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Quantum Phase Transitions
Week 5, 7th-11th Febuary, 2005
Blogger: Piers Coleman.
Thursday discussion by Mike Norman.
Edited by Mike Norman.
Another week of
lively discussion. On Monday, Vasily Shaginyan
described the arguments for "Fermion condensation". Our
experimentalist of the week was Neil Harrison from the high field lab
in Los Alamos, who presented a very
exciting set of results showing de Haas van Alphen measurements that
chart the evolution of the Fermi surface in CeIn3 as it
approaches a
field-induced quantum phase transition. On Thursday, we
discussed the
enigma posed by the non-Fermi liquid phases of MnSi and Dietrich Belitz
described his theory with Sumanta Tewari of "quantum blue fog" :
the
idea that the low pressure non Fermi liquid phase of MnSi is an
electronic cholesteric, analagous to the blue fog phases of cholesteric
liquid crystals. Read on to hear how these discussions evolved.
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This is my last week as
Blogger. It has been an exciting five weeks. I
feel that some of the issues - e.g. the change in Fermi surface
volume at a QPT may be much closer to theoretical resolution than they
were at the start of the program. Many new
theoretical developments have been seeded by the past few weeks.
On my
way back to LA on Friday evening, we encountered the most
studendous rainbow near Point Magu, along highway 1. I shall take it
as good omen for all the good
research results to expect in the coming
weeks.
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| Rainbow at Point
Magu 2/11/05. Photo: J Rech. |
Piers Coleman |
Participants
Blackboard Seminar
Main Seminar
Thursday Discussion
Participants
present.
Click on participant to read questions that they have posed
Belitz, Dietrich
Carbotte, Jules
Coleman, Piers
Dzero, Maxim
Ingersent, Kevin
Gan, Jinwu
Harrison, Neil
Nicole, Elizabeth
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Norman, Michael
Paul, Indranil
Pepin, Catherine
Shaginyan Vasily
Tewari, Sumanta
Vojta, Thomas
Young, Peter
Zwicknagl, Gertrude
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Blackboard
Discussion. 10am Monday, February 7th.
Dr. Vasily Shaginyan
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Fermion
Condensation[Aud][Cam] |
(a) Single particle
spectrum in Fermion condensate after Khodel et al, cond-mat/0502292,
(b) occupancies n(p) showing region where 0<n(p)<1 (c) group
velocity.
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Vasily presented a
talk in two parts:
- In which he proposed that the properties
of heavy electron systems near quantum criticality can be understood in
terms of a field or temperature-dependent generalization of Landau
Fermi liquid theory.
- In which he introduced the concept of "Fermion
condensation" as a way to understand the underlying order which
develops at a heavy electron quantum critical point. One of the
interesting features of this state, is that the effective mass of the
quasiparticles is infinite over the entire Fermi surface - consistent
with the apparent divergence of the quasiparticle effective mass
near a quantum critical point.
The main points of this talk can also be found in the paper by
Shaginyan et al, cond-mat/0501093.
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In the first part of the talk, Dr Shaginyan
argued that in the Landau
Fermi liquid that arises either side of a heavy electron quantum
critical point, one could, as a good approximation, ignore the lattice
and then invoke Landau's relationship for the effective mass of a
quasiparticle.
In Landau's original
derivation, this relationship depends on Gallilean invariance.
Shaginyan argues that in a finite field and temperature, one can
replace m* by a field and temperature dependent quantity m*(T,B),
where m*(T,B) is determined by this same equation, with the
zero field, zero temperature interaction parameters, i.e,
Near the
QCP, the 1/m*(0,0) term is argued to be negligible, and by expanding
the occupancy term to
order T2 (m*/m)2 or b2
(m*/m)2 , where b=|B-Bc|, the
relationship m* ~ 1/
max(T,b)2/3
is derived. Over
a higher temperature, a similar relationship with a 1/2 exponent was
derived. This power-law behavior is reminiscent of the observed
properties of
heavy fermions.
