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Quantum Phase Transitions
Week 4. Jan 31-Feb 4, 2005.
Blogger(s): Piers Coleman and Michael
Norman.
Mike Norman arrived this
week, and we now have a slow leadership transfer! Since
neither Mike nor I were present on Monday, our black-board discussion
moved to Wednesday. It was a very lively week -
I was sorry to miss the Directors Lunch by John Kogut on Monday, where
I gather he suggested that much of the language in the lattice gauge
theory and QPT programs is similiar - and that interesting links might
be found. On Wednesday, Brian Maple gave us a marvellous overview
of the impurity and spin-glass route to non Fermi liquid behavior in
the heavy electron systems including the work on Uranium Paladium 3
that started the interest in the early nineties.
On Wednesday, we continued our discussion of new zero modes at
the heavy electron quantum critical point. Catherine Pepin offered the
first bet of the meeting (see below!). Coleman and Pepin offered two
views about how spinless, charged fermion modes might play a very
important role a the heavy electron quantum critical point. There seem
to be three related ideas here -
- A new phase with Fermi
surface of spinless fermions that becomes almost gapless at the QCP.
- Deconfinement of holons and spinons at the QCP, with free holons
developing above a gap energy that goes to zero at the QCP.
- Development of a zero energy fermi surface of excitations at
the QCP, forming something I don't yet understand, called a "Fermion
condensate".
These are ideas in their infancy - and it was great to see the
community discussing the new ideas in an friendly, yet critical mode.
Participants
Blackboard Seminar
Directors Lunch
Main Seminar
Thursday Discussion
Participants
present.
Click on participant to read questions that they have posed
Belitz, Dietrich
Coleman, Piers
Dzero, Maxim
Ingersent, Kevin
Mydosh John
Maple, Brian
Norman, Michael
Paul, Indranil
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Pepin, Catherine
Shaginyan Vasily
Sushkov, Oleg
Tewari, Sumanta
Vojta, Thomas
Young, Peter
Zwicknagl, Gertrude
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Seminar,
12.30 Tuesday, February 3rd.
| Dr. M. Brian
Maple UCSD |
Non-Fermi Liquid
Behavior Near Magnetic Quantum Critical Points in Uranium-Based Systems[Aud][Cam] |
E/T scaling in
Sc1-xUxPd3 after cond_mat/0501004
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This talk came
in two parts - the first about non-Fermi liquid behavior in Scandium
doped UPd_3, the second about the induction of ferromagnetism in
Rhenium-doped
URu2Si2.
Brian began with a historical overview of non-Fermi liquid behavior. He
reminded us that there are probably two routes to non-Fermi liquid
behavior in heavy electron behavior:
- Single ion mechanisms
- Lattice mechanisms.
He took us back to the early nineties, when work on Yttrium doped
UPd_3 led to the first papers on non-Fermi liquid behavior in heavy
electron systems. The first experiments on this material at that time
showed that as Y was doped in on the U site, the Kondo temperature
dropped dramatically and non-Fermi liquid behavior, characterized by a
logarithmic temperature dependence
- Cv/T ~ b R/T0
ln
(T0/T) with a low temperature upturn that is
consistent with a
zero-point entropy.
- Chi ~ Chi0 ( 1 -
c (T/T0)1/2
)
These properties appear to accompany the approach to a spin glass
phase, which nestles between a full fledged antiferromagnet, for
dopings of about x = 0.3 to x=0.55.
These two observations led to two famous papers -
- Seaman et al, PRL 67, 2882 1991 proposing
that two channel physics was the origin
- Andraka and Tsvelik, PRL 67,2886 1991
proposing that there was a quantum critical point.
At the time, there were material difficulties with the Y doping. Brian
now introduced us to th e sister compound, Sc1-xUxPd3
where the Scandium enters the
material in a far more homogenious fashion. Its properties are
basically identical with the Y material. The two properties mentioned
above - are consistent with the original two-channel scenario -
but the resistivity in the range x=0.1 - 0.3 is inconsistent, and
follows the behavior
rho(T) ~ (1 - a
(T/T0)) a ~ 0.23 (exponent 1.1
essentially unity)
whereas T3/2 is expected in the two-channel physics. Brian also showed
a set of fascinating thermopower measurements, which showed that
when the spin glass develops around x=0.3, the thermopower switches
from a large positive value, consistent with Kondo behavior, to a
negative value suggesting that the kondo effect dies in the spin glass.
