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Quantum Phase Transitions
Week 3. 24-28 January, 2004
Blogger: Piers Coleman
Finally the conference
is over, and the workshop fully underway. All of us felt very
excited by the conference the week before, and the morning coffee
discussions have been very animated.
On the first day of the week, Nikolai Prokofiev gave a beautiful
black-board talk on his studies of the 2D x-y model, and in particular
the transition from Neel state to valence bond solid. To learn
more - read below! On Tuesday, Bill Buyers described the latest
neutron measurements on under-doped cuprate materials. Remarkably
- they see no sign of any incipient magnetism in the nodal state,
despite the suppression of Tc almost to zero. There was much
discussion associated with both talks.
Participants
Blackboard Seminar
Directors Lunch
Main Seminar
Thursday Discussion
Participants
present.
Click on participant to read questions that they have posed
Belitz, Dietrich
Buyers, Bill
Coleman, Piers
Ingersent, Kevin
Kirkpatrick, Ted
Mydosh, John
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Paul, Indranil
Pepin, Catherine
Prokoviev, Nikolai
Sushkov, Oleg
Vojta, Thomas
Young, Peter
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Blackboard
Discussion. 10am Monday, 24th
January.
Dr. Nikolai Prokofiev
University Massachusetts |
Superfluid-Solid
Quantum Phase Transitions[Aud][Cam] |
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This talk was
motivated by the numerical work on the x-y model with ring exchange by
Sandvik et al, and the proposal, inspired by this work, of
"deconfined criticality" recently made by Senthil et al.
Prokofiev described their efforts to study the 2D quantum x-y model
with ring exchange by modelling it as hard core bosons moving along
worldlines in a 2+1 dimensional space. In this approach the
superfluid (magnetic) phase is described by a world-line configuration
with at least one path that winds around the system. Insulators
must have highly ordered paths that either do not wrap around or have
paths of compensating winding number. Prokofiev used this idea to
argue that the ground-state must always have some kind of order, and so
the set of order parameters should be regarded as a vector of fixed
length. In such a picture, the valence bond solid and the neel
antiferromagnet are vectors at "ninety degrees" to one -another, and
without a Hamiltonian of high symmetry, one would then expect first
order transitions between the neel state and the valence bond
solid.
An alternative, might be that the effective
temperature was large enough to "melt" this effective vector - in
which case one might imagine a second order transition between the
antiferromagnet and the valence bond solid.
Prokofiev described how they were able to map the 2+1 xy
model onto a classical stat mech problem, and then examine the phase
diagram for lattices
up to 128^3. For smaller lattices, the neel-vbs transition looked
second order.
However, as in the work by Sandvik et al, the stiffness increased as
the lattice size increased, and in the largest simulations, it became
clear that the transition was first
order. They also looked for "spinon" configurations by
looking for 1/2 jumps in the temporal evolution of the winding number.
These were never seen. Despite efforts to vary the Hamiltonian,
they find that the Neel-VBS transition is always first order, with no
spinons.
The last part of the talk discussed how, with long
range gauge forces in an x-y model, one might be able to recover the
deconfined scenario.
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Directors
Lunch 12.30 Monday, 24th
January.
Dr. Piers
Coleman
Rutgers |
Quantum
Phase Transitions - A New Challenge in the Cryostat and Possibly Also
the Cosmos[Aud][Cam] |
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Coleman gave a black-board talk based on the review article written
with Andrew Schofield to commemorate the Einstein Centennial, in Nature.
Quantum
Criticality: nature review article
Tony Zee asked why given the close link betwen the cosmos and the
cryostat, condensed matter physicists had not discovered a laboratory
realization of Einsteins Gravity.
Coleman responded that Volovik has made proposals that liquid He-3 can
provide a realization of gravity.
Mathew Fisher pointed out that many of the current efforts to
understand quantum phase transitions are based on gauge theory ideas,
and that there is room for much exchange between the lattice gauge
theory program and the quantum phase transition program running
concurrantly at the KITP.
