A simple anisotropic three-dimensional quantum spin liquid with fracton topological order

TL;DR

The authors construct a simple 3D spin model on an $L_x\times L_y\times L_z$ lattice, anisotropic in $z$, that hosts fracton topological order by coupling adjacent toric-code planes into eight-spin cube operators. The Hamiltonian $H=-\sum_c\hat{Q}_c$ yields immobile fractons at cube corners and two-dimensional string-ends that form dimension-2 excitations, along with dimension-1 excitations propagating along $\hat{z}$; the ground-state degeneracy is $\mathcal{D}=2^{L_xL_y-2+2L_z}$, decomposable into a topological part $2^{2L_z}$ and a non-topological part $2^{L_xL_y-2}$. The model exhibits mutual semion statistics between the dimension-2 flavors $e$ and $m$, and the excitations can be combined into mobile composites whose motion is restricted to either straight lines along $\hat{z}$ or the $xy$ plane. Stability analysis via a tunable coupling to decoupled toric-code planes shows fracton order is a fine-tuned point, while open boundaries generate characteristic surface modes: zero-energy modes on $x$/$y$ boundaries and no such modes on $z$ boundaries, explained by the structure of the composite excitations. Together, the work clarifies how a simple, anisotropic 3D lattice model can realize fracton order and illuminates the rich boundary physics that accompanies bulk fracton topological phases.

Abstract

We present a three-dimensional cubic lattice spin model, anisotropic in the $\hat{z}$ direction, that exhibits fracton topological order. The latter is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground state degeneracy: On an $L_x\times L_y\times L_z$ three-torus, it has a $2^{2L_z}$ topological degeneracy, and an additional non-topological degeneracy equal to $2^{L_xL_y-2}$. The fractons can be combined into composite excitations that move either in a straight line along the $\hat{z}$ direction, or freely in the $xy$ plane at a given height $z$. While our model draws inspiration from the toric code, we demonstrate that it cannot be adiabatically connected to a layered toric code construction. Additionally, we investigate the effects of imposing open boundary conditions on our system. We find zero energy modes on the surfaces perpendicular to either the $\hat{x}$ or $\hat{y}$ directions, and their absence on the surfaces normal to $\hat{z}$. This result can be explained using the properties of the two kinds of composite two-fracton mobile excitations.

Figures (9)

• Figure 1: Operator definitions for the 3D model. The cube operator (a) $\hat{Q}_c$ is the product of the two plaquette operators at adjacent $xy$ planes: (b) $\hat{Q}_{xy}(z)$ and (c) $\hat{Q}_{xy}(z+1)$.
• Figure 2: Starting with a system in a ground state, acting on the center spin with $\hat{Z}$ creates four $Q_c=-1$ excitations: excited cubes (dark grey and light grey) are diagonal from each other in the $xy$ plane and stacked one on top of the other along the $\hat{z}$ direction.
• Figure 3: (a) Dimension-2 particles (AB and CD, pairs of cubes with $Q_c=-1$ sharing an $xy$ face) are located at the ends of an open string operator (dotted line), which is realized as a product of $\hat{X}$ and $\hat{Z}$ indicated by filled and empty circles respectively. (b) Pairs of cubes making up dimension-2 particles can be split into individual fractons (A, B, C, and D) by stacking open strings in adjacent $xy$ planes such that their endpoints are directly above one another. The open strings can be freely deformed apart from their endpoints, and are indicated by the dotted lines. (c) Dimension-1 particles are pairs of cubes, diagonal from one another, at the same height $z$: AC and BD.
• Figure 4: (a) Two dimension-2 particles of different flavors, each consisting of a pair of adjacent excited cubes, are shown. The red ($m$ flavored) and blue ($e$ flavored) strings schematically indicate the presence of a second dimension-2 particle of the same flavor at the other end of the string. (b) The red dotted string indicates that the $m$ flavored dimension-2 particle has been moved around the $e$ particle in a closed loop. The green cross designates the crossing of the $e$ and $m$ strings. (c) Both $m$ (red) and $e$ (blue) dimension-2 particles take the form of rods extended along the $\hat{z}$ direction. (d) The four dotted red cycles stacked on top of one another along $\hat{z}$ depict moving of the $m$-flavored rod in space in a closed loop.This winding process leads to three $e$-$m$ string crossings, indicated by green crosses.
• Figure 5: Upper panel : Partial lifting of the ground state degeneracy of Eq. (\ref{['eq:Hdecoupled']}) as a function of $\lambda$ for a system with $L_x\times L_y=20$. The energy level shown in red corresponds to the $2^{2L_z}$ ground states of the fracton model (\ref{['eq:H3d']}) with all $Q_{xy}$ quantum numbers equal to $+1$. The gap to the other states indicated in blue closes at the fracton point, $\lambda=0$. Inset shows how the first three excited energy levels of the fracton model are connected to the lowest five excited energy levels of the decoupled toric code planes. Lower panel : Various examples of excited states. Excited cubes are in red and excited plaquettes have a red cross. Red crosses connected by a string lie in the same $xy$ plane. (b) and (c) have four excited cubes (corresponding to $E(\lambda=0)=4$) while having a different number of excited plaquettes corresponding respectively to $E(\lambda=1)=2$ and $E(\lambda=1)=4$. (d) and (e) have six excited cubes (corresponding to $E(\lambda=0)=6$) but have a different number of excited plaquettes corresponding respectively to $E(\lambda=1)=4$ and $E(\lambda=1)=6$.