High T
c
Superconductors &
QED
3
theory
of
the cuprates
Tami Pereg

Barnea
UBC
tami@physics.ubc.ca
outline
High T
c
–
Known and unknown
Some experimental facts and phenomenology
Models
Attempts to solve the problem
The inverted approach
QED
3
–
formulations and consequences
Facts
The parent compounds are AF insulators.
2D layers of CuO
2
Superconductivity is the condensation of
Cupper pairs with a D

wave pairing
potential.
The cuprates are superconductors of
type II
The “normal state” is a non

Fermi liquid,
strange metal.
YBCO
microwave conductivity
BSCCO
ARPES
Underdoped Bi2212
Neutron scattering
–
(
p,p
) resonance in
YBCO
Phenomenology
The superconducting state is a D

wave
BCS superconductors with a Fermi liquid
of nodal quasiparticles.
The AF state is well described by a Mott

Hubbard model with large U repulsion.
The pseudogap is strange!
Gap in the excitation of D

wave symmetry but no
superconductivity
Non Fermi liquid behaviour
–
anomalous power laws
in verious observables.
Phase diagram
AF
AF
AF
Theoretical approaches
Starting from the Hubbard model at ½
filling.
Slave bosons SU(2) gauge theories
Spin and charge separation
Stripes
Phenomenological
SO(5) theory
DDW competing order
The inverted approach
Use the phenomenology of d

SC as a
starting point.
“Destroy” superconductivity without closing
the gap and march backwards along the
doping axis.
The superconductivity is lost due to
quantum/thermal fluctuations in the phase
of the order parameter.
Vortex Antivortex unbinding
Emery & Kivelson Nature 374, 434 (1995)
Franz & Millis PRB 58, 14572 (1998)
Phase fluctuations
Assume
D
0
= 
D
= const.
Treat exp{i
f
(r)} as a quantum number
–
sum
over all paths.
Fluctuations in
f
are smooth (spin waves) or
singular (vortices).
Perform the Franz Tešanović transformation

a
singular gauge transformation.
The phase information is encoded in the
dressed fermions and two new gauge fields.
Formalism
Start with the BdG
Hamiltonian
FT transformation
–
in
order to avoid branch
cuts.
The transformed Hamiltonian
The gauge field a
m
couples
minimally
m
a
m
The resulting partition
function is averaged over all
A, B configurations and the
two gauge fields are coarse
grained.
The physical picture
RG arguments show that
v
m
is massive
and therefore it’s interaction with the
Toplogical fermions is irrelevant.
The
a
m
field is massive in the dSC phase
(irrelevant at low E) and massless at the
pseudogap.
The kinetic energy of
a
m
is Maxwell

like.
Quantum “Electro” Dynamics
Linearization of the theory
around the nodes.
Construction of 2
4

component Dirac
spinors.
Dressed QP’s
QED
3
Spectral function
Optimally doped BSCCO
Above T
c
T.Valla
et al
. PRL (’00)
Chiral Symmetry Breaking
AF order
The theory of Quantum electro dynamics has an
additional symmetry, that does not exist in the
original theory.
The Lagrangian
is invariant under the global transformation
where
G
is a linear combination
of
The symmetry is broken spontaneously through
the interaction of the fermions and the gauge
field.
The symmetry breaking (mass) terms that are
added to the action, written in the original nodal
QP operators represent:
Subdominant d+is SC order parameter
Subdominant d+ip SC order parameter
Charge density waves
Spin density waves
Antiferromagnetism
The spin density wave is described by:
where
,
labels denote nodes.
The momentum transfer
is Q, which spans two
antipodal nodes.
At ½ filling, Q → (
,
)
–
commensurate
Antiferromagnetism.
_
Summary
Inverted approach: dSC → PSG → AF
View the pseudogap as a phase disordered
superconductor.
Use a singular gauge transformation to
encode the phase fluctuation in a gauge field
and get QED
3
effective theory.
Chirally symmetric QED
3
Pseudogap
Broken symmetry
Antiferromagnetism
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