Beyond Majorana Fermions: Parafermions and Other Exotic Excitations [email protected] (University of Cologne, Institute of Theoretical Physics) Primer on Majorana Fermions One 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 † j c j +1 +c † j +1 c j )-μc † j c j +Δc j c j +1 + ¯ Δc † j +1 c † j Two fundamentally different topological phases: H = i ∑ j a j b j H = i ∑ j b j a j +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: ia i a j ib i b j ia i b j a i a j b i b j Hermitean Time-reversal invariant T :(a i , b i ) 7→ (a i , -b i ) Effect of interactions: Reduction Z → Z 8 Explanation in terms of group theory [2, 3] What are Parafermions? Prominent features of Z N parafermions: • Generalized Pauli exclusion principle: Each state occupied with up to N -1 parafermions... • Non-abelian braid statistics • Symmetry Z N ⊂ U (1), generated by ω = exp ( 2πi N ) • Duality with the “Z N clock model” (chiral Potts model) • Reduction to Majorana case for N =2 Same site j Different sites j<k Algebraic structure: (χ j ) N =1 χ j χ k = ωχ k χ j (ψ j ) N =1 ψ j ψ k = ωψ k ψ j χ j ψ j = ωψ j χ j χ j ψ k = ωψ k χ j Known: Phase with parafermionic edge zero modes exists Relevant Hamiltonian for N =3 [4]: H = if X j h χ † j ψ j - ψ † j χ j i + iJ X j h ψ † j χ j +1 - χ † j +1 ψ j i 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. Pictures taken from [5] Pictures taken from [6] Suggested verification: Josephson and tunneling measurements Manipulation: Interface proximity tunneling Goals 1. 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 applications Conformal Field Theory Experimental realization Group 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.