The following questions were raised:
- The derivation is relevant to the electron gas,
yet Monte Carlo calculations show one has to reach a very high value of
r_s before an instability occurs, and the instability is probably first
order into a Wigner crystal. (Zwicknagl,Fisher)
- Heavy electron materials are crystal lattices -
how can one apply a result that depends so essentially on Gallilean
invariance? (Coleman)
- Surely, at the lowest temperatures, m* must
become constant as the Fermi liquid settles down into its
ground-state. Over what range then, does the power-law
relationship apply? (Norman)
- Is the mass relationship consistent with
detailed diagrammatics? (Pepin)
In the second part of the talk, Dr Shaginyan presented the idea
of fermion condensation. The essence of this idea, is that there are
new solutions to the Landau condition,
(Notice that this expression depends on the use of the entropy formula
In the conventional Landau solution, as the temperature T
goes to zero,
the log goes to zero. When the dispersion of the bare energy
becomes negative, it is
argued that solutions become possible where the logarithm is finite,
and the effective dispersion vanishes. This is called a "fermion
condensate". Shaginyan argued that it was like a superconductor in
which the order parameter has been sent to zero. One of the main
questions about this state to arise was
- If the zero temperature occupancies are neither
zero nor unity, isn't the entropy of the fermion condensate
finite? (Coleman)
Shaginyan argued that at T=0, the entropy is zero, even though
the
traditional Landau expression for the entropy (-n Log(n) - (1-n)
Log(1-n)) is
clearly finite in this state.
The plight of the fermion condensate is however, one that is shared by
any model of the quantum critical point that purports to develop
localized fermionic modes. How can one produce a dispersionless band
without a zero-point entropy?
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Seminar,
12.30 Tuesday 8th February
Dr. Neil
Harrison
(NHMFL, Los Alamos)
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Fermi Surface
Hot Spots[Slides][Aud][Cam] |
Regions of large
mass renormalization on CeIn3 Fermi surface.
The 111 orbit
exhibits a strong field dependence, whereas the 100 orbit has a field
independent mass. This is attributed to the anisotropic mass
renormalization on the Fermi surface.
Routes between the magnetic and paramagnetic state.
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Neil Harrison
presented a detailed study of the evolution of the de
Haas van Alphen in CeIn3 is field-tuned towards a
quantum
critical point at about 60 Tesla.
One of the fascinating aspects of these studies, is that they show the
orbits around the 111 direction have a much stronger mass dependence on
field than in the 100 direction. The effective mass of a dHvA orbit is
given by
Harrison argued that the anisotropies in field dependence occur because
certain "hot" regions experience strong virtual spin fluctuations which
produce strong field dependence in the renormalized mass. At 40T,
the mass in the "hot" region was 30 me whereas the mass in
the cold region was estimated to be about 2 me .
Norman worried whether the observed results might be a consequence of
magnetic breakdown, but Harrison explained that magnetic
breakdown
would not be expected to produce the same effect as a mass
renormalization.
The term "hot spot" caused much confusion because theorists think of a
"hot spot" as a region of strong scattering. In this context, it
implies a region where there is a large amount of virtual scattering,
affecting the real part of the self energy.
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Harrison presented a very nice summary of the possible routes from
localized, to itinerant f-electron behavior. He expressed the
view
that in pressure experiments, the f-s delocalize, but in field
driven expts,
they remain localized - converting from antiferromagnetically, to
ferromagnetically polarized local moments.
The following questions were raised:
- Can the large mass renormalizations
around the FS be accounted for in a model of almost localized magnetism
or do we need to invoke the Kondo effect?
- How big is the Kondo temperature?
- Are the moments fully polarized?
The close vicinity to a
pressure induced transition suggests that they may be substantially
reduced by the Kondo effect. Harrison did not agree with this point.
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Discussion:
4.30pm Thursday, February 10th, Founders Room.
