High T c superconductivity and the QED 3 theory of the cuprates

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