Entanglement renormalization of fractonic anisotropic $\mathbb{Z}_N$ Laplacian models

TL;DR

This paper explores the entanglement renormalization of anisotropic $ ext{Z}_N$ Laplacian models, revealing how their fracton phases and ground state properties depend on lattice geometry and the value of N.

Contribution

It provides a detailed analysis of ERG flows and ground state degeneracies for various lattice geometries in anisotropic $ ext{Z}_N$ Laplacian models, highlighting their sensitivity to N and lattice structure.

Findings

Models exhibit bifurcating ERG behaviors with distinct flows.

Triangular and honeycomb lattice models are equivalent when N is coprime to 3.

Kagome lattice model is robust against perturbations only if N is coprime to 6.

Abstract

Gapped fracton phases constitute a new class of quantum states of matter which connects to topological orders but does not fit easily into existing paradigms. They host unconventional features such as sub-extensive and robust ground state degeneracies as well as sensitivity to lattice geometry. We investigate the anisotropic $\mathbb{Z}_N$ Laplacian model [1] which can describe a family of fracton phases defined on arbitrary graphs. Focusing on representative geometries where the 3D lattices are extensions of 2D square, triangular, honeycomb and Kagome lattices into the third dimension, we study their ground state degeneracies and mobility of excitations, and examine their entanglement renormalization group (ERG) flows. All models show bifurcating behaviors under ERG but have distinct ERG flows sensitive to both $N$ and lattice geometry. In particular, we show that the anisotropic $\mathbb{Z}_N$ Laplacian models defined on the extensions of triangular and honeycomb lattices are equivalent when $N$ is coprime to $3$. We also point out that, in contrast to previous expectations, the model defined on the extension of Kagome lattice is robust against local perturbations if and only if $N$ is coprime to $6$.