Kohtaro Kato

Despite recent advances in the lattice representation theory of (generalized) symmetries, many simple quantum spin chains of physical interest are not included in the rigid framework of fusion categories and weak Hopf algebras. We demonstrate that this problem can be overcome by relaxing the requirements on the underlying algebraic structure, and show that general matrix product operator symmetries are described by a pre-bialgebra. As a guiding example, we focus on the anomalous $\mathbb Z_2$ symmetry of the XX model, which manifests the mixed anomaly between its $U(1)$ momentum and winding symmetry. We show how this anomaly is embedded into the non-semisimple corepresentation category, providing a novel mechanism for realizing such anomalous symmetries on the lattice. Additionally, the representation category which describes the renormalization properties is semisimple and semi-monoidal, which provides a new class of mixed state renormalization fixed points. Finally, we show that up to a quantum channel, this anomalous $\mathbb Z_2$ symmetry is equivalent to a more conventional MPO symmetry obtained on the boundary of a double semion model. In this way, our work provides a bridge between well-understood topological defect symmetries and those that arise in more realistic models.

Abstract We introduce the entanglement bootstrap program, a powerful new approach to study topologically ordered quantum many-body systems. In this program, we posit that local reduced density matrices of the underlying system obey a set of simple constraints that are motivated by the entanglement entropy calculations in the literature. We show that these constraints lead to a highly nontrivial set of identities that can be interpreted as the basic axioms of the emergent theory describing the topological charges in such systems. The surprising nature of our work lies in the fact that these basic axioms of the emergent theory, which are typically assumed in the literature, can actually be derived from a simple entanglement property of the ground state. Furthermore, we apply this line of reasoning to systems with gapped domain walls and derive a hitherto unknown set of topological charges and identities that those topological charges need to satisfy. Thus, our work establishes a deep connection between entanglement and the exotic behaviors of topological charges in topologically ordered systems.

A special feature of the ground state of a topologically ordered phase is large-scale correlations depending on the topology of the regions. These correlations can be detected by the topological entanglement entropy, or by a measure called the irreducible correlation. In this work, we show that the two measures coincide for states obeying an area law and having zero-correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols.