Slide 1

Introduction to topological superconductivity and Majorana fermionsTejas Deshpande

(17 February 2013)1Introduction

Majorana fermions in p-wave superconductorsRepresentation in terms of fermionic operatorsNon-abelian statisticsMajorana qubits and topological quantum computation

Proximity-induced superconductivity in spin-orbit semiconductors

Induced p-wave-like gap in semiconductors

Conclusions and outlookOutline2

Kitaev 1-D ChainSpinless p-wave superconductorTight-Binding Hamiltonian

Defining Majorana Operators

Anticommutation relations for Majorana OperatorsSpecial case: left Majoranas on different sites

Majoranas on same site:

3

Kitaev 1-D ChainHamiltonian in terms of Majorana OperatorsSimple case

Recall anticommutation

4

Kitaev 1-D ChainAlternative pairing of Majorana fermionsRecall 1-D Majorana Hamiltonian

Define

Diagonalized Hamiltonian

5

Role of pairing in Kitaev 1-D ChainWhat is the nature of pairing?Recall the tight-Binding Hamiltonian

Why is this a p-wave superconductor?For the so-called s-, p-, d- or f-wavesuperconductorPairing in real spaceHow to visualize cooper pairs?Lattice model for (conventional) s-wave

On-site particle number operatorThis bosonic blob still at the same site

6Role of pairing in Kitaev 1-D ChainPairing in conventional superconductivityRecall lattice model for conventional superconductor

Applying Wicks theorem

The mean field Hamiltonian

Pairing in unconventional superconductivity

On-site pairingNearest-neighbor pairing7Role of pairing in Kitaev 1-D ChainPairing in unconventional superconductivity2-D lattice model mean field Hamiltonian (lattice constant = 1)

Diagonalize

8Properties of Majorana FermionsAre Majoranas hard-core balls?Majorana mode is a superposition of electron and hole statesIs this like a bound state? e.g. exciton, hydrogen atom, positronium?Can count them by putting them in bins?Sure, define a number operatorGarbage! Okay, counting doesnt make sense!Regular fermion basisWe can count regular fermionsWe can pair Majoranas into regular fermions and measure themHow to chose? Number of pairings:

Overlap between states

To observe the state of the system we need to fuse two Majoranas

9Properties of Majorana FermionsNon-abelian statisticsA system of 2N well separated Majoranas has a 2N degenerate ground state. Think of N independent of 1-D Kitaev chainsExchanging or braiding connects two different ground statesWhat is nonabelian about them?

if one performs sequential exchanges, the final state depends on the order in which they are carried out

Consider the exchange of two Majoranas

10Properties of Majorana FermionsNon-abelian statisticsExchange of two Majoranas

Define braiding operator

11Properties of Majorana FermionsNon-abelian statisticsExchange of two Majoranas

Effect on number states

12

Define Pauli matrices for rotations on the Bloch sphere

Braiding as rotationsProperties of Majorana FermionsNon-abelian statisticsExchange of four Majoranas

Effect on number states

13Ingredients for observing Majoranas?Key ingredientsMechanism for pairing of regular fermionsSpin degree of freedom must be suppressedAdditionally we needp-wave pairing symmetrySpin-triplet stateTools that provide these ingredientsPairing in superconductors or proximity effectSuppress spin break time-reversal symmetry or polarize a bandFew important approaches/proposalsEngineer systems with strong spin-orbit coupling and superconductorsInduced triplet p-wave pairing in non-centrosymmetric superconductorsDiscover Time Reversal Invariant topological superconductors!

14Artificial topological superconductorsSpinless p-wave superconductors2nd quantized Hamiltonian

Pairing Hamiltonian:

Nambu spinor

s-wave singlet pairing inherited from superconductor:

15Artificial topological superconductorsSpinless p-wave superconductorsRecall 1st and 2nd quantized Hamiltonians

Define

Then total Hamiltonian is given by

where

16Artificial topological superconductorsObtaining Bogoliubov-de Gennes (BdG) equation

Compare

17Artificial topological superconductorsArtifacts of the BdG formalism? Particle-hole symmetryAction on operators

Relation between particle and hole eigenstates

Therefore, Majorana fermionsStructure of the Nambu spinor

18

Artificial topological superconductorsSpinless p-wave superconductors2nd quantized Hamiltonian

Assume on-site pairingFirst considerBlock diagonal 2 2 Hamiltonians

where

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Artificial topological superconductorsSpinless p-wave superconductorsDiagonalizing 2 2 matrices

Eigenvalues

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Artificial topological superconductorsSpinless p-wave superconductorsRecall eigenvalues

For no Zeeman field

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Artificial topological superconductorsSpinless p-wave superconductorsNow, slowly turn on the pairing

Total Hamiltonian becomes

Brute force diagonalization

The gap vanishes at

22

Artificial topological superconductorsSpinless p-wave superconductorsThe gap vanishes at

and

Consider the limit

where

23Conclusions and OutlookOverviewHow to obtain Majorana fermionsNon-abelian statisticsEngineering/finding systems that host Majorana zero modesExperimental progressKouwenhoven group first to see zero bias conductance peak (ZBCP) in InSb nanowiresOther groups confirmed existence of ZBCP with different experimental parametersExperimental to-dosVerify non-abelian statisticsTest more platforms for hosting Majorana fermionsAccomplish reliable quantum computation

24ReferencesMartin Leijnse and Karsten Flensberg, Introduction to topological superconductivity and Majorana fermions, Semiconductor Science and Technology, vol. 27, no. 12, p. 124003, 2012

Jason Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Reports on Progress in Physics, vol. 75, no. 7, p. 076501, 201225Thanks for listening!Questions?26