A Generalization of Non-Abelian Anyons in Three Dimensions

Abstract

We introduce both an exactly solvable model and a coupled-layer construction for an exotic, three-dimensional phase of matter with immobile topological excitations that carry a protected internal degeneracy. Unitary transformations on this degenerate Hilbert space may be implemented by braiding certain point-like excitations. This provides a new way of extending non-Abelian statistics to three-dimensions.

Figures (7)

• Figure 1: The Model: We consider layers of of complex fermions on the sites of a square lattice ($f$), coupled to fermions that lie on the links ($c$) which play the role of a static $Z_{2}$ gauge field. Within each layer, the hopping and pairing interaction of the fermions are mediated by the fermions on the links, which interact at each plaquette as shown. The gauge symmetry of the Hamiltonian (\ref{['eq:H']}) is implemented by the operator $G_{j,\ell} = \Gamma_{j, \ell -1}^{(f)}\,\mathcal{O}_{j,\ell}\,\Gamma_{j,\ell+1}^{(f)}$ which couples adjacent layers, with $\mathcal{O}_{j,\ell}$ as defined in the main text. The ground-state realizes an exotic phase with immobile, point-like excitations that carry a protected internal degeneracy.
• Figure 2: Coupled 2D $G$ gauge theories: Intersecting layers of 2D gauge theories for finite the group $G$, stacked in the $xy$, $yz$ and $xz$ planes as shown in (a). "Condensing" the excitation in (b) -- a composite of four $[g]$ fluxes -- can produce a fracton topological phase, where the fracton excitation inherits certain properties from the non-Abelian $[g]$ flux as shown in (c) and described in the main text.
• Figure 3: Checkerboard Model: Majorana fermions are placed on the sites of a three-dimensional cubic lattice. At each cube, the fermion parity is defined by the product of the eight Majorana fermions at the vertices. The checkerboard model is given by summing this operator over the blue cubes on the lattice, which form a checkerboard array. Adapted from Ref. Vijay1.
• Figure 4: Mobile Fractionalized Excitations: Pairs of fractons -- appearing (a) at the ends of a line-like operator or at the end of (b) a "short" membrane -- are fractionalized excitations that with reduced mobility. The excitation in (a) can move along the line defined by the line-like operator while the one in (b) is free to move in a plane orthogonal to the short membrane. Adapted from Ref. Vijay1.
• Figure 5: Braiding Transformations: By applying membrane-like operators, we may exchange pairs of fractons in nearby layers to affect unitary transformations on the degenerate states in the Hilbert space.