Computing data for Levin-Wen with defects

Abstract

We demonstrate how to do many computations for doubled topological phases with defects. These defects may be 1-dimensional domain walls or 0-dimensional point defects.Using Vec⁡(S3) as a guiding example, we demonstrate how domain wall fusion and associators can be computed using generalized tube algebra techniques. These domain walls can be both between distinct or identical phases. Additionally, we show how to compute all possible point defects, and the fusion and associator data of these. Worked examples, tabulated data and Mathematica code are provided.

Matrix product operator symmetries and intertwiners in string-nets with domain walls

“…The PEPS representations for boundaries and domain walls in string-net models introduced in this work can then be used to investigate mechanisms of anyon condensation and to characterise the properties of excitations living on these domain walls. While this has already been understood in the abstract diagrammatic language of category theory [9,13], tensor networks allow for the numerical simulation of such systems. This is especially relevant in the context of error-correcting codes based on string-net models, where the explicit computation of properties such as error thresholds has proven to be difficult using traditional methods [39][40][41].…”

“…The assignments made for the tensors in Equations (13) and (24) have been chosen in such a way that the various consistency conditions on those tensors all amount to some pentagon equation for a pair of fusion categories C and D together with a (C, D)-bimodule category M. In this section we explain how the specific form of these tensors can be derived from a Turaev-Viro state-sum construction of a 3d TFT. More precisely, the PEPS tensor network can be shown to be an instance of such a 3d Turaev-Viro TFT on a particular three-manifold with a choice of skeleton.…”

“…Using the more general PEPS representations for string-nets mentioned above, we define explicit tensor network representations for these boundaries and domain walls. While these features have been fully understood abstractly in the setting of category theory [9][10][11][12][13], tensor networks allow to devise actual tensors with all required properties and put those to work on the computer. This situation is similar to the difference between 6j symbols and 3j symbols in group theory: while 6j symbols are sufficient to think about the structure of representations, we need the Clebsch-Gordan coefficients to build effective models realizing those symmetries.…”

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

“…In this context, (bulk) anyonic excitations, defined as a region whose energy is higher than that of the ground state, are described in terms of the so-called twisted quantum double of the group, whose irreducible representations provide the simple objects of the Drinfel'd centre of the category of G-graded vector spaces [39,40]. Gapped boundaries are found to be labelled by a simple set of data, namely a subgroup of the input group and a 2-cochain that is compatible with the input 3-cocycle [12,14,[41][42][43], and their excitations have been considered for instance in [16,18,29,44,45].…”

“…The tube algebra approach can be adapted in order to study excitations on defects and gapped boundaries, and has been employed in some specific cases in [10,44,45,53]. In this context, the tube possesses two kinds of boundary: a physical gapped boundary that corresponds to the one of the spatial manifold, and a boundary obtained by removing a local neighbourhood of an excitation incident on the boundary of the spatial manifold.…”