Hong Kong Forum of Condensed Matter Physics: Past, Present, and Future December 20, 2006

1. Dirac Quasiparticles in Condensed Matter Physics Adam Durst Department of Physics and Astronomy Stony Brook University Hong Kong Forum of Condensed Matter Physics: Past, Present, and Future December 20, 2006

2. Dirac Quasiparticles in Condensed Matter Physics (mostly d-wave superconductors) Adam Durst Department of Physics and Astronomy Stony Brook University Hong Kong Forum of Condensed Matter Physics: Past, Present, and Future December 20, 2006

3. Outline • Background • d-Wave Superconductivity • Universal Limit Thermal Conductivity (w/ aside on Graphene) • Quasiparticle Transport Amidst Coexisting Charge Order • Quasiparticle Scattering from Vortices • Summary

4. Dirac Fermions Relativistic Fermions (electrons) Massless Relativistic Fermions (neutrinos)

5. What does this have to do with Condensed Matter Physics? • We need only non-relativistic quantum mechanics and electromagnetism • But in many important cases, the low energy effective theory is described by Dirac Hamiltonian and Dirac energy spectrum • Examples include: • Quasiparticles in Cuprate (d-wave) Superconductors • Electrons in Graphene • etc… • Low energy excitations are two-dimentional massless Dirac fermions

6. T AF dSC x High-Tc Cuprate Superconductors

7. s-Wave Superconductor Fully gapped quasiparticle excitations

8. d-Wave Superconductor Quasiparticle gap vanishes at four nodal points Quasiparticles behave more like massless relativistic particles than normal electrons

9. Anisotropic Dirac Cone d-Wave Superconductivity Quasiparticle Excitation Spectrum Two Characteristic Velocities

10. Density of States N(w) Disorder-Induced Quasiparticles w Disorder-Induced Quasiparticles L. P. Gorkov and P. A. Kalugin, JETP Lett.41, 253 (1985)

11. Universal Limit Transport Coefficients Disorder generates quasiparticles Disorder-independent conductivities Disorder scatters quasiparticles Disorder-dependent Disorder-independent P. A. Lee, Phys. Rev. Lett.71, 1887 (1993) M. J. Graf, S.-K. Yip, J. A. Sauls, and D. Rainer, Phys. Rev. B53, 15147 (1996) A. C. Durst and P. A. Lee, Phys. Rev. B62, 1270 (2000)

12. Low Temperature Thermal Conductivity Measurements YBCO: BSCCO: Taillefer and co-workers, Phys. Rev. B62, 3554 (2000)

13. Graphene Single-Layer Graphite

14. 2 15 devices 1 max(h/4e2) 0 0 4,000 8,000  (cm2/Vs) Universal Conductivity? Bare Bubble: Missing Factor of p!!! Novosolov et al, Nature, 438, 197 (2005) Can vertex corrections explain this? Shouldn’t crossed (localization) diagrams be important here?

15. Low Temperature Quasiparticle Transport in a d-Wave Superconductor with Coexisting Charge Density Wave Order (with S. Sachdev (Harvard) and P. Schiff (Stony Brook)) Checkerboard Charge Order in Underdoped Cuprates T x underdoped STM from Davis Group, Nature430, 1001 (2004)

16. Hamiltonian for dSC + CDW Current Project: Doubles unit cell Future:

17. CDW-Induced Nodal Transition Nodes survive but approach reduced Brillouin zone boundary Nodes collide with their “ghosts” from 2nd reduced Brillouin zone Nodes are gone and energy spectrum is gapped K. Park and S. Sachdev, Phys. Rev. B64, 184510 (2001)

18. Thermal Conductivity Calculation Green’s Function 4×4 matrix Disorder Heat Current Kubo Formula

19. Analytical Results in the Clean Limit

20. Beyond Simplifying Approximations Realistic Disorder 4×4 matrix • Self-energy calculated in presence of dSC+CDW • 32 real components in all (at least two seem to be important) Vertex Corrections • Not clear that these can be neglected in presence of charge order Work in Progress with Graduate Student, Philip Schiff

21. 2R H x Scattering of Dirac Quasiparticles from Vortices (with A. Vishwanath (UC Berkeley), P. A. Lee (MIT), and M. Kulkarni (Stony Brook)) Scattering from Superflow + Aharonov-Bohm Scattering (Berry phase effect) Two Length Scales

22. Ps R r R Model and Approximations • Account for neighboring vortices by cutting off superflow at r = R • Neglect Berry phase acquired upon circling a vortex • - Quasiparticles acquire phase factor of (-1) upon circling a vortex • - Only affects trajectories within thermal deBroglie wavelength of core • Neglect velocity anisotropy vf = v2

23. Single Vortex Scattering Momentum Space Coordinate Space

24. Cross Section Calculation • Start with Bogoliubov-deGennes (BdG) equation • Extract Berry phase effect from Hamiltonian via gauge choice • Shift origin to node center • Separate in polar coordinates to obtain coupled radial equations • Build incident plane wave and outgoing radial wave • Solve inside vortex (r < R) and outside vortex (r > R) to all orders in linearized hamiltonian and first order in curvature terms • Match solutions at vortex edge (r = R) to obtain differential cross section Small by k/pF

25. Contributions to Differential Cross Section from Each of the Nodes

26. YBa2Cu3O6.99 Calculated Thermal Conductivity Experiment (Ong and co-workers (2001)) Calculated

27. What about the Berry Phase? Should be important for high field (low temperature) regime where deBroglie wavelength is comparable to distance between vortices Over-estimated in single vortex approximation Branch Cut Better to consider double vortex problem Elliptical Coordinates Work in Progress with Graduate Student, Manas Kulkarni

28. Summary • The low energy excitations of the superconducting phase of the cuprate superconductors are interesting beasts – Dirac Quasiparticles • Cuprates provide a physical system in which the behavior of these objects can be observed • In turn, the study of Dirac quasiparticles provides many insights into the nature of the cuprates (as well as many other condensed matter systems)