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arXiv:2603.29390

Nonlinear hydrodynamic response of a quantum Hall system

Hiroki Isobe

cond-mat.mes-hallcond-mat.str-elphysics.flu-dynquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026
arXivHTMLPDF

Abstract

The quantum Hall effect realizes a quantized Hall resistance $R_{xy} = h/(νe^2)$ whereas the longitudinal resistance vanishes. The quantized value consists of the fundamental physical quantities, the elementary charge $e$ and the Planck constant $h$, along with an integer or fractional constant $ν$. High precision measurements of $R_{xy}$ allude to a linear relation between the applied current $I$ and the Hall voltage $V_\mathrm{H}$. Here, we argue that a nonlinear relation between $I$ and $V_\mathrm{H}$ could arise when the electric field is spatially inhomogeneous. We first discuss that the linear $I$-$V_\mathrm{H}$ 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.

Journal reference
Newton 2, 100466 (2026)
DOI
10.1016/j.newton.2026.100466
Comments
8+10 pages, 2+1 figures

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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.
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.
A nonlinear single-particle model is introduced, which captures the characteristic of systems in the quantum Hall regime. The model involves the magnetic Schrödinger equation with spatially variable magnetic flux density. The distribution of flux is prescribed via the postulates of the mesoscopic mechanics (MeM) introduced in my previous articles (cf. [9, 10]). The model is found to imply exact integer and fractional quantitzation of the Hall conductance. In fact, Hall resistance is found to be R H = (h/e 2)(M/N) at the filling factor value N/M. The assumed geometry of the Hall plate is rectangular. Special properties of the magnetic Schrödinger equation with the mesoscopic feedback loop allow us to demonstrate quantization of Hall resistance as a direct consequence of charge and flux quantization. I believe results presented here shed light at the overall status of the MeM in quantum physics, confirming its validity.
A linear response expression for the Hall conductance ${\ensuremath{\sigma}}_{\mathrm{xy}}$ of interacting two-dimensional electrons in a strong magnetic field is derived. When the system is dissipationless, ${\ensuremath{\sigma}}_{\mathrm{xy}}$ can be written solely in terms of the density-density response function ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow}) of the electron liquid. Thus, a specific behavior of ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow}) is required on a Hall plateau. This behavior is ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow})\ensuremath{\propto}q2{q}^{2}q2 for small wave vector q, with a proportionality constant given by the Hall quantum number.
Measurement techniques based upon the Hall effect are invaluable tools in condensed-matter physics. When an electric current flows perpendicular to a magnetic field, a Hall voltage develops in the direction transverse to both the current and the field. In semiconductors, this behavior is routinely used to measure the density and charge of the current carriers (electrons in conduction bands or holes in valence bands)--internal properties of the system that are not accessible from measurements of the conventional resistance. For strongly interacting electron systems, whose behavior can be very different from the free electron gas, the Hall effect's sensitivity to internal properties makes it a powerful tool; indeed, the quantum Hall effects are named after the tool by which they are most distinctly measured instead of the physics from which the phenomena originate. Here we report the first observation of a Hall effect in an ultracold gas of neutral atoms, revealed by measuring a Bose-Einstein condensate's transport properties perpendicular to a synthetic magnetic field. Our observations in this vortex-free superfluid are in good agreement with hydrodynamic predictions, demonstrating that the system's global irrotationality influences this superfluid Hall signal.
When vortices are displaced in Bose-Einstein condensates (BECs), the Magnus force gives the system a momentum transverse in the direction to the displacement. We show that BECs in long channels with vortices exhibit a quantization of the current response with respect to the spatial vortex distribution. The quantization originates from the well-known topological property of the phase around a vortex; it is an integer multiple of $2\ensuremath{\pi}$. In a way similar to that of the integer quantum Hall effect, the current along the channel is related to this topological phase and can be extracted from two experimentally measurable quantities: the total momentum of the BEC and the spatial distribution. The quantization is in units of m/2hm/2hm/2h, where mmm is the mass of the atoms and hhh is Planck's constant. We derive an exact vortex momentum-displacement relation for BECs in long channels under general circumstances. Our results present the possibility that the configuration described here can be used as a novel way of measuring the mass of the atoms in the BEC using a topological invariant of the system. If an accurate determination of the plateaus are experimentally possible, this gives the possibility of a topological quantum mass standard and precise determination of the fine structure constant.
We study the quantum dynamics in response to time-dependent external potentials of the edge modes of a small fractional quantum Hall fluid composed of few particles on a lattice in a bosonic Laughlin-like state at filling ν=1/2. We show that the nonlinear chiral Luttinger liquid theory provides a quantitatively accurate description even for the small lattices that are available in state-of-the-art experiments, away from the continuum limit. Experimentally accessible data related to the quantized value of the bulk transverse Hall conductivity are identified both in the linear and the non-linear response to an external excitation. The strong nonlinearity induced by the open boundaries is responsible for sizable quantum blockade effects, leading to the generation of nonclassical states of the edge modes.