Fracton physics of spatially extended excitations. II. Polynomial ground state degeneracy of exactly solvable models

TL;DR

This paper analyzes exactly solvable fracton models with spatially extended excitations, deriving polynomial formulas for ground state degeneracy that reveal complex geometric and topological features of higher-dimensional systems.

Contribution

It introduces systematic methods to compute ground state degeneracy in fracton models, revealing polynomial dependencies on system size and linking them to higher-dimensional topological structures.

Findings

Derived explicit GSD formulas with polynomial dependence on system size

Identified diverse polynomial patterns indicating complex topological features

Extended understanding of fracton models with spatially extended excitations

Abstract

Generally, ``fracton'' topological orders are referred to as gapped phases that support \textit{point-like topological excitations} whose mobility is, to some extent, restricted. In our previous work [Phys. Rev. B 101, 245134 (2020)], a large class of exactly solvable models on hypercubic lattices are constructed. In these models, \textit{spatially extended excitations} possess generalized fracton-like properties: not only mobility but also deformability is restricted. As a series work, in this paper, we proceed further to compute ground state degeneracy (GSD) in both isotropic and anisotropic lattices. We decompose and reconstruct ground states through a consistent collection of subsystem ground state sectors, in which mathematical game ``coloring method'' is applied. Finally, we are able to systematically obtain GSD formulas (expressed as $\log_2 GSD$) which exhibit diverse kinds of polynomial dependence on system sizes. For example, the GSD of the model labeled as $[0,1,2,4]$ in four dimensional isotropic hypercubic lattice shows $ 12L^2-12L+4$ dependence on the linear size $L$ of the lattice. Inspired by existing results [Phys. Rev. X 8, 031051 (2018)], we expect that the polynomial formulas encode geometrical and topological fingerprints of higher-dimensional manifolds beyond toric manifolds used in this work. This is left to future investigation.