Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases

TL;DR

This work constructs exactly solved commuting projector Hamiltonians for all known 2+1d fermionic SPTs with symmetry $G_f = G \times \mathbb{Z}_2^f$, introducing a discrete spin-structure via a Kasteleyn orientation to place models on arbitrary genus surfaces. The $G=\mathbb{Z}_2$ instance realizes all eight interacting SPT phases, including odd $\\nu$ beyond group supercohomology, using a decorated-domain-wall framework where domain walls bind Majorana chains and are encoded as dimers on a Majorana graph. A crucial ingredient is the Kasteleyn orientation, which ensures fermion parity conservation during domain-wall fluctuations and links to a spin-structure on the spatial manifold. The results generalize to arbitrary finite $G$, showing that all 2+1d fermionic SPTs admit commuting projector representations, with the full classification captured by a combination of a root phase and stacking with group supercohomology data, providing a robust framework for studying interacting fermionic SPTs on complex geometries.

Abstract

We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group $G_f = G \times \mathbb{Z}_2^f$, where $G$ is finite and $\mathbb{Z}_2^f$ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1d. A natural ingredient in our construction is a discrete form of the spin structure of the 2d spatial surface $M$ on which our model is defined, namely a `Kasteleyn' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all $8$ members of the $\mathbb{Z}_8$ classification of 2d fermionic SPTs with $G = \mathbb{Z}_2$.

Figures (12)

• Figure 1: Graphical representation of the degrees of freedom in our model. There is one spinless fermion per link, represented by two Majorana operators, drawn as two red dots. There is also an Ising spin $\frac{1}{2}$ degree of freedom on each plaquette, represented by a blue arrow. We will discuss various spin configurations in the $\sigma^z$ basis. The yellow region represents a spin up domain, the purple region a spin down domain; the two are separated by a domain wall.
• Figure 2: Trivalent lattice on the surface of a torus. The red dots represent Majorana operators, as discussed in the text. Although our subsequent discussion is illustrated only on the hexagonal lattice, it applies to general trivalent lattices on arbitrary genus $g$ 2d surfaces $M$.
• Figure 3: Alternative representation of the degrees of freedom in terms of a graph $\Lambda$. The vertices are the Majorana modes, still represented by the same red dots as before. Each site of the original lattice $L$ has now been replaced by a small triangular face. Although the dynamical Ising spin $\frac{1}{2}$ degrees of freedom are located only on the non-triangular faces, which we refer to as plaquettes, we find it useful to extend each spin configuration to a spin configuration over the triangles as well. This extension is determined uniquely by majority rule: each triangle spin points in the same direction as the majority of its neighbors. The edges of $\Lambda$ carry a Kasteleyn orientation that preserves the translational symmetry and some of the translational symmetry of $\Lambda$.
• Figure 4: Constructing a non-vanishing vector field (purple) with only even singularities, given a Kasteleyn orientation. The vector field is defined to point towards the red vertices in their vicinity, and is then extended over the edges. This extension is dictated by the Kasteleyn orientation of each edge (black arrow): if the edge is oriented clockwise around a face - a triangle in this case - the purple vector in the middle of the edge points into the face, and otherwise it points out. Note that this rule assumes an orientation on $M$. Finally the vector field is extended into the interior of the face, resulting in at most an even singularity due to the Kasteleyn property, namely an odd number of clockwise pointing arrows around the face.
• Figure 5: Dimer covering of $\Lambda$ associated to a particular configuration of spins. Away from the domain walls, dimers form across the type I edges, i.e. ones that connect different triangles, whereas along domain walls the dimers form on intra-triangular type II edges. Note that the end of a domain wall carries an unpaired Majorana mode, illustrated here as an unpaired red dot.