Anyons and matrix product operator algebras

TL;DR

This work develops MPO-injective PEPS by encoding virtual topological symmetry in PMPOs and extracting topological data from the resulting C*-algebra of central idempotents. It provides a constructive framework to build anyon wavefunctions, determine fusion and braiding, and compute the S matrix and topological spins directly from virtual data, without system-size scaling. The approach is illustrated through discrete gauge theories and string-net models, reproducing known anyon structures (e.g., toric code, Fibonacci, Ising) and connecting to fusion category concepts such as the pentagon equation and Ocneanu rigidity. The results offer a scalable, entanglement-based route to nonchiral topological order within tensor networks, with potential extensions to chiral phases, fermionic systems, and symmetry-enriched topological orders.

Abstract

Quantum tensor network states and more particularly projected entangled-pair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of matrix product operators. In this paper, we present a systematic study of those matrix product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix product operators we construct a C*-algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S matrix, fusion and braiding relations can readily be extracted from the idempotents. As the matrix product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.

Figures (14)

• Figure 1: (a) MPO tensor $B^{ij}$. (b) Injective MPO tensor $B^{ij}_a$. (c) Left hand side of equation \ref{['blocks']}.
• Figure 2: Property of MPO and fusion tensors that follows from associativity of the multiplication of $O^L_a$, $O^L_b$ and $O^L_c$.
• Figure 3: Two paths giving rise to the pentagon equation \ref{['pentagoneq']}.
• Figure 4: (a) A MPO-injective PEPS on a 2 by 2 square lattice with open boundaries. We assigned an orientation to every edge as indicated by the black arrows and an orientation to the internal MPO index represented by the red arrow. (b) A tensor $A$ that can be used to complete the PEPS on the square lattice.
• Figure 5: Acting with pseudo-inverse $B^+$ on the MPO tensor $B$ gives a projector on block-diagonal matrices with $\mathcal{N}$ blocks of dimension $\chi_a$.