1. Symmetry breaking in the Pseudogap state and Fluctuations about it Schematic Universal phase diagram of high-T c superconductors MarginalFermi-liquid Fermi liquid T x (doping) T* Crossover QCP I II SC III AFMAFM “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.

2. 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.

3. Antiferromagnetism Marginal Fermi liquid x (doping) T* Crossover CP Q T F Phenomenology (1989): Properties in Region I follow if there exists a Quantum Critical Point with scale invariant fluctuations given by I II SC From approximate Inversion of ARPES and Optical conductivity: Pairing glue has spectrum consistent with this. Deriving these fluctuations may be considered the central problem.

4. II I Antiferromagnetism Superconductivity Pseudogapped metal Pseudogapped metal Marginal Fermi liquid T x (doping) T* Crossover CP Q Broken Symmetry? 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.

5. Microscopic Model: cu o o Cannot be reduced to a Hubbard Model because the ionization energy of Cu is nearly the same as the ionization energy of oxygen. Why are Cuprates Unique? (1987)

6. Look for symmetry breaking not ruled out by Experiments Preserve translational symmetry: severely limits possible phases; Bond Decomposition of near neighbor interactions.

7. Only Possible States not changing Translational and Spin-Rotational symmetries have order parameters: Time-Reversal and some Reflection Symmetries lost.

8. 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).

9. Kaminski et al., Nature (2002) Dichroism in BISCCO

10. Fauques et al. (2005): Polarized Elastic Neutron Scattering in underdoped and overdoped Y(123)

11. Fauques et al. (2005): Polarized Neutron diffraction in YBCuO Magnetic Diffraction Pattern consistent with Loop Current Phase II just as Dichroic ARPES

12. 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.

13. Phase Diagram of the Ashkin-Teller Model (Baxter, Kadanoff) Gaussian line

14. Specific Heat for the relevant region: (Hove and Sudbo)

15. Antiferromagnetism Superconductivity Marginal Fermi liquid x (doping) T* Crossover CP Q T F Quantum critical Fluctuations : Fluctuations of the order parameter which condenses to give broken symmetry in Region II. I II SC Very simple but peculiar Phenomenology:

16. 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.

17. 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.

18. 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.

19. Steps in the derivation: lives on the bonds of the lattice 2. 1. 3. i i+x i+y

20. .. Velocity field due to : decreases as 1/r. Velocity field due to : spatially independent Time-independent Time-dependent

21. 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. A remarkable simplification which allows a solution!

22. RG equations for fugacity y for instantons and for, similar to flows in the Kondo problem or the KT problem: Instanton field does the disordering:

23. Calculate order parameter correlation functions: At Associate variation of with change in doping. This is then a theory of critical 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 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. Crossover for

24. Coupling of Fluctuations to Fermions and pairing vertex kk+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 repulsive D-wave and X-S are attractive, 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. Inversion of ARPES indicates a broad featureless spectrum is the glue.

25. Antiferromagnetism Superconductivity Marginal Fermi liquid x (doping) T* Crossover CP Q T F 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 other contexts. A Possible Theory for the Cuprates if the symmetry breaking in Region II is further confirmed.

26. 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.

27. 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?

28. 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

29. 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.

30. 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.

31. 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.