Beyond Majorana Fermions:Parafermions and Other Exotic ExcitationsThomas.Quella@uni-koeln.de (University of Cologne, Institute of Theoretical Physics)

Primer on Majorana FermionsOne physical fermion Two Majorana (=real) fermions

c, c† ⇔ a = c + c†

b = i(c− c†)⇒ One Majorana fermion = half a physical fermion

Isolating Majorana Fermions

Hamiltonian for non-interacting spinless fermions [1]:

H =∑

j

[−w(c†jcj+1+c

†j+1cj)−µc

†jcj+∆cjcj+1+∆̄c†j+1c

†j

]Two fundamentally different topological phases:

H = i∑j ajbj H = i

∑j bjaj+1

⇒ Isolated Majorana fermions and as dangling edge modes

Duality with the Ising model: Ordered vs. disordered phase

Time Reversal Protected Topological Phases

Realization of multiple Majorana fermions (no interactions):

# of chains

{⇒ Z-invariant

Time reversal invariance provides mechanism of protection:

iaiajibibj

iaibjaiajbibj

HermiteanTime-reversal invariantT : (ai, bi) 7→ (ai,−bi)

Effect of interactions: Reduction Z→ Z8

Explanation in terms of group theory [2, 3]

What are Parafermions?

Prominent features of ZN parafermions:

• Generalized Pauli exclusion principle:Each state occupied with up to N−1 parafermions...

• Non-abelian braid statistics

• Symmetry ZN ⊂ U(1), generated by ω = exp(

2πiN

)• Duality with the “ZN clock model” (chiral Potts model)

• Reduction to Majorana case for N = 2

Same site j Different sites j < k

Algebraic structure: (χj)N = 1 χjχk = ω χkχj

(ψj)N = 1 ψjψk = ω ψkψj

χjψj = ω ψjχj χjψk = ω ψkχj

Known: Phase with parafermionic edge zero modes exists

Relevant Hamiltonian for N = 3 [4]:

H = if∑

j

[χ†jψj − ψ

†jχj

]+ iJ

∑j

[ψ†jχj+1 − χ

†j+1ψj

]Expectation: N distinct topological phases. Their nature?!?

Physical Relevance

• Theoretical prediction of exotic quasi-particles

• Development of detection and manipulation techniques

• Ultimate goal: Universal topological quantum computation

Suggested Experimental Realization

Realization in devices involving fractional quantum Hall samples(e.g. at filling ν = 1/m) and s-wave superconductors [5, 6].Alternative: Fractional topological insulators.

Pic

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sta

ken

from

[5]

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from

[6]

Suggested verification: Josephson and tunneling measurementsManipulation: Interface proximity tunneling

Goals1. Classification of symmetry protected topological phases

• Effect of space-time symmetries (time-reversal, inversion, ...)

• Investigation of ladder systems

2. Construction of chains based on more exotic quasi-particles• Usage of non-abelian quantum Hall states

• Dualities to more complicated models of statistical physics

3. Exploration of applications and experimental realizations

4. Extension to higher dimensions

Methods

• Group theory and group cohomology

• Conformal Field Theory (e.g. to model FQHE wave functions)

• Dualities and generalized Jordan-Wigner transformations

Aspired Role in the SPP 1666

Expertise provided Expertise sought

Mathematical & Theoretical Theoretical & Experimental

Topological Phases of Matter Potential applicationsConformal Field Theory Experimental realizationGroup Theory & Mathematics

Literature[1] A. Kitaev, Physics Uspekhi 44 (2001) 131, arXiv:cond-mat/0010440.

[2] L. Fidkowski and A. Kitaev, Phys. Rev. B83 (2011) 075103, arXiv:1008.4138.

[3] A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B83 (2011) 075102, arXiv:1008.4346.

[4] P. Fendley, arXiv:1209.0472.

[5] D. J. Clarke, J. Alicea, and K. Shtengel, arXiv:1204.5479.

[6] N. H. Lindner, E. Berg, G. Refael, and A. Stern, arXiv:1204.5733.