Coleman pointed out that Scott Thomas's idea of emergent supersymmetry
may apply near the quantum critical point, and that lattice gauge
theorists should be able to test this for lattice models of N=1
supergravity in 3D.
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Seminar,
12.30 Tuesday 25th January.
Dr. Bill Buyers
National Research Council |
Near the
Superconducting Edge: Spin Confinement and Central Modes[Slides][Aud][Cam] |
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Buyers presented the results of an extensive
neutron scattering study of high quality underdoped crystals of
YBCO_6+x. There is a lot of wonderful stuff in their data!
Highlights include:
- Observation that the Tc and the position of the
resonant gap are not linear in doping, but vanish like a soft mode, i.e
Tc ~ (x-xc)^1/2.
- The observation that the cone of high-energy
damped spin-wave excitations is isotropic - does not contain a "box" as
might be expected if one had static stripes.
- The discovery of a large almost elastic feature
that grows at low temperatures, and seems to represent
slowly fluctuating slabs of spin, with a correlation length of about 8
lattice spacing.
- The discovery that the chi''(omega) at
Q=(pi,pi) contains much more low-energy weight than is expected on the
basis of a BCS superconductor. What is the origin of these low energy
spin degrees of freedom?
- There was no sign of an ordered
antiferromagnet, no d-density wave peak either.
There was much discussion about point 4. Mathew Fisher very
enthusiastic about the possibility that this might be a nodal spin
fluid.
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Discussion:
4.30pm Thursday, 27th
January, Founders Room.
Present at the discussion were Oleg Sushkov, Nikolai
Prokofiev, Bill Buyers, Catherine Pepin, Deitrich Belitz, Peter Young,
Jerome Rech, Piers Coleman, Thomas Vojta and Bahman Roostaei. It
was a wide-ranging discussion in which Nikolai and Bill, who are
leaving this week, were asked to comment on their work and discussions
during the workshop. Many discussions took place and I welcome further
input to this blog from fellow participants. It is Thursday night as I
write this, and I hope to finish up the details
over the weekend. Watch this space for the improved version on
Monday.
Here are the topics as they appeared on the blackboard:
1 :
Prokofiev Self Dual X-Y models and the screening and generation of long
range interactions.
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Nikolai presented
some of the work he had completed while at the KITP. He has been
interested in the possible equivalence between long range and
short-range x-y models. This interest has been, I gather, spurred on by
his earlier, unsuccessful efforts to confirm the "deconfined
criticality" scenario in the x-y/ring exchange model. Prokofiev
showed how the 1/r^2 3D x-y model forms a family of models, where under
duality, the magnetic coupling J(q) and J*(q) of the dual models
transform as
J(q) ~A/q + B
<-------> J*(q)~ A' /q + B'
so that for one value
of A, it is self-dual. As A is varied, the scaling dimensions D l
and Dj
x-y model has loop and bond dimensions which lie on the same
trajectory. Could this, Nikolai suggests, mean that the
short-range model develops 1/r^2 long-range forces, whereas the dual
1/r model is screened to a shorter range 1/r model is partially
screened to develop shorter range 1/r^2 forces - and does this mean
that there might be spinons hanging around the critical point.
What has this to do with real quantum phase transitions ?
Prokofiev felt it was a model that might teach us about the development
of long-range gauge forces in 2+1D models. These processes might be
particularly important near a quantum phase transition, he speculates.
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2. Fisher: why we need to extend
duality to include the electrons.
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Mathew Fisher felt that the work of Prokofiev was related to the
problem of applying such dual transforms to mixed combinations of
superconductor and fermions, such as in the nodal fluid that develops
in under-doped cuprates. Mathew asked - what happens
to an electron in a Cooper pair condensate, when you take it around a
superconducting vortex? Somehow this is related to the problem of
duality transforms for electrons and Cooper pairs.
