From a phase fluctuating
superconductor to normal modes of a
Cooper pair Wigner crystal
cond-mat/0604559
High Tc Superconductivity
Tami Pereg-Barnea (University of Texas @ Austin)
In collaboration with
Marcel Franz (University of British Columbia)
Texas A&M Condensed – April 25, 2006
1
High Tc Cuprates
2
The pseudogap
MDC along the
Nodal direction
EDC at the
nodal point
T.Valla et al. PRL 85, 828 (2000)
3
Two Paradigms
Competing order
Phase fluctuations
SDW
CDW
DDW
Emery & Kivelson, Nature 374, 434 (1995).
No off diagonal order
4
Kosterlitz-Thouless
“CHEAP and FAST vortices”- P. A. Lee, 2002
5
What’s so special about d-wave
superconductivity?
• The SC gap changes sign
on the Fermi surface – four
nodal points.
• This results in low E QPs.
• Quasi particles are
sensitive to the presence
of vortices.
+
-
-
+
+
+
+
+
6
Singular gauge transformation
• Want to attach ½ of the phase of the order
parameter to each fermion
.
• Problem – The fermion wave function is not well
defined in the presence of vortices.
• Workaround - FT singular gauge transformation.
M. Franz & Z. Tešanović PRL 84, 554 (2000).
Ψ
-Ψ
7
The transformed Hamiltonian
• Two auxiliary gauge fields:
• At Low energy
– v is gapped due to the Meissner effect.
– The theory is formally equivalent to QED3.
8
Low energy sector
QED3
• Elegant!
• Nodal quasiparticles couple to a gauge
field and become strongly interacting, in
good agreement with ARPES data.
• Chiral (BdG) symmetry breaking leads to
antiferromagnetism at low dopings.
M. Franz and Z. Tesanovic, PRL 87 257003 (2001)
9
Cuprates
under the ‘Microscope’
• STM experiments provide a
closer look into the
electronic structure.
• Fourier transformed atomic
resolution scans reveal
charge modulations of two
types:
dispersing & non-dispersing
10
Dispersing features
Local density of states N(x)
Fourier transformed density of states N(q)
J. E. Hoffman Science 297 1148 (2002)
11
Interference of Quasiparticles
below Tc
• The dispersing features are seen at low energy
(below the gap maximum) => quasiparticles
• The patterns appear due to impurity scattering.
Q.-H. Wen & D.-H. Lee
PRB 67 20511 (2003)
• The patterns are peak-like due to the limited
phase space for scattering and the BdG
coherence factors.
TPB & M.Franz PRB 68, 180506(R) (2003)
12
QPI in the pseudogap
• The phase fluctuations
scenario predicts similar
patterns in the pseudogap,
maybe washed out.
• Competing order scenarios
predict different patterns.
Generally not peak-like.
TPB & M.Franz
IJMP B 19, 731 (2005)
13
Checkerboard Patterns
In vortex cores
•
Predicted by Arovas,
Berlinsky, Kallin and Zhang,
PRL 79, 2871 (1997).
•
Seen by J. Hoffman et al.
Science 295, 466 (2002).
•
Associated with
antiferromagnetism.
14
Checkerboard patterns
in underdoped cuprates
• Seen close to and at the
pseudogap phase.
• Two wavelength structure.
• Often incommensurate
with the ionic lattice.
• Coexists with dispersing
features at low energy.
• No antiferromagnetism.
K. McElroy et al. Nature 422, 592 (2003)
M. Vershinin et al. Science 303, 1995 (2004)
T. Hanaguri et al. Nature 430, 1001 (2004)
15
What’s going on?
• A competing order may exist at higher
energies.
• However, it must affect the low energy
quasi-particles.
• A Wigner crystal of Cooper pairs produces
checkerboard patterns and leaves the low
energy QPs intact.
16
Why should pairs crystallize?
•
:
If the phase (of Cooper pairs) is disordered the
number (density of Cooper pairs) may order.
• More rigorously – use a duality transformation to
describe everything in terms of vortices.
• Unlike Cooper pairs, vortices are well defined
real space objects.
17
Vortex point of view
• In a dual picture vortices see the electric charge
as a source of a dual magnetic field with
average field (1-x)/2 per plaquette.
• This is a Hofstadter
problem, the result is
supermodulation
with wave vectors
which depend on the
doping, x.
D. R. Hofstadter, PRB 14 2239 (1976)
18
Comparison
Z . Tes
anovic
Phys.
& A. M
Rev. B
elikya
71, 21
The d
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19
Another clue – thermal conductivity
measurements
• Measurements of the thermal conductivity,
κ, found a mysterious bosonic mode that
appears in the pseudogap.
L. Taillefer Unpublished.
• The thermal conductivity is defined by
κ=Q/∇T(L/A)
• Normally κ=Cv Vs l / 3
20
Thermal conductivity
• In the cuprates:
κ(T) = cT + β T3
Fermions
Phonon?
Magnon?
Bosons
• For any boson with linear dispersion:
• Could the bosonic mode be a pair Wigner crystal
vibration?
21
Finding the interaction
• Start with GL theory:
t ic
kine
Coupling to
external field
Potential
elec
trom
agn
etic
• Fix the magnitude of the order parameter:
• Dualize and get a GL theory:
22
Dual Vortices = Cooper pairs
• Vortices in the dual model are Cooper pairs.
• The dual model can be mapped onto a Coulomb
gas
• The interaction is a screened Coulomb potantial
in 2D:
• λTF is the Thomas Fermi screening length.
• λd is the dual penetration depth.
23
Normal Modes
• Due to the long-range interaction
the longitudinal mode is gapped.
• The transverse mode is acoustic
• The sound velocity depends on the interaction
strength and vortex mass.
• The vortex mass is renormalized by
wavefunction overlap.
• The sound velocity is anisotropic:
24
Dual Vortex Lattice
25
Results
• The dual penetration depth is given by the STM
measured δN/N. It is about 5-10 lattice
constants.
• The Thomas-Fermi screening length is similar to
that of a metal.
• We assume sample boundary scattering,
l~1mm.
• The estimated β is between
0.02 and 10 mW / K4 cm.
• The measured β is 0.5 mW / K4 cm.
• β is proportional to 1/λd2 , this means that at the
transition β vanishes.
26
Summary & Conclusions
• The picture of phase fluctuations is useful in
describing the low E properties of the
pseudogap. It agrees with STM and ARPES
data.
• At high energies phase fluctuations lead to a
pair Wigner crystal – in agreement with static
charge modulations seen by STM.
• The lattice vibrations may explain the extra
bosonic mode seen in thermal conductivity
measurement.
27