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Chiral Topological Elasticity and Fracton Order

Andrey Gromov

TL;DR

The paper shows that conventional higher-rank fracton gauge theories are not gauge-invariant in curved space, signaling a geometric origin for fracton phenomenology. It introduces an APD-invariant, vielbein-based theory of chiral topological elasticity that reduces to the higher-rank model in the flat limit and remains well-defined on arbitrary manifolds. Key results include a non-linear glide constraint for dislocations, a disclination–dislocation coupling, and a torsional Hall viscosity response, all framed within teleparallel/ Riemann-Cartan geometry. This geometric perspective suggests fracton order may be fundamentally tied to the structure of space itself and opens new directions for quantization, boundary physics, and lattice realizations.

Abstract

We analyze the "higher rank" gauge theories, that capture some of the phenomenology of the Fracton order. It is shown that these theories lose gauge invariance when arbitrarily weak and smooth curvature is introduced. We propose a resolution to this problem by introducing a theory invariant under area-preserving diffeomorphisms, which reduce to the "higher rank" gauge transformations upon linearization around a flat background. The proposed theory is geometric in nature and is interpreted as a theory of chiral topological elasticity. This theory exhibits some of the Fracton phenomenology. We explore the conservation laws, topological excitations, linear response, various kinematical constraints, and canonical structure of the theory. Finally, we emphasize that the very structure of Riemann-Cartan geometry, which we use to formulate the theory, encodes some of the Fracton phenomenology, suggesting that the Fracton order itself is geometric in nature.

Chiral Topological Elasticity and Fracton Order

TL;DR

The paper shows that conventional higher-rank fracton gauge theories are not gauge-invariant in curved space, signaling a geometric origin for fracton phenomenology. It introduces an APD-invariant, vielbein-based theory of chiral topological elasticity that reduces to the higher-rank model in the flat limit and remains well-defined on arbitrary manifolds. Key results include a non-linear glide constraint for dislocations, a disclination–dislocation coupling, and a torsional Hall viscosity response, all framed within teleparallel/ Riemann-Cartan geometry. This geometric perspective suggests fracton order may be fundamentally tied to the structure of space itself and opens new directions for quantization, boundary physics, and lattice realizations.

Abstract

We analyze the "higher rank" gauge theories, that capture some of the phenomenology of the Fracton order. It is shown that these theories lose gauge invariance when arbitrarily weak and smooth curvature is introduced. We propose a resolution to this problem by introducing a theory invariant under area-preserving diffeomorphisms, which reduce to the "higher rank" gauge transformations upon linearization around a flat background. The proposed theory is geometric in nature and is interpreted as a theory of chiral topological elasticity. This theory exhibits some of the Fracton phenomenology. We explore the conservation laws, topological excitations, linear response, various kinematical constraints, and canonical structure of the theory. Finally, we emphasize that the very structure of Riemann-Cartan geometry, which we use to formulate the theory, encodes some of the Fracton phenomenology, suggesting that the Fracton order itself is geometric in nature.

Paper Structure

This paper contains 10 sections, 25 equations, 1 figure.

Table of Contents

  1. Introduction.---
  2. "Higher rank" gauge theory.---
  3. Area-preserving diffeomorphisms.---
  4. Chiral Topological Elasticity.---
  5. Operator content.---
  6. Glide constraint.---
  7. Disclinations, Curvature and Fractons.---
  8. Linear response.---
  9. Conclusions.---
  10. Note added.---

Figures (1)

  • Figure 1: Chiral topological elastic medium, $\hat{e}^A_\mu$, can be visualized on top of a classical lattice that is used to probe the system, $e^A_\mu$. Here the classical background lattice is drawn with solid lines, while the quantum lattice is dashed. The lattice constant of the quantum lattice is twice smaller than the one of the background, corresponding to $k=2$. Dislocations of a quantum lattice carry a fractional (in the units of primitive lattice vectors of the classical lattice) Burgers vector. When $k=1$ two lattices coincide. Another possibility (for $k=1$) is the that the background lattice is actually the dual lattice paulose2015topological. If $k$ is not an integer then the two lattices are incommensurate.