Isotropic Layer Construction and Phase Diagram for Fracton Topological Phases

TL;DR

This work presents an isotropic, three-dimensional construction in which stacks of two-dimensional toric codes are coupled to realize, via condensation of composite excitations, both conventional 3D $Z_{2}$ topological order and the fracton X-cube phase. A loop-gas perspective is developed for the fracton ground state, and a natural $Z_{N}$ generalization of the X-cube phase is introduced. A duality to (3+1)-D $Z_{N}$ lattice gauge theory explains the phase structure, revealing an intermediate dual-Coulomb phase for $N \ge 5$ and suggesting possible continuous transitions to the fracton phase along certain lines. The paper also provides a solvable projector model describing confinement from the X-cube phase to a trivial confined phase, highlighting the rich interplay between layered topological orders and emergent fracton dynamics with potential routes to continuum field theories for fracton phases.

Abstract

Starting from an isotropic configuration of intersecting, two-dimensional toric codes, we construct a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point- like topological excitations ("fractons"), and degenerate ground-states on the torus that are locally indistinguishable. Our proposal leads to a simple description of the fracton excitations and of the ground-state as a "loop" condensate, and provides a basis for building new 3D topological orders such as a natural, $Z_{N}$ generalization of this fracton phase, which we introduce. We describe the rich phase structure of our layered $Z_{N}$ system. By invoking a lattice duality, we demonstrate that when $N \ge 5$, there is an intermediate phase that appears between the decoupled, layered system and the fracton topologically-ordered state, which opens the possibility of a continuous transition into the fracton topological phase. We conclude by presenting a solvable model, that interpolates between the fracton phase and a confined phase in which the phase transition is first-order.

Figures (10)

• Figure 1: Intersecting Layers of 2D Toric Codes: A stack of two-dimensional, square-lattice toric codes in the $xy$ (green), $yz$ (red) and $xz$ (blue) directions, which intersect at sites. The resulting three-dimensional cubic lattice has two spins per link ($\sigma$, $\mu$) as shown. A single layer of the square-lattice toric code is shown as well, with the "star" and "plaquette" operators defined as shown.
• Figure 2: $Z_{2}$ charge & flux operators for the decoupled layers: The locations of the (a) $Z_{2}$ charge and (b) $Z_{2}$ flux operators at each site of the three-dimensional cubic lattice for the decoupled, intersecting layers of two-dimensional toric codes; each charge or flux operator is oriented along the $xy$ (green), $yz$ (red), or $xz$ (blue) planes.
• Figure 3: Composite Charge Condensation: The operator $\sigma^{z}_{\boldsymbol{s}\boldsymbol{s}'}\mu^{z}_{\boldsymbol{s}\boldsymbol{s}'}$, when acting on the ground-state $\ket{\Psi_{\mathrm{decoupled}}}$ of the decoupled, two-dimensional toric codes, creates four electric charge excitations, two in each of the orthogonal planes that meet at the link, as shown. Condensing this composite charge excitation leads to a three-dimensional $Z_{2}$ topological phase.
• Figure 4: "Gluing" Loops: Condensing the composite charge excitation has the effect of "gluing" electric charge loops in adjacent, orthogonal layers. The resulting wavefunction is that of the 3D $Z_{2}$ topological phase, as described in Sec. I.
• Figure 5: Composite Flux Loop Condensation: The operator $\sigma^{x}_{\boldsymbol{s}\boldsymbol{s}'}\mu^{x}_{\boldsymbol{s}\boldsymbol{s}'}$, when acting on the decoupled layers of the two-dimensional $Z_{2}$ phase, creates four flux excitations in orthogonal planes, as shown. Condensing this composite flux leads to the "X-cube" fracton topological phase, with fracton operator $\mathcal{O}_{c}$ as given in the main text, and shown in Fig. \ref{['fig:Z_N_Xcube']}.