Quantum Phase Transitions


Week 6, 14th-18th February,  2005

Blogger:  Mike Norman 




    A change of the guard.  Piers has left, but Hilbert has arrived.  On Monday, we had a great introduction to the physics of spin glasses by Peter Young, followed by Mike's blackboard lunch talk on Fermi surfaces in quantum critical systems.  Then we heard about some key experimental results on magnetic quantum critical systems from Hilbert on Tuesday.  This was topped off by the Thursday discussion session.


Participants
Blackboard Seminar
Blackboard Lunch
Experimental Seminar
Thursday Discussion
Participants present. Click on participant to read questions that they have posed
Bedell, Kevin
Belitz, Dietrich
Carbotte, Jules
Dzero, Maxim
Ingersent, Kevin
Larkin, Anatoli
Lavagna, Mireille
Marenko. Maxim
Nicol, Elisabeth
Norman, Michael 		Paul, Indranil
Pepin, Catherine
Shaginyan Vasily
Tewari, Sumanta
Vojta, Thomas
von Lohneysen, Hilbert
Woefle, Peter
Ye, Jinwu
Young, Peter
Zwicknagl, Gertrud

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Blackboard Discussion. 10am Monday, February 14th.

Dr. A. Peter Young, UC Santa Cruz
 Spin Glasses[Aud][Cam]
     



. 
Aging in a spin glass (above).  On a temperature sweep down, a wait is performed, causing a dip in the susceptibility.  On continuation, the susceptibility rapidly recovers to the reference curve.  But on a subsequent sweep up, the dip is recoverved.  From Jonason et al, PRL 81, 3243 (1998).

Divergence of the non-linear susceptibility for a spin glass (left).  ai is the Hi coefficient of the magnetization, M.  Shown in a recent talk by Peter.

Spin glasses are characterized by a cusp in the imaginary part of the AC susceptibility versus temperature with the cusp location moving as a function of frequency.  A divergence is found at the so-called spin glass temperature in the static non-linear susceptibility (coefficient of H3 in an expansion of the magnetization with respect to H).  This implies that the spin glass temperature is indeed a transition temperature, but to a state whose order parameter is a multi-spin correlator.  The AC susceptibility also relaxes as a function of time below the spin glass temperature.  Most unusual is that the system "remembers" its history.  If one sits at a particular temperature point during a sweep down in temperature, and then waits (so the susceptibility changes with time), then when the temperature is further lowered, the susceptibility returns rapidly to what it would have been if the wait had not occurred.  On the other hand, during a subsequent temperature sweep up without a wait, the susceptibility deviates to follow the behavior it did on the temperature down sweep with the wait.  This "memory and rejuvination" effect has yet to be understood theoretically.

Peter then went on to discuss the standard Edwards-Anderson model based on a distribution of exchange couplings with infinite range, and subsequent work by Sherrington and Kirkpatrick, and by Parisi.  Then he turned to short range models, which have been addressed numerically.  In work he has done, the spin glass temperature is found by performing a finite sized scaling analysis, where the correlation length for different sized systems all intersect at the spin glass transition temperature.  This transition temperature vanishes for 2D systems but is finite for 3D systems, regardless of whether the spins are described by a Heisenberg, XY, or Ising model.  He then related the on-going debate about whether the Parisi treatment (based on replica symmetry breaking) is appropriate in the short range case versus the alternate "droplet" model proposed by Daniel Fisher and David Huse.  Neither picture at the present time seems to be consistent with all of the available experimental data.

Peter cautioned us that one must be careful about what one calls a spin glass.  His feeling was that unless the divergence is seen in the non-linear susceptibility, it was premature to call a system a "spin glass".

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Blackboard Lunch, 12.15 Monday, 14th February

Dr.  Michael Norman
(Argonne National Laboratory)
 Fermi Surfaces and Quantum Critical Points [Aud][Cam]

Mike's talk was intended to educate others at the KITP about some of the issues we are grappling with in the program.   He started out with the Ginsburg-Landau free energy function, which is expressed in terms of order parameter degrees of freedom.  This theory has been remarkably successful in explaining classical phase transitions in solids.  But its extension to quantum phase transitions by Hertz and Millis has not been as successful.  In such theories which include dynamics, the role of the dimension is replaced by an effective dimension which is the sum of d and z, where z is a dynamical critical exponent equal to 2 for an antiferromagnet, and 3 for a ferromagnet.  As such, this implies that for most cases of interest, one is above the upper critical dimension, so mean field theory should be applicable.  As we know, this is not the case experimentally, and Mike gave two examples of this:

1.  The observation of omega/T scaling in the dynamical susceptibility, a phenomenon known in the field theory community as hyperscaling, which can only take place below the upper critical dimension.

