Fermions and nontrivial loop-braiding in a three-dimensional toric code

We study an exactly solvable toric code type of Hamiltonian in three dimensions, defined on the diamond lattice with spin-$1/2$ degrees of freedom at each site. The Hamiltonian is a sum of mutually commuting plaquette operators $B_p$, all of which have eigenvalue +1 in the ground state. The excitations are “fluxes,” which are plaquettes with $B_p=-1$. Due to certain local kinematic constraints, fluxes form loops. The elementary flux-loop excitations are fermions, in contrast to other solvable spin-$1/2$ models in three dimensions, where the excitations are bosons. Furthermore, the flux loops braid nontrivially, giving rise to Abelian anyonlike statistics.

Figure 1

The diamond lattice. Green, red, black, and blue lines respectively denote links with labels $a$, $b$, $c$, and $d$.

Figure 2

The shaded region passing through the sites $I,J,K,L$ is a part of the surface on which the global surface operator $M_x$ is defined.

Figure 3

Black sites and green links form $\mathcal{L}$, and red sites and blue links form ${\mathcal{L}}_p$. Plaquette $\{1,2,3,4,5,6\}$ in $\mathcal{L}$ corresponds to the link $l$ in ${\mathcal{L}}_p$.

Figure 4

A particle hop between connected sites $i$ and $j$ with amplitude ${\widehat{t}}_{ij}$.

Figure 5

The four links at a site in $\mathcal{L}$, each link represents a six-loop in ${\mathcal{L}}_p$.

Figure 6

Schematic representation of a six-loop braiding through a bigger loop.