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.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.

cond-mat.mes-hallCross-listedarXiv:2603.29390Apr 1, 2026

Theory of the Fractional Quantum Hall Effect

Allan H. MacDonald

Recent Progress in Many-Body TheoriesJan 1, 19880 citations

STATISTICAL AND GROUP PROPERTIES OF THE FRACTIONAL QUANTUM HALL EFFECT

B. G. Wybourne

International School on Symmetry and Structural Properties of Condensed Matter (SSPCM)May 1, 20010 citations

Classification of fractional quantum Hall states with spatial symmetries

Naren Manjunath, Maissam Barkeshli

Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$ -crossed braided tensor categories ( $G\times$ BTCs) for SETs. We consider systems with $U(1)$ charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all $5$ orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains $9$ terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the $G\times$ BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the $G\times$ BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs) for SETs. We consider systems with $U(1)$ charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all $5$ orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains $9$ terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the $G\times$BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the $G\times$BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.

arXiv:2012.11603Dec 21, 202014 citations

Emergence and mechanism in the fractional quantum Hall effect

Jonathan Bain

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsNov 1, 201613 citations

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.

G

-crossed braided tensor categories (

G\times

BTCs) for SETs. We consider systems with

U(1)

charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all

5

orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains

9

terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the

G\times

BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the

G\times

BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.