Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex-vortex interactions are shown to lead to the lifting of the topological degeneracy and the energy is discovered to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice model of Majorana fermions. We find necessary conditions for it to approximate the spectrum of the honeycomb lattice model and show that bi-partite interactions are responsible for the degeneracy lifting also in many vortex systems.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  Fractional Quantum Hall Effect and Nonabelian Statistics , 1992, hep-th/9202001.

[3]  Flux phase of the half-filled band. , 1994, Physical review letters.

[4]  N. Read,et al.  Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect , 1999, cond-mat/9906453.

[5]  A. Kitaev.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.

[6]  M. Lukin,et al.  Controlling spin exchange interactions of ultracold atoms in optical lattices. , 2002, Physical review letters.

[7]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[8]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[9]  Electronic transport in an array of quasiparticles in theν=5∕2non-Abelian quantum Hall state , 2005, cond-mat/0511670.

[10]  P. Zoller,et al.  A toolbox for lattice-spin models with polar molecules , 2006 .

[11]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.

[12]  Matthias Troyer,et al.  Interacting anyons in topological quantum liquids: the golden chain. , 2007, Physical review letters.

[13]  Zero modes of two-dimensional chiral p -wave superconductors , 2006, cond-mat/0610094.

[14]  J. Pachos,et al.  Why should anyone care about computing with anyons? , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  C. P. Sun,et al.  Mosaic spin models with topological order , 2007, 0708.0676.

[16]  Three-body interactions with cold polar molecules , 2007, cond-mat/0703688.

[17]  T. Xiang,et al.  Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions. , 2007, Physical review letters.

[18]  Hong Yao,et al.  Exact chiral spin liquid with non-Abelian anyons. , 2007, Physical review letters.

[19]  Zhenghan Wang,et al.  On Classification of Modular Tensor Categories , 2007, 0712.1377.

[20]  J. Pachos.  The wavefunction of an anyon , 2006, quant-ph/0605068.

[21]  S. Simon,et al.  Non-Abelian Anyons and Topological Quantum Computation , 2007, 0707.1889.

[22]  Yue Yu,et al.  An exactly soluble model with tunable p-wave paired fermion ground states , 2007, 0708.0631.

[23]  T. Stitt,et al.  Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model , 2007, 0712.1164.

[24]  J. Vala,et al.  Topological degeneracy and vortex manipulation in Kitaev's honeycomb model. , 2008, Physical review letters.

[25]  S. Dusuel,et al.  Perturbative approach to an exactly solved problem: Kitaev honeycomb model , 2008, 0809.1553.

[26]  L. Fu,et al.  Superconducting proximity effect and majorana fermions at the surface of a topological insulator. , 2007, Physical review letters.

[27]  Zohar Nussinov,et al.  Exact results of the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations , 2007, cond-mat/0703633.

[28]  Shou-Cheng Zhang,et al.  Algebraic spin liquid in an exactly solvable spin model. , 2008, Physical review letters.

[29]  Jiannis K. Pachos,et al.  Non-Abelian statistics as a Berry phase in exactly solvable models , 2009, 0901.3674.

[30]  Meng Cheng,et al.  Splitting of Majorana-fermion modes due to intervortex tunneling in a p(x) + ip(y) superconductor. , 2009, Physical review letters.

[31]  Parsa Bonderson.  Splitting the topological degeneracy of non-Abelian anyons. , 2009, Physical review letters.

[32]  L. N. Pfeiffer,et al.  Measurement of filling factor 5/2 quasiparticle interference with observation of charge e/4 and e/2 period oscillations , 2008, Proceedings of the National Academy of Sciences.

[33]  Matthias Troyer,et al.  Collective states of interacting anyons, edge states, and the nucleation of topological liquids. , 2008, Physical review letters.

[34]  M. Baraban,et al.  Numerical analysis of quasiholes of the moore-read wave function. , 2009, Physical review letters.

[35]  J. Vala,et al.  Description of Kitaev’s honeycomb model with toric-code stabilizers , 2009, 0903.5211.

[36]  J. Pachos,et al.  Topological phase transitions driven by gauge fields in an exactly solvable model , 2010, 1003.1086.

[37]  C. Beenakker,et al.  Majorana bound states without vortices in topological superconductors with electrostatic defects. , 2010, Physical review letters.

[38]  S Das Sarma,et al.  Generic new platform for topological quantum computation using semiconductor heterostructures. , 2009, Physical review letters.

[39]  S. Sarma,et al.  Tunneling of anyonic Majorana excitations in topological superconductors , 2010, 1006.0452.

[40]  J. Vala,et al.  Zero energy and chiral edge modes in a p-wave magnetic spin model , 2010, 1006.4951.

[41]  T. Mizushima,et al.  Splitting and oscillation of Majorana zero modes in the p -wave BCS-BEC evolution with plural vortices , 2010, 1005.4738.

[42]  C. Kane,et al.  Topological Insulators , 2019, Electromagnetic Anisotropy and Bianisotropy.

[43]  Parsa Bonderson,et al.  Implementing arbitrary phase gates with Ising anyons. , 2009, Physical review letters.

[44]  S. Vishveshwara,et al.  Topological phases, Majorana modes and quench dynamics in a spin ladder system , 2011, 1102.0824.

[45]  J. Vala,et al.  Kaleidoscope of topological phases with multiple Majorana species , 2010, 1012.5276.

[46]  Matthias Troyer,et al.  Two-dimensional quantum liquids from interacting non-Abelian anyons , 2010, 1003.3453.

[47]  Xiao-Liang Qi,et al.  Possible proximity of the Mott insulating iridate Na2IrO3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field , 2011, 1101.1145.

[48]  Dexter Kozen,et al.  New , 2020, MFPS.