Elsevier

Annals of Physics

Volume 321, Issue 1, January 2006, Pages 2-111
Annals of Physics

Anyons in an exactly solved model and beyond

https://doi.org/10.1016/j.aop.2005.10.005Get rights and content

Abstract

A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

Section snippets

Comments to the contents: what is this paper about?

Certainly, the main result of the paper is an exact solution of a particular two-dimensional quantum model. However, I was sitting on that result for too long, trying to perfect it, derive some properties of the model, and put them into a more general framework. Thus many ramifications have come along. Some of them stem from the desire to avoid the use of conformal field theory, which is more relevant to edge excitations rather than the bulk physics. This program has been partially successful,

The model

We study a spin-1/2 system in which spins are located at the vertices of a honeycomb lattice, see Fig. 3A. This lattice consists of two equivalent simple sublattices, referred to as “even” and “odd” (they are shown by empty and full circles in the figure). A unit cell of the lattice contains one vertex of each kind. Links are divided into three types, depending on their direction (see Fig. 3B); we call them “x-links,” “y-links,” and “z-links.” The Hamiltonian is as follows:H=-Jxx-linksσjxσkx-Jy

A general spin-fermion transformation

Let us remind the reader some general formalism pertaining to Fermi systems. A system with n fermionic modes is usually described by the annihilation and creation operators ak, ak (k = 1, …, n). Instead, one can use their linear combinations,c2k-1=ak+ak,c2k=ak-aki,which are called Majorana operators. The operators cj (j = 1,  , 2n) are Hermitian and obey the following relations:cj2=1,cjcl=-clcjifjl.Note that all operators cj can be treated on equal basis.

We now describe a representation of a spin

Quadratic Hamiltonians

In the previous section, we transformed the spin model (4) to a quadratic fermionic Hamiltonian of the general formH(A)=i4j,kAjkcjck,where A is a real skew-symmetric matrix of size n = 2m. Let us briefly state some general properties of such Hamiltonians and fix the terminology.

First, we comment on the normalization factor 1/4 in Eq. (18). It is chosen so that[-iH(A),-iH(B)]=-iH[A,B].Thus, the Lie algebra of quadratic operators −iH(A) (acting on the 2m-dimensional Fock space) is identified with s

The spectrum of fermions and the phase diagram

We now study the system of Majorana fermions on the honeycomb lattice. It is described by the quadratic Hamiltonian Hu = H (A), where Ajk=2Jαjkujk, ujk = ±1. Although the Hamiltonian is parametrized by ujk, the corresponding gauge-invariant state (or the state of the spin system) actually depends on the variables wp, see (15).

First, we remark that the global ground state energy does not depend on the signs of the exchange constants Jx, Jy, Jz since changing the signs can be compensated by changing the

Properties of the gapped phases

In a gapped phase, spin correlations decay exponentially with distance, therefore spatially separated quasiparticles cannot interact directly. That is, a small displacement or another local action on one particle does not influence the other. However, the particles can interact topologically if they move around each other. This phenomenon is described by braiding rules. (We refer to braids that are formed by particle worldlines in the three-dimensional space-time.) In our case the particles are

The conic singularity and the time-reversal symmetry

Phase B (cf. Fig. 5) carries gapped vortices and gapless fermions. Note that vortices in this phase do not have well-defined statistics, i.e., the effect of transporting one vortex around the other depends on details of the process. Indeed, a pair of vortices separated by distance L is strongly coupled to fermionic modes near the singularity of the spectrum, |q  q*|  L−1. This coupling results in effective interaction between the vortices that is proportional to ε (q)  L−1 and oscillates with

Edge modes and thermal transport

Remarkably, any system with nonzero Chern number possesses gapless edge modes. Such modes were first discovered in the integer quantum Hall effect [51]; they are chiral, i.e., propagate only in one direction (see Fig. 8). In fact, left-moving and right-moving modes may coexist, but the following relation holds [52]:νedge=def#ofleft-movers-#ofright-movers=ν.In the absence of special symmetry, counterpropagating modes usually cancel each other, so the surviving modes have the same chirality. A

Non-Abelian anyons

We continue the study of phase B in the magnetic field. Now that all bulk excitations are gapped, their braiding rules must be well-defined. Of course, this is only true if the particles are separated by distances that are much larger than the correlation length associated with the spectrum (51). The correlation length may be defined as follows: ξ = |Im q|−1, where q is a complex solution to the equation ε (q) = 0. Thusξ=3JΔJ3hxhyhz.The braiding rules for vortices depend on the spectral Chern

The 16-fold way

Let us again consider the theory with Z2-vortices and free fermions whose spectrum is gapped and characterized by the Chern number ν. The properties of anyons in this model depend on ν mod 16. In the previous section we studied the case of odd ν; now we assume that ν is even.

For even ν, the vortices do not carry unpaired Majorana modes (see Appendix C), therefore a vortex cannot absorb a fermion while remaining in the same superselection sector. Thus, there are actually two types of vortices,

Odds and ends

What follows are some open questions, as well as thoughts of how the present results can be extended.

  • 1.

    Duan et al. [42] proposed an optical lattice implementation of the Hamiltonian (4). It would be interesting to find a solid state realization as well. For example, the anisotropic exchange could be simulated by interaction of both lattice spins with a spin-1 atom coupled to a crystal field.

  • 2.

    The weak translational symmetry breaking in the Abelian phase has some interesting consequences. A

Acknowledgments

During the years this work was in progress I received genuine interest and words of encouragement from John Preskill, Michael Freedman, Mikhail Feigelman, Grigory Volovik, and many other people. I thank Andreas Ludwig for having convinced me in the importance of the chiral central charge in the study of topological order. A conversation with Dmitri Ivanov was essential for understanding the difference between non-Abelian anyons and Majorana vortices in two-dimensional p-wave superconductors. I

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