Symmetry breaking in the Pseudogap state and Fluctuations about it
1. Symmetry and Topology in Region II of the phase diagram?
Why no specific heat singularity at T*(x)?
2. Quantum critical fluctuations in Region I. (with Vivek Aji)
3. D-wave pairing.
T
T*
II
A
F
M
Marginal
Fermi-liquid
Crossover
I
“PseudoGapped”
Fermi liquid
III
SC
QCP
x (doping)
Schematic Universal phase diagram of high-Tc superconductors
Two Principle Themes in the work:
1. Fluctuations due to a Quantum critical point determine
the normal state properties as well as leads to
superconductivity.
2. Cuprates are unique and this is due to their unique solid
state Chemistry. A microscopic theory should be built
on a model which represents this solid state chemistry.
Phenomenology(1989): Properties in Region I follow if there exists a
Quantum Critical Point with scale invariant fluctuations given by
sgn( ) , for c
T.
T*
Antiferromagnetism



Marginal
Fermi liquid

I
II
Crossover
TF
Fermi liquid
SC
QCP
x (doping)
From approximate Inversion of ARPES and Optical conductivity: Pairing glue has spectrum
consistent with this. Deriving these fluctuations may be considered the central problem.
Quantum Critical Point in high Tc crystals
If there is a QCP, there might be an ordered phase emanating from
it on one side and a Fermi-liquid below another line emanating from it.
T
Pseudogapped metal
Broken Symmetry?
Antiferromagnetism
T*
Marginal
Fermi liquid
Crossover
Fermi liquid
I
II
QCP
x (doping)
Superconductivity
Microscopic Model:
Why are Cuprates Unique? (1987)
o
o
cu
Cannot be reduced to a Hubbard Model because the ionization energy
of Cu is nearly the same as the ionization energy of oxygen.
Look for symmetry breaking not ruled out by Experiments
Preserve translational symmetry: severely limits possible phases;
Bond Decomposition of near neighbor interactions.
Only Possible States not changing Translational and Spin-Rotational symmetries
have order parameters:
Time-Reversal and some Reflection Symmetries lost.
Experiments to look for time-reversal breaking in the pseudogap phase;
Dichroism in Angle-Resolved Photoemission:
Experiment by Kaminski et al. (2002);
Direct Observation by Polarized neutron Diffraction
(Bourges et al. 2005).
Kaminski et al., Nature (2002)
Dichroism in BISCCO
Fauques et al. (2005): Polarized Elastic Neutron Scattering in
underdoped and overdoped Y(123)
Fauques et al. (2005):
Polarized Neutron diffraction in YBCuO
Magnetic Diffraction Pattern consistent with
Loop Current Phase II just as Dichroic ARPES
Why no specific heat singularity at T*(x)?
Classical Stat. Mech. Model for the observed Loop Current Phase
Time-reversal and 3 of four reflections broken:
Two Ising degrees of freedom per unit-cell
Four states per unit-cell.
Ashkin-Teller Model :
Phase diagram obtained by Baxter; Kadanoff et al.
Observed broken symmetry for -1 < J4/J2 < 1.
Gaussian line
Phase Diagram of the Ashkin-Teller Model (Baxter, Kadanoff)
Specific Heat for the relevant region: (Hove and Sudbo)
Quantum critical Fluctuations : Fluctuations of the order parameter
which condenses to give broken symmetry in Region II.
Very simple but peculiar Phenomenology:
sgn( ) , for c
T.
T*
Antiferromagnetism



