Fractons with twisted boundary conditions and their symmetries

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 $\ensuremath{\tau}={\ensuremath{\tau}}_{1}+i{\ensuremath{\tau}}_{2}$ has rational ${\ensuremath{\tau}}_{1}=k/m$. Then, in systems based on $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized ${\mathbb{Z}}_{m}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{m}$, 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 ${\mathbb{Z}}_{gcd(N,m)}$.