Quadrupole-conserving dynamics in the non-commutative plane

TL;DR

This work introduces a new dynamical universality class arising from area-preserving dynamics on the non-commutative plane, governed by the symmetry group $G=\mathrm{SL}(2,\mathbb{R})\rtimes \mathbb{R}^2$ that enforces dipole and quadrupole conservation. It develops both non-dissipative and dissipative effective field theories, rooted in polygon (notably triangle) area invariants, and analyzes lattice realizations to reveal a consistent $\omega\propto k^3$ phonon dispersion and the breakdown of linear hydrodynamics due to relevant nonlinearities. The authors demonstrate, via a Trotter-split numerical scheme that exactly preserves conserved charges, that the long-wavelength dynamics flow to a universality with dynamical exponent $z\approx 3$, independent of initial energy density over accessible times. While experimental realization remains challenging, the study outlines potential quantum Hall or vortex-fluid platforms and toy classical implementations that could realize this quadrupole-conserving fracton-like dynamics, offering a new lens on fracton hydrodynamics in two dimensions.

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.

Figures (7)

• Figure 1: Summary of conserved quantities, their Hamiltonian vector fields, and geometric interpretations.
• Figure 2: Triangular lattice with the ground state $A_s = A_0$ for all triangles $s$. Each vertex is a particle.
• Figure 3: Sub-system symmetries for the (a) triangular lattice and (b) square lattice. Notice in figure (b) of the square lattice, the yellow and purple arrows indicate two local transformations. The leftmost square represents the four subsystem symmetries one would naively guess at first glance. Figure (c) shows the change in area of the square adjacent to a square in which a local transformation (shown in figure (b)) is applied. The green shaded area is the area lost and the orange is the area gained. At first order in $\epsilon$, these areas are equal in magnitude, and the net change in area is $\Delta A \sim \epsilon^2$. Thus the transformation leaves both areas invariant and explains the proliferation of zero modes within linear response.
• Figure 4: (a) Additional right triangle motifs that can be built on the square lattice. (b) The $k_{x,y}\rightarrow 0$ limit of the dispersion relation coming from a Hamiltonian $H$. The Hamiltonian $H_{\Delta_1}$ corresponds to the sum of $\Delta_1$-type triangles [from (a)] in each unit cell; $H_{\square}$ corresponds to the square unit cell.
• Figure 5: Configuration of triangles that enter Hamiltonian \ref{['eq:Hamiltonian_triangle']} that has no subsystem symmetries.