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

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
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
d-Wave Superconductivity
Quasiparticle Excitation Spectrum
Two Characteristic Velocities
10
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. B 53, 15147 (1996) A. C. Durst and P.
A. Lee, Phys. Rev. B 62, 1270 (2000)
12
Low Temperature Thermal Conductivity Measurements
YBCO
BSCCO
Taillefer and co-workers, Phys. Rev. B 62, 3554
(2000)
13
Graphene
Single-Layer Graphite
14
Universal Conductivity?
Bare Bubble
Missing Factor of p!!!
Novosolov et al, Nature, 438, 197 (2005)
Can vertex corrections explain this?
Shouldnt 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, Nature 430, 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. B 64, 184510
(2001)
18
Thermal Conductivity Calculation
Greens Function
44 matrix
Disorder
Heat Current
Kubo Formula
19
Analytical Results in the Clean Limit
20
Beyond Simplifying Approximations
Realistic Disorder
44 matrix

• Self-energy calculated in presence of dSCCDW
• 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
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
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 lt R) and outside vortex (r
  gt 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
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)