Continuum Fractons: Quantization and the Few Body Problem
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.