Majorana Tutorial

Tutorial for students, part I, 6/2/2013@PKU 1/36

Introduction to Majorana fermions: part I

Fa Wang ( 王垡 )

ICQM & School of Physics, Peking University


References

● Main references:

– A.Y. Kitaev, Phys.Usp. 44, 131 (2001); arXiv:cond-mat/0010440

– J.Alicea, Rep.Prog.Phys. 75, 076501 (2012); arXiv:1202.1293

● Theoretical references:

– C.Nayak et al., Rev.Mod.Phys. 80, 1083 (2008); arXiv:0707.1889

– R.M. Lutchyn et al., Phys.Rev.Lett. 105, 077001 (2010).

– Y.Oreg et al., Phys.Rev.Lett. 105, 177002 (2010).

– J.Alicea et al., Nat.Phys. 7, 412 (2011); arXiv:1006.4395

– K.T.Law et al., Phys.Rev.Lett. 103, 237001 (2009).

– L.Jiang et al., Phys.Rev.Lett. 107, 236401 (2011).

– L.Fu & C.L.Kane, Phys.Rev.Lett. 102, 216403 (2009).


References

● Experimental references:

– V.Mourik et al., Science 336, 1003 (2012).

– A.Das et al., Nat. Phys. 8, 887 (2012).

– L.P.Rokhinson et al., Nat. Phys. 8, 795 (2012).

● Reading materials:

– F.Wilczek, “Majorana returns”, Nat.Phys. 5, 614 (2009).


Trivia about Majorana (fermion)

● About Ettore Majorana

– Aug. 5, 1906: born in Catania, Sicily

– March 1938: disappeared in Palermo, Sicily

● About Majorana fermion

– Ref.: E. Majorana, Nuovo Cimento 14, 171 (1937).

– Real solutions of Dirac equation, are their own anti-particles.

– Elementary particles: Neutrinos?

– 2D condensed matter systems: ν=5/2 FQHE state? Sr2RuO

4? ...

– 1D condensed matter systems: semiconductor nanowire?

• Evidence: V.Mourik et al., Science 336, 1003 (2012).


§1: Basics about Majorana fermion


Basics of Majorana fermion: preview

● Majorana fermions are “real-valued” fermion modes

– “real-valued”: they are their own anti-particles: =in contrast to “complex” fermions:

e.g. electrons vs. positrons.

– “fermion”: different Majoranas anti-commute.

– Majorana fermions may obey non-Abelian statistics: Ising anyons.

• might be used for quantum computation.

• c.f. Nayak RMP'08


Defining Majorana fermion

● Consider one fermion mode:

– 2d Hilbert space spanned by unoccupied and occupied states:

– Fermion creation/annihilation operators

– Anti-commutation relation:



● Majorana fermion operators from one fermion mode:

– “Real-valuedness”:

– Anti-commutation relation: esp.

– Fermion number:

– Fermion number parity:



● Two fermion modes:

– 4d Hilbert space: tensor product of two 2d Hilbert space.

– Basis: tensor products of single fermion basis




● Two fermion modes: (cont'd)



● Majorana fermions from two fermion modes:



– Fermion number:




● N fermion modes:

– -dim'l Hilbert space:

– basis:

● 2N Majorana fermions: c.f. Jordan-Wigner transformation

– Fermions from Majoranas:



Properties of Majorana fermion

● “Real-valuedness”:● Anti-commutation relation:

– Majorana has no “vacuum”:

● “Basis change” of Majorana fermions:

– Real orthogonal transformation:

– May/May not change fermion parity:

– Includes particle-hole transformation of fermions:example, two Majoranas from one fermion



● Non-locality: example

– affects/depends on many sub-Hilbert spaces.

– Products of odd # of Majoranas have similar property.

– NOTE: Hamiltonian can only contain products of even number of Majorana fermion operators. Examples:

– NOTE: Hamiltonian preserves fermion parity:

– NOTE: Hilbert space of 2N Majorana fermions divides into even&odd fermion number sectors, each is of dimension



● Non-locality(cont'd):

– Fermions have the similar property: Hamiltonian cannot contain products of odd # of fermions operators.

– However there is a non-trivial bosonic hermitian operator (observable) from a single fermion mode:

– NOTE: There is no non-trivial bosonic hermitian operator from a single Majorana:

– Non-trivial observables must contain two or more Majoranas (information is stored non-locally).



● Non-Abelian statistics: c.f. Nayak et al. RMP'08

– Abelian statistics: with certain # of fermions at fixed positions,

• the Hilbert space is 1dim'l,

• exchanges of fermion pairs just change the phase of wavefunction. Different fermion pair exchanges commute.

– Non-Abelian statistics: with 2N Majoranas at fixed positions,

• the Hilbert space is -dim'l,

• different Majorana pair exchange/braiding do not commute: represented as non-commuting matrices.



● Non-Abelian statistics(cont'd):

– Braiding of Majorana fermion:

– corresponding unitary transformation on Hilbert spacesatisfies

– Exercise: check

– Non-Abelian statistics: exercise


Summary #1

● Basics of Majorana fermion:


• Equal weight superposition of particle and hole.

– Anti-commutation relation (Clifford algebra):

• Majorana has no “vacuum”:

• Basis changes should be real orthogonal transformations.

