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Quantum Phase Transitions
Week 9, 7th February-11th March, 2005
Blogger: Mike Norman (Monday), Dirk Morr (Tuesday), Andrey
Chubukov (Thursday)
We started our week with a nice talk by
Matthew Fisher on Unconventional Quantum Criticality. On Tuesday, Girsh
Blumberg presented a nice review on the physics of the electron-doped
cuprate superconductors. We ended the week with a lively
discussion of various aspects of strongly correlated electron systems.
Participants
Blackboard Seminar
Experimental
Seminar
Thursday Discussion
Participants
present.
Click on participant to read questions that they have posed
Abrahams, Elihu
Blumberg, Girsh
Chubukov, Andrey
Efetov, Kostya
Ingersent, Kevin
Geshkenbein, Vadim
Larkin, Anatoli
Lavagna, Mireille
Marenko. Maxim
Monien, Hartmut
Morr, Dirk
Norman, Michael |
Pepin, Catherine
Shankar, Ramamurti
Turlakov, Misha
Weng, Zheng-Yu
Yakovenko,Victor
Yang, Kun
Ye, Jinwu
Young, Peter
Zarand, Gergely
Zlatic, Veljko
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Blackboard
Discussion. 10am Monday, February 28
Matthew
Fisher - Monday, March 7, 10 am, KITP Auditorium
Unconventional Quantum Criticality
Matthew reviewed some
of the work he and his collaborators (Senthil and Balents) have done on
quantum phase transitions. Their approach is conceptually
different from the standard Hertz-Millis approach, and provides a clear
contrast to several of the talks which had gone before during the
workshop.
Matthew started out by emphasizing the critical nature of the Fermi
liquid at T=0 due to the presence of a Fermi surface, and the fact that
there is no obvious "order parameter" for such a phase (and perhaps for
other phases of interest as well, like the pseudogap).
He then launched into a review of some work in 2D quantum magnets,
starting by noting the existence of Berry phases, and then introducing
the concept of emergent symmetry. To illustrate the latter, he
looked at an XY Hamiltonian, -J Sum_i,j cos(phi_i-phi_j), supplemented
by an anisotropy term of the form -lambda Sum_i cos(4*phi_i). For
lambda=0 one has a phase transition at a critical value of J separating
an XY ferromagnet from a paramagnet. For non-zero lambda, the RG
trajectory for the ferromagnet flows from the XY point to a ferromagnet
with discrete symmetry. But at the critical J_c, lambda flows to
zero, so the critical point retains its U(1) symmetry, and thus has a
symmetry "enlarged" from that of the Hamiltonian with the symmetry
breaking term.
This led to a discussion of emergent gauge symmetry, which occurs when
one represents the spin operator using slave particles. He
emphasized that the gauge field associated with the redundancy in the
definition of the slave particles is compact, meaning that in
principle, monopole events (representing a change in flux through a
plaquette by 2*pi) could not be ignored. When such monopoles
proliferate, this leads to confinement, where the slave particles
representing the spin recombine with those representing the charge,
leading to just an ordinary electron operator. So, the challenge
is getting deconfined behavior, which at the minimum would require
gapless slave particles (i.e., gapless spinons).
He illustrated this by
taking the example of a frustrated magnet, where one can tune from a
Neel ordered phase to a valence bond solid (again, assuming an XY
model). One can rewrite this Hamiltonian in quantum rotor form,
with H_rotor = -J Sum_i,j cos(phi_i - phi_j) + U Sum_i (n_i - 1/2)^2
where S^+ = exp(i*phi) and S^z = (n-1/2). The "dual" Hamiltonian
for such a system involves "up" and "down" spin vortices interacting
with a logarithmic potential, and a gauge field which corresponds to a
"pi" flux per plaquette, this being the Berry phase associated with the
spins in the original Lagrangian. In the "dual" theory, the gauge
field is now non-compact. The resulting RG flow is similar to the
one discussed for the anisotropic XY case previously, with the role of
lambda now played by a term lambda_4 involving an interaction term
between up and down vortices. The effect of the lambda_4 term is
to cause the spin liquid phase at zero lambda_4 to flow to a valence
bond crystal.
So, what relevance
does all of this have to heavy fermions? In this context, he
mentioned the recent work of Senthil, Sachdev, and Vojta, based on
similar considerations to those above, where they postulate two types
of Fermi liquid phases, one of these being the ordinary Kondo phase,
the other a phase where the local moments form their own (spinon) Fermi
surface. At lower temperatures, the latter phase is unstable to
long range magnetic order. The physics has some similarity to
that proposed by Coleman and Pepin, where the spin is also written as a
composite object. It will be interesting to see how far these
ideas go in explaining the novel behaviors associated with heavy
fermion quantum critical points.
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Experimental
Seminar,
12.30 Tuesday, March 1
Dr.
Girsh Blumberg
(Bell Labs)
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Electron
Doped Cuprates [Slides][Aud][Cam] |
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In
the first
part of his talk, Girsh Blumberg presented a general overview of some
of the most
interesting aspects of the electron doped cuprates.
