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Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model arXiv:0712.1164

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, 2006J.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, 2:341-347,2005.

Motivation

1) Phase space geometry

2) Non-abelian character: zero modes

3) Interactions 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 zi 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 Ji 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=1/2Full-vortex: J=1/√2Sparse: 1/2 ≤ J ≤ 1/√2

(Jz = 1 and J = Jx = Jy )

The fermion gap:

vortex-free full-vortex

Temperature - vortex density: Tunable parameter that induces phase transition

2) Zero modes

The fermion gap Δ2v 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

Δ2v,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:

2n-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: 1st excited state twofold degenerate

6-vortex: 1st and 2nd 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 → 1Occupied 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 << ΔE2v) and fermionic spectrum (T << Δ0)

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:0712.1164