Detecting topological order in a ground state wave function

TL;DR

A way to detect a kind of topological order using only the ground state wave function which directly measures the total quantum dimension D= Sum(id2i).

Abstract

A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, d_i, F^{ijk}_{lmn}, δ_{ijk}). We describe a way to detect this kind of topological order using only the ground state wave function. The method involves computing a quantity called the ``topological entropy'' which directly measures the quantum dimension D = \sum_i d^2_i.

Figures (6)

• Figure 1: One can detect topological order in a state $\Psi$ by computing the von Neumann entropies $S_1, S_2, S_3, S_4$ of the above four regions, $A_1, A_2, A_3, A_4$, in the limit of $R,r \rightarrow \infty$. Here the four regions are drawn in the case of the honeycomb lattice. The geometry ensures that the number of sites $n_1,n_2,n_3,n_4$ along the boundaries of the $4$ regions obey the relation $n_1 - n_2 = n_3 - n_4$.
• Figure 2: The state $\Psi$ contains nonlocal correlations originating from the fact that closed strings always cross a closed curve $C$ an even number of times. These correlations can be measured by a string operator $W(C)$ (thin blue curve). For more general states, a fattened string operator $W_{\text{fat}}(C)$ (light blue region) is necessary.
• Figure 3: A simply connected region $R$ in the honeycomb lattice. We split the sites on the boundary links into two sites labeled $\v{i_m}$ and $\v{j_m}$, where $m=1,...,n$.
• Figure 4: A typical string-net state on the honeycomb lattice. The empty links correspond to spins in the $i=0$ state. The orientation conventions on the links are denoted by arrows.
• Figure 5: The basic string-net configurations (a) $X_{\{q,s\}}$ for inside $R$ and (b) $Y_{\{r,t\}}$ for outside $R$.