An anyonic superselection sector is defined to be an equivalence class of local excitations under local operations. A non-trivial anyon (i.e. a non-trivial superselection sector) is a "local excitation which cannot be created locally".

Here is a precise statement of this notion which works for systems with zero correlation length, where anyons can be exactly localized into a finite region. The main point is that to make the notion of superselection sector precise, you have to consider an infinite system. In a *finite* system with periodic boundary conditions, the overall state always has trivial anyonic charge.

Let $\mathcal{L}$ be the algebra of operators acting on finite sets of spins within an infinite spin lattice $\mathbb{Z}^d$. A **state** is defined to be a function mapping operators to their expectation values, i.e. a linear map $\omega : \mathcal{L} \to \mathbb{C}$, such that $\omega(A) \in \mathbb{R}$ if $A = A^{\dagger}$, $\omega(A) \geq 0$ if $A$ is positive-semidefinite, and $\omega(\mathbb{I}) = 1$. (If the system were finite, we could define $\omega(A) = \mathrm{Tr}(A\rho)$, where $\rho$ is the density matrix representing the state of the system, and the three conditions stated would correspond to $\rho = \rho^{\dagger}$, $\rho \geq 0$ and $\mathrm{Tr} \rho = 1$ respectively. However, it is not possible to define a sensible notion of density matrix in an infinite system.)

A **pure state** is a state $\omega$ which cannot be expressed as a convex linear combination of other states: $$\omega = p\omega_1 + (1-p) \omega_2, p \in [0,1] \implies \omega_1 = \omega_2 = \omega.$$ Again, in a finite system this would be equivalent to saying that the density matrix is a pure state $\rho = |\Psi\rangle \langle \Psi|$.

Let us fix one particular pure state $\omega_{GS}$ which represents the ground state of our topologically ordered system. A **local excitation** is any pure state $\omega$ which "looks the same as the ground state far away", that is there exists some finite region $R \subseteq \mathbb{Z}^d$ such that $\omega(A) = \omega_{\mathrm{GS}}(A)$ for any operator $A$ that acts trivially inside of $R$. EDIT: We also have to require that the support of $A$ does not enclose $R$ -- otherwise we could detect the presence of an anyonic charge through a Wilson loop.

We can define an equivalence relation on local excitations: $\omega \sim \omega^{\prime} $ iff $\omega^{\prime}$ can be created from $\omega$ locally, by acting with a unitary $U$ on a finite region, so that $\omega^{\prime}(A) = \omega(U^{\dagger} A U)$ for all $A$. The superselection sectors are precisely the equivalence classes of local excitaions under this equivalence relation.

This post imported from StackExchange Physics at 2015-05-31 13:06 (UTC), posted by SE-user Dominic Else