Variational methods for characterizing matrix product operator symmetries

Abstract

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological $F$-symbols of the underlying fusion category, which unequivocally identify the topological order of the ground state. We bring the $F$-symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological $S$ and $T$ matrices encoding mutual- and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of $Z_3$, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of $S_3$ and $\mathrm{Rep}(S_3)$ string net models.

References in corpus (16)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Identifying Topological Order by Entanglement Entropy
• Spin liquid phase of the $S=\frac{1}{2}$ $J_1-J_2$ Heisenberg model on the triangular lattice
• Characterizing topological order by studying the ground states of an infinite cylinder
• Global Phase Diagram of Competing Ordered and Quantum Spin Liquid Phases on the Kagomé Lattice
• Gauging quantum states: from global to local symmetries in many-body systems
• Weak plaquette valence bond order in the $S=1/2$ honeycomb $J_1-J_2$ Heisenberg model
• Distinct Spin Liquids and their Transitions in Spin-1/2 XXZ Kagome Antiferromagnets
• Assembling Fibonacci Anyons From a ${\mathbb{Z}}_3$ Parafermion Lattice Model
• Gapless Coulomb state emerging from a self-dual topological tensor-network state
• Competing Abelian and non-Abelian topological orders in $\nu =1/3+1/3$ quantum Hall bilayers
• Entanglement order parameters and critical behavior for topological phase transitions and beyond
• Quantum phases of the frustrated XY models on the honeycomb lattice
• Constructing topological models by symmetrization: A PEPS study
• Nature of continuous phase transitions in interacting topological insulators
• Determining topological order from infinite projected entangled pair states