PC asked him if you can really do a duality transformation for
electrons and Cooper pairs. Mathew said yes, but that one had to
use some kind of slave particle representation. Thomas Vojta
asked whether such transformations, being topological, are restricted
to 2+1 dimensions. Mathew joked that in 3+1 D it would be a
string theory, and perhaps some other representation was preferable.
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3.
Buyers: my impressions of the workshop - the problem of top down
versus bottom up and the need to close the divide between theory and
experiment.
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Bill Buyers started by complaining that he would have prefered more
participants around - though he confessed that he had learnt the most
on the week with the least participants! He said he'd learnt a
lot about the language and concepts of quantum criticality.
Bill Buyers lamented that he had not, as yet been successful in
convincing theorists to work directly on his data. Oleg said he'd
started!
What ensued was a whole discussion about "top down" versus "bottom up"
theoretical approaches. Buyers fel that model building was
largely irrelevant, and that theorists should start modelling real data
and stay closer to the real world.
MF argued that you had to go to models that were even more
conceptualized than Hubbard Models, using a "top down" approach, since
bottom up had not worked.
Catherine Pepin expressed the view that there are some models that one
can almost solve - like the spin fermion model, but that it was a long
way from the models we'd really like to solve, like the Hubbard or t-J
model.....
Others felt that tackling the cuprates head on, may be difficult
because of the existence of many overlapping, conceptual challenging
problems. The heavy electrons, low dimensional
antiferromagnets, and the ladder compounds were cited as simpler
systems, that may give new insights, as emphasized in Marston's talk
last week.
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4.
Discussion on the nature of the central mode seen in YBCO_x (x=6.35)
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Central Peak:
Discrepency between neutrons and NMR.
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(Buyers then
admitted that he was teasing us. )What ensued was a long
discussion about the nature of the central mode that Buyers' team has
found develops around (pi,pi) in underdoped YBCO(x=6.35). This
low
energy mode has all of its weight below 0.07 meV, and is
essentially
elastic to neutrons. It has a correlation length of about 8 unit cells,
but they know its dynamic because it is not seen in muons.
The
central peak starts to grow at about 50K, and saturates at 10K.
They
think it describes large fluctuating domains of antiferromagnetism.
Buyers pointed out that neutrons really don't see the pseudogap in
underdoped cuprates - if you compare 1/(T1T ) from NMR, it shows a
pseudo gap starting to develop around 150K, but at this temperature,
the corresponding neutron signal at (pi,pi) and 12 meV is still growing
as the temperature drops.
Does this mean that there are very slow fluctuations?
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5. Discussion on the nature of the
metallic, nodal fluid that is seen in underdoped YBCO. What is
the nature of the long spin correlations, the origin of the metallic
behavior, and enhanced Wiedemann Franz ratio?
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After a passionate argument between Oleg Sushkov and Bill Buyers about
which is the more representative material - YBCO with its commensurate
spin structure, which dissappears long before optimal doping, or LSCO,
with its inccommensurate spin correlations that last up to optimal
doping - the discussion ended on the nature of underdoped YBCO,
which is thought to
- be a non-magnetic metal with an enhanced
Wiedemann Franz Ratio
- exhibits strong low-frequency spin
fluctuations, with a long, but finite correlation length
- probably retain the nodal excitation structure
of the superconductor.
What is the nature of this extroordinary metal? Mathew Fisher pointed
out that it was quite significant that the experiments showed it was
not a magnet or spin density wave. He seemed to think it
might be some kind of nodal spin liquid. PC asked - how then is it a
metal too?
Here are some questions that seem to have arisen from our discussion:
- Is it a spin-charge decoupled system with
nodal spin fermions?
- Is it better regarded as a d-wave
superconductor which has lost its long-range phase coherence?
- Why does it conduct more heat than
charge?
- What is the nature of the quantum phase
transition by which this state emerges from the superconductor?
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