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arXiv:2309.05755 · SciPost Physics

Internal Levin-Wen models

Vincentas Mulevičius, Ingo Runkel B, Thomas Voß

Published Sep 24, 2024• 3 citations
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Abstract

Levin–Wen models are a class of two-dimensional 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} 𝒞 . The lattice system is defined in terms of an orbifold datum \mathbb{A} 𝔸 in \mathcal{C} 𝒞 , from which we construct a state space and a commuting-projector Hamiltonian H_{\mathbb{A}} H 𝔸 acting on it. The topological phase of the degenerate ground states of H_{\mathbb{A}} H 𝔸 is characterised by a modular fusion category \mathcal{C}_\mathbb{A} 𝒞 𝔸 defined directly in terms of \mathbb{A} 𝔸 . By choosing different \mathbb{A} 𝔸 ’s for a fixed \mathcal{C} 𝒞 , one obtains precisely all phases which are Witt-equivalent to \mathcal{C} 𝒞 . As special cases we recover the Kitaev and the Levin–Wen 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} 𝒞 .

DOI
10.21468/scipostphys.17.3.088

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Levin–Wen models are a class of two-dimensional 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} 𝒞 . The lattice system is defined in terms of an orbifold datum \mathbb{A} 𝔸 in \mathcal{C} 𝒞 , from which we construct a state space and a commuting-projector Hamiltonian H_{\mathbb{A}} H 𝔸 acting on it. The topological phase of the degenerate ground states of H_{\mathbb{A}} H 𝔸 is characterised by a modular fusion category \mathcal{C}_\mathbb{A} 𝒞 𝔸 defined directly in terms of \mathbb{A} 𝔸 . By choosing different \mathbb{A} 𝔸 ’s for a fixed \mathcal{C} 𝒞 , one obtains precisely all phases which are Witt-equivalent to \mathcal{C} 𝒞 . As special cases we recover the Kitaev and the Levin–Wen 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} 𝒞 .
We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category C\mathcal{C}C, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center Z(C)Z(\mathcal{C})Z(C). We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.
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.
Abstract The Levin-Wen model is a general construction that uses an input planar diagram algebra to construct a Hamiltonian with a topologically ordered ground state (the Drinfeld double of the input category). The ground state (code) space consists of a sum over all diagrams weighted by the evaluation of the diagram. We work through the example of the doubled Fibonacci model. Here the input category is the Fibonacci fusion rules and F-matrices. We work with continuum diagrams and construct the full tube algebra of the model. We identify the quasiparticle types to include both left- and right-handed Fibonacci anyons. In the case that the input category admits a modular braiding, much of the analyis simplifies and the output is two copies of the input category of opposite chiralities (both left- and right-handed). We finally turn to the construction of the explicit Hamiltonian on the lattice, and construction of quasiparticle string (or ribbon) operators. We return to the reconsider the toric code and doubled semion models as simple examples.
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.
This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category C\mathcal{C}C on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided C∗\rm C^*C∗-tensor category of superselection sectors SSS\mathsf{SSS}SSS. By constructing explicit isomorphisms between the fusion spaces of SSS\mathsf{SSS}SSS and those of the Drinfeld center Z(C)Z(\mathcal{C})Z(C), we show that these two categories have isomorphic FFF- and RRR-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center,  SSSf≃Z(C).\,\mathsf{SSS}_f \simeq Z(\mathcal{C}).SSSf​≃Z(C). This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.
Topological orders are a class of phases of matter that beyond the Landau symmetry breaking paradigm. The two (spatial) dimensional (2d) topological orders have been thoroughly studied. It is known that they can be fully classified by a unitary modular tensor category (UMTC) and a chiral central charge c. And a class of 2d topological orders whose boundary are gappable can be systematically constructed by Levin-Wen model whose ground states are string-net condensed states. Previously, the three spatial dimensional topological orders have been classified based on their canonical boundary described by some special unitary fusion 2-category, 2VecGω2\mathcal{V}ec_G^ω2VecGω​ or an EF 2-category. However, a lattice realization of a three spatial dimensional topological orders with both canonical boundary and arbitrary boundaries are still lacking. In this paper, we construct a 3d membrane-net model based on spherical fusion 2-category, which can be used to systematically study all general 3d topological order with gapped boundary. The partition function and lattice Hamiltonian of the membrane-net model is constructed based on state sum of spherical fusion 2-category. We also construct the 3d tube algebra of the membrane-net model to study excitations in the model. We conjecture all intrinsic excitations in membrane-net model have a one-to-one correspondence with the irreducible central idempotents (ICI) of the 3d tube algebra. We also provide a universal framework to study mutual statistics of all excitations in 3d topological order through 3d tube algebra. Our approach can be straightforwardly generalized to arbitrary dimension.