Fractons with Twisted Boundary Conditions and Their Symmetries

TL;DR

This paper investigates the effects of twisted boundary conditions on fracton models like the X-cube, revealing how geometry influences ground state degeneracy and uncovering new global symmetries related to subsystem symmetries.

Contribution

It introduces a framework for understanding how twisted boundary conditions affect fracton models and their symmetries, providing new insights into ground state degeneracy and global symmetry structures.

Findings

Ground state degeneracy depends on geometrical parameters and continuum limit.

A new global symmetry emerges from subsystem symmetries under twist.

Ground state degeneracy is related to the modular parameter of the twisted torus.

Abstract

We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the $xy$-plane. The ground state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in $2+1$ and $3+1$ dimensional systems. It originates from the underlying subsystem symmetry, but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus $\tau = \tau_1 + i \tau_2$ has rational $\tau_1 = k / m$. Then, in systems based on $U(1)\times U(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, the new symmetry is a projectively realized $\mathbb{Z}_m\times \mathbb{Z}_m$, which leads to an $m$-fold ground state degeneracy. In systems based on $\mathbb{Z}_N$ symmetries, like the X-cube model, each of these two $\mathbb{Z}_m$ factors is replaced by $\mathbb{Z}_{\gcd(N,m)}$.