Physical review. B, Condensed matter

Linear response and the quantized Hall effect

T. McMullen

Published Jul 15, 1985• 6 citations

Abstract

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}${q}^{2}$ for small wave vector q, with a proportionality constant given by the Hall quantum number.

DOI: 10.1103/physrevb.32.1415

Related papers

Field-theory approach to the quantum Hall effect

A. Cabo, Масуд Чайчиан

Physical review. B, Condensed matterNov 15, 19916 citations

Conductivity and Hall effect in the two-dimensional Hubbard model

Puru Voruganti, A. A. Golubentsev, Sajeev John

Physical review. B, Condensed matterJun 15, 199258 citations

Hall response of locally correlated two-dimensional electrons at low density

Giacomo Morpurgo, Louk Rademaker, Christophe Berthod +1 more

Physical Review ResearchJan 29, 20243 citations

Near Linearity of the Macroscopic Hall Current Response in Infinitely Extended Gapped Fermion Systems

Marius Wesle, Giovann Marcelli, Tadahiro Miyao +2 more

arXiv:2411.06967Communications in Mathematical PhysicsJul 2, 20253 citations

Nonlinear hydrodynamic response of a quantum Hall system

Hiroki Isobe

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

A note on Hall conductance and Hall conductivity in interacting Fermion systems

Stefan Teufel, Marius Wesle

arXiv:2506.13581Journal of Mathematical PhysicsFeb 1, 20260 citations

Physical review. B, Condensed matter

Linear response and the quantized Hall effect

T. McMullen

Published Jul 15, 1985• 6 citations

Publisher

Abstract

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}

{q}^{2}

for small wave vector q, with a proportionality constant given by the Hall quantum number.

DOI: 10.1103/physrevb.32.1415

Related papers

Field-theory approach to the quantum Hall effect

A. Cabo, Масуд Чайчиан

Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu} ${\mathit{e}}^{2}$ /h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu}${\mathit{e}}^{2}$/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.

Physical review. B, Condensed matterNov 15, 19916 citations

Conductivity and Hall effect in the two-dimensional Hubbard model

Puru Voruganti, A. A. Golubentsev, Sajeev John

A path-integral field-theoretic derivation of electromagnetic linear response for the two-dimensional Hubbard model is given. We find, remarkably, that changes in the Fermi-surface topology associated with incommensurate planar spin-density-wave saddle points induce a change in sign of the Hall coefficient at dopings ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$=0.02--0.5 for U/t=2--10. The change in sign is not affected by short-range magnetic domains. We delinate from first principles an anomalous temperature dependence of the Hall carrier density at dopings close to ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$. An additional anisotropic component to the usual dc conductivity is nonvanishing for certain types of spirals. The paper extends the Bloch-Boltzmann theory to the case of untraditional Fermi liquids where the damping of the quasiparticles is \ensuremath{\Gamma}(\ensuremath{\epsilon})\ensuremath{\sim}max( ${\mathit{k}}_{\mathit{B}}$ T,\ensuremath{\epsilon}).A path-integral field-theoretic derivation of electromagnetic linear response for the two-dimensional Hubbard model is given. We find, remarkably, that changes in the Fermi-surface topology associated with incommensurate planar spin-density-wave saddle points induce a change in sign of the Hall coefficient at dopings ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$=0.02--0.5 for U/t=2--10. The change in sign is not affected by short-range magnetic domains. We delinate from first principles an anomalous temperature dependence of the Hall carrier density at dopings close to ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$. An additional anisotropic component to the usual dc conductivity is nonvanishing for certain types of spirals. The paper extends the Bloch-Boltzmann theory to the case of untraditional Fermi liquids where the damping of the quasiparticles is \ensuremath{\Gamma}(\ensuremath{\epsilon})\ensuremath{\sim}max(${\mathit{k}}_{\mathit{B}}$T,\ensuremath{\epsilon}).

Physical review. B, Condensed matterJun 15, 199258 citations

Hall response of locally correlated two-dimensional electrons at low density

Giacomo Morpurgo, Louk Rademaker, Christophe Berthod +1 more

We study the Hall constant in a homogeneous two-dimensional fluid of correlated electrons immersed in a perpendicular magnetic field, with special focus on the regime of low carrier density. The model consists of a one-band tight-binding model and a momentum-independent causal self-energy, representing interaction-induced correlations effects that are restricted to be local in space. We write general gauge-invariant equations for the conductivity tensor at first order in the magnetic field and solve them numerically—–analytically when possible—in a minimal model of anisotropic square lattice with constant self-energy. Our results show that deviations from the universal behavior of the Hall constant, as observed in the semiclassical regime, appear upon entering the quantum regime, where the Fermi energy and interaction are comparable energy scales. Published by the American Physical Society 2024We study the Hall constant in a homogeneous two-dimensional fluid of correlated electrons immersed in a perpendicular magnetic field, with special focus on the regime of low carrier density. The model consists of a one-band tight-binding model and a momentum-independent causal self-energy, representing interaction-induced correlations effects that are restricted to be local in space. We write general gauge-invariant equations for the conductivity tensor at first order in the magnetic field and solve them numerically—–analytically when possible—in a minimal model of anisotropic square lattice with constant self-energy. Our results show that deviations from the universal behavior of the Hall constant, as observed in the semiclassical regime, appear upon entering the quantum regime, where the Fermi energy and interaction are comparable energy scales. Published by the American Physical Society 2024