• has eigenvalues b/c

– Non-locality: information is stored in pairs of Majoranas.

– Non-Abelian statistics.


• -dim'l Hilbert space divides into even&odd subspaces, each is of dimension


§2: Model realization of Majorana fermion


Model realization: the goal

● To realize well-separated localized Majorana zero modes in a system with bulk gap

– “Majorana zero modes”: , γs do no appear in H.Action of these Majoranas do not change energy.

– 2n Majorana zero modes: -fold degenerate ground states.Majorana zero modes act non-trivially in this subspace.

– Bulk gap: clear separation b/w ground & excited states.

– Localized and well-separated: local perturbations will not lift the “topologically protected”ground state degeneracy, b/c it cannot involve more than one Majorana mode. ... ground states

bulk excitations

bulk gap

Energy spectrum


Model realization: 1D p-wave “superconductor”

● 1D spinless fermion chain with p-wave pairing

– Reference: Kitaev, Phys.Usp.'01.

– t, Δ, μ are real parameters. As an example, N is assumed even.

i=1 2 NN-13 N-2...



● Rewrite the Hamiltonian in terms of Majoranas

– a tight-binding model of Majorana fermions: exercise

...

i=1 i=2 i=3 i=N



● Special case #1: trivial phase

– t=Δ=0, μ<0:

• sum of N mutually commuting terms

– Unique ground state: all namely

– Bulk excitations of energy : one of namely

i=1 i=2 i=3

...

i=N



● Special case #2: non-trivial phase

– t=−Δ>0, μ=0:

• sum of N-1 mutually commuting terms

• Ground states: all

• Bulk excitations of energy : one of

i=1 i=2 i=3

...

i=N



● Special case #2: non-trivial phase (cont'd)

– NOTE: do no appear in H.

• “Majorana zero modes”:

– Two-fold degeneracy:

• Action of switches b/w the two degenerate states.

i=1 i=2 i=3

...

i=N



● Special case #2: non-trivial phase (cont'd)

– Fermion # parity:

– Explicit form of (un-normalized) ground states:

• Use projectors



● Less-special case #3: non-trivial phase

– t>−Δ>0, μ=0: two Majorana chains with alternating hoppings.

– The “weak-strong-...-strong-weak” chain have two zero energy edge modes in N→∞ limit.

...

i=1 i=2 i=3 i=N



● Less-special case #3: non-trivial phase (cont'd)

– To see , rewrite H of upper chain as

– Characteristic length coherent length of pairing, when |Δ|<<|t|.

...

amplitude ∝



● Generic case: criterion for “non-trivialness”:

– Ref.: Alicea,RepProgPhys'12

– Rewrite H of periodic chain into Bogoliubov-de Gennes form.

– under the mapping: the image of Brillouin zone 0≤k<2π is a closed loop, winding around the origin

odd(non-trivial) or even(trivial) number of times.

εk

Δk

εk

ΔkNon-trivial: |μ|<|2t| Trivial: |μ|>|2t|


Model realization: braiding in 1D

● Ref.: Alicea et al., Nat.Phys. 7, 412 (2011); arXiv:1006.4395

– move Majorana fermions by gating(tuning local μ)

– Braiding/exchange in “1D” without “collision” by sidetracks.

1 2 3 4 1

2

3 4 1 43

2

1 3

2

4

1 3 2 4

gates

trivial,large μ

non-trivial,small μ

1D spinless p-wave superconductor


Summary #2

● Prototypical model of Majorana zero modes:

– 1D spinless fermion with p-wave pairing, in the non-trivial “topological superconductor” phase.

– Majorana zero modes localized on the ends (boundaries between trivial & non-trivial regions).Characteristic length ~ coherent length of pairing.


§3: Experimental realization and detection


Experimental realization

● Realization of 1D “spinless” fermion with p-wave pairing:

– Semiconductor wire with spin-orbit coupling + Zeeman field + proximity to s-wave superconductor: c.f. Oreg PRL'10

– Experiments: V.Mourik et al., Science 336, 1003 (2012);A.Das et al., Nat. Phys. 8, 887 (2012);L.P.Rokhinson et al., Nat. Phys. 8, 795 (2012).

+ spin-orbit coupling + Zeeman field

EF

Ek


Experimental realization

● Realization of 1D “spinless” fermion with p-wave pairing:

– Edge states of 2D topological insulator in proximity of s-wave superconductor: c.f. Fu&Xu, Phys. Rev. B, 81, 134435 (2010)

● 2D realizations: c.f. Tutorial part 0&II.

Ek

EF

bulk states

bulk states

EF

“trivial” region

“non-trivial” region


Experimental detection

● Zero-bias tunneling conductance: c.f. KTLaw PRL'09

– Perfect Andreev reflection, conductance = 2e2/h.

– Exp.: Mourik et al. Science'12; Das et al. NatPhys'12

● Fractional Josephson effect: c.f. Kitaev PhysUsp'01

– Josephson current vs. flux (phase difference) is h/e-periodic instead of h/2e.

– Exp.: Rokhinson et al. NatPhys'12

● Interferometry: c.f. Fu PRL'09

– For 2D realizations, c.f. Tutorial part 0&II

Mourik et al. Science'12

c.f. Alicea RepProgPhys'12


The End.

Majorana Tutorial

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