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Quantum Phase Transitions
Week 14.
April 11th - 15th, 2005
Bloggers: Dirk Morr, Piers Coleman, Andrey Chubukov.
We're into the last three
weeks of the program! On Monday, Karyn Le Hur gave us an extensive
review of local quantum critical models in "the road to quantum
criticality", emphasizing some interesting links between the world of
quantum dots, and heavy electron quantum criticality. Greg Stewart has
arrived as our new resident experimentalist, and on Tuesday, gave us
some interesting insights into some of the difficulties and pitfalls of
over-interpretation of experimental results in this field. On
Thursday, we had an extremely lively, and at some points, heated
discussion, with Ping Sun from Rutgers and Qimiao Si from Rice
explaining two different viewpoints on local quantum criticality, in
the midst of which, Joerg Schmalian made a loud and hilarious bet.
These Thursday discussions have turned into a minor highlight of the
week, and are always very enjoyable.
Participants
Blackboard Seminar
Main Seminar
Thursday Discussion
Participants
present.
Click on participant to read questions that they have posed
Chubukov, Andrey
Coleman,
Piers
Eshrig,
Mathias
Hanke,
Werner
Ho, Andrew
Khodel,
Victor
Krotkov, Pavel
Le Hur, Karyn
Morr,
Dirk
Pepin,
Catherine
Posazhennikova, Anna
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Ramazashvili, Revaz
Schofield,
Andrew
Schmalian,
Joerg
Si, Qimiao
Sun, Ping
Vekhter, Ilya
von
Lohneysen, Hilbert
Weng, Zheng-Yu
Yakovenko,Victor
Zhu, Lijun
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Blackboard
Discussion. 10am Monday, 11th April.
Dr. Karyn Le Hur
Universite de Sherbrooke, Canada |
The Road to
Local Quantum Criticality[Aud][Cam] |
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Karyn began her presentation by reviewing the phase diagram of the
Bose-Fermi Kondo model introduced by Anirvan Sengupta, Qimiao Si
et al . She discussed
the RG-flow for a system in which the impurity spin couples to a
bosonic mode. In this case,
there are two fixed points anSU(2) fixed point and an xy-fixed
point,
describing a multi channel boson problem. When the impurity spin is in
addition coupled to a fermionic bath, the RG flow diagram now contains
three fixed points
(FP): a stable SU(2)-Bose FP, a stable SU(2)-Fermion FP, and a
metastable SU(2) Bose-Fermion FP. The later FP appears to order
epsilon, where epsilon determines the
spectral function of the bosonic bath which is given by
omega^(1- epsilon).
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Karyn then discussed the question of how a finite value of epsilon (i.e
sub-ohmic bath) can be obtained in a realistic system. As an
example, she mentioned the Luttinger-liquid, in which
epsilon>0 naturally emerges and is given by epsilon=1-(k rho).
Another example which has been proposed by Q. Si et al. is a
two-dimensional
itinerant ferromagnet where the spectral density of the bosonic bath
(ferromagnetic spin waves) has a square-root dependence on the
frequency. Andrey Chubukov was extremely sceptical about this
example, but the issue was resolved after the talk.
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Discussion then
turned to a specific Bose-Fermion Kondo-model
motivated by a noisy mesocopic qubit (or quantum dot). When the tunnel
coupling with
the reservoir lead is small, the average number of electrons increases
in a step-like
manner when the number of holes is varied. Karyn demonstrated that near
such a
step, the system can be mapped onto a two-level charge system with a
transverse
Kondo-coupling. It is also capacitively coupled to some bosonic modes
via fluctuations in the
gate voltage. The bosons represent the "electromagnetic
noise" in the gate voltage stemming from the finite resistance R in the
gate lead for example, and
the
spin-boson coupling reflects the effect of the voltage noise on the
charge fluctuations of the qubit. The coupling with the bosons is
controlled
by the
resistance R of the gate lead in the setup.
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Karyn then discussed in more detail the theoretical model she used to
describe such a system. In particular, she considered a Bose-Fermion
Kondo-model with
a transverse coupling to the fermionic bath (which is one-dimensional),
and an Ising
coupling to the bosonic bath (with epsilon=0). She derived the
RG-flow
equations, and argued that this model belongs to the same universality
class as the
Caldeira-Legett model. |
By
bosonizing the fermionic bath, Karyn derived the Hamiltonian of the
impurity spin coupled to two bosonic baths. She demonstrated that the
dynamic part of
the effective action is given by |omega| phi^2 for both bosonic baths.
