagram for undoped CeCu₂Si₂that emerged from earlier studies, is somewhat similar to CePd₂Si₂ -- it contains an antiferromagnetic phase and a superconducting phase. The main difference is that the superconducting phase extends to larger pressures. Frank argued that this phase diagram is misleading. He presented the new data for doped CeCu₂(Si_1-xGe_x)₂ and demonstrated that there are in fact two different superconducting phases. One phase forms a dome on top of antiferromagnetic QCP, another phase is unrelated to the magnetic QCP and occurs at higher pressures. Frank speculated that this is connected a quantum critical end point associated with a jump in the Cerium valence.
(Bloggers aside: in the following week, Gil Lonzarich presented similar data for CeCu₂Ge₂. Gil argued that there is additional CDW transition at higher pressures, and the second superconducting region surrounds that QCP). Frank then discussed resistivity and specific heat data near the antiferromagnetic QCP. He argued that some data are reasonably well described by 3D SDW Hertz-Millis theory, but some data, e,g., an upturn in the specific heat coefficient, are inconsistent with SDW scenario. He also mentioned that the small ordered moment (0.1 µ_B) that they observe in the A- phase of CeCu₂Si₂is inconsistent with the gapping of the substantial portion (40%) of the Fermi surface. Frank speculated that this may indicate the presence of the hidden order.

He then went on to discuss two possible scenarios of quantum criticality -- SDW scenario and ``local scenario'' by Coleman, Si and others, in which Kondo scale vanishes at the same pressure at T_N . He argued that while in CePd₂Si₂ and CeCu₂Si₂ the Kondo scale likely remains finite at criticality, the data for YbRh₂(Si_1-xGe_x)₂, which he presented next, cannot be described within finite $T_K$ SDW scenario. He speculated that local scenario may be the clue for the understanding the data for YbRh₂(Si_1-xGe_x)₂.
	To support this idea, Frank presented numerous data for quantum-critical behavior in undoped and Ge doped YbRh₂(Si_1-xGe_x)₂ in zero field and in the field. His main point is that there are two characteristic temperature scales near QCP, one around 20-30K and a second around 0.3K. He noted that the resistivity is linear in T from 10K downwards and does not see any of the subsequent low temperature scales. and the specific heat coefficient Cv/T scales as log T from 10K downwards, but below 0.3K starts to rise as T^--1/3. The spin susceptibility behaves roughly as T^--3/4 between 0.3K and 1K crossing over to Curie behavior at higher temperatures. Frank argued that this two-scale behavior is observed at the field induced quantum critical point in YbRh₂Si₂, and in the Germanium doped sample at approximately zero field. Bloggers aside: This is clearly a very deep mystery. Why is it that the thermodynamics and the spin susceptibility are sensitive to these low energy scales, yet the resistivity is linear over three decades?
Conventional Gruneisen parameter for CeNi₂Ge₂. "unconventional" Gruneisen parameter for YbRh₂(Si_1-xGe_x)₂	At the end of his talk, he discussed the Grunesien parameter , the ratio of the thermal expansion coefficient to specific heat). He argued that for CeNi₂Ge₂, is proportional to T, which is consistent with 3D SDW scenario. However, in YbRh₂(Si_1-xGe_x)₂, scales as T^0.7 at the lowest T. primarily due to the singular behavior of the specific heat coefficient. Frank's conclusion is that quantum critical behavior in CePd₂Si₂ ,CeCu₂Si₂ and YbRh₂(Si_1-xGe_x)₂ show marked differences, and all data for YbRh₂(Si_1-xGe_x)₂ indicate that the QC behavior is highly unconventional. Answering one of the questions, he also presented susceptibility data near QCP, which indicate that the uniform susceptibility diverges faster than the effective mass. This likely indicates that ferromagnetic fluctuations are enhanced near QCP, despite the fact that magnetic ordering is antiferromagnetic.

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Discussion: 4.30pm Thursday, 14th April. Founders Room.

In this penultimate discussion, Andy Ho from Birmingham University, gave an inspirational review on the physics of atom traps, with the view towards possible areas with quantum phase transitions. In the second half of the discussion, we turned to discuss issues arising out of Frank Steglich's Wednesday seminar.

