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arXiv:2412.12535

Breakdown of hydrodynamics in a Galilean quantum Hall crystal

Xiaoyang Huang, Andrew Lucas

Published Dec 17, 2024• 0 citations
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Abstract

We construct a nonlinear fluctuating hydrodynamic effective field theory for Galilean-invariant quantum Hall systems with spontaneously broken translational symmetry. Neglecting the role of energy conservation in a low-temperature regime, the hydrodynamic mode is a magnetophonon with quartic attenuation: $ω\sim \pm k^2-\mathrm{i} k^z$ with $z=4$. However, this linear response theory is unstable, and flows to a non-trivial dynamical universality class with $z\approx 3$. We observe this scaling in numerical simulations of many-body classical Hamiltonian dynamics, in a model of an electronic crystal in the lowest Landau level. Observing this magnetophonon decay rate in a quantum Hall crystal represents a promising setting to detect an analogue of a "fractonic dynamical universality class" in a solid-state system, e.g. using microwave impedance microscopy.

DOI
10.48550/arxiv.2412.12535

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We construct a nonlinear fluctuating hydrodynamic effective field theory for Galilean-invariant quantum Hall systems with spontaneously broken translational symmetry. Neglecting the role of energy conservation in a low-temperature regime, the hydrodynamic mode is a magnetophonon with quartic attenuation: ω∼±k2−ikzω\sim \pm k^2-\mathrm{i} k^zω∼±k2−ikz with z=4z=4z=4. However, this linear response theory is unstable, and flows to a non-trivial dynamical universality class with z≈3z\approx 3z≈3. We observe this scaling in numerical simulations of many-body classical Hamiltonian dynamics, in a model of an electronic crystal in the lowest Landau level. Observing this magnetophonon decay rate in a quantum Hall crystal represents a promising setting to detect an analogue of a "fractonic dynamical universality class" in a solid-state system, e.g. using microwave impedance microscopy.
Strongly interacting quantum Hall liquids can spontaneously break translational symmetry, forming a quantum Hall (Wigner) crystal. Here, the authors investigate its finite-temperature out-of-equilibrium dynamics, and show that nonlinear fluctuations drive the system to an unusual dynamical universality class, analogous to a ``fracton fluid''. The authors provide supporting numerical evidence and propose experimental observation of this phenomenon in high-quality GaAs heterostructures using microwave impedance microscopy.
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.
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.
The quantum Hall effect realizes a quantized Hall resistance Rxy=h/(νe2)R_{xy} = h/(νe^2)Rxy​=h/(νe2) whereas the longitudinal resistance vanishes. The quantized value consists of the fundamental physical quantities, the elementary charge eee and the Planck constant hhh, along with an integer or fractional constant ννν. High precision measurements of RxyR_{xy}Rxy​ allude to a linear relation between the applied current III and the Hall voltage VHV_\mathrm{H}VH​. Here, we argue that a nonlinear relation between III and VHV_\mathrm{H}VH​ could arise when the electric field is spatially inhomogeneous. We first discuss that the linear III-VHV_\mathrm{H}VH​ relation holds with Galilean invariance. Then we consider a hydrodynamic description of a quantum Hall liquid to deal with an axially symmetric electric field. It reveals a nonlinear electronic response arising from the centrifugal force exerted on a curved flow and the density gradient invoked by vorticity.
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.
The low-energy dynamics of many-body systems is governed by gapless modes whose properties are dictated by symmetry. Their existence follows from Goldstone's theorem, while their effective description at zero temperature is determined by the pattern of symmetry breaking. At finite temperature, an analogous role is played by hydrodynamics, which describes the universal behavior of many-body systems over long times and large distances. These principles are well understood for uniform symmetries, which act homogeneously in spacetime and lead to a correspondence between massless Goldstone modes and broken generators, as well as gapless hydrodynamic modes and conserved charges. However, this simple picture changes in the presence of nonuniform symmetries, whose generators do not commute with spacetime translations. The low-energy implications of these symmetries remain less understood, as they do not introduce additional gapless modes but instead constrain the dynamics of the existing degrees of freedom. In this thesis, we develop a unified framework for many-body systems with nonuniform symmetries and show that their effects can be understood in terms of additional fields that are not independent at low energies and can be eliminated, leading to kinematic constraints that reshape the infrared dynamics. In systems with spontaneous symmetry breaking, this mechanism modifies the effective theory and often softens the dispersion relations of the remaining modes. At finite temperature, it manifests in hydrodynamics as constraints on macroscopic currents. As a result, nonuniform symmetries give rise to qualitatively new physical phenomena, including modified spectra of collective excitations, exemplified by transverse Tkachenko oscillations in quantum vortex crystals, and unconventional transport phenomena, such as anomalously slow diffusion and softened sound modes in dipole-conserving systems.