By
performing aunitary transformation, one obtains a local action
describing the
impurity spin that
is coupled to a single bosonic bath, and thus corresponds to the
Caldeira-Legett model. Finally Karyn discussed the RG-phase diagram,
which contains
(i) a ferromagnetic Kondo phase, in which the moment is
unscreened,
and anantiferromagnetic Kondo-phase which is a Fermi liquid. These two
phases are separated by a Kosterlitz-Thouless transition. Karyn
mentioned that this model was analyzed using NRG. The main important
result is that one can have access to the whole phase diagram
considering the noisy qubit above by tuning the resistance R of the
gate lead and temperature for example. |
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Seminar,
12.30 Tuesday 12th April
Dr. Greg Stewart
University of Florida |
Materials
Problems in QPT Systems: Their Non-Negligible Effect and the Struggle
to Avoid/Minimize Them[Slides][Aud][Cam] |
(Piers introducing Greg. Note Andrey running out of the Seminar
Room.)
Greg In the seminar.
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Before starting this talk, Greg warned us that this was going to be a
tough talk for theorists - but - he said that like the "Cod Liver
Oil" we had as kids - we would hate it, but it would be good for
us! As you can see from the photo, some theorists simply couldn't take
it!
Greg Stewart used his talk to discuss some of the very dangerous
pitfalls that he worries we might be making in the interpretation of
data on quantum critical systems. His talk was wide
ranging, spanning the following topics:
- The hazards of extracting specific heat
exponents without ultra-low temperature data and more than a decade of
temperature range.
- The importance of sample purity.
- The dangers of inhomogeniety,
particularly in UCu4Pd.
- The inconsistencies that sometimes appear
between field tuning, and pressure/doping through the quantum phase
transition.
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| Greg started the
talk by pointing out that the interpretation of specific heat at
quantum critical points is a delicate matter. He told us that
theorists must always insist on data below 1.4K as an absolute
requirement before believing any fits. Beyond this, he pointed out the
well-known fact that log-specific heats and power-law specific heats
with low exponents, are almost indistinguishable. |
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Greg's next lesson was the importance of sample purity. Quantum
critical materials are particularly sensitive to disorder. Greg gave
the impressive example of Strontium Ruthenate (SR3Ru2O7), where early
data suggested a "phase" surrounding the quantum critical point, with a
T^3 specific heat. Higher quality samples showed te dangers of early
interpretation - instead of a separate phase, the higher quality data
showed that the residual resistivity is ultra-sensitive to the distance
from the critical point.
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The mysterious phase at the quantum critical end point in Strontium
Ruthenate is later revealed to be..... |
due to an acute impurity sensitity near the quantum critical end point |
Greg then turned to
the issue of field tuning versus pressure/doping tuning of quantum
critical points. Greg pioneered field tuning and found in early
experiments, that the effect of field tuning CeCu_6-Silver produces a
conventional Moriya-Hertz type specific heat,
with no divergence, in contrast to the doping tuing, which von
Lohneyson et al have shown leads to a log divergence in Cv/T. The
table across summarizes some of these differences.
Note that in YbRh_2Si_2, field tuning and doping give very similar
results, though the range of the linear resistivity is much smaller in
the Ge doped materials.
At this meeting there has been a wide discussion about the relationship
between possible Moriya-Hertz type QPT and more localized
transitions. Can field tuning and pressure tuning take us
through different types of QCP, without the nature of the AFM changing
somehow? (See discussion last week).
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In the last part of
the talk, Greg talked about sample inhomogeniety. He
discussed Yttrium and Scandium doped UPd_3, where the doping
induces a magnetic quantum critical point. Around the critical
doping (eg 0.2 in Scandium doped), one sees a log specific heat
and powerlaw resistivity. However, the material also shows
inhomogenieties on a scale of 20µm. Greg also mentioned
CeNi_2Ge_2 and UCu_4Pd as systems with serious problems with
homogeniety. The former system shows non fermi liquid behavior that
have been interpreted in terms of an SDW transition (gamma ~ const -
Sqrt(T)), but the homogeniety issue has become so severe, that the
community has largely stopped working on it. EXAFS measurements by
Booth et al have revealed that the latter system , UCu_4Pd
suffers from large-scale disordering in the Pd sublattice. Greg argued
that we need to look for systems with more homogenious disorder, (such
as CeCu_6-xAu_x), if we are to use disorder as a tuning parameter in
QCP research.
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Discussion:
4.30pm Thursday, 14th April. Founders Room.
Extended Dynamical Mean Field theory: how to do it right.
Presentation by Dr Ping Sun, Rutgers University with rowdy
participation from the croud.
Qimiao Si (left) discusses with Ping Sun (right) in the Founders room
of the KITP.
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Today's discussion focussed predominanly on the extended dynamical mean
field approach to quantum critical points. This is an area of work that
was initiated by Si,
Rabello, Ingersent and Smith and more recently,
with Zhu and
Grempel, as a core constituent of their locally quantum
critical
theory of heavy electron quantum phase transitions. In a separate
development Sun
and Kotliar have considered the Anderson lattice model
in using the same approach. Ping Sun came from Rutgers to discuss
their different
point of view. There was a lively exchange of
viewpoints, and this blog attempts to capture the main points that were
made.
