Quadrupole-conserving dynamics in the non-commutative plane

TL;DR

This paper introduces a new universality class of dynamics arising from local area-preserving symmetries in the non-commutative plane, with implications for fracton-like behavior and breakdown of hydrodynamics.

Contribution

It identifies a novel dynamical universality class based on non-commutative geometry and analyzes its effects on lattice dynamics and hydrodynamic descriptions.

Findings

Symmetry group $ ext{SL}(2, ext{R}) times ext{R}^2$ governs the dynamics.

Hydrodynamics breaks down due to nonlinearities in this symmetry class.

Numerical simulations align with effective field theory predictions.

Abstract

Inspired by ``fracton hydrodynamic" universality classes of dynamics with unusual conservation laws, we present a new dynamical universality class that arises out of local area-preserving dynamics in the non-commutative plane. On this symplectic manifold, the area-preserving spatial symmetry group $\mathrm{SL}(2,\mathbb{R})\rtimes \mathbb{R}^2$ is a symmetry group compatible with non-trivial many-body dynamics. The conservation laws associated to this symmetry group correspond to the dipole and quadrupole moments of the particles. We study the unusual dynamics of a crystal lattice subject to such symmetries, and argue that the hydrodynamic description of lattice dynamics breaks down due to relevant nonlinearities. Numerical simulations of classical Hamiltonian dynamical systems with this symmetry are largely consistent with a tree-level effective field theory estimate for the endpoint of this instability.