Fracton topological order, generalized lattice gauge theory, and duality

We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, pointlike topological excitations, and subextensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical tool set for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.

Figure 1

The fundamental excitations of the X-cube model are shown in (a) and (b). Acting on the ground state of the X-cube model with a product of ${\sigma }^z$ operators along the colored red links that lie within a flat, rectangular region $\mathcal{M}$ generates four fracton cube excitations $e_a^{\left(0\right)}$ at the corners of the region. A straight Wilson line of ${\sigma }^x$ operators acting on the blue links in (b) isolates a pair of quasiparticles $(m_a^{\left(1\right)}$ or $m_b^{\left(1\right)})$ at the ends that are only free to move along the line. Attempting to move these quasiparticles in any other direction by introducing a corner in the Wilson line creates a topological excitation at the corner, as shown in (b).

Figure 2

A “domain wall” in the ground state of the plaquette Ising model $H_{\mathrm{plaq}}$ is depicted by coloring the plaquette interactions that have been flipped by the action of a spin-flip transformation along a planar region $\mathrm{\Sigma }$. The F-S duality implies that the ground state for the X-code fracton phase is given by an equal superposition of a dual representation of these domain walls.

Figure 3

The nexus charge is a fracton only if there is no operator $W$ that can create an isolated pair of excitations when acting on the ground state of $H_{\mathrm{fracton}}$, as in (a). Equivalently, a dual representation of the operator, given as a product of the interaction terms in the quantum dual as shown in (b), cannot create an isolated pair of spin flips when acting on the paramagnetic state $|{\mathrm{\Psi }}_{\mathrm{para}}\rangle \equiv |\to \cdots \to \rangle$. An example is given in (c) and (d); a straight Wilson line acting on the ground state of $H_{\mathrm{X}\text{−}\mathrm{Cube}}$ in (c) admits a dual representation as a product of four-spin plaquette interactions along a line, as shown in (d). No product of interaction terms in the plaquette Ising model can produce an isolated pair of spin flips. As a result, the nexus charge in $H_{\mathrm{X}\text{−}\mathrm{Cube}}$ must be a fracton.

Figure 4

(a) Schematic phase diagram of the spin-nexus Hamiltonian (6). The Higgs phase for the nexus field is smoothly connected to the phase reached by condensing the nexus charge. The checkerboard model coupled to Ising matter fields admits an additional self-duality under the exchange of the nexus charge and flux; as a result, the phase diagram is as shown in (b).