The string operator is a way to study the quasiparticle excitations in the string-net model http://arxiv.org/abs/cond-mat/0404617.

It is claimed in the above reference (Eq.(19), p.9) that for string operator $W(P)$ defined on an arbitrary path $P = I_1,\dots,I_N$ on the honeycomb lattice, which is a product of simple string operators $W(P) = W_{s_1}(P)\dots W_{s_m}(P)$, and $n_s$ the non-negative integers characterizing the action of $W(P)$ on the vacuum: $W(P)|0> = \sum_{s}n_s |s> $, one can show that the matrix elements of $W(P)$ between an initial spin state $i_1,\dots,i_N$ and final spin state $i_1^{\prime},\dots,i_N^{\prime}$ are always of the form \begin{equation} W_{i_1,\dots,i_N}^{i_1^{\prime},\dots,i_N^{\prime}}(e_1e_2,\dots,e_N) = \sum_{\{ s_k\}}(\prod_{k=1}^N F_{s_k^* i_{k-1}^{\prime}i_{k}^{\prime *}}^{e_k i_k^{*}i_{k-1}})\text{Tr}(\prod_{k=1}^N \Omega_k(i_k,i_k^{\prime},s_k,s_{k+1})) \ \ \ (1) \end{equation} where $\Omega_k$ is some $n_{s_k}\times n_{s_{k+1}}$ complex matrix.

The matrix element of a type$-s$ simple string operator is \begin{equation} W_{s,i_1,\dots,i_N}^{\ i_1^{\prime},\dots,i_N^{\prime}}(e_1e_2,\dots,e_N) = (\prod_{k=1}^N F_{s^* i_{k-1}^{\prime}i_{k}^{\prime *}}^{e_k i_k^{*}i_{k-1}})(\prod_{k=1}^N \omega_k (i_k,i_k^{\prime},s)) \ \ \ (2) \end{equation} where $\omega_k$ is some complex number.

I failed to show the above claim.

I tried inserting $m$ identity operators $I =\sum_{i_1^{(l)},\dots,i_N^{(l)}} |i_1^{(l)},\dots,i_N^{(l)}><i_1^{(l)},\dots,i_N^{(l)}|, 1\leq l \leq m$, between $<i_1,\dots,i_N|W_{s_1}(P)\dots W_{s_m}(P)|i_1^{\prime},\dots,i_N^{\prime}>$, and then applying Eq.$(2)$, and simplifying the resultant expression.

One could also try using the graphical representation of the string operators illustrated in Appendix D. In particular, the effect of applying $W_{s_1}(P)\dots W_{s_m}(P)$ on some string-net state is to add $m$ simple strings $s_1,\dots,s_m$ in the fattened honeycomb lattice along the path $P = I_1,\dots,I_N$. One can then apply rules Eq.$(D1)$ (for simple strings) to merge these additional strings with the original strings in the unfattened lattice, and write the state as a linear combination of the string-net state on the unfattened lattice.

However, in both approaches, it is not immediately clear to me that the matrix elements of $W = W_{s_1}\dots W_{s_m}$ is of the form Eq.$(1)$, for some complex matrices $\Omega_k$ of appropriate dimensions.


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