Generalized string-net models: A thorough exposition

TL;DR

The paper extends Levin–Wen string-net theory by introducing generalized string-nets that relax planar isotropy and tetrahedral symmetry, enabling realization of a wider array of 2D topological orders. Ground states are defined via local rules using $F^{abc}_{def}$, $ ilde{F}^{abc}_{def}$, and $Y^{ab}_c$, while a commuting-projector Hamiltonian on the honeycomb lattice implements these states and yields gapped excitations. The authors construct quasiparticle string operators, derive braiding statistics through the $S$-matrix and monodromy, and establish precise isotropy conditions on the plane and sphere; they also relate the generalized framework to the original Levin–Wen construction via a data dictionary. Through concrete abelian and non-abelian examples (including $\Z_2$, $\Z_3$, $\Z_4$, Fibonacci, and TY$_3$), the work demonstrates how generalized data can realize TRS-breaking topological orders and broadens the toolbox for exact lattice realizations of topological phases with potential implications for symmetry-protected and symmetry-enriched topological phenomena.

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 $Φ$ 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, $Φ(X_1) \neq Φ(X_2)$. 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.

Figures (7)

• Figure 1: A typical example of a string-net with string types $\{1,2\}$, with dual string types defined by $\bar{1} = 1$ and $\bar{2} = 2$, and branching rules $\{(1,2; 2), (2,1;2), (2,2;1), (2,2; 2)\}$. Bivalent vertices are marked with dots for clarity. Note that other (unmarked) corners are not bivalent vertices, but rather kinks in the piecewise differentiable strings.
• Figure 2: Two different ways to relate the amplitude of (a) to the amplitude of (c). Consistency requires the two sequences of operations give the same result.
• Figure 3: The string-net Hamiltonian (\ref{['hsn0']}). The $Q_I$ operator acts on 3 spins around each vertex (blue dots). The $B_p$ operator acts on 12 spins adjacent to the plaquette $p$ (red dots).
• Figure 4: The $S$ matrix is computed by by comparing the action of $W_\alpha(P_3) W_\beta(P_2) W_\alpha(P_1)$ and $W_\alpha(P_3) W_\alpha(P_1) W_\beta(P_2)$ where $P_1\cup P_3$ forms a closed loop. Specifically, $S_{\alpha \beta} = \frac{d_\alpha d_\beta}{ D} M_{\alpha\beta}$.
• Figure 5: Self-consistency requires the conditions (\ref{['eq2a']}).