Tensor-product representations for string-net condensed states

TL;DR

This work demonstrates that general string-net condensed states, which realize PT-symmetric topological orders, can be efficiently captured by tensor-product states built from local tensors. By constructing explicit TPS representations for the $Z_2$ gauge model (toric code) and the double-semion model, and extending to general string-net models, the authors provide a local, tensor-network description of long-range entanglement and topological order. They introduce the tensor-entanglement renormalization group (TERG) and prove that string-net states are fixed points under TERG, implying a vanishing correlation length and a robust fixed-point structure. The framework enables efficient computation of observables and offers a path to study quantum phase transitions between distinct topological orders using local tensor parameters.

Abstract

We show that general string-net condensed states have a natural representation in terms of tensor product states (TPS) . These TPS's are built from local tensors. They can describe both states with short-range entanglement (such as the symmetry breaking states) and states with long-range entanglement (such as string-net condensed states with topological/quantum order). The tensor product representation provides a kind of 'mean-field' description for topologically ordered states and could be a powerful way to study quantum phase transitions between such states. As an attempt in this direction, we show that the constructed TPS's are fixed-points under a certain wave-function renormalization group transformation for quantum states.

Figures (16)

• Figure 1: $Z_2$ gauge model on a square lattice. The dots represent the physical states which are labeled by $m$. The above graph can also be viewed as a tensor-network, where each dot represents a rank-3 tensor $g$ and each vertex represents a rank-4 tensor $T$. The two legs of a dot represent the $\alpha$ and $\beta$ indices in the rank-3 tensor $g^m_{\alpha\beta}$. The four legs of a vertex represent the four internal indices in the rank-4 tensor $T_{\alpha\beta\gamma\lambda}$. The indices on the connected links are summed over which define the tensor trace tTr.
• Figure 2: The double-semion model on the honeycomb lattice. The ground state wavefunction (\ref{['semWF']}) has a TPS representation given by the above tensor-network. Note that $T$ and $g$ has a double line structure. Note that the vertices form a honeycomb lattice which can divided into A-sublattice and B-sublattice.
• Figure 3: Using the fusion rules, we can represent the coherent states $\left|t,s,u,\cdots\right\rangle$ in terms of the orthogonal string-net states.
• Figure 4: The graphic representation for the tensor $G^{\al\bt\ga}_{tsu}$.
• Figure 5: A tensor-complex formed by vertices, links, and faces. The dashed curves are boundaries of the faces. The links that connect the dots carry index $\al,\bt,...$ and the faces carry index $u,s,...$. Each trivalent vertex represents a $T$-tensor. The vertices on A-sublattice (red dots) represents $T_{\al\bt\ga;t,s,u}$. The vertices on B-sublattice (blue dots) represents $T'_{\al\bt\ga;t,s,u}= T_{\al^*\bt^*\ga^*;t,s,u}$. The dots on the links represent the $g^m$-tensor $g^m_{\al,\bt}$. In the weighted tensor trace, the $\al,\bt,...$ indices on the links that connect the dots are summed over, and the $u,s,...$ indices on the closed faces are summed over with a weighting factor $a_ua_s...$.

Theorems & Definitions (2)

• proof
• proof