What was new however, is that at the critical doping xc=0.3, the
resistivity does develop the square root behavior expected for a two
channel Kondo problem. The neutron measurements also show that
the spin-correlations obey E/T scaling. Summarizing - at the spin glass
quantum critical point of the Scandium doped UPd3,
- rho ~ (1 - a (T/T0)1/2)
- chi'' (E,T)~ T-1/5f(E/T)
I find this very curious. It raises the fascinating question:
- Does the approach to a QCP into a spin
glass phase produce a
local quantum critical point with properties that are strongly
reminscent (zero point entropy, log in Cv/T, square root in chi, square
root component to resistivity) of the
two-channel physics?
The last part of the talk discused Rhenium doped URu2Si2. The Maple
group find that when the Rhenium doping exceeds 0.3, a spin glass
develops as a precursor to full fledged ferromagnetism. At small
doping, the resistivity has a quadratic temperature dependence, whereas
at x=0.3, the temperature dependence depends linearly on the
temperature. Clearly - a lot more work is needed to understand
the heavy electron- spin glass quantum phase transition.
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Blackboard
Discussion. 10am Wednesday, 2nd February. (The blackboard
seminars took place on Wednesday this week. )
| Dr. Catherine Pepin |
Fractionization
and Fermi Surface Volume in Heavy Electron |
Dr Catherine Pepin
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Catherine Pepin's
talk discussed the conjecture that the zero modes of a heavy electron
quantum critical point involve massless fermion excitations.
After writing down her proposed Lagrangian, she made an open bet - to
quote
"I'm offering a
bottle of (French?) Champagne for anyone who can reproduce the
experimental data of YbRh_2Si_2 without invoking a massless fermion a
the QCP."
After an extensive review of the experiments -
Dr Pepin described her phenomenological Lagrangian. One of the
important aspects of the data that motivates her work is the
Cv/T ~ max(B,T)-1/3
seen in the data of Custers et al.. The form of the Pepin
Lagrangian is motivated by a Schwinger boson decoupling of the Kondo
-Heisenberg lattice model. This model contains two features:
- Mobile spin 1/2 boson excitations that become
critical at the QCP
- A Fermi surface of charged, spinless fermions
that can not admix with the heavy electron Fermi surface in the
paramagnetic state.
Dr Pepin explained how the data, together with her RG procedure suggest
that the Fermi surface of spinless fermions must have a vanishing
velocity, with an energy that depends as the cube power of the
deviation in momentum space from the Fermi surface.
E(k) ~ (k-kF)3
This is an idea that is in someways, quite close to the idea of
"fermion condensation" due to Khodel and Shaginyan, that will be
discussed by Shaginyan next week. Pepin recognized that the
massless fermi surface was extremely finely tuned - but advocated that
it was strongly implicated by the power laws and that it might
have a hitherto undiscovered symmetry origin.
This prompted a very lively discussion. Here are some of the
questions that arose:
- (Mathew Fisher) Surely, there are gauge
fluctuations of the bond variables associated with the Schwinger Bosons
that are important for the non-Fermi liquid behavior. Pepin
expressed the view that the most important properties can be extracted
without recourse to examining these fluctuations
- (Coleman) What is the origin of the finely
tuned Fermi surface?
- Is the state with the charged spinless fermions
a phase of matter - and does it form via a
quantum phase transition.
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Dr Piers Coleman
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Towards a
Large N Description of the Magnetic and Paramagnetic Phases of the
Kondo Lattice[Aud][Cam] |
Dr Piers Coleman
Specific heat - Schwinger boson
approach to Kondo model.
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Coleman discussed the technical difficulties associated with the
Schwinger boson approach to the Kondo model, outlining recent work
carried out in collaboration with Jerome Rech, Olivier Parcollet and
Gergely Zarand.
He began by emphasizing that perhaps our failure to understand quantum
criticality in the heavy electrons stems from our inability to
construct a mean field theory that spans the magnetic quantum critical
point, from the
local moment antiferromagnet, to the heavy electron paramagnet.
There are two kinds of large N approaches for spins - bosonic, based on
the Schwinger boson, and fermionic, based on the Abrikosov
pseudo-fermion. The latter is well suited to the paramagnet, and
ill-suited to antiferromagnetism. The former is well suited to
describing low dimensional and fluctuating magnets, but ill-suited to
the Kondo problem.
Coleman described how the Parcollet Georges approach can be adapted to
the Kondo lattice. How the problem of a small phase shift can be solved
by using a Ward Identity. This approach permits the Arovas
Auerbach method to be unified with the Kondo problem.
Results were presented for the single impurity, where the bosons
develop a confinement gap at low temperatures and the two-impurity,
where the development of a singlet bond between the spins gives rise to
a magnetically correlated Fermi liquid, which at large Heisenberg
coupling, gives rise to the Varma Jones fixed point.