Girsh began by reviewing the ARPES experiments by Armitage et al. which
in the weakly
doped Nd(2-x)Ce(x)CuO(4) (NCCO) observe a bandstructure consistent with
the presence of a spin-density wave (SDW). Girsh pointed out that the
electronic structure of hole- and electron-doped cuprate
superconductors is similar, despite the fact that in the former, the
doped holes reside primarily on the oxygen site, while in the latter,
the doped electrons are predominantly located at the Cu site.
Recent ARPES experiments by Matsui et al. observe evidence for an SDW
transition even at x=0.13.
Girsh also reviewed (i) optical conductivity experiments by Homes et
al. which find non-Drude behavior, and (ii) conductivity experiments by
Onose et al conclude that a pseudo-gap region is present in the
electron-doped cuprates at temperatures above the Neel
transition.
Finally, Girsh discussed recent Hall-effect measurements by Dagan et
al. which show that the Hall coefficient changes sign as a function of
temperature between x=0.16 and x=0.17.
Girsh then turned to the controversy over the pairing symmetry in the
electron-doped materials,
and the question whether this symmetry changes as a function of
doping. Early Raman experiments by Stadlober et al. indicated that the
pairing symmetry was anisotropic s-wave. In contrast, scanning SQUID
measurement by Tsuei and Kirtley made the case for a d-wave symmetry in
the electron-doped cuprates. On the basis of penetration depth
experiments, Skinta et al. argued that the electron-doped systems
undergo a transition from s- to d-wave symmetry around optimal doping.
After giving a short tutorial on Raman spectroscopy, Girsh presented
the results of Raman experiments in B1g and B2g symmetry (the former
probing the antinodal regions of the Fermi surface, the latter the
diagonal parts) on NCCO with a Tc of 22K. In the B1g channel, the 2
Delta peak is located at 8.3 meV, while in the B2g it is located at 6.5
meV. This suggests (a) that the superconducting gap in NCCO does not
follow a simple [cos(kx)-cos(ky)] form, and (b) that is it
non-monotonic along the Fermi surface. Support for this conclusion and
the presence of a non-monotonic gap also comes from recent ARPES
measurements by Matsui et al. Finally, recent Raman measurements in the
overdoped cuprates show the same energy for 2 Delta excitations in the
B1g and B2g channel.
Finally, Girsh discussed magnetic field effects in NCCO. He presented
results which show that the 2 Delta peak rapidly shifts to lower
energies with increasing magnetic field. Thus, in contrast to
conventional superconductors, or hole-doped superconductors, the
magnetic field very effectively suppresses the SC order.
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Discussion:
4:30 pm Thursday, March , 10thFounders Room.
On Thursday afternoon, we had a
lively blackboard discussion in a packed Founders room on various
aspects of the physics of strongly correlated electron systems.
First, Kostya Efetov discussed a
series of his recent works on multilayer s-wave superconductor-ferromagnet
structures. He cited recent
experiments which seem to indicate that a superconductivity penetrates
through a ferromagnetic layer. It would be impossible
for a conventional s-wave superconductivity to penetrate through a
ferromagnet. Kostya argued, however, that an inhomogeneity of a
magnetization near a boundary with a superconductor (a domain wall)
generates an odd-frequency, triplet harmonics of the gap, and this odd
frequency harmonics can easily penetrate through a ferromagnet. He
discussed an
experimental setup in which one could accurately and elegantly measure
this effect. The discussion, following his talk, focused on whether the
existing experiments support his theory, or can be explained in a more
``conventional'' way.
Second, R. Shankar discussed his recent
work with G. Murthy on
deconfiment at quantum criticality. It is an extension of recent ideas
that there exist quantum phase transitions in which the
critical point is described by variables that are confined except at
criticality and different from those that describe the phases on either
side. He considered the massive Schwinger model in 1+1 dimension:
electrons of mass m and
charge e
interacting via a linear Coulomb potential, and placed in an external
electric
field 0 < theta < 2 pi. Coleman had shown that at theta =pi
certain half- asymptotic particles (which had to be created in an
alternating
array of fermion followed by antifermion) are liberated and that these
disappear at strong coupling. Shankar's main points were: (1) in the e/m versus theta -pi the transition
can be described by a soft spin version of the Ising model, with e/m ~ T and theta -pi ~ h.,
(2) the truly liberated particles were Majorana fermions and not the
half-asymptotic ones of Coleman, (3) there existed (Ising spin)
variables which describe the physics both at and away from a critical
point within the
Landau framework (4) the transition did not have to be second order for
deconfinement. However the mapping between the original Dirac fermions
and the final Ising or
Majorana variables was shown to be very complicated.
Finally, Kun Yang briefly described
his work on the derivation of the effective bosonic theory near a QCP,
using a multidimensional bosonization. He argued that in his approach
to a transition at q=0, there
is no Landau damping
(omega/q) term in the effective action. Instead,
he argued that omega^2/q^2 term is present, and that the dynamical
exponent is z=2 rather than z=3. We had a brief but intense
discussion after his talk on the relation between his approach and a
more conventional, Hertz-Millis-type approach which gives rise to the
Landau damping term in the effective action for low-energy bosonic
degrees of freedom.
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