A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations

We introduce exactly solvable models of interacting (Majorana) fermions in $d\ge 3$ spatial dimensions that realize a new kind of fermion topological quantum order, building on a model presented by S. Vijay, T. H. Hsieh, and L. Fu [Phys. Rev. X 5, 041038 (2015)]. These models have extensive topological ground-state degeneracy and a hierarchy of pointlike, topological excitations that are only free to move within submanifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a “geometric” framework for the emergence of topological order.

Figure 1

Majorana cubic model. The Majorana cubic model is defined on a cubic lattice, as in (a), with a single Majorana fermion per lattice site (colored red). The operator ${\mathcal{O}}_n$ is the product of the 8 Majorana fermions at the vertices of a cube. The Hamiltonian is a sum of these local operators over every other cube (colored blue in (a)) in a checkerboard pattern. As any pair of operators either share exactly one edge or none, all operators mutually commute. We choose to label the cubic operators $A,B,C$, and $D$ as shown in (b). Acting with a single Majorana operator ${\gamma }_j$ at the indicated site creates these four excitations.

Figure 2

Dimension-1 particle. Excitations (colored) may be created by acting with Wilson line operators. In (a), a straight Wilson line creates pairs of dimension-1 particles at the end points. The dimension-1 particle may hop freely in the direction of the Wilson line, by acting with Majorana bilinear terms. Remarkably, the dimension-1 particle cannot hop in any other direction without creating additional excitations. Introducing a “corner” in the Wilson line, as in (b), creates an additional topological excitation localized at the corner.

Figure 3

Dimension-2 anyon. Acting with two adjacent Wilson line operators ${\widehat{W}}_1$ and ${\widehat{W}}_2$ creates pairs of excitations at the end points of the same type ($AA,BB,CC$, or $DD$). These two-fracton excitations are free to move in a two-dimensional plane orthogonal to the shortest line segment connecting the pair of Wilson lines. Furthermore, in (b) we may detect a fracton (colored blue) by braiding a dimension-2 anyon around a closed loop enclosing the fracton. As the braiding operator, a pair of closed Wilson line operators ${\widehat{W}}_1{\widehat{W}}_2$ is equal to the product of the enclosed cube operators as shown above. Therefore, the braiding produces an overall minus sign if an odd number of fractons are enclosed.

Figure 4

Membrane operator and fracton excitations. Acting with a product of Majorana operators on a surface $\mathrm{\Sigma }$ creates localized excitations at the corners of the boundary $\partial \mathrm{\Sigma }$ as shown above.

Figure 5

The Majorana plaquette model, as studied in [12]. Consider a honeycomb lattice with a single Majorana fermion on each lattice site. We define an operator ${\mathcal{O}}_p$ as the product of the six Majorana fermions on the vertices of a hexagonal plaquette $p$, as shown in (a). The colored plaquettes in (b) correspond to the three distinct bosonic excitations with mutual semion statistics ($A, B$, or $C$) that may each be created in pairs by acting with Wilson line operators.