Hyperbolic Fracton Model, Subsystem Symmetry and Holography III: Extension to Generic Tessellations

TL;DR

This work generalizes the Hyperbolic Fracton Model (HFM) from the {5,4} tessellation to arbitrary {p,q} hyperbolic tessellations, uncovering a geometry-driven structure for subsystem symmetries and fracton mobility. The authors develop an inflation-based framework, culminating in closed-form growth laws via the inflation matrix, and derive expressions for ground-state degeneracy, residual entropy, and black-hole entropy that scale with layer structure and horizon length. Despite increased geometric complexity, key holographic features persist: Rindler reconstruction enables bulk recovery from boundary data, and a discrete Ryu–Takayanagi relation links mutual information to wedge boundary length. Additionally, fracton excitations exhibit exponential-in-layer growth and geometry-sensitive propagation rules, revealing rich, globally constrained dynamics on hyperbolic lattices. Overall, the paper demonstrates robust holographic duality in fracton models across generic tessellations, offering a fertile platform for exploring generalized symmetries, holography, and error-correcting perspectives in curved lattices.

Abstract

We generalize the Hyperbolic Fracton Model from the $\{5,4\}$ tessellation to generic tessellations, and investigate its core properties: subsystem symmetries, fracton mobility, and holographic correspondence. While the model on the original tessellation has features reminiscent of the flat-space lattice cases, the generalized tessellations exhibit a far richer and more intricate structure. The ground-state degeneracy and subsystem symmetries are generated recursively layer-by-layer, through the inflation rule, but without a simple, uniform pattern. The fracton excitations follow exponential-in-distance and algebraic-in-lattice-size growing patterns when moving outward, and depend sensitively to the tessellation geometry, differing qualitatively from both type-I or type-II fracton model on flat lattices. Despite this increased complexity, the hallmark holographic features -- subregion duality via Rindler reconstruction, the Ryu-Takayanagi formula for mutual information, and effective black hole entropy scaling with horizon area -- remain valid. These results demonstrate that the holographic correspondence in fracton models persists in generic tessellations, and provide a natural platform to explore more intricate subsystem symmetries and fracton physics.

Figures (12)

• Figure 1: Layer-by-layer construction of a $\{5,4\}$ hyperbolic tessellation. Each new layer's generation follows an inflation rule. $(a)$ one layer, $(b)$ two layers, $(c)$ three layers. The different types of polygons ($\alpha, \beta, \sigma$) and vertices ($X, Y$) are labeled according to their connectivity to the previous layer. The recursive inflation rules generate a self-similar structure where the number of polygons grows exponentially with the number of layers $l$.
• Figure 2: The subsystem symmetries of the Plaquette Ising Model. Starting from an all-spins-up ground state and flipping spins along $(a)$ horizontal, $(b)$ vertical lines, or a combination of both $(c)$ will result in another ground state. These operations form the basis of the model's sub-extensive ground state degeneracy.
• Figure 3: Fracton excitations in the Plaquette Ising Model. $(a)$ A single fracton excitation (represented by a star) is immobile due to the need to flip an infinite number of spins. $(b)$ A line of spin flips creates a bound pair of fractons that can propagate collectively along that line. $(c)$ A single spin flip creates four bound fractons, which can propagate freely.
• Figure 4: Vertex operators $\mathcal{O}_v$ in the Hyperbolic Fracton Model for various tessellations. The operator is a product of the spins on the polygons meeting at that vertex. $(a)$ A $\{5,4\}$ tessellation has 4-spin operators. $(b)$ A $\{5,5\}$ tessellation has 5-spin operators. $(c)$ A $\{4,5\}$ tessellation also has 5-spin operators.
• Figure 5: Illustration of the counting procedure on a $\{5,4\}$ tessellation. $(a)$ An initial random configuration. $(b)$ A single spin's value (green polygon) is chosen, introducing one DOF. Its value, combined with the $\mathcal{O}_v=1$ constraints (yellow dots) at adjacent vertices, $(c)$ iteratively fixes the values of neighboring spins (blue polygons). $(d)$ Upon layer closure, a periodic boundary condition removes one DOF. Subsequent layers inherit constraints from prior ones via inflation rules, ensuring consistency across the lattice. The total number of DOF is given by counting the spins left to fluctuate.