Fracton Topological Order, Generalized Lattice Gauge Theory and Duality

TL;DR

Problem: Fracton topological order lacks a unifying framework; paper introduces a generalized lattice gauge theory and a duality (F-S) to connect fracton phases to subsystem-symmetry breaking. Approach: construct nexus fields mediating multi-spin interactions; derive gauge constraints, provide explicit X-Cube example, and develop an algebraic stabilizer-map framework with Buchsbaum-Eisenbud criterion to guarantee fracton order. Contributions: shows sub-extensive ground-state degeneracy, immobile fracton excitations, and phase diagram with Higgs and confinement regimes; provides codimension and fracton conditions and demonstrates how to generate a family of fracton models from classical spin systems. Significance: offers systematic, algebraic method to identify, classify, and potentially realize fracton topological order in quantum materials and lattice models.

Abstract

We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, point-like topological excitations, and sub-extensive 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 toolset for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.

Figures (7)

• 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 ($e^{(0)}_{a}$) 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 ($m^{(1)}_{a}$ or $m^{(1)}_{b}$) 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 $\Sigma$. 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 $\ket{\Psi_{\mathrm{para}}} \equiv \ket{\rightarrow\cdots\rightarrow}$. 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: Schematic phase diagram of (a) the spin-nexus Hamiltonian (\ref{['eq:H_ss']}). 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).
• Figure 5: Dual representation of the plaquette Ising model in the presence of a transverse field. We place nexus spins at the center of each four-spin plaquette interaction, so that $\sigma^{x}_{p}\equiv \prod_{i\in\partial p}\tau^{z}_{i}$. The product of four adjacent four-spin interactions that wrap around the cube is equal to the identity (e.g. the product of plaquette interactions $p$, $q$, $r$ and $u$). In the dual representation, this leads to the indicated constraints at each cube. Only two of the three constraints are independent.