1. Three Discoveries in Underdoped Cuprates “Thermal metal” in non-SC YBCO Sutherland et al., cond-mat/050124 Giant Nernst effect Z. A. Xu et al., Nature 406, 486 (2000) T. Hanaguri, C. Lupien et al., Nature 430, 1001 (2004) “Electron Crystal” in STM M. Vershinin et al.,, 1995 (2004); Science 303, 1995 (2004);

2. “Normal” state appears strange and non-Fermi liquid in nature. Strong correlations among its fermionic excitations are apparent. In contrast, the superconducting state appears BCS-like. Its low energy fermionic excitations are protected by a large d-wave (pseudo)gap. Pairing d-wave Pseudogap in Underdoped Cuprates Construct a theory in which a correlated d-wave superconductor serves as a reference state and the basis for exploration of the pseudogap. As x is reduced, quantum vortex-antivortex pairs are progressively admixed into the ground state. Key questions: i)How does such superconductor become “normal”? ii)What are the quantum ground states in natural proximity to such superconductor? What is the “pseudogap” state? iii) What is the low energy effective theory within the pseudogap whose role parallels that of a Fermi liquid in conventional metals? ? dSC

3. As doping decreases, the number of quantum vortex-antivortex pairs admixed into dSC ground state increases (n s << n 0 ). QED 3 Phase Diagram

4. Nernst Effect in HTS BiSrCaCuO Courtesy of N.P. Ong Vortex fluctuations in the pseudogap state !! Z. A. Xu et al., Nature 406, 486 (2000)

5. HTS are Nodal d-wave Superconductors (Phase sensitive exps., ARPES, FT-STS, etc.) FT-STS measured  ARPES  Mesot - PRL 83,840 (1999) Courtesy of J.C. Davis Tsuei & Kirtley, 1997 FT-STS from Davis’ group, Nature 422, 520 (2003).

6. Sutherland et al., cond-mat/050124 (2005) YBCO : non-superconducting state is a thermal metal H = 0 p < p SC LSCO : non-superconducting state is an insulator “Thermal Metal” in YBCO Courtesy of M. Sutherland

7. As doping decreases, the number of quantum vortex-antivortex pairs admixed into dSC ground state increases (n s << n 0 ). QED 3 Phase Diagram Quantum vortex-antivortex pairs unbind and ODLRO is destroyed (n s = 0). Still, BdG chiral symmetry of d-wave nodal qparticles remains, protecting gapless fermionic excitations !! Chiral symmetry  d-wave amplitude stiffness

8. BdG chiral symmetry is finally broken, nodal fermions are gapped   AF/SDW As doping decreases, the number of quantum vortex-antivortex pairs admixed into dSC ground state increases (n s << n 0 ). QED 3 Phase Diagram Quantum vortex-antivortex pairs unbind and ODLRO is destroyed (n s = 0). Still, BdG chiral symmetry of d-wave nodal qparticles remains protecting gapless fermionic excitations !! Chiral symmetry  d-wave amplitude stiffness

9. “Electron Crystal” in Underdoped Cuprates M. Vershinin et al., Science 303, 1995 (2004); T. Hanaguri, C. Lupien et al., Nature 430, 1001 (2004) Courtesy of J.C. Davis Underdoped Ca 2-x Na x CuO 2 Cl 2 x=0.08 (I), 0.10 (SC), 0.12 (SC)

10. Vershinin et al. Science 303, 1995 (2004) 35 pS 150 pS 500Å x 200 Å Cu-O 100K Pseudogap State Modulation along the Cu-O bond direction K-Space Courtesy of A. Yazdani

11. Dual theory supplies the connection between two phenomena. It leads to Hofstadter “magic fractions” f = p/q for Cooper pair DW at special x: f = p/q = (1 – x)/2 !! Hofstadter Butterfly and Cooper Pair Density-Wave in HTS Hofstadter “Butterfly” Spectrum “Electron Crystal” in HTS (Vershinin, et al., Hanaguri, Lupien, et al.)

12. Pseudogap: Charge Insulator, Spin Conductor Charge current cancels: J c = 0. Charge center-of-mass is pinned to the lattice: Abrikosov-Hofstadter dual vortex solid. Spin current J s is finite. Pseudogap is a spin (semi) metal: Spinful BdG nodal fermions with long range gauge field correlations -- chiral QED 3. e charge current spin current The system consists of electrons paired in spin singlets (Cooper pairs) + unpaired electrons (BdG nodal fermions). Both carry charge (2e vs. e) but only BdG nodal fermions carry spin. As Cooper pairs and nodal fermions screen each other charges, the spin remains “free”  Wiedemann-Franz law is violated  Wiedemann-Franz law is violated !!

13. Phase Diagram from Dual Theory of a d-wave Superconductor Antiferromagnet/SDW (Mott ins.) Pairing pseudogap (chiral QED 3 ) dSC Nodal pair DW (chiral QED 3 ) Supersolid

14. QED 3 Phase Diagram