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 τ=τ1+iτ2 has rational τ1=k/m. Then, in systems based on U(1)×U(1) subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized Zm×Zm, which leads to an m-fold ground state degeneracy. In systems based on ZN symmetries, like the X-cube model, each of these two Zm factors is replaced by Zgcd(N,m).