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arXiv:2004.07045 · Physical review. B./Physical review. B

Generalized string-net model for unitary fusion categories without tetrahedral symmetry

Alexander Hahn, Ramona Wolf

Published Sep 24, 2020• 69 citations
arXivHTMLPDF

Abstract

The Levin-Wen model of string-net condensation explains how topological phases emerge from the microscopic degrees of freedom of a physical system. However, the original construction is not applicable to all unitary fusion categories since some additional symmetries for the $F$-symbols are imposed. In particular, the so-called tetrahedral symmetry is not fulfilled by many interesting unitary fusion categories. In this paper, we present a generalized construction of the Levin-Wen model for arbitrary multiplicity-free unitary fusion categories that works without requiring these additional symmetries. We explicitly calculate the matrix elements of the Hamiltonian and, furthermore, show that it has the same properties as the original one.

DOI
10.1103/physrevb.102.115154

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The Levin-Wen model of string-net condensation explains how topological phases emerge from the microscopic degrees of freedom of a physical system. However, the original construction is not applicable to all unitary fusion categories since some additional symmetries for the FFF-symbols are imposed. In particular, the so-called tetrahedral symmetry is not fulfilled by many interesting unitary fusion categories. In this paper, we present a generalized construction of the Levin-Wen model for arbitrary multiplicity-free unitary fusion categories that works without requiring these additional symmetries. We explicitly calculate the matrix elements of the Hamiltonian and, furthermore, show that it has the same properties as the original one.
The Levin-Wen string-net formalism provides a canonical mapping from spherical fusion categories to local Hamiltonians defining Topological Quantum Field Theories (TQFTs). While the topological invariance of the ground state is guaranteed by the pentagon identity, the realization of the model on a physical Hilbert space requires the category to be unitary. In this work, we investigate the obstructions arising when this construction is applied to non-unitary spherical categories, specifically the Yang-Lee model (the non-unitary minimal model M(2,5)\mathcal{M}(2,5)M(2,5)). We first validate our framework by explicitly constructing and verifying the Hamiltonians for rank-3 (Rep(D3)\text{Rep}(D_3)Rep(D3​)), rank-5 (TY(Z4)\text{TY}(\mathbb{Z}_4)TY(Z4​)), and Abelian (Z7\mathbb{Z}_7Z7​) unitary categories. We then apply this machinery to the non-unitary Yang-Lee model. Using a custom gradient-descent optimization algorithm on the manifold of FFF-symbols, we demonstrate that the Yang-Lee fusion rules admit no unitary solution to the pentagon equations. We explain this failure analytically by proving that negative quantum dimensions impose an indefinite metric on the string-net space, realizing a Krein space rather than a Hilbert space. Finally, we invoke the theory of PT\mathcal{PT}PT-symmetric quantum mechanics to interpret the non-Hermitian Hamiltonian, establishing that the obstruction is intrinsic to the fusion ring and cannot be removed by unitary gauge transformations.
Abstract Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the Drinfeld center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.
The Levin--Wen model provides a lattice realization of topological orders associated with a given unitary fusion category. A longstanding open problem is to characterize Levin--Wen ground-state wave functions intrinsically, without assuming a priori categorical symmetry data or a Hamiltonian. We address this by proposing six axioms on a family of wave functions defined on lattices at multiple scales. These axioms allow us to reconstruct the underlying unitary fusion category and prove that the resulting wave functions map to nonzero Levin--Wen ground-state vectors of the emergent category.
Symmetry protected (SPT) and symmetry enriched topological (SET) phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin–Wen (LW) model for spin/boson systems is an important rigorously solvable model for studying 2D topological phases. The input data for the LW model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the LW model, and show that the extended LW model is a natural playground for the theoretical study of SPT and SET phases of matter.
We introduce a class of 2d2d2d quantum spin systems based on a unitary spherical fusion category with a chosen object. These models have frustration-free, reflection positive, but generally non-commuting local interactions, generalizing the Levin-Wen string-net models. Based on the theory developed in arXiv:2510.20662, we explicitly realize the Temperley-Lieb-Jones algebra, and more generally fusion spin chains as their boundary algebras.
Generalized string-net models have been recently proposed in order to enlarge the set of possible topological quantum phases emerging from the original string-net construction. In the present work, we do not consider vertex excitations and restrict to plaquette excitations, or fluxons, that satisfy important identities. We explain how to compute the energy-level degeneracies of the generalized string-net Hamiltonian associated to an arbitrary unitary fusion category. In contrast to the degeneracy of the ground state, which is purely topological, that of excited energy levels depends not only on the Drinfeld center of the category, but also on internal multiplicities obtained from the tube algebra defined from the category. For a noncommutative category, these internal multiplicities result in extra nontopological degeneracies. Our results are valid for any trivalent graph and any orientable surface. We illustrate our findings with nontrivial examples.