School of something FACULTY OF OTHER Quantum Information Group School of Physics and Astronomy Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model arXiv: Ville Lahtinen Jiannis K. Pachos Graham Kells, Jiri Vala (Maynooth)‏ Ringberg 12/07

Kitaev's honeycomb lattice model: A.Y. Kitaev, Annals of Physics, 321:2, 2006 J.K. Pachos, Annals of Physics, 6:1254,2006 An exactly solvable 2D spin model. Two different topological phases (anyon models): Toric code (abelian) and SU(2) 2 (non-abelian). Can be used to study a topological phase transition. Could be constructed in a lab. Motivation Micheli et al. Nature Physics, 2: ,2005.

Motivation 1) Phase space geometry 2) Non-abelian character: zero modes 3) I nteractions between vortices 4) Energy gaps: experimental stability Why study the spectrum:

The honeycomb lattice model H commutes with the plaquette operators

Represent spins by Majorana fermions by Solving the model Define a (M,N)-unit cell with respect to which the configuration of u's is periodic Fix the eigenvalues u on all links - fix the vortex configuration

Solving the model Fourier transform with respect to the unit cell (MN-vectors)‏ (MN x MN matrices)‏ Site i inside the (M,N)-unit cell labeled by: k=(m,n) indicates a particular z-link λ=1,2 indicates a black or a white site Lattice basis vectors:

Strategy Fix the eigenvalues u=±1 on all links to create a vortex configuration. Diagonalize the Hamiltonian and obtain where z i are n “zero modes” possibly appearing in the spectrum. Study the relative ground state energies and fermion gaps of different vortex configurations as a function of J i and K

Sparse 2-vortex configurations u=-1 on all z-links along the strings. u=1 on all other links. Study relative ground state energies and fermion gaps as a function of s. Attribute the behavior to vortex- vortex interactions. We use (20,20)-unit cell. Create a 2-vortex configuration with vortex separation s:

1) Phase space geometry A phase always gapped (toric code) B phase gapped when K>0 (SU(2) 2 )‏ Phase boundaries at Δ → 0 depend on underlying vortex configuration Boundaries: Vortex-free: J=1/2 Full-vortex: J=1/√2 Sparse: 1/2 ≤ J ≤ 1/√2 (J z = 1 and J = J x = J y )‏ The fermion gap: vortex-free full-vortex Temperature - vortex density: Tunable parameter that induces phase transition

2) Zero modes The fermion gap Δ 2v above a 2-vortex configuration Insensitive to s in the abelian phase Vanishes with s in the non-abelian phase Twofold degenerate ground state at large s Degeneracy lifted at small s Δ 2v,2 insensitive to s One zero mode per two vortices

2) Zero modes 4-vortex 6-vortex Degeneracy at large s: 4-vortex: fourfold degeneracy 6-vortex: eightfold degeneracy p-wave superconductors: 2 n -fold degeneracy in the presence of 2n well separated SU(2) 2 vortices Lifting of degeneracy at short ranges due to vortex interactions Degeneracy at small s: 4-vortex: 1 st excited state twofold degenerate 6-vortex: 1 st and 2 nd excited states threefold degenerate

2) Zero modes 4-vortex SU(2) 2 fusion rules: Identify: ψ ~ fermion mode, σ ~ vortex Interpret: Unoccupied zero mode ~ σxσ → 1 Occupied zero mode ~ σxσ → ψ E.g. } Four fusion channels:

3) Vortex gap Asymptotically: Abelian phase (J=1/3)‏ Very weak attractive interaction Strong attractive interaction Vanishes exponentially Interaction neglible when s>2 Non-abelian phase (J=1)‏ For a particular s:

The low energy spectrum For large vortex separations (s>2)‏ For small vortex separations (s<2)‏ Consist of only vortices Applies for arbitrary number of well separated vortices Agrees with the SU(2) 2 prediction In the absence of vortices spectrum purely fermionic Vacuum fusion channels tend towards ground state Fermion fusion channels tend towards higher energies (J=1 and K=1/5)‏

Summary Experimental perspectives Future work We demonstrate: Interactions between vortices Zero modes in the presence of SU(2) 2 vortices Exact agreement with numerics Stability of vortex sectors (T << ΔE 2v ) and fermionic spectrum (T << Δ 0 )‏ Energy level splitting at close ranges (read-out of information, non-topological gates for universality)‏ Temperature as a tunable parameter to induce phase transition between non-abelian and abelian phases Berry phases (holonomy) and non-abelian statistics arXiv: