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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
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Spin Glasses[Aud][Cam] |
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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). a i is the H i coefficient of the
magnetization, M. Shown in a recent talk by Peter.
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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
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.
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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. |
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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. |
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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)
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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"?
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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.
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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)
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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.
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2. Disorder in Ferromagnetic QCPs
(Paul);
Nature of the Magnetism in MnSi (once again)
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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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