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Interactions + Spin-orbit = ?. Tami Pereg-Barnea McGill University. CAP Congress, June 16, 2014. Collaborators. Gil Refael (Caltech). Rosa Rodriguez. Marcel Franz (UBC). Jan Borchmann. Aaron Farrell. Shunji Matsuura. KunWu Kim (Caltech). Maxime Beaudry. Ying- Jer Kao (NTU).

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Tami pereg barnea mcgill university

Interactions + Spin-orbit = ?

Tami Pereg-Barnea

McGill University

CAP Congress, June 16, 2014


Collaborators
Collaborators

Gil Refael(Caltech)

Rosa Rodriguez

Marcel Franz (UBC)

Jan Borchmann

Aaron Farrell

Shunji Matsuura

KunWu Kim

(Caltech)

MaximeBeaudry

Ying-Jer Kao (NTU)


Paradigm shift in cm physics
Paradigm shift in CM physics

  • Symmetry

  • Topology


2 level system 1d
2 level system1d

|

π/2

0

Chern # = how many times the spin winds around the unit circle

±π

Winding is important

-π/2


2 level system 2 d
2 level system2d

|

Chern # = how many times the spin covers the unit sphere


Bulk edge
Bulk → Edge

  • Topological invariant /Chern number – non-local operator, integer.

  • Cannot change smoothly → gap closes

Δ(x)


Majorana fermions wanted since 1937
Majorana FermionsWanted since 1937!

Majorana Fermion

Fermion

Majorana


Majorana statistics
Majorana statistics

  • Non-Abelian statistics:

  • Bosons : , Fermions Anyons

  • Non-abelianAnyons

  • A fermion: ;

  • Is it useful?


Topological superconductors
Topological superconductors

  • Pairing order parameter

  • The topology is a k-space vortex in the order parameter.


Topo i nsulator t opo superconductor
TopoInsulator → Topo Superconductor

s-wave

Topological super-conductor

2D Dirac cone

3D TI

  • Fu and Kane, PRL 2008





Proximity effect driven superconductivity
Proximity effect driven superconductivity

  • Proximity effect → topological superconductor

Sau, Lutchyn, Tewari, and Das Sarma, PRL 2010.

Alicea PRB 2010


1d topo superconductivity
1D topo-superconductivity

Theory: Oreg, Refael, vonOppen, PRL 105, 177002 (2010)Cook, Vazifeh and Franz, PRB 86, 155431 (2012)

Experiments: Mouriket al. , Science 336, 1003 (2012)

Daset al., Nature Physics8, 887(2012)


Interaction driven superconductivity
Interaction driven superconductivity?

  • Interaction induced superconductivity?

    • Can e-e interactions replace the proximity effect?

Aaron Ferrell and TPB

PRB 87 214517 (2013)



Interaction driven topological superconductivity1
Interaction driven topological superconductivity

Aaron Ferrell and TPB

PRB 87 214517 (2013)



Strong coupling treatment
Strong coupling treatment

  • The interaction isn’t weak - expand in a t/U fashion.

  • Up to second order – t-J model generalization

Spin-spin

Zeeman

hopping


At half filling
At half filling

  • No hopping

  • Unconventional spin Hamiltonian

  • Jδis anisotropic, non diagonal.

  • Dzjaloshinskii-Moriya and Compass anisotropy



At half filling1
At half filling

  • Incommensurate spin density wave

Aaron Ferrell, P.-K. Wu, Y-J Kao and TPB

arXiv:1402.4093


Ansatz vs monte carlo
Ansatz vs. Monte-Carlo


Away from filling gutzwiller projected variational wavefunction
Away from ½ filling:Gutzwillerprojected variationalwavefunction

  • Variational study

  • Gutzwiller projected mean field wave function

  • Estimate the energy and minimize:

  • Evaluated by Monte-Carlo


Gutzwiller approximation
Gutzwiller Approximation

  • Parameters get renormalized

  • Evaluate the man field energy


Strong coupling treatment1
Strong coupling treatment

Aaron Ferrell and TPB

Phys. Rev. B 89, 035112 (2014)


Interacting topological systems
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
Entanglement entropy

  • Density Matrix

  • Von-Neumann entropy

  • Reduced density matrix

  • A measure of entanglement

  • Sensitive to topology

A

B



Entanglement spectrum
Entanglement spectrum

  • Defined as the spectrum of

  • Different from the physical spectrum

  • Contains edge modes predominantly


Summary and outlook
Summary and Outlook

  • Closer to a topological superconductor

  • Majorana fermions are closer than ever!

    • Still need - characterization, control

  • Developing new tools to study strongly interacting topological systems

  • New types of topological systems in the strongly correlated regime?


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