Symmetry breaking in the Pseudogap state and Fluctuations about it
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Symmetry breaking in the Pseudogap state and Fluctuations about it
Schematic Universal phase diagram of high-Schematic Universal phase diagram of high-TTcc superconductorssuperconductors
MarginalMarginalFermi-liquidFermi-liquid
Fermi liquidFermi liquid
TT
x (doping)x (doping)
T*T*CrossoverCrossover
QCP
III
SCIII
AFM
“Pseudo- Gapped”
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.
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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.
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AntiferromagnetismAntiferromagnetismMarginal Marginal
Fermi liquidFermi liquid
Fermi liquidFermi liquid
x (doping)x (doping)
T*T*CrossoverCrossover
CPQ
TF€
€
∝ sgn(ω) , for ωc ff ω ff T.
Phenomenology(1989): Properties in Region I follow if there exists a Quantum Critical Point with scale invariant fluctuations given by
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I
II
SC
From approximate Inversion of ARPES and Optical conductivity: Pairing glue has spectrumconsistent with this. Deriving these fluctuations may be considered the central problem.
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II I
AntiferromagnetismAntiferromagnetism
SuperconductivitySuperconductivity
Pseudogapped metalPseudogapped metal
Marginal Marginal Fermi liquidFermi liquid
Fermi liquidFermi liquid
TT
x (doping)x (doping)
T*T*CrossoverCrossover
CPQ
Broken Symmetry?
Quantum Critical Point in high Tc crystals
If there is a QCP, there might be an ordered phase emanating fromit on one side and a Fermi-liquid below another line emanating from it.
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Microscopic Model:
cu
o
o
Cannot be reduced to a Hubbard Model because the ionization energyof Cu is nearly the same as the ionization energy of oxygen.
Why are Cuprates Unique? (1987)
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Look for symmetry breaking not ruled out by ExperimentsPreserve translational symmetry: severely limits possible phases;
Bond Decomposition of near neighbor interactions.
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Only Possible States not changing Translational and Spin-Rotational symmetries have order parameters:
Time-Reversal and some Reflection Symmetries lost.
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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).
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Kaminski et al., Nature (2002)
Dichroism in BISCCO
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Fauques et al. (2005): Polarized Elastic Neutron Scattering in underdoped and overdoped Y(123)
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Fauques et al. (2005):
Polarized Neutron diffraction in YBCuO
Magnetic Diffraction Pattern consistent withLoop Current Phase II just as Dichroic ARPES
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Classical Stat. Mech. Model for the observed Loop Current Phase
Four states per unit-cell.
Why no specific heat singularity at T*(x)?
Time-reversal and 3 of four reflections broken:
Two Ising degrees of freedom per unit-cell
Ashkin-Teller Model :
Observed broken symmetry for -1 < J4/J2 < 1.
Phase diagram obtained by Baxter; Kadanoff et al.
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Phase Diagram of the Ashkin-Teller Model (Baxter, Kadanoff)
Gaussian line
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Specific Heat for the relevant region: (Hove and Sudbo)
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AntiferromagnetismAntiferromagnetism
SuperconductivitySuperconductivity
Marginal Marginal Fermi liquidFermi liquid
Fermi liquidFermi liquid
x (doping)x (doping)
T*T*CrossoverCrossover
CPQ
TF€
€
∝ sgn(ω) , for ωc ff ω ff T.
Quantum critical Fluctuations : Fluctuations of the order parameter which condenses to give broken symmetry in Region II.
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I
II
SC
Very simple but peculiar Phenomenology:
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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 abovethe 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.
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Critical Region:Need not consider anisotropy term.For simplicity keep only the xy-term.
Fourier transformed dissipation: Derivable from elimination of current-current 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.
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Previous work on the dissipative xy model:
Nagaosa (1999); Tewari, Chakravarti, Toner (2003),…
Below a critical value of , dissipation destroys long range orderat T=0. But no calculation of correlation functions, connection with vortex fluctuations or connection with the classical transition.
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Steps in the derivation:
lives on the bonds of the lattice
2.
1.
3.
i i+x
i+y
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. .
Velocity field due to: decreases as 1/r.
Velocity field due to: spatially independent
Time-independent Time-dependent
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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-dependentpart and a time-dependent part. Problem transforms to a K-T problem in space and (mathematically) a Kondo problem in time.
A remarkable simplification which allows a solution!
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RG equations for fugacity y for instantons and for , similar toflows in the Kondo problem or the KT problem:
Instanton field does the disordering:
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Calculate order parameter correlation functions: At
Associate variation of with change in doping. This is then a theory ofcritical fluctuations at x=x_c as a function of temperature.
Gaussian Model : No corrections?
For , proliferates and disorders the velocity field.
This was the Phenomenological Spectrum proposed in 1989 to explainThe anomalous normal state (Marginal Fermi-liquid) and suggested asthe glue for pairing. Fluctuations are of current loops of all sizes and directions.
Crossover for
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Coupling of Fluctuations to Fermions and pairing vertex
k k+q
g(k, k+q)
Leading deviations from MFT allow this calculation:From this calculate Pairing Vertex:
g g
Decompose into different IR’s:S-wave and p-wave are repulsiveD-wave and X-S are attractive,Just as in the old calculation(Miyake, Schmitt-Rink, cmv) for AFMFluctuations.Right energy scale and coupling constant for Tc.Answers why self-energy ind. of q but d-wavePairing.
Inversion of ARPES indicates a broad featureless spectrum is the glue.
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AntiferromagnetismAntiferromagnetism
SuperconductivitySuperconductivity
Marginal Marginal Fermi liquidFermi liquid
Fermi liquidFermi liquid
x (doping)x (doping)
T*T*CrossoverCrossover
CPQ
TF€
€
∝ sgn(ω) , for ωc ff ω ff T.
€ €
I
II
SC
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 othercontexts. A Possible Theory for the Cuprates if the symmetry breaking in Region II is further confirmed.
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Spectra and thermodynamics in the underdoped cuprates.
BUT, A time-reversal violating state with a normal Fermi-surface is not possible: (PRL (99); PR-B(06))
Quasi-particle velocity ---->
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
Time-reversal violation alone does not lead to observed properties.
Observed Phase must be accompanied by a Fermi-surface Instability.
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To see what can happen, look at the same issue as it arisesin another context.
Pomeranchuk Expansion of the free-energy for distortions of theFermi-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?
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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
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Have found a stable state (PRL-99, PRB- 06)with
Ground state has only four fermi-points. No extra change in symmetry, just in topology, (Lifshitz Transition).
is coupling of flucts. at q = 0 to the fermions at the F.S.
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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.
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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.