Ping Sun began by outlining the main differences between the two groups
approaches. Here they are:
- Model. Si et al consider a 2D Kondo model
whereas Sun et al consider a 2D Anderson lattice model. The local
dynamics that the two groups use is essentially identical.
- The method of identifying the antiferromagnetic
instability is different between the two groups. This, rather than the
difference of models is probably the heart of the difference between
the two groups.
- Si, Grempel et al find a second-order
transition, whereas Kotliar and Sun find a first order phase transition
- according to Sun and Kotliar, there is no local quantum critical
transition and to make progress, one has to consider intersite
interactions of Kondo spins.
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groups agree, that the main difference in their approaches concerns the
"induced interactions". At the local level, both groups follow
exactly the same procedure. Ping Sun went on to explain
that differences lie in the way they locate the instability in
the phase diagram - the philosophy of the Sun-Kotliar group is that you
must take account of how the electrons in the bulk modify the
q-dependent interactions, and when you do so, this kills the second
order transition. |
Andrey Chubukov
pushed for more clarity, and at this point Qimiao Si asked if he could
try to summarize the way EDMFT identifies the point of magnetic
instability.
At this point, Qimiao Si stepped to the (left-hand) blackboard......
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Chubukov requests clarification!
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So
here's how it is according to Qimiao Si. Magnetism demands that one
seek a divergence in 
(at zero frequency),
which is given by the expression
Here,  is the intersite
interactions of the local moments that are already included at the
Hamiltonian level. All the action is in the 
term, and it is here that the two groups differ fundamentally.
This term takes into account the change in the RKKY interaction due to
the environment, and formally its given by the difference of two
inverse susceptibilities -
where the lattice
susceptibility is
computed as a convolution of two lattice Green's functions, and the
local susceptibility is computed as a convolution of two local Green's
functions.
Explicitly,
with no vertex corrections, and
is a convolution of the local Green's functions. |
At this
point
Qimiao pointed out that there are three possibilities:
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| Scheme
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Philosophy
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Paramagnet
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Antiferromagnet
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I
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"No
momentum is
special: don't mess with " |
=0 |
=0 |
| II |
"Interactions
change
in the AFM"
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=0 |
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III
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Interactions
change
in the paramagnet too
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Sun and Kotliar adopt scheme II, whereas Grempel and Si adopt scheme I.
The underlying philosophies of the two groups are different -
Grempel and Si take the view that there is no special momentum in
extended dynamical mean field theory - and consequently, one should not
favor any particular momentum and should average over all
- which averages to zero. Qimiao argued that if the
only consistent schemes are scheme I or scheme III. In these schemes,
he argued, the RKKY interactions on the two sides of the
transition are incorporated on an equal footing, so that there is no
artificial extra energy gain introduced preferentially into one, or the
other phases. Qimiao claimed that scheme II is doomed to produce first
order transitions, for any itinerant system, even for SDW transitions
in models without Kondo physics.
By contrast, Ping argued that the
magnetic ordering Q-vector
is special - and one should follow the scheme from DMFT,
taking the effect of the fermions on the non-local interactions into
account. He also pointed out that scheme II is needed if one is
to correctly reproduce the Hartree effects in the antiferromagnet.
This prompted a loud and hilarious response from Joerg Schmalian,
who announced defiantly that
He,
i.e Joerg Schmalian would bet a $25 bottle of wine that the
nature of the magnetic condensate would be different for
different "choices", but that antiferromagnetism, would occur in
either scheme I or scheme II.
We take to mean that if magnetism occurs in only scheme II,
Ping collects, but that if it occurs in both schemes, Joerg
collects. We aren't sure whether making a profit like this is legal
under NSF guidelines but the organizers would indeed like to know the
ultimate outcome of this bet.
Whereas the two groups could not agree on why one scheme is better
than the
other, they do apparently agree that this is the difference
between the two
approaches. It would perhaps be useful if one had a systematic
derivation of
these schemes, starting for example from a Kadanoff Baym generating
functional. Ping Sun claims this is exactly what has been done in
their recent work.
There was not enough time to pursue this point further.
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At the
end of the discussion, Ping turned to discuss his recent work with
Kotliar using a two impurity
cluster DMFT to describe the Kondo lattice. One of the nice things
about this work, is that it permits one to compute the temperature
dependence of the onsite and nearest neighbor spin correlators.
Ping and Kotliar find a number of interesting features in their
solution -
- The specific heat is found to follow a T
Log(T0/T) behavior.
- The intersite spin susceptibility is found to
change sign at low temperatures, suggesting the growth of ferromagnetic
spin correlations near the quantum critical point.
We could have used more time to hear this discussion in more detail,
but we were all exhausted, and closed the discussion at six pm.
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