Continuum Fractons: Quantization and the Many Body Problem

TL;DR

This work develops a continuum quantum framework for non-relativistic fractons with dipole conservation, revealing a sharp spectral transition controlled by the edge decay $K(x)\sim|x - x_{\mathrm{edge}}|^{\theta}$, with a discrete spectrum for $\theta<2$ and a continuous spectrum for $\theta>2$ in the two-fracton problem. The authors map the problem to a Sturm–Liouville form on $(-1,1)$, apply a Liouville transform to connect to standard Schrödinger dynamics, and use a lattice regularization to study the three-fracton sector, uncovering attractor-like localization and a likely $\theta_c \approx 2$ transition in the continuum. They demonstrate that wavepackets either reflect or pile up at edges depending on $\theta$, and that three-fracton states localize along classical attractor lines with tunneling between permutation sectors, suggesting quantum analogs of classical ergodicity breaking. The results illuminate how dipole-conserving fractons escape ergodicity even after quantization, connect to lattice fragmentation ideas, and motivate continuum-field theory formulations that account for edge-dependent physics beyond ultra-local limits.

Abstract

We formulate a continuum quantum mechanics for non-relativistic, dipole-conserving fractons. Imposing symmetries and locality results in novel phenomena absent in ordinary quantum mechanical systems. A single fracton has a vanishing Hamiltonian, and thus its spectrum is entirely composed of zero modes. For the two-body problem, the Hamiltonian is perfectly described by Sturm--Liouville (SL) theory. The effective two-body Hamiltonian is an SL operator on $(-1,1)$ whose spectral type is set by the edge behavior of the pair inertia function $K(x)\sim \lvert x -x_\mathrm{edge} \rvert^θ$. We identify a sharp transition at $θ=2$: for $θ<2$ the spectrum is discrete and wavepackets reflect from the edges, whereas for $θ>2$ the spectrum is continuous and wavepackets slow down and, dominantly, squeeze into asymptotically narrow regions at the edges. For three particles, the differential operator corresponding to the Hamiltonian is piecewise defined, requiring several "matching conditions" which cannot be analyzed as easily. We proceed with a lattice regularization that preserves dipole conservation, and implicitly selects a particular continuum Hamiltonian that we analyze numerically. We find a spectral transition in the three-body spectrum, and find evidence for quantum analogs of fracton attractors in both eigenstates and in the time evolution of wavepackets. We provide intuition for these results which suggests that the lack of ergodicity of classical continuum fractons will survive their quantization for large systems.

Figures (11)

• Figure 1: (a): A classical trajectory for two particles generically spreads out to $\left|x_1 - x_2\right| = \pm 1$. (b): Quantum dynamics for two particles in reduced coordinates $x_1 - x_2$. Remarkably, for $K(x)$ vanishing as $K \sim (1 - \left|x\right|)^\theta$ with $\theta > 2$, the quantum fractons behave classically: at late times, the wavefunction piles up at the edge!
• Figure 2: The reduced coordinates for (a) two particles and (b) three particles. Any square-integrable wave function with support in the blue shaded region corresponds to an exact zero-energy eigenstate of the Hamiltonian. This region is unbounded. Finite energy eigenstates are supported in the unshaded complement.
• Figure 3: \ref{['eq:K_theta']} for various values of $\theta$. The spectral transition in the two-fracton problem occurs at $\theta = 2$ with $\theta<2$ hosting a discrete spectrum and $\theta>2$ hosting a continuous spectrum.
• Figure 4: Wave reflection in the two regimes $\theta > 2$ (continuous spectrum) and $\theta < 2$ (discrete spectrum). (a): Incoming wave $u_+$ and reflection $r(E)u_-$, in the original fracton $x$-coordinates. (b): Liouville map onto $\xi \in (-\infty, \infty)$ for $\theta > 2$. The effective potential $V$ decays and behaves nicely with no singularities. An incoming wave with high enough energy is expected to pass through with suppressed reflection. (c): Liouville map for $\theta < 2$, with $x=0$ mapping onto a finite value of $\xi$. The effective potential becomes infinite at $\xi = 0$, so that any incoming wave will totally reflect.
• Figure 5: Numerical measurements for the wavepacket reflection time for $K = (1-x^2)^3$, i.e., $\theta=3$, where the Hamiltonian spectrum is continuous. The reflection time is defined as the time for $\langle \left|x\right|^2 \rangle$ to peak. The initial wavepacket is taken to be $\psi(x, t=0) \sim \exp(-x^2/\sigma^2) \exp(i q x)$, with $\sigma=0.05$ and $q = 30$. In numerical simulations of wavepacket dynamics, the wavepacket piles up at one of the edges $\pm 1$, before reflecting. The time to reflect increases with the number of lattice discretization steps, implying that no reflections are present in the continuum limit.