X-Cube Fracton Model on Generic Lattices: Phases and Geometric Order

TL;DR

This work shows that fracton order, exemplified by the X-Cube model, is inherently sensitive to lattice geometry. By introducing intersecting-surfaces (i-surfaces) lattices, the authors generalize the X-Cube model to arbitrary 3D geometries and demonstrate that mobility, braiding, and ground-state degeneracy depend on curvature and i-surface topology. They reveal geometry-driven phase structure, proposing a coarse phase equivalence based on generalized local unitaries combined with quasi-isometries, effectively redefining fracton phases as geometric orders. The results imply a rich landscape of fracton phases across diverse lattices and hint at deeper connections to geometric group theory and gravity-like behavior in gapped fracton systems.

Abstract

Fracton order is a new kind of quantum order characterized by topological excitations that exhibit remarkable mobility restrictions and a robust ground state degeneracy (GSD) which can increase exponentially with system size. In this paper, we present a generic lattice construction (in three dimensions) for a generalized X-cube model of fracton order, where the mobility restrictions of the subdimensional particles inherit the geometry of the lattice. This helps explain a previous result that lattice curvature can produce a robust GSD, even on a manifold with trivial topology. We provide explicit examples to show that the (zero temperature) phase of matter is sensitive to the lattice geometry. In one example, the lattice geometry confines the dimension-1 particles to small loops, which allows the fractons to be fully mobile charges, and the resulting phase is equivalent to (3+1)-dimensional toric code. However, the phase is sensitive to more than just lattice curvature; different lattices without curvature (e.g. cubic or stacked kagome lattices) also result in different phases of matter, which are separated by phase transitions. Unintuitively however, according to a previous definition of phase [Chen, Gu, Wen 2010], even just a rotated or rescaled cubic lattice results in different phases of matter, which motivates us to propose a new and coarser definition of phase for gapped ground states and fracton order. The new equivalence relation between ground states is given by the composition of a local unitary transformation and a quasi-isometry (which can rotate and rescale the lattice); equivalently, ground states are in the same phase if they can be adiabatically connected by varying both the Hamiltonian and the positions of the degrees of freedom (via a quasi-isometry). In light of the importance of geometry, we further propose that fracton orders should be regarded as a geometric order.

Figures (13)

• Figure 1: Geometric braiding operators for the (a) X-cube model on a cubic lattice VijayXCubeMaLayersfractonQFT, (b) X-cube model on a stacked kagome lattice, and (c) Chamon model ChamonModelBravyi2011 of fracton order. The surfaces are membrane operators that braid fractons around the edge of the membrane by exchanging other excitations in the interior of the membrane fractonQFT. The lines are string operators that braid dimension-1 particles. The operators are products of $X$, $Y$, and $Z$ Pauli operators, which are colored red, green, and blue. These operators can be used to detect subdimensional particles. This figure exemplifies that in fracton orders, the braiding paths of the subdimensional excitations take rigid geometric shapes that depend on the model and lattice.
• Figure 2: (a) For every cube, the X-cube model on a cubic lattice (Eq. (\ref{['eq:Xcube H']})) has a 3-cell operator which is a product of 12 Pauli $Z$ operators along the edges of the cube: $\prod_{\ell \in \text{\mancube}} Z_\ell$. (b) At each vertex, there are three cross operators: one for each of the three planes that intersect the vertex. The operators are a product of four $X$ operators on the four links within the plane that neighbor the vertex: $\prod_{\ell \in +} X_\ell$. (Regarding daggers in figure: foot:daggers.)
• Figure 3: (a) A cubic lattice and (b) a stack of kagome lattices constructed from intersecting surfaces (colored planes), which we refer to as i-surfaces. Pauli operators live on the links (black) of the lattice, which are placed where two i-surfaces intersect. The Hamiltonian (Eq. (\ref{["eq:Xcube H'"]})) consists of three cross operators at each vertex (Fig. \ref{['fig:Xcube']}), and a 3-cell operator at each 3-cell (3D volume enclosed by i-surfaces) which is a product of $Z$ operators on the edges of the 3-cell. The stack of kagome lattices has two kinds of 3-cells: a triangular prism (Fig. \ref{['fig:3cell']}) and a hexagonal prism.
• Figure 4: New kinds of X-cube Hamiltonian terms in Eq. (\ref{["eq:Xcube H'"]}). (a) An example of a 3-cell operator on a triangular prism, which is a 3-cell in a stack of kagome lattices (Fig. \ref{['fig:cubicKagome']}). The 3-cell operator ($\prod_{\ell \in \text{\mancube}} Z_\ell$) on a 3-cell ($\text{\mancube}$) is a product of $Z$ operators on the links around the edges of the 3-cell. (b) A product of $Z$ operators on the links around a loop: $\prod_{\ell \in \ocircle} Z_\ell$. (c) A product of $X$ operators on links connecting two parallel loops: $\prod_{\ell \perp \circledcirc} X_\ell$. (Regarding daggers in figure: foot:daggers.)
• Figure 5: Order-4 pentagonal tiling of the hyperbolic plane. A stack of this lattice can be created from the intersecting surface construction. The dimension-1 particles move along the lines of the above lattice, which are geodesics of the hyperbolic plane. As a result, the braiding operators in Fig. \ref{['fig:braid']} will curve with the lattice geometry. The geometry of a hyperbolic 3-space can be constructed from an order-4 dodecahedral honeycomb dodecahedralHoneycomb.