Fractons on Graphs and Complexity

Abstract

We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 U (1) gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and graph theory quantities. For instance, the ground state degeneracy of the matter theory equals the number of spanning trees of the spatial graph, which is a common measure of complexity in graph theory ("GSD = complexity"). The discrete global symmetry is identified as the Jacobian group of the graph. In the gauge theory, superselection sectors of fractons are in one-to-one correspondence with the divisor classes in graph theory. In particular, under mild assumptions on the spatial graph, the fracton immobility is proven using a graph-theoretic Abel-Jacobi map.

Anisotropic higher rank $\mathbb{Z}_N$ topological phases on graphs

“…[17] for a study on 3D fracton topological phases on the Cayley trees and Ref. [18] for analysis on the Lifshitz theory on a graph and complexity. )…”

Global dipole symmetry, compact Lifshitz theory, tensor gauge theory, and fractons

“…2 This discretization has the advantage of being well-defined on other spatial lattices, including general graphs. (See [37] for a discussion of exotic lattice models on graphs based on the discrete Laplacian operator.) Alternatively, one could first integrate 1 An even simpler version of this theory in 1+1d is analyzed in [21].…”

“…The duality in Section 3.2.1 between φi and (A τ , A ij ) is the lattice version of the fracton-elasticity duality of [54,56,57], while the duality in Section 2.2.1 between the dipole φ-theory and ( Âτi , Âij ) is the lattice version of the lineon-elasticity duality of [53]. Right: The Laplacian φ-theory is self-dual [13,37] (see Section 2.1.1). The Laplacian φ-theory Higgses the Laplacian gauge theory (A τ , A), which has no duality.…”

“…The Laplacian φ-theory and gauge theory were recently introduced in [37] on a general spatial graph. In this paper, we will place them on a 2d torus, and compare them with their dipole counterparts: the dipole φ-theory and the scalar charge theory.…”

“…In this paper, we will place them on a 2d torus, and compare them with their dipole counterparts: the dipole φ-theory and the scalar charge theory. We will make use of the results from [37] throughout.…”