Three-dimensional topological lattice models with surface anyons

TL;DR

This work analyzes three-dimensional topological lattice models built from Walker-Wang constructions, focusing on a simple yet representative 3D semion model that exhibits bulk confinement but rich surface topological order. By solving these models exactly on closed and boundary-bearing manifolds, the authors connect lattice properties to continuum field theories via $bF$ and $bF+bb$ descriptions, illustrating how surface anyons encode the bulk topology. They show a unique bulk ground state on orientable closed manifolds for confined WW models, while boundaries host deconfined chiral anyons and a boundary-dependent ground-state degeneracy. The study further links these 3D lattice theories to surface fractional quantum Hall states (bosonic $ u=1/2$ and fermionic $ u=1/3$), and extends the framework to general MTC-based WW models, highlighting the central role of boundary physics in classifying 3D topological phases of matter.

Abstract

We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang [arXiv:1104.2632v2]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $ν=1/2$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of $bF$ theories. Our special example of the 3D semion model captures much of the behavior of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum hall effect. We find that in these models a fermion is always deconfined in the three dimensional bulk.

Figures (33)

• Figure 1: (Color online) This figure shows a region of the lattice for the 2D models, where we have represented the spin degree of freedom on each edge with a black dot. Bold black edges represent $\sigma^{z}=-1$ spin configurations, while dashed edges represent $\sigma^{z}=+1$. The figure also indicates the edges involved in the definition of $B_v$; the three green edges in $s(v)$ are acted on with $\sigma^z$ matrices as shown. The edges involved in $B_p$ differ between the toric code and the DSem model. In the toric code $B_p$ acts on the six red edges in $\partial p$ with $\sigma^x$ matrices as shown, and does not act on the six blue edges in $s(p)$ i.e. $f(\sigma^z)=1$. In the DSem model, in addition to acting on the red edges with $\sigma^x$, the plaquette operator also acts on the six blue edges in $s(p)$ with $f\left( \sigma^z\right) = i^{(1 - \sigma^z)/2}$.
• Figure 2: This figure shows the graphical rules for the toric code and DSem models. Row (a) represents the fact that the ground state only involves vertices with $B_v=1$. The diagrams in (b)-(d) serve two purposes. Firstly, they tell us the relative amplitudes of loop gas configurations in the ground state e.g. row (c) tells us that configurations related by removing a closed loop occur with the same amplitude in the toric code ground state, but with a relative minus sign in the DSem ground state. Second, these diagrams provide a neat graphical mnemonic for the definitions of string operators.
• Figure 3: This diagram shows a specific small region of the lattice for four distinct spin configurations. The configurations are taken to be the same everywhere outside this region. For the toric code, all of these configurations occur in the ground state wave function with the same amplitude. This is because (b), (c), and (d) can all be made to look like (a) using loop deformation, loop collapsing and fusion respectively. In the DSem model, (a) and (b) occur with the same phase because they are related by deforming a loop. (a) and (c) occur with a relative minus sign because they differ by the presence of a single closed loop. (a) and (d) also occur with a relative minus sign because they are related by a single fusion (Fig. \ref{['tcdsem']}(d)).
• Figure 4: This figure shows the 4 independent ground state sectors in the toric code with periodic boundary conditions in both directions (i.e., on a torus). The 4 "canonical" kets can be taken to be the four spin configurations on the far left side labeled (a)-(d). Each for the 4 ground states is a superposition of all spin configurations which can be made to look like the canonical ket by using the equivalence rules shown in Fig. \ref{['tcdsem']}b-d. For the toric code the relative coefficient is always $+$. The ground state structure for the DSem model is similar except the kets marked with a $\pm$ appear with negative coefficients, a fact that can easily be checked using the rules in Fig. \ref{['tcdsem']}.
• Figure 5: (Color online) This figure shows the different types of string operators in the simple 2D lattice models. For the toric code, the vertex type string operator simply flips the spin on every edge of a path with $\sigma^x$. The plaquette type string operator is represented as a line living on the dual lattice, and every edge that crosses the line is acted on with $\sigma^z$. For the DSem model, the plaquette type string operator is precisely the same. The vertex type string operator, however, includes additional phases which depend on the spins touching the path. A phase $(\pm i)^{(1-\sigma^z)/2}$ is associated with the dashed (green) edges labelled $R$ which lie just to the right of the path; the choice of $\pm$ determines the chirality of the string. Vertices labelled $L$ are attached to an edge lying on the left of the path, and these vertices are associated with a phase $(-1)^{\frac{1}{4}(1-\sigma^z_i)(1+\sigma^z_j)}$ where $i/j$ are the edges on the path just before/after the L-vertex.