Marginal
Fermi liquid

I
II
Crossover
TF
Fermi liquid
SC
QCP
x (doping)
Superconductivity
Quantum Critical Fluctuations:
Vivek Aji, cmv (Preprint soon)
AT model:
is equivalent by
to
Replace constraint with a four-fold anisotropy term
Same classical criticality as AT model. Constraint irrelevant above
the critical line and relevant below.
Add quantum-mechanics: Moment of Inertia plus damping
due to Fermions.
Model is related to 2+1 dim. Quantum xy models with dissipation.
Critical Region:
Need not consider anisotropy term.
For simplicity keep only the xy-term.
Fourier transformed dissipation: Derivable from elimination of currentcurrent coupling of collective modes to fermions:
Without dissipation model is 3d xy ordered at T=0.
We also assume J such that it is ordered at finite T of interest.
Wish to examine region where dissipation disorders the phase.
Previous work on the dissipative xy model:
Nagaosa (1999); Tewari, Chakravarti, Toner (2003),…
Below a critical value of , dissipation destroys long range order
at T=0.
But no calculation of correlation functions, connection with vortex
fluctuations or connection with the classical transition.
Steps in the derivation:
1.
i+y
lives on the bonds of the lattice
2.
3.
i
i+x
.
Velocity field due to
: decreases as 1/r.
Time-independent
.
Velocity field due to
: spatially independent
Time-dependent
A remarkable simplification which allows a solution!
Action in terms of
, (schematically):
+ terms which are not singular when integrated over k and omega.
Partition function splits into a product of a space-dependent
part and a time-dependent part.
Problem transforms to a K-T problem in space and
(mathematically) a Kondo problem in time.
Instanton field
does the disordering:
RG equations for fugacity y for instantons and for , similar to
flows in the Kondo problem or the KT problem:
For
field.
,
proliferates and disorders the velocity
Calculate order parameter correlation functions: At
Gaussian Model : No corrections?
This was the Phenomenological Spectrum proposed in 1989 to explain
The anomalous normal state (Marginal Fermi-liquid) and suggested as
the glue for pairing. Fluctuations are of current loops of all sizes and
directions.
Associate variation of
with change in doping. This is then a theory of
critical fluctuations at x=x_c as a function of temperature.
Crossover for
Coupling of Fluctuations to Fermions and pairing vertex
Inversion of ARPES indicates a broad featureless spectrum is
the glue.
g(k, k+q)
k
k+q
Leading deviations from MFT allow this calculation:
From this calculate Pairing Vertex:
g
Decompose into different IR’s:
S-wave and p-wave are repulsive
D-wave and X-S are attractive,
g Just as in the old calculation
(Miyake, Schmitt-Rink, cmv) for AFM
Fluctuations.
Right energy scale and coupling constant for Tc.
Answers why self-energy ind. of q but d-wave
Pairing.
Summary: It is possible to understand different regions of the
phase diagram of the cuprates with a single idea.
Interesting Quantum criticality. Probably relevant in several other
contexts. A Possible Theory for the Cuprates if the symmetry
breaking in Region II is further confirmed.
sgn( ) , for c
T.
T*
Antiferromagnetism



Marginal
Fermi liquid

I
II
Crossover
TF
Fermi liquid
SC
QCP
x (doping)
Superconductivity
Spectra and thermodynamics in the underdoped cuprates.
Time-reversal violation alone does not lead to observed properties.
BUT, A time-reversal violating state with a normal Fermisurface is not possible: (PRL (99); PR-B(06))
For fluctuations of non-conserved discrete quantities,
damping of fluctuations and their coupling of
fermions to fluctuations is finite for
.
This leads to single-particle self-energy
Quasi-particle velocity ---->
Observed Phase must be accompanied by a Fermi-surface Instability.
To see what can happen, look at the same issue as it arises
in another context.
Pomeranchuk Expansion of the free-energy for distortions of the
Fermi-surface
But
So, no symmetry change possible in the
channel
But something must happens since specific heat cannot be
allowed to be negative. Look at
0<z<1. Therefore instability due to diverging velocity.
What cures the instability?
Approach to the
Instability
Suggests a state with an anisotropic gap at the chemical potential:
No change in Symmetry, only change in Topology of the Fermi-surface
Have found a stable state (PRL-99, PRB- 06)with
is coupling of flucts. at q = 0 to the fermions at the F.S.
Ground state has only four fermi-points. No
extra change in symmetry, just in topology,
(Lifshitz Transition).
Kanigel et al. (2006) : Define “Fermi-arc length” as the set
of angles for which at any , the spectral function peaks
at the chemical potential for compounds with different x.
The data for 6 underdoped BISCO samples scales with
T*(x) and shows four fermi-points as T --->0.
Compare calculated “Fermi-arc length” with
Experiments(Lijun Zhu and cmv- 2006)
using the spectrum derived plus self-energy calculated using
only kinematics. Same D_0/T_g gives the
measured Specific Heat and Magnetic Susceptibility.