Matrix product operator symmetries and intertwiners in string-nets with domain walls

TL;DR

The paper develops a comprehensive framework connecting matrix product operator (MPO) symmetries in PEPS representations of string-net models to bimodule-category data, showing that MPO consistency conditions are captured by six coupled pentagon equations and enabling classification of equivalent PEPS representations via MPO intertwiners.It provides explicit tensor-network realizations of boundaries and domain walls using bimodule associators, and demonstrates these constructions through concrete examples such as the toric code and the $S_3$ quantum double, including explicit MPO intertwiners between different PEPS representations.The work further situates these tensor-network results within Turaev–Viro topological field theory, showing that the PEPS can be viewed as a TV state-sum on a three-manifold with physical boundaries, thereby furnishing a mathematically rigorous footing and enabling TFT-based reasoning for MPO injectivity and Morita equivalence.Together, the results offer a path toward a general, explicit tensor-network realization of topological phases with boundaries and defects, with potential applications to quantum error correction, topological quantum computation, and numerical simulations, while outlining avenues for future extensions to fermionic or higher-categorical settings.

Abstract

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.

Figures (7)

• Figure 1: (a) Two PEPS representations for the ground state of the same string net model $\mathcal{D}$, determined by module categories $\mathcal{M}_1$ and $\mathcal{M}_2$. (b) For the particular case that $\mathcal{M}_1 \,{=}\, \mathcal{D}$ and $\mathcal{M}_2 \,{=}\, \mathcal{M}$, explicit tensors can be constructed in terms of the $F$-symbols of $\mathcal{M}$ as a $(\mathcal{C},\mathcal{D})$-bimodule category.
• Figure 2: (a) A domain wall between two string-net models $\mathcal{D}_1$ and $\mathcal{D}_2$, described by $\text{PEPS}_{\mathcal{M}_1,\mathcal{D}_1}$ and $\text{PEPS}_{\mathcal{M}_2,\mathcal{D}_2}$ respectively. (b) For the particular case that $\mathcal{M}_1 \,{=}\, \mathcal{D}_1$ and $\mathcal{M}_2 \,{=} \mathcal{M}$, explicit tensors can be constructed in terms of $F$-symbols of a $(\mathcal{D}_1,\mathcal{D}_2)$-bimodule category $\mathcal{M}$.
• Figure 3: (a) A region of the three-manifold $M_\Sigma$, where the surface $\Sigma$ is endowed with a honeycomb-lattice cell decomposition. The physical boundary is depicted in green, the gluing boundary is white. (b) The assignment of state-sum variables to the two types of plaquettes with their orientation, as well as the vector spaces $V_{e_0}$ and $V_{e_0}^*$ associated to half-edges of $M_\Sigma$.
• Figure 4: Application of the evaluation map on a vertex of the physical boundary (a) leads to a tetrahedral diagram (b) that upon evaluation yields the PEPS tensor.
• Figure 5: Application of the evaluation map to the intersection of the boundary Wilson line $\Omega$ with an edge on the physical boundary (a) leads to a tetrahedral diagram (b) that in special cases evaluates to the MPO symmetry or MPO intertwiner tensors. Note that the lines labeled by $\Omega$ and $\alpha$ do not cross, as is evident from (a).