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arXiv:2003.09429 · Physical Review Research

Fracton hydrodynamics

Andrey Gromov, Andrew Lucas, Rahul Nandkishore

Published Jul 22, 2020• 234 citations
arXivHTMLPDF

Abstract

We introduce new classes of hydrodynamic theories inspired by the recently discovered fracton phases of quantum matter.Fracton phases are characterized by elementary excitations (fractons) with restricted mobility.The hydrodynamic theories we introduce describe thermalization in systems with fractonlike mobility constraints, including fluids where charge and dipole moment are both locally conserved, and fluids where charge is conserved along every line or plane of a lattice.Each of these fluids is subdiffusive and constitutes a new universality class of hydrodynamic behavior.There are infinitely many such classes, each with distinct subdiffusive exponents, all of which are captured by our formalism.Our framework naturally explains recent results on dynamics with constrained quantum circuits, as well as recent experiments with ultracold atoms in tilted optical lattices.We identify crisp experimental signatures of these novel hydrodynamics and explain how they may be realized in near term ultracold atom experiments.

DOI
10.1103/physrevresearch.2.033124

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We introduce new classes of hydrodynamic theories inspired by the recently discovered fracton phases of quantum matter.Fracton phases are characterized by elementary excitations (fractons) with restricted mobility.The hydrodynamic theories we introduce describe thermalization in systems with fractonlike mobility constraints, including fluids where charge and dipole moment are both locally conserved, and fluids where charge is conserved along every line or plane of a lattice.Each of these fluids is subdiffusive and constitutes a new universality class of hydrodynamic behavior.There are infinitely many such classes, each with distinct subdiffusive exponents, all of which are captured by our formalism.Our framework naturally explains recent results on dynamics with constrained quantum circuits, as well as recent experiments with ultracold atoms in tilted optical lattices.We identify crisp experimental signatures of these novel hydrodynamics and explain how they may be realized in near term ultracold atom experiments.
We examine the hydrodynamics of systems with spontaneously broken multipolar symmetries using a systematic effective field theory. We focus on the simplest nontrivial setting: a system with charge and dipole symmetry, but without momentum conservation. When no symmetries are broken, our formalism reproduces the quartic subdiffusion ($\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}i{k}^{4}$) characteristic of ``fracton hydrodynamics'' with conserved dipole moment. Our formalism also captures spontaneous breaking of charge and/or dipole symmetry. When charge symmetry is spontaneously broken, the hydrodynamic modes are quadratically propagating and quartically relaxing ($\ensuremath{\omega}\ensuremath{\sim}\ifmmode\pm\else\textpm\fi{}{k}^{2}\ensuremath{-}i{k}^{4}$). When the dipole symmetry is spontaneously broken but the charge symmetry is preserved, then we find quadratically relaxing (diffusive) transverse modes, plus another mode which, depending on parameters, may be either purely diffusive ($\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}i{k}^{2}$) or quadratically propagating and quadratically relaxing ($\ensuremath{\omega}\ensuremath{\sim}\ifmmode\pm\else\textpm\fi{}{k}^{2}\ensuremath{-}i{k}^{2}$). Our work provides concrete predictions that may be tested in near-term cold atom experiments, and also lays out a general framework that may be applied to study systems with spontaneously broken multipolar symmetries.
We show that the simplest universality classes of fracton hydrodynamics in more than one spatial dimension, including isotropic theories of charge and dipole conservation, can exhibit hidden quasiconservation laws, in which certain higher multipole moments can only decay due to dangerously irrelevant corrections to hydrodynamics. We present two simple examples of this phenomenon. First, an isotropic dipole-conserving fluid in the infinite plane conserves an infinite number of harmonic multipole charges within linear response; we calculate the decay or growth of these charges due to dangerously irrelevant nonlinearities. Second, we consider a model with xy and x^{2}-y^{2} quadrupole conservation, in addition to dipole conservation, which is described by isotropic fourth-order subdiffusion, yet has dangerously irrelevant sixth-order corrections necessary to relax the harmonic multipole charges. We confirm our predictions for the anomalously slow decay of the harmonic conserved charges in each setting by using numerical simulations, both of the nonlinear hydrodynamic differential equations, and in quantum automaton circuits on a square lattice.
The realization of synthetic gauge fields for charge neutral ultracold atoms and the simulation of quantum Hall physics have witnessed remarkable experimental progress. Here, we establish key signatures of fractional quantum Hall systems in their nonequilibrium quantum dynamics. We show that in the lowest Landau level the system generically relaxes subdiffusively. The slow relaxation is understood from emergent conservation laws of the total charge and the associated dipole moment that arises from the effective Hamiltonian projected onto the lowest Landau level, leading to subdiffusive fracton hydrodynamics. We discuss the prospect of rotating quantum gases as well as ultracold atoms in optical lattices for observing this unconventional relaxation dynamics.
Abstract We construct a theory of hydrodynamic transport for systems with conserved dipole moment, U(1) charge, energy, and momentum. These models have been considered in the context of fractons, since their elementary and isolated charges are immobile by symmetry, and have two known translation-invariant gapless phases: a “p-wave dipole superfluid” phase where the dipole symmetry is spontaneously broken and a “s-wave dipole superfluid” phase where both the U(1) and dipole symmetries are spontaneously broken. We argue on grounds of symmetry and thermodynamics that there is no transitionally-invariant gapless fluid with unbroken dipole symmetry. In this work, we primarily focus on the hydrodynamic description of p-wave dipole superfluids, including leading dissipative corrections. That theory has, in a sense, a dynamical scaling exponent z = 2, and its spectrum of fluctuations includes novel subdiffusive modes ω ∼ −ik4 in the shear sector and magnon-like sound mode ω ∼ ±k2 − ik2. By coupling the fluid to background fields, we find response functions of the various symmetry currents. We also present a preliminary generalization of our work to s-wave dipole superfluids, which resemble z = 1 fluids and feature sound waves and diffusive shear modes, as in an ordinary fluid. However, the spectrum also contains a magnon-like second-sound mode ω ∼ ±k2 ± k4 − ik4 with subdiffusive attenuation.
Fracton phases constitute a new class of quantum state of matter. They are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics or mobile only in certain directions. These phases do not wholly fit into any of the existing paradigms but connect to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective and allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at nonzero density before concluding with a brief survey of some open problems.
We construct infinite families of new universality classes of fracton hydrodynamics with momentum conservation, both with multipole conservation laws and/or subsystem symmetry. We explore the effects of broken inversion and/or time-reversal symmetry at the ideal fluid level, along with momentum relaxation. In the case of one-dimensional multipole-conserving models, we write down explicit microscopic Hamiltonian systems realizing these new universality classes. All of these hydrodynamic universality classes exhibit instabilities and will flow to new nonequilibrium fixed points. Such fixed points are predicted to exist in arbitrarily large spatial dimensions.