Generalized string-net models: A thorough exposition

TL;DR

This paper introduces generalized string-net models that extend the original framework to realize a broader class of 2D topologically ordered phases, allowing for anisotropic ground states and more diverse anyon excitations.

Contribution

It develops a comprehensive framework for constructing and analyzing generalized string-net models without symmetry constraints, expanding the scope of topological phases accessible.

Findings

Constructed generalized string-net models with anisotropic ground states

Derived conditions for isotropic ground state wave functions

Provided methods to create and analyze anyon excitations

Abstract

We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of 2D topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different "string-net configurations", where each string-net configuration is a trivalent graph with labeled edges, drawn in the $xy$ plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground state wave function $\Phi$ is "isotropic": i.e. in the generalized setup, two string-net configurations $X_1, X_2$ that can be continuously deformed into one another can have different ground state amplitudes, $\Phi(X_1) \neq \Phi(X_2)$. As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground state wave functions on the plane or the sphere -- a property that may be useful in some applications.