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Quantum Phase Transitions
Week 8, 28th February-4th March, 2005
Blogger: Mike Norman
Thursday Discussion: Andrey Chubukov
Andrey, Elihu, and Kostya have arrived, so the
program continues to evolve. Thomas, before he departed, gave us
an illuminating overview of the relevance of disorder near QCPs.
This was followed on Tuesday by Juan Carlos' review of photoemission
data on cuprates. We ended the week with a detailed discussion of
the nature of superconductivity near ferromagnetic QCPs.
Participants
Blackboard Seminar
Experimental
Seminar
Thursday Discussion
Participants
present.
Click on participant to read questions that they have posed
Abrahams, Elihu
Bedell, Kevin
Belitz, Dietrich
Chubukov, Andrey
Efetov, Kostya
Ingersent, Kevin
Larkin, Anatoli
Lavagna, Mireille
Marenko. Maxim
Norman, Michael |
Pepin, Catherine
Turlakov, Misha
Vojta, Thomas
Woelfle, Peter
Ye, Jinwu
Young, Peter
Zlatic, Veljko
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Blackboard
Discussion. 10am Monday, February 28
Dr. Thomas Vojta,
University Missouri - Rolla
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Disorder Effects
Near QCPs [Aud][Cam] |
To proceed further, one notes that the scaling dimension of the
disorder interaction is 4-d. Analyzing the RG flow, one finds
that although a new random fixed point exists, most RG flows have large
excursions before reaching this point, indicating an unstable flow.
He then turned to discussion of the contribution of rare regions
(Griffiths effect). The probability for the existence of these
regions is exponential in the size of the region, W ~ exp(-c*Ld),
where c is the concentration. The question is whether order
parameter fluctuations are strong enough to compensate for this.
Considering first transitions of continuous symmetry, Thomas showed
that for finite T, and for T=0 and z < 2, the order parameter
fluctuations are power law in nature, and thus cannot compensate
W. On the other hand, at T=0 and z=2, the theory is marginal, so
the order parameter fluctuation is logarithmic, thus giving an
exponential correction to the tuning parameter, epsilon ~ 1/L2
exp(-bLd). That is, the two effects are of the same
order. Thus, the disorder averaged local susceptibility is of the
form Td/z'-1 where z'=c/b, with z' increasing as one
approaches the dirty critical point. Finally, Thomas mentioned
the Ising case, where one can show that the effect of disorder is to
smear the transition.
In summary, Thomas emphasized that rare region effects are far more
important for quantum phase transitions than for classical ones.
One can thus construct three regimes where deff = d+z and
dc- is the lower critical dimension
(understanding that for a rare
region, d=0)
deff < dc-
-- Classical Griffiths behavior
deff = dc-
-- power law quantum Griffiths behavior
deff > dc-
-- disorder smeared transition
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Thomas gave a very
illuminating overview concerning the relevance of disorder near quantum
critical points.
The basic idea is to allow the tuning parameter to be spatially
dependent. By doing a real space scaling analysis, one can easily
show that "clean" physics is nominally o-kay as long as d*nu > 2,
where nu is the critical exponent associated with the tuning
parameter. For instance, for an antiferromagnet, nu=1/2, so this
so-called Harris criterion is violated for d < 4.
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Experimental
Seminar,
12.30 Tuesday, March 1
Dr. Juan
Carlos Campuzano
(University of Illinois at Chicago and Argonne National Laboratory)
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Photoemission
in Cuprates [Slides][Aud][Cam] |
Phase diagram of
the cuprates.
Variation of the spectrum at (pi,0) versus temperature for an overdoped
sample in the normal state. Note the disappearance of coherent
bonding and antibonding peaks as the temperature is raised.
Anisotropy of the spectral lineshape around the Fermi surface for an
optimally doped sample above T c. N is the node and A
the antinode.
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Juan Carlos gave a rather thorough overview of the nature of the
cuprate phase diagram as revealed by angle resolved photoemission data.
JC first emphasized that spectra at fixed energy as a function of
momentum (MDCs) are sharper than spectra at fixed momentum as a
function of energy (EDCs). This is due to the strong energy
dependence of the electron self-energy. In fact, the MDCs are
rather straightforward to interpret, as their width is simply the
imaginary part of the self-energy at that energy divided by the bare
Fermi velocity. He then showed that in the normal state of
cuprates, this imaginary part is of the form a + b*E, indicating
marginal behavior. On the other hand, in the superconducting
state, the imaginary part drops at low energies, indicating the onset
of coherence.
JC then turned to the phase diagram itself. As mentioned above,
the superconducting state is characterized by sharp spectral peaks in
the EDCs, indicating coherent behavior. Above Tc at
lower
dopings, there is a pseudogap phase where a partial gap appears.