Andy explained that there were two ways of looking at atom trap systems:

as quantum simulators of models from strong correlation ("analog quantum computers")
as an opportunity for quantum engineering

He listed the following pros and cons:

Pros	Cons
Tunability. The ability to continuously tune parameters in the lattice, such as "t" and "U"	Finite size of the system.
Clean	Inhomogeniety. An atom trap has a trapping potential- not uniform except perhaps in chip traps
New Field, many great new ideas.	Phase co-existence

Andy pointed out that it is comparatively easy to produce 1D and 3D lattices, but that 2D lattices are trickier, but might be possible with the new idea of a "chip-trap", whereby an engineered chip provides the external potential. Ilya asked what the actual potential was- Andy pointed out that it derives from the polarizeability of atoms, proportional to the square of the field E². In a Hubbard lattice, t, and U are tuneable, with an approximate dependence on the tunnelling amplitude V and field E given by

and

. The exponential dependence of the hopping on V meant, he said, that one could reach regimes where the temperature is much smaller than the hopping. One of the big challenges, he said, is the development of

new probes. There is no thermodynamic probe, there are no leads and one can not take advantage of the atom-light interaction.

Indeed, at present, there is little more one can do than time-of-flight measurement of the momentum distribution function.

As an example of a problem which has attracted some interest as a possible connection with strongly correlated systems, Andy mentioned the recent work of Powell, Sachdev and Buchler on Bose-Fermi mixtures containing fermions, bosons and fermion-boson molecule bound-states. The model Hamiltonian looks a lot like a slave-boson model without a lattice:

and can be treated in a similar vein. Qimiao pointed out the the Fermi surface volume transitions that take place are like a kind of Lifshitz transition. This work was also discussed at the QPT conference by Subir Sachdev in January.

We then switched subjects. Frank took the podium and reminded us about the paradox we face in interpreting the resistivity and susceptibility of YbRh₂(Si_1-xGe_x)₂. Why is the resistivity linear to the lowest temperatures, yet at the same time, the susceptibility develops a "Curie Law" at low temperatures, reminding us that at low T,

On the other hand, he pointed out, we know that the entropy is much less thatn Rln(2), so the entropy of local moments is not present. He expressed worry that the interpretation in terms of a Curie Weiss law may be in doubt. He shared with us data(over) that shows that dM/dT= dS/dB=\xi satisfies a scaling law

At this point, a lot of discussion broke out. Victor Khodel suggested that many of the properties of YbRh₂(Si_1-xGe_x)₂can be explained using the ideas of the fermion condensate.

Andrey Chubukov asked Frank Steglich whether the T2 resisitivty expected from the Moriya-Hertz-Millis theory has ever been observed? Qimiao Si suggested that, effectively, the predictions of the SDW scenario are realized at the QCP in CrVa. Frank said that perhaps YbIr₂Si₂might be an example - but was not sure whether one had yet reached the QCP. A discussion followed about what was expected from Achim Rosch's model for the transport from hot-spots.

Finally, discussion turned to whether the interpretation of YbRh₂Si₂as an almost ferromagnetic metal, was correct. Here the issue is the following - is the enhanced uniform susceptibility an indication that Q=0 is preferentially enhanced, or whether there may be many other soft regions of momentum space. Qimiao Si argued that the remarkable observation that the modified, "kadowaki Woods ratio

is constant, (the conventional Kadowaki Woods Ratio is not constant) as one tunes at low temperatures, within the Fermi liquid, towards the QCP, suggests that the large susceptibility at Q=0 is tracking a large susceptibility at some other Q responsible for relaxing the current. Q=0 fluctuations, he pointed out, don't relax the current. He proposed that the Curie energy scale in the Schoeder-Aeppli scaling form for the susceptibilty (*) must be small in many regions of momentum space other than Q=0.

Schroeder, Aeppli et al (2001).

On Thursday evening, a group photo was taken at a get-together hosted by Qimiao Si. Participants present are

Back row: Martin Einhorn, Victor Yakovenko

Middle Row: Andrey Vorontsov, Joerg Schmalian, Ilya Vekhter, Piers Coleman, Frank Steglich, Qimiao Si, Anna Posazhennkova, Karyn Le Hur, Kamel, Pavel Krotkov

Front Row: Andrew Ho, Stephan Haas

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