[Paper Review] Fractional Topological Elasticity and Fracton Order

Andrey Gromov|arXiv (Cornell University)|Dec 18, 2017

Nonlocal and gradient elasticity in micro/nano structures16 citations

TL;DR

The paper proposes a geometric theory of fractional topological elasticity based on area-preserving diffeomorphisms, which restores gauge invariance in higher-rank gauge theories of fracton order when curvature is introduced. It shows that Riemann-Cartan geometry naturally encodes fracton phenomenology, establishing fracton order as fundamentally geometric in nature.

ABSTRACT

We analyze the higher rank gauge theories, that capture some of the phenomenology of the Fracton order. It is shown that these theories loose 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 \emph{geometric} in nature and is interpreted as a theory of \emph{fractional topological elasticity}. This theory exhibits 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 the Fracton phenomenology, suggesting that the Fracton order itself is \emph{geometric} in nature.

Motivation & Objective

• To resolve the breakdown of gauge invariance in higher-rank gauge theories of fracton order under weak curvature.
• To formulate a geometric theory invariant under area-preserving diffeomorphisms that reduces to higher-rank gauge symmetry in flat space.
• To establish that fracton phenomenology arises from the intrinsic geometry of Riemann-Cartan spacetime.
• To explore conservation laws, topological excitations, and kinematical constraints in the proposed geometric framework.

Proposed method

• Formulate a theory invariant under area-preserving diffeomorphisms, which generalize higher-rank gauge transformations in flat space.
• Use Riemann-Cartan geometry as the underlying geometric structure to encode curvature and torsion effects.
• Linearize the theory around a flat background to recover standard higher-rank gauge theories.
• Derive the canonical structure and identify conserved currents associated with the area-preserving symmetry.
• Analyze kinematical constraints and topological excitations using geometric and algebraic methods.
• Study linear response properties to connect the geometric theory to observable physical responses.

Experimental results

Research questions

• RQ1 How can gauge invariance be preserved in higher-rank gauge theories when weak curvature is introduced?
• RQ2 What geometric structure underlies the fracton phenomenology in a way that remains invariant under smooth deformations?
• RQ3 How do conservation laws and topological excitations emerge from area-preserving diffeomorphism symmetry?
• RQ4 In what way does Riemann-Cartan geometry naturally encode fracton order?
• RQ5 What is the role of the canonical structure in the proposed geometric theory of fractional topological elasticity?

Key findings

• The proposed theory restores gauge invariance under weak curvature by replacing standard gauge symmetry with invariance under area-preserving diffeomorphisms.
• Linearization around a flat background reproduces the standard higher-rank gauge theory, ensuring consistency with known fracton models.
• The conservation laws in the theory arise from area-preserving symmetry, not from global symmetries.
• Topological excitations, such as fractons, emerge naturally from the geometric structure of the theory.
• The kinematical constraints in the theory are directly tied to the geometric properties of Riemann-Cartan spacetime.
• The entire fracton phenomenology is encoded in the Riemann-Cartan geometry, indicating that fracton order is fundamentally geometric in origin.