Various points were raised:
(1) whether the phase where the chi fermions are mobile has a fermi
surface in the lattice, and whether this is a "new phase of
matter".
(2) whether in the overscreened two impurity Kondo problem
one sees that J_H is always relevant.
(3) whether one can really examine the Fermi surface expansion
if the change in Fermi surface volume is a 1/N effect?
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Discussion:
4.30pm Thursday, February 3rd, Founders Room.
Discussion
Session –
4:30 pm, Thursday, February 3, Founders Room
Participants –
Dietrich
Belitz, Maxim Dzero, John Mydosh, Mike Norman, Indranil Paul,
Catherine Pepin, Jerome Rech, Oleg Suskov, Sumanta Tewari, Thomas
Vojta, PeterYoung, Gertrud Zwicknagl
Topic – Nature of the
Quantum
Phase Transition in Heavy Fermion Antiferromagnets
This discussion session
focussed on one of the key questions that was raised in the blog from
week one.
The discussion was started
by Dietrich
Belitz, who asked whether there were any known examples for heavy
fermions where the behavior near an antiferromagnetic quantum
critical point was described by the Hertz-Millis theory. Although
there was a claim for CeNi2Ge2, it seems that
in the cases best studied and of most interest, such as CeCu6-xAux
and YbRh2Si2, the answer was no. Unknown was
the situation for CeIn3, which has a simple cubic
structure, and whose experimental facts will be discussed next week
when Neal Harrison gives the experimental talk.
A number of questions were
raised by
various participants during the discussion in connection with this
topic.
-
Is the nature of the
antiferromagnetic phase understood? Although this is often assumed to
be a local moment state, in reality it seems that in many cases, the
nature of this state is poorly understood. Oftentimes, the moment value
is very small (like for YbRh2Si2), and the
specific heat coefficient is very large, which would argue against a
local moment state. Catherine Pepin claims that partial Kondo screening
could explain why the moment value is so small and the effective mass
is high, even if one uses a “local moment” description for this state.
And she argued that this could even be consistent with the large Hall
number changed observed at the quantum critical point by the Dresden
group.
-
What does the small
moment in materials like YbRh2Si2 mean? In URu2Si2,
for instance, the moment is small, but the specific heat anomaly at the
transition is large, arguing for the presence of a hidden order
parameter (as presented earlier in the workshop by John Mydosh). On the
other hand, the entropy involved in the specific heat anomaly in YbRh2Si2
does seem to be consistent with the small moment. Although this elastic
component gets quenched at the quantum critical point, there is
evidence, as remarked by Catherine Pepin, that the fluctuating moment
remains sizable.
-
Is locality necessary to
invoke to understand the dynamic susceptibility and the observed E/T
scaling? Such scaling is easiest to invoke for d=0, and this is a key
component as well of the marginal Fermi liquid conjecture for high Tc
cuprates. As Catherine Pepin reiterated, available evidence in Au doped
CeCu6 points to the same scaling behavior for all q
points, and as Mike Norman pointed out, a similar behavior has been
advocated for UCu5-xPdx by Osborn and Aronson.
-
What is the significance
of the sublinear exponents observed in the dynamic susceptibility? That
is, the susceptibility has the form X-1 = f(q)+Taf(E/T)
where a=3/4 for Au doped CeCu6, a=1/3 for UCu5-xPdx, and
a=1/5
for Sc1-xUxPd3. John Mydosh raised the
question of the influence of disorder on these exponents. Catherine
Pepin raised the question of logarithmic corrections that may not have
been identified yet by the experimentalists.
-
Are there corrections to
scaling in the quantum critical regime? Peter Young argued that
although corrections to scaling have been heavily investigated for
classical phase transitions, much less attention has been paid to this
for quantum phase transitions. Even for the antiferromagnetic side of
the critical point, the nature of the classical critical regime has not
been studied in detail for any of the heavy fermion cases, at least to
the participants’ knowledge.
-
And, finally, what is
the role of spin glass formation? In many of these systems, one
observes a spin glass behavior once the antiferromagnetism is
suppressed, as discussed by Brian Maple during his talk this week. This
has also been seen by Panagapolous in cuprates as he reported at the
conference. Is this important for some of the strange behavior observed
near the quantum critical point?
The
overall
conclusion of the discussion was that more experimental data on well
characterized systems would be needed to answer these questions. On
a final note, several of us indicated that in principle, close enough
to the quantum critical point, one would expect the long wave length
modes associated with the order parameter to eventually dominate the
singular behavior. The question is why this has not been seen in
many systems of interest. Does this mean that one has not gotten
close enough to the critical point with tuning parameter, or gone to
low enough temperature?
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