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?