beyond majorana fermions: parafermions and other exotic excitations
TRANSCRIPT
Beyond Majorana Fermions:Parafermions and Other Exotic [email protected] (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
ture
sta
ken
from
[5]
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ture
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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.