Physical Review ResearchJan 29, 20243 citations

Near Linearity of the Macroscopic Hall Current Response in Infinitely Extended Gapped Fermion Systems

Marius Wesle, Giovann Marcelli, Tadahiro Miyao +2 more

Abstract We consider an infinitely extended system of fermions on a d-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength $\varepsilon$ ε is of order $\mathcal {O}(\varepsilon ^\infty )$ O ( ε ∞ ) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in $\varepsilon$ ε up to terms of order $\mathcal {O}(\varepsilon ^\infty )$ O ( ε ∞ ) . The proportionality factor $\sigma _\textrm{H}$ σ H is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for $d=2$ d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with $\sigma _{\textrm{H}}$ σ H in expectation up to $\mathcal {O}(\varepsilon ^\infty )$ O ( ε ∞ ) , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.Abstract We consider an infinitely extended system of fermions on a d-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength $$\varepsilon $$ ε is of order $$\mathcal {O}(\varepsilon ^\infty )$$ O ( ε ∞ ) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in $$\varepsilon $$ ε up to terms of order $$\mathcal {O}(\varepsilon ^\infty )$$ O ( ε ∞ ) . The proportionality factor $$\sigma _\textrm{H}$$ σ H is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for $$d=2$$ d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with $$ \sigma _{\textrm{H}}$$ σ H in expectation up to $$\mathcal {O}(\varepsilon ^\infty )$$ O ( ε ∞ ) , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.

arXiv:2411.06967Communications in Mathematical PhysicsJul 2, 20253 citations

Nonlinear hydrodynamic response of a quantum Hall system

Hiroki Isobe

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

A note on Hall conductance and Hall conductivity in interacting Fermion systems

Stefan Teufel, Marius Wesle

In this note we consider lattice fermions on Z2 with a locally unique gapped ground state and show how to apply the non-equilibrium almost stationary state (NEASS) approach to linear response to derive a formula for the Hall conductance in terms of the ground state expectation of a commutator of modified step functions. This formula is usually derived by a charge pumping argument going back to Laughlin. Here we show that it can also be obtained as the linear response coefficient of the microscopic current response to an adiabatic increase of the chemical potential on a half plane (or more generally on any cone-like region). Indeed, in a manner reminiscent of the bulk-boundary correspondence, we show that raising the chemical potential in any cone-like region gives rise to a current that flows along its boundary and is nearly linear in the increase in chemical potential. We also discuss the connection with the double commutator formula with modified position operators for the Hall conductivity derived in [Wesle et al., Commun. Math. Phys. 406, 199 (2025)] as the linear response coefficient of the macroscopic current response to the adiabatic application of a constant electric field.In this note we consider lattice fermions on Z2 with a locally unique gapped ground state and show how to apply the non-equilibrium almost stationary state (NEASS) approach to linear response to derive a formula for the Hall conductance in terms of the ground state expectation of a commutator of modified step functions. This formula is usually derived by a charge pumping argument going back to Laughlin. Here we show that it can also be obtained as the linear response coefficient of the microscopic current response to an adiabatic increase of the chemical potential on a half plane (or more generally on any cone-like region). Indeed, in a manner reminiscent of the bulk-boundary correspondence, we show that raising the chemical potential in any cone-like region gives rise to a current that flows along its boundary and is nearly linear in the increase in chemical potential. We also discuss the connection with the double commutator formula with modified position operators for the Hall conductivity derived in [Wesle et al., Commun. Math. Phys. 406, 199 (2025)] as the linear response coefficient of the macroscopic current response to the adiabatic application of a constant electric field.

arXiv:2506.13581Journal of Mathematical PhysicsFeb 1, 20260 citations

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}

{q}^{2}

for small wave vector q, with a proportionality constant given by the Hall quantum number.

{\mathit{e}}^{2}

/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.

{\mathit{k}}_{\mathit{B}}

T,\ensuremath{\epsilon}).

\varepsilon

ε is of order

\mathcal {O}(\varepsilon ^\infty )

O ( ε ∞ ) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in

\varepsilon

ε up to terms of order

\mathcal {O}(\varepsilon ^\infty )

O ( ε ∞ ) . The proportionality factor

\sigma _\textrm{H}

σ H is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for

d=2

d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with

\sigma _{\textrm{H}}

σ H in expectation up to

\mathcal {O}(\varepsilon ^\infty )

O ( ε ∞ ) , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.

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.