Present at the discussion were Dietrich Belitz,
Jules Carbotte, Piers Coleman, Maxim Dzero,
Neil Harrison, Kevin Ingersent, Dirk Morr, Elisabeth Nicol, Mike
Norman, Indranil Paul, Catherine Pepin, Jerome Rech, Bahman Roostaei,
Vasily Shaginyan, Sumanta Tewari, Thomas Vojta, Peter Young, Veljko
Zlatic, and Gertrud Zwicknagl.
Here were the topics discussed:
1 :
Luttinger's Theorem Near Heavy Fermion Quantum Critical Points
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As Piers was leaving
at the end of this week, he offered to spend the first half hour
summarizing the work he had done while here at the KITP. His
project was to use Ward identities to generalize Luttinger's theorem
concerning Fermi surface volume to more exotic circumstances where one
might have spinless fermion degrees of freedom, as in the "chi fermion"
approach advocated by Catherine Pepin and in related work by him and
Olivier Parcolet to describe the quantum critical point in heavy
fermion metals (see the blackboard talks given by Piers and Catherine
in the previous week). From these relations, Piers claims to have
derived a simple formula for the Fermi surface volume which involves
the chi fermions, and thus in some sense "violates" the traditional
Luttinger's theorem considered early on for heavy fermion systems by
Richard Martin, where the Fermi volume would either count just the
conduction electrons (as in a localized phase), or the f electrons plus
conduction electrons (as in the heavy Fermi liquid phase).
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2. Nature of the Magnetism in MnSi
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After this, the
discussion session turned to the main topic at hand, which is the
nature of MnSi. A detailed summary of MnSi was given by Dietrich
Belitz, with some clarifying remarks offered by Thomas Vojta. The
experimental situation was also treated in a talk given by Christian
Pfleiderer during the QPT conference the second week of the program,
from which we have taken the figures for this blog. This
transition metal compound (whose crystal structure has no inversion
symmetry) has an ordered magnetic phase which is helical in nature,
with the transition being first order. Pressure suppresses the
transition temperature, leading to second order behavior, and
eventually to a quantum critical point. Beyond this point, there
is a large range in pressure where non Fermi liquid behavior is
observed, indicated by a T3/2 resistivity.
Inelastic neutron scattering studies reveal a rather interesting story.
In the ordered phase, one finds a sphere of scattering, with the sphere
radius
equal to the magnitude of the helix (2*pi/q is about 170 Angstroms at
zero
pressure). The sphere radius appears to be resolution
limited. Embedded
on the surface of this sphere are magnetic Bragg vectors along the
<111>
directions (each direction corresponding to a different domain of the
sample).
As pressure is applied and one enters the non-ordered phase, these
<111>
spots on the surface of the sphere "smear out", and at higher pressures
eventually rotate to be along the <110> directions instead.
From the data,
a temperature T0 can be defined in the non Fermi liquid
phase where this
unusual behavior in the neutron scattering sets in.
Dietrich then tried to make an analogy of MnSi with certain liquid
crystal
systems known as "blue" phases. His idea is that the ordered
helical phase would be the analogue of a chiral solid, and the non
Fermi liquid phase beyond this a chiral liquid. He termed this
chiral liquid phase the "blue quantum fog". Chiral fluctuations
are strongly evident in neutron scattering studies, as remarked by
several of the participants. Dietrich then remarked that in the
isotropic case, the inverse of the order parameter correlator would go
like qz2 + (qx2+qy2)2,
with q relative to the ordering wavevector. Interestingly, as
remarked by Mike Norman, this is the same form assumed by Revaz
Ramazashivili as a model of the inelastic neutron scattering data near
the quantum critical point of Au doped CeCu6, which shows a
butterfly pattern for the scattering (rather than the sphere seen in
the MnSi case). Revaz used this
model as a possible explanation of the non Fermi liquid behavior
observed in
that system. Dietrich is currently working out the details of the
blue fog in collaboration with another program participant, Sumanta
Tewari.
All the participants then agreed that the advent of the "blue quantum
fog" was a good point to stop the discussion and move on to the more
important matter of dinner, which turned out to be an enjoyable night
out for all those involved.
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