In the pseudo-gapped regions of the Brillouin zone, the spectra are
completely incoherent. The remaining part forms a gapless arc of
excitations (centered at the d-wave node), with well defined spectral
peaks, though these are quite broad and not coherent as in the
superconducting state. For higher dopings, the pseudogap fills
in, leading to the arc expanding out and recovering the full underlying
Fermi surface. Then in the overdoped limit, this strange metal
phase gives way to a more coherent Fermi liquid like phase, where sharp
peaks begin to appear in the spectra, giving way to even sharper peaks
once the superconducting phase is reached.
JC then mentioned recent work of his looking at the anisotropy of the
spectral lineshape as a function of momentum, where he commented that
only in the Fermi liquid like phase is the lineshape isotropic,
otherwise, strong momentum anisotropy is seen, which is obviously
connected to the existence of d-wave pairing.
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Discussion:
4:30 pm Thursday, March 3rd, Founders Room.
Present at the discussion were Elihu Abrahams, Kevin
Bedell, Dietrich Belitz, Andrey Chubukov, Kostya Efetov, Kevin
Ingersent, Mireille Lavagna, Mike
Norman, Catherine Pepin, Bahman Roostaei, Thomas Vojta, Peter Wolfle,
Maxim Marenko, Misha Turlakov, Jinwu Ye, and Peter Young.
We had an informal blackboard discussion on superconductivity near a
ferromagnetic quantum critical point.
Dietrich
Belitz gave a nice introduction to the subject. He argued that
the physics community was excited about superconductivity at the edge
of ferromagnetism about 25 years ago (ErNi2B2C was one of the mostly
studied materials at that time), and now there is a revival of interest
in this problem.
Dietrich cited current studies of three materials: UGe2, URhGe and
ZrZn2, all of which display superconductivity near a ferromagnetic
transition. He discussed in some length the early work by Fay and
Appel, who found p-wave superconductivity on both sides of a
ferromagnetic transition, with Tc vanishing at criticality, and argued
that experiments are in conflict with this result as the
superconductivity is only visible at the ordered side of the
transition.
He then proceeded to discuss his own work done with collaboration with
Ted Kirkpatrick, Thomas Vojta and others. He argued that the p-wave
superconductivity is indeed enhanced in the ferromagnetic phase because
for d <4, the longitudinal spin susceptibility, which accounts for
the pairing, is enhanced due to coupling to Goldstone transverse spin
fluctuations.
Dietrich also discussed the difference between Tc for spin-up and
spin-down pairing, the structure of the Goldstone modes in the
superconducting state, and anomalously strong fluctuations of the
vortex lattice.
Kevin
Bedell was next at the blackboard, summarizing his works on
superconductivity near a ferromagnetic instability. He speculated that
the pairing in the ferromagnetic phase can actually be s-wave. He
argued that a ferromagnetic ordering changes the sign of the s-wave
component of the scattering amplitude. In the paramagnetic phase, the
s-wave scattering amplitude is positive (i.e., repulsive), but once 1 +
F crosses zero and the system enters into a ferromagnetic state, the
sign of A changes due to a feedback effect from spontaneous
magnetization.
He also argued that, within self-consistent theory, they found both
s-wave and p-wave pairing in the ferromagnetic phase. They also found
that the ferromagnetic transition becomes first order at the lowest T.
Finally, Andrey
Chubukov entertained the audience with his heavy Russian
accent. He discussed the work done in collaboration with A.
Finkelstein, D. Morr and R. Haslinger on the p-wave pairing in the
immediate vicinity of the ferromagnetic instability (assuming that the
magnetic transition remains second order down to T=0). Previous works
all found that superconducting Tc vanishes at the quantum critical
point. Andrey argued that this vanishing is due to strong
pair-breaking effect from classical fluctuations which for spin-triplet
pairing act in the same way as magnetic impurities in s-wave
superconductors. When the magnetic correlation length becomes infinite,
these pair-breaking fluctuations diverge at any finite T, and this
brings Tc to zero. At the same time, at T=0, classical pair-breaking
fluctuations vanish, and the solution of the nonlinear gap equations
yields a finite p-wave gap even at criticality.
Chubukov argued that the actual superconducting transition at the
ferromagnetic boundary is first order: coming from the normal
state, the onset of the pairing occurs at vanishingly low T
because of classical pair-breaking fluctuations. At the same time,
increasing T starting from T=0 does not immediately destroy the
superconducting state, as in the presence of the spin-triplet gap,
longitudinal spin fluctuations are gapped, and this suppresses
classical pair-breaking effects. Andrey argued that the actual
first-order Tc is finite at criticality, although numerically Tc is
rather small.
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