Phase space fractons

Abstract

Perhaps the simplest approach to constructing models with sub-dimensional particles or fractons is to require the conservation of dipole or higher multipole moments. We generalize this approach to allow for moments in phase space and classify all possible classical fracton models with phase-space multipole conservation laws. We focus on a new self-dual model that conserves both dipole and quadrupole moments in position and momentum; we analyze its dynamics and find quasi-periodic orbits in phase space that evade ergodic exploration of the full phase space.

Figures (2)

• Figure 1: Top: A sample trajectory with $\textbf{m}=\textbf{n}=2$ for $N=3$ particles. Bottom: A sample trajectory with $N=4$ particles. For both plots, dashed lines show the strict bounds enforced by ${\mathcal{Q}}_2 = \textrm{const}$ and ${\mathcal{P}}_2 = \textrm{const}$. These simulations use $f(R) = 1/(R^2 + 1)$.
• Figure 2: Trajectories for $N=5$ with identical global conserved quantities, but otherwise random initial conditions. The bounding ellipse is identical, but trajectories evidently explore different regions of phase space, even at the single particle level. Hence the trajectories are non-ergodic in phase space. The global conserved quantities are taken to be $Q_2=4$, $P_2=4$, $\Pi_{11}=2$, $Q_1=0$, and $P_1=0$.