Shivaji Sondhi

Abstract:

We formulate a continuum quantum mechanics for nonrelativistic, 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 $K\left(x\right)\sim |x-x_{\mathrm{edge}}{|}^{\theta }$ . We identify a sharp transition at $\theta =2$ : for $\theta <2$ the spectrum is discrete and wave packets reflect from the edges, whereas for $\theta >2$ the spectrum is continuous and wave packets 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 wave packets. We provide intuition for these results which suggests that the lack of ergodicity of classical continuum fractons will survive their quantization for large systems.

Abstract:

Perhaps the simplest approach to constructing models with subdimensional 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 quasiperiodic orbits in phase space that evade ergodic exploration of the full phase space.