Classical Non-Relativistic Fractons

TL;DR

This paper explores the classical mechanics of non-relativistic fractons, revealing how locality induces a Machian dynamics where particles become immobile or form oscillating clusters, with implications for understanding fracton behavior.

Contribution

It introduces a family of classical fracton models with conserved dipole moments and analyzes their dynamics, revealing novel immobility and cluster formation phenomena.

Findings

Particles become immobile if separated beyond a certain distance.

Two particles within inertial reach tend to become immobile over time.

Three or more particles form oscillating clusters with limit cycles.

Abstract

We initiate the study of the classical mechanics of non-relativistic fractons in its simplest setting - that of identical one dimensional particles with local Hamiltonians characterized by by a conserved dipole moment in addition to the usual symmetries of space and time translation invariance. We introduce a family of models and study the $N$ body problem for them. We find that locality leads to a ``Machian" dynamics in which a given particle exhibits finite inertia only if within a specified distance of at least another one. For well separated particles this leads to immobility, much as for quantum models of fractons discussed before. For two or more particles within inertial reach of each other at the start of motion we get an interesting interplay of inertia and interactions. Specifically for a solvable ``inertia only" model of fractons we find that $N=2$ particles always become immobile at long times. Remarkably $N =3$ particles generically evolve to a late time state with one immobile particle and two that oscillate about a common center of mass with generalizations of such ``Machian clusters" for $N > 3$ . Interestingly, Machian clusters exhibit physical limit cycles in a Hamiltonian system even though mathematical limit cycles are forbidden by Liouville's theorem.