Machian fractons, Hamiltonian attractors and non-equilibrium steady states

TL;DR

This paper investigates classical fracton systems, revealing that their nonlinear dynamics lead to non-ergodic, non-equilibrium steady states with broken translational symmetry, challenging traditional equilibrium theorems.

Contribution

It introduces a novel understanding of fracton dynamics showing late-time attractors and non-equilibrium states, expanding the paradigm of Hamiltonian many-body physics.

Findings

Fracton systems exhibit late-time attractors in position-velocity space.

These attractors violate ergodicity and lead to non-equilibrium steady states.

Translational symmetry is broken even in low-dimensional systems.

Abstract

We study the $N$ fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all $N$, despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to non-equilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. While a full understanding of the many-body nonlinear problem is a formidable and incomplete task, we provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of `order-by-disorder' borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a new paradigm for Hamiltonian dynamics and non-equilibrium many-body physics.