tami pereg-barnea mcgill university cap congress, june 16, 2014
TRANSCRIPT
Tami Pereg-BarneaMcGill University
Interactions + Spin-orbit = ?
CAP Congress, June 16, 2014
Collaborators
Gil Refael (Caltech)
Aaron Farrell
Shunji Matsuura
Rosa Rodriguez Jan Borchmann
Ying-Jer Kao (NTU)
Marcel Franz (UBC)
KunWu Kim(Caltech)
Maxime Beaudry
Paradigm shift in CM physics
• Symmetry • Topology
2 level system1d
±π
π/2
-π/2
0
|
Chern # = how many times the spin winds around the unit circle
E=±∨h⃗ (𝑘 )∨¿
Winding is important
2 level system2d
|
Chern # = how many times the spin covers the unit sphere
E=±∨h⃗ (𝑘 )∨¿
Bulk → Edge
• Topological invariant /Chern number – non-local operator, integer.
• Cannot change smoothly → gap closes
Δ(x)
Majorana FermionsWanted since 1937!
FermionMajorana Fermion
Majorana
Majorana statistics
• Non-Abelian statistics:
• Bosons : , Fermions Anyons
• Non-abelian Anyons • A fermion: ;• Is it useful?
Topological superconductors
• Pairing order parameter • The topology is a k-space vortex in the order
parameter.
•
Topo Insulator → Topo Superconductor
• Fu and Kane, PRL 2008
3D TI
2D Dirac cones-waveTopological
super-conductor
Dirac point in 2d
kx
ky
Superconductivity + Dirac
} 2Δ
Spin-orbit semiconductors
Proximity effect driven superconductivity
• Proximity effect → topological superconductor
Sau, Lutchyn, Tewari, and Das Sarma, PRL 2010.
Alicea PRB 2010
1D topo-superconductivity
Theory: Oreg, Refael, vonOppen, PRL 105, 177002 (2010)Cook, Vazifeh and Franz, PRB 86, 155431 (2012)
Experiments: Mourik et al. , Science 336, 1003 (2012) Das et al., Nature Physics 8, 887(2012)
Interaction driven superconductivity?
• Interaction induced superconductivity?– Can e-e interactions replace the proximity effect?
Aaron Ferrell and TPBPRB 87 214517 (2013)
Interaction driven topological superconductivity
Interaction driven topological superconductivity
Aaron Ferrell and TPBPRB 87 214517 (2013)
Phase diagram
Strong coupling treatment
• The interaction isn’t weak - expand in a t/U fashion.
• Up to second order – t-J model generalization
Zeemanhopping
Spin-spin
At half filling
• No hopping• Unconventional spin Hamiltonian• Jδ is anisotropic, non diagonal.• Dzjaloshinskii-Moriya and Compass
anisotropy
½ filling phase diagram
At half filling• Incommensurate spin density wave
Aaron Ferrell, P.-K. Wu, Y-J Kao and TPB arXiv:1402.4093
Ansatz vs. Monte-Carlo
Away from ½ filling:Gutzwiller projected variational wavefunction
• Variational study• Gutzwiller projected mean field wave function
• Estimate the energy and minimize:
• Evaluated by Monte-Carlo
Gutzwiller Approximation
• • Parameters get renormalized
• Evaluate the man field energy
Strong coupling treatment
Aaron Ferrell and TPB Phys. Rev. B 89, 035112 (2014)
Interacting topological systems
• Chern # = momentum integral on Berry curvature. states involved.
• Well defined in non-interacting systems.• Alternative definition includes Green’s
function (require the full spectrum)• Entanglement entropy,
entanglement spectrum.
Entanglement entropy
• Density Matrix • Von-Neumann entropy • Reduced density matrix
• A measure of entanglement• Sensitive to topology
A
B
Signatures of topology in
Entanglement spectrum
• Defined as the spectrum of
• Different from the physical spectrum• Contains edge modes predominantly
Summary and Outlook
o Closer to a topological superconductoro Majorana fermions are closer than ever!
o Still need - characterization, control o Developing new tools to study strongly
interacting topological systemso New types of topological systems in the
strongly correlated regime?