2.  The fact that the quantum critical regime is a non Fermi liquid, with the temperature dependence of the resistivity and specific heat typically different from what one would from Hertz-Millis theory.

So, what's the problem?  This bosonic theory is based on an underlying theory with fermions.  The question is whether one has thrown the baby out with the bath water in the process of integrating out the fermions.

Mike then remarked that for a superconductor (or a nested antiferromagnet), all the "constants" in the Ginzburg-Landau theory actually diverge in the T=0 limit.  This divergence is a consequence of the step function nature of the Fermi Dirac distribution at zero temperature.  The same step function behavior is also responsible for the divergence which characterizes the Kondo effect.  This indicates that the underlying nature of the fermionic system must be taken into account in the quantum case.

He then turned to the question of the Fermi surface itself, which is known to be different in the two low temperature phases (magnetically ordered versus quantum disordered).  Perhaps part of the reason for the non Fermi liquid nature of the quantum critical phase is the schizophrenic character of the electrons trying to decide which Fermi surface they should form.

Many of these issues are covered in the review article by Coleman, Pepin, Si, and Ramazashivili (J Phys Cond Matter 13, R723 (2001)).  As they discuss there, one can imagine two scenarios in the antiferromagnetic case, one where the Fermi surface change is confined to hot lines on the Fermi surface (separated by the ordering vector, Q), and another where the Fermi surface radically changes (say, due to the f electrons decoupling from the Fermi surface).  This would be evident in the Hall number.  They speculated that in the first case, the change in the Hall number would be proportional to the square of the order parameter, in the second case, the Hall number would jump discontinuously.



This "jump" was found in vanadium doped chromium.  But in this case, one has Fermi surface nesting.  If the Fermi surface was flat, one would indeed have a jump in the Hall number.  But even with warping, the Hall number change would be proportional to Delta, not to Delta2 as assumed before (where Delta is the energy gap due to magnetism).  This Delta dependence has been found by careful pressure studies, as presented by Tom Rosenbaum during the conference.


Mike then turned to the case of the bilayer ruthenates, where a field induced critical region is found.  Again, he related the story by Christoph Bergemann during the conference where he speculated that this critical region is characterized by a Pomeranchuk instability of the Fermi surface caused by the proximity of a van Hove singularity, where the Fermi surface breaks lattice translational symmetry.  Again, Mike emphasized the presence of Fermi surface nesting in these materials.


As a final example, Mike mentioned the poster child of quantum criticality, the heavy fermion metal YbRh2Si2.  Despite the small value of the ordered moment (0.01 Bohr magnetons), one finds a change of the Hall number by one carrier unit at the field induced quantum critical point, as related by Bergemann in his conference talk (the work of Silke Paschen which she will present on March 18 at the KITP).  This would seem to imply the two phases differ by an f hole, which is surprising given the fact that the ordered moment is two orders of magnitude smaller than this.

Mike mentioned two possibilities here.  One is that something truly novel is going on, and he mentioned the "chi fermion" ideas that Catherine Pepin and in a related vein Piers Coleman are pursuing (as given in their talks two weeks ago).  But another possibility was offered.  Band calculations can reproduce this large change by small shifts in the f electron level energies, with the resulting phases which reproduce the observed values of the Hall number only differing by 0.03 f electrons.  This is due to the very complicated Fermi surface expected to be present in this system.

David Gross then thanked Mike for summarizing the "confusion" of the field.  Things concluded with a rather interesting exchange with our fellow members of the other program concerning the relative merits of our respective fields.  All agreed that these talks should lead to more interaction between the two programs.

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Seminar, 12.30 Tuesday, 15th February

Dr.  Hilbert von Lohneysen
(U. Karlsruhe)
 Magnetic Quantum Phase Transitions [Slides][Aud][Cam]


Hilbert lectures in the new KITP auditorium.

     

Momentum dependence of the dynamic magnetic susceptibility in Au doped CeCu6, forming a double Y or "butterfly" structure.



Variation of the Hall coefficient with  doping in Au doped CeCu6.  Note the sign change at x=0.05.



The phase diagram of Pd doped UCu5, with an antiferromagnetic phase to the left, and a spin glass phase to the right.



