Fractons with twisted boundary conditions and their symmetries

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 $\text{U}\left(1\right)\times \text{U}\left(1\right)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this 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 ${\mathbb{Z}}_{gcd(N,m)}$.

Figure 1

The fundamental domain of the spatial torus.

Figure 2

The $X$ and $\tilde{X}$ cycles.

Figure 3

The surface integral over the twisted torus in terms of the $x$ and $y$ integrals. The contour of the integral $\oint dy$ runs along the blue lines, and the $\int dx$ integral runs over the red segment of length ${\ell }_x^{\text{eff}}$.