SciPost Phys. 17, 088 (2024) ·
published 24 September 2024

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LevinWen models are a class of twodimensional lattice spin models with a Hamiltonian that is a sum of commuting projectors, which describe topological phases of matter related to Drinfeld centres. We generalise this construction to lattice systems internal to a topological phase described by an arbitrary modular fusion category $\mathcal{C}$. The lattice system is defined in terms of an orbifold datum $\mathbb{A}$ in $\mathcal{C}$, from which we construct a state space and a commutingprojector Hamiltonian $H_{\mathbb{A}}$ acting on it. The topological phase of the degenerate ground states of $H_{\mathbb{A}}$ is characterised by a modular fusion category $\mathcal{C}_\mathbb{A}$ defined directly in terms of $\mathbb{A}$. By choosing different $\mathbb{A}$'s for a fixed $\mathcal{C}$, one obtains precisely all phases which are Wittequivalent to $\mathcal{C}$. As special cases we recover the Kitaev and the LevinWen lattice models from instances of orbifold data in the trivial modular fusion category of vector spaces, as well as phases obtained by anyon condensation in a given phase $\mathcal{C}$.
SciPost Phys. 15, 207 (2023) ·
published 27 November 2023

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This paper is a followup to [J. High Energy Phys. 06, 025 (2020)] in which twodimensional conformal field theories in the presence of spin structures are studied. In the present paper we define four types of CFTs, distinguished by whether they need a spin structure or not in order to be welldefined, and whether their fields have parity or not. The cases of spin dependence without parity, and of parity without the need of a spin structure, have not, to our knowledge, been investigated in detail so far. We analyse these theories by extending the description of CFT correlators via threedimensional topological field theory developed in [Nucl. Phys. B 646, 353 (2002)] to include parity and spin. In each of the four cases, the defining data are a special Frobenius algebra $F$ in a suitable ribbon fusion category, such that the Nakayama automorphism of $F$ is the identity (oriented case) or squares to the identity (spin case). We use the TFT to define correlators in terms of $F$ and we show that these satisfy the relevant factorisation and singlevaluedness conditions. We allow for world sheets with boundaries and topological line defects, and we specify the categories of boundary labels and the fusion categories of line defect labels for each of the four types. The construction can be understood in terms of topological line defects as gauging a possibly noninvertible symmetry. We analyse the case of a $\mathbb{Z}_2$symmetry in some detail and provide examples of all four types of CFT, with BershadskyPolyakov models illustrating the two new types.