Anyons in an exactly solved model and beyond
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:
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, (k = 1, …, n). Instead, one can use their linear combinations,which are called Majorana operators. The operators cj (j = 1, … , 2n) are Hermitian and obey the following relations: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 formwhere 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 thatThus, the Lie algebra of quadratic operators −iH(A) (acting on the 2m-dimensional Fock space) is identified with
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 , 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]: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. ThusThe braiding rules for vortices depend on the spectral Chern
The 16-fold way
Let us again consider the theory with -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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