A blue phase of a liquid crystal.  Is MnSi a magnetic analogue?  A "blue quantum fog"?
Hilbert, as well as being a co-organizer, was our "experimentalist for the week".  He gave a review of magnetic quantum critical points, looking at three different cases:











1.  Au doped CeCu6 - He emphasized the "double Y" structure of the critical wavevectors in momentum space, often referred to as the "butterfly".  This structure is also seen in pure CeCu6, so is not a disorder effect.  He then reviewed the well known results of Schroeder et al on the omega/T scaling of the dynamic susceptibility, and emphasized the two proposed scenarios for the quantum critical point:  one where the Kondo temperature vanished at the critical point, the other where it did not.

He then showed new results concerning the doping dependence of the Hall number, where a sign change is seen as a function of Au doping at a concentration of 0.05.  This indicates a continuous evolution of the Fermi surface with doping.

Hilbert next turned to the specific heat, and demonstrated that at the critical point, the specific heat was identical, regardless of whether it was reached by pressure or by doping.  He then showed some results that indicated that with an applied magnetic field, these materials seem to more resemble what would be expected from the "standard" Hertz-Millis-Moriya theory than what is observed at zero field.












2.  Pd doped UCu5 - This material, which is heavily disordered, has a line of zero temperature fixed points which connect a magnetic phase at low doping to a spin glass phase at higher doping.  Again, omega/T scaling is observed, but with a smaller exponent (1/3) than that observed for CeCu6 (3/4).  C/T appears to have a power law divergence whose coefficient varies weakly as a function of doping.







3.  MnSi - This was a powerpoint version of the discussion offered by Dietrich Belitz last week.  Hilbert showed that the <111> magnetic Bragg vectors evolved as a function of pressure to form a sphere of scattering, whose "hot spots" eventually rotated to the <110> directions, indicating a tendency for the magnetic helix to become depinned.  He denoted this as "partial order" where the helical order starts to melt.  He then made the analogy with liquid crystal "blue" phases as Dietrich had done, reminding us of the nice review article  done by Wright and Mermin on these phases (RMP 61, 385 (1989)).  Hilbert informed us that a blue phase was not stable for a chiral ferromagnet, but speculated that in the quantum critical case, where deff = d+z, such a blue phase might be stabilized.
Hilbert concluded by noting that many of the quantum critical systems of interest showed an excess of low temperature entropy.  He felt this "softness" of the system might be playing a key role in the novel physics of quantum phase transitions.

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Discussion: 4.30pm Thursday, February 17th, Founders Room.

Present at the discussion were Dietrich Belitz, Jules Carbotte, Maxim Dzero, Kevin Ingersent, Elisabeth Nicol, Mike Norman, Indranil Paul, Catherine Pepin, Bahman Roostaei, Vasily Shaginyan, Sumanta Tewari, Thomas Vojta, Peter Young, Veljko Zlatic, Hilbert von Lohneysen, Mireille Lavagna, Peter Wolfle, Kevin Bedell, Anatoli Larkin, Jinwu Ye, Maxim Marenko and Gertrud Zwicknagl.

Here were the topics discussed:

1 : Superconductivity in Doped Semiconductors (Dzero);
     Electronic Structure of Uranium Systems (Zwicknagl)
Maxim Dzero discussed work he did at the KITP in collaboration with Jeorg Schmalian concerning Tl doped PbTe.  This material, despite its low doping, exhibits superconductivity.  It also has an unusual linear T resistivity followed at low temperatures by an upturn before superconductivity sets in.  Their idea is that the Tl dopants act like negative U centers, and so one gets a charge analogue of the Kondo effect.  Mike Norman pointed out that similar behavior has been observed in Nb doped strontium titanate.

Gertrud Zwicknagl next discussed the work she had done on the electronic structure of uranium systems.  She argued that these materials are always in the mixed valent regime, but that Hunds rule effects are so strong, they must be incorporated into the electronic structure.  By looking at small clusters, they surmised the possibility of an orbitally selective Mott transition, where two of the f electrons become localized, and the others remain itinerant.  She is currently investigating a mean field theory of this physics by using a slave boson scheme.
2.  Disorder in Ferromagnetic QCPs (Paul);
     Nature of the Magnetism in MnSi (once again)
Indranil Paul then discussed the work he had finished at the KITP looking at disorder effects near a 2D ferromagnetic quantum critical point.  They derived four regimes as a function of separation from the critical point and temperature, where the Altshuler-Aronov corrections to transport either show a linear T correction, a log T corrction, a T1/3 correction, or a ln2T correction (see photo).  He was also engaged in a few other projects he did not have time to elaborate on.

Finally, the discussion inevitably turned back to the "quantum blue fog" of interest in last week's debate, and the question whether such a state could be stabilized in a chiral ferromagnet such as MnSi.  At that point, people were glad to call it quits, and head off to dinner.

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