We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.
We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories, but is formalised and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism represent discrete path integrals in Euclidean space-time. This language is more directly related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We introduce our formalism by simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.
The tensor network representation of many-body quantum states, given by local tensors, provides a promising numerical tool for the study of strongly correlated topological phases in two dimensions. However, the representation may be vulnerable to instabilities caused by small variations in the local tensors. For example, the topological order in the tensor network representations of the toric code ground state has been shown in Chen, Zeng, Gu, Chuang, and Wen, Phys. Rev. B 82, 165119 (2010) to be unstable if the variations break certain
Z2 symmetry of the tensor. In this work, we ask whether other types of topological orders suffer from similar kinds of instability and if so, what is the underlying physical mechanism and whether we can protect the order by enforcing certain symmetries on the tensor. We answer these questions by showing that the tensor network representations of all string-net models are indeed unstable, but the matrix product operator (MPO) symmetries of the tensors identified in \ifmmode \mbox{\c{S}}\else \c{S}\fi{}ahino\ifmmode \breve{g}\else \u{g}\fi{}lu, Williamson, Bultinck, Mari\"en, Haegeman, Schuch, and Verstraete, arXiv:1409.2150 can help to protect the order. In particular, we show that a subset of variations that break the MPO symmetries lead to instability by inducing the condensation of bosonic quasiparticles, which destroys the topological order in the wave function. Therefore such variations must be forbidden for the encoded topological order to be reliably extracted from the local tensors. On the other hand, if a tensor network based variational algorithm is used to simulate the phase transition due to boson condensation, such variation directions may prove important to access the continuous transition correctly.
We use a recently proposed class of tensor-network states to study phase transitions in string-net models. These states encode the genuine features of the string-net condensate such as, e.g., a nontrivial perimeter law for Wilson loops expectation values, and a natural order parameter detecting the breakdown of the topological phase. In the presence of a string tension, a quantum phase transition occurs between the topological phase and a trivial phase. We benchmark our approach for
Z2 string nets and capture the second-order phase transition which is well known from the exact mapping onto the transverse-field Ising model. More interestingly, for Fibonacci string nets, we obtain first-order transitions in contrast with previous studies but in qualitative agreement with mean-field results.
We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string-net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows one to carry over to these string-net models the representation-theoretic classification of the excitations in quantum double models as well as define them in arbitrary lattices and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalizing the map to broader classes of string nets is considered.
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.