Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractons with Twisted Boundary Conditions and Their Symmetries

Tom Rudelius, Nathan Seiberg, Shu-Heng Shao

TL;DR

This work analyzes fracton-like phases on a twisted two- and three-torus, revealing that boundary twists convert subsystem symmetries into a nontrivial clock- and shift-type algebra. The authors develop a unified framework using transition functions and lattice regularization to count ground-state degeneracies in both $\mathbb{Z}_N$ and $U(1)$ tensor gauge theories across 2+1D and 3+1D, with GSDs that depend sensitively on the twisted geometry via effective lengths and the torsion parameter $m$, and a natural rational-tau continuum limit. A central finding is that the twist yields a noncommuting $\mathbb{Z}_m\times\mathbb{Z}_m$ symmetry in the $\phi$-theory, which generalizes to $\gcd(N,m)$ copies in the $\mathbb{Z}_N$ tensor theories, thereby encoding topological sector structure tied to torsion cycles. The paper also demonstrates multiple dual descriptions (including $\phi$-$A$ and BF-type formulations) that consistently reproduce the same ground-state degeneracies, highlighting the deep interplay between geometry, transition functions, and subsystem symmetries in fracton-like systems with twisted boundaries.

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 $τ= τ_1 + i τ_2$ has rational $τ_1 = k / m$. 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 $\mathbb{Z}_{\gcd(N,m)}$.

Fractons with Twisted Boundary Conditions and Their Symmetries

TL;DR

This work analyzes fracton-like phases on a twisted two- and three-torus, revealing that boundary twists convert subsystem symmetries into a nontrivial clock- and shift-type algebra. The authors develop a unified framework using transition functions and lattice regularization to count ground-state degeneracies in both and tensor gauge theories across 2+1D and 3+1D, with GSDs that depend sensitively on the twisted geometry via effective lengths and the torsion parameter , and a natural rational-tau continuum limit. A central finding is that the twist yields a noncommuting symmetry in the -theory, which generalizes to copies in the tensor theories, thereby encoding topological sector structure tied to torsion cycles. The paper also demonstrates multiple dual descriptions (including - and BF-type formulations) that consistently reproduce the same ground-state degeneracies, highlighting the deep interplay between geometry, transition functions, and subsystem symmetries in fracton-like systems with twisted boundaries.

Abstract

We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the -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 and 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 has rational . Then, in systems based on subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, the new symmetry is a projectively realized , which leads to an -fold ground state degeneracy. In systems based on symmetries, like the X-cube model, each of these two factors is replaced by .

Paper Structure

This paper contains 31 sections, 206 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The twisted torus
  3. A new, surprising symmetry
  4. Outline
  5. Geometry
  6. Continuum geometry
  7. Lattice geometry
  8. Winding on the twisted two-torus
  9. Transition functions and winding charges
  10. Winding configurations of $\phi$
  11. Special configurations
  12. More general configurations
  13. Comments about the $2+1$-dimensional $\phi$-theory
  14. $2+1$-dimensional $\mathbb{Z}_N$ tensor gauge theory
  15. Special $\mathbb{Z}_N$ configurations
  16. ...and 16 more sections

Figures (3)

  • 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 $\ell_x^{\textrm{eff}}$.