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.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.

Physical review. B, Condensed matterJul 15, 19856 citations

Prediction of a layer nonlinear Hall effect in bilayer nonmagnetic or antiferromagnetic systems

Zehou Li, Shenda He, Pan Zhou +3 more

Nonlinear Hall effects provide a powerful probe of quantum geometry in solids and enable rectification phenomena beyond the constraints of linear response. In this Letter, we predict a \emph{layer nonlinear Hall effect} (LNHE) in stacked bilayer systems composed of nonmagnetic or antiferromagnetic materials with a vanishing linear Hall conductivity. In such systems, the second- or third-order nonlinear Hall responses are intrinsically layer odd: the contributions from the two constituent layers have equal magnitude but opposite sign, resulting in exact cancellation under layer-exchange symmetry. An out-of-plane electric field $E_z$ can break this symmetry, thereby unveiling the hidden response and converting it into a switchable macroscopic nonlinear Hall signal. Using a minimal $k\!\cdot\!p$ model, we demonstrate that the LNHE can originate from the Berry curvature dipole mechanism. A systematic symmetry analysis of all 80 layer groups further yields a complete classification of the symmetry constraints and stacking configurations that allow for this type of LNHE. Beyond this mechanism, additional symmetry analysis reveals that LNHE may also arise from quantum metric dipole or inversed mass dipole. Remarkably, even in cases where second-order nonlinear Hall responses are symmetry forbidden, a third-order LNHE can still survive in certain stacked bilayer configurations. First-principles calculations on representative bilayers---nonmagnetic 1T $'$ -WTe $_2$ and 1T $'$ -ReS $_2$ ---explicitly demonstrate electrically reversible second-order LNHE, in full agreement with our symmetry-based predictions. Overall, our results establish LNHE as a universal phenomenon in a wide range of layered quantum materials and provide a robust route toward electrically tunable nonlinear transport.Nonlinear Hall effects provide a powerful probe of quantum geometry in solids and enable rectification phenomena beyond the constraints of linear response. In this Letter, we predict a \emph{layer nonlinear Hall effect} (LNHE) in stacked bilayer systems composed of nonmagnetic or antiferromagnetic materials with a vanishing linear Hall conductivity. In such systems, the second- or third-order nonlinear Hall responses are intrinsically layer odd: the contributions from the two constituent layers have equal magnitude but opposite sign, resulting in exact cancellation under layer-exchange symmetry. An out-of-plane electric field $E_z$ can break this symmetry, thereby unveiling the hidden response and converting it into a switchable macroscopic nonlinear Hall signal. Using a minimal $k\!\cdot\!p$ model, we demonstrate that the LNHE can originate from the Berry curvature dipole mechanism. A systematic symmetry analysis of all 80 layer groups further yields a complete classification of the symmetry constraints and stacking configurations that allow for this type of LNHE. Beyond this mechanism, additional symmetry analysis reveals that LNHE may also arise from quantum metric dipole or inversed mass dipole. Remarkably, even in cases where second-order nonlinear Hall responses are symmetry forbidden, a third-order LNHE can still survive in certain stacked bilayer configurations. First-principles calculations on representative bilayers---nonmagnetic 1T$'$-WTe$_2$ and 1T$'$-ReS$_2$---explicitly demonstrate electrically reversible second-order LNHE, in full agreement with our symmetry-based predictions. Overall, our results establish LNHE as a universal phenomenon in a wide range of layered quantum materials and provide a robust route toward electrically tunable nonlinear transport.

cond-mat.mtrl-sciCross-listedarXiv:2608.20724Aug 24, 2026

Inside the quantum Hall effect

Wolfgang Ketterle

Nature PhysicsFeb 1, 20158 citations

Hall conductivity reveals the nature of quantum coherence in strongly correlated metals

Emily Z. Zhang, Thomas Devereaux

Linear-in-temperature resistivity is a hallmark for strange metallic transport, and appears universally in many strongly correlated electron systems. However, the focus on the longitudinal channel often overshadows the profound microscopic insights contained within the transverse response. Here, we utilize numerically exact determinantal quantum Monte Carlo simulations of the doped Hubbard model in a magnetic field to calculate longitudinal and transverse transport. We demonstrate that while the resistivity is robustly $T$ -linear across parameter sets, the Hall response is highly sensitive to particle-hole asymmetry, Fermi surface topology, and many-body correlation effects. Specifically, the combination of these effects determine a crossover scale in which the system becomes quantum-coherent, and is reflected in the Hall conductivity. Our results demonstrate that while the $T$ -linearity in resistivity appears universal, the Hall response reveals a crossover from semi-classical to quantum-coherent transport otherwise masked in the longitudinal channel.Linear-in-temperature resistivity is a hallmark for strange metallic transport, and appears universally in many strongly correlated electron systems. However, the focus on the longitudinal channel often overshadows the profound microscopic insights contained within the transverse response. Here, we utilize numerically exact determinantal quantum Monte Carlo simulations of the doped Hubbard model in a magnetic field to calculate longitudinal and transverse transport. We demonstrate that while the resistivity is robustly $T$-linear across parameter sets, the Hall response is highly sensitive to particle-hole asymmetry, Fermi surface topology, and many-body correlation effects. Specifically, the combination of these effects determine a crossover scale in which the system becomes quantum-coherent, and is reflected in the Hall conductivity. Our results demonstrate that while the $T$-linearity in resistivity appears universal, the Hall response reveals a crossover from semi-classical to quantum-coherent transport otherwise masked in the longitudinal channel.

arXiv:2606.12665Jun 10, 20260 citations

{q}^{2}

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

E_z

can break this symmetry, thereby unveiling the hidden response and converting it into a switchable macroscopic nonlinear Hall signal. Using a minimal

k\!\cdot\!p

model, we demonstrate that the LNHE can originate from the Berry curvature dipole mechanism. A systematic symmetry analysis of all 80 layer groups further yields a complete classification of the symmetry constraints and stacking configurations that allow for this type of LNHE. Beyond this mechanism, additional symmetry analysis reveals that LNHE may also arise from quantum metric dipole or inversed mass dipole. Remarkably, even in cases where second-order nonlinear Hall responses are symmetry forbidden, a third-order LNHE can still survive in certain stacked bilayer configurations. First-principles calculations on representative bilayers---nonmagnetic 1T

'

-WTe

_2

and 1T

'

-ReS

_2

---explicitly demonstrate electrically reversible second-order LNHE, in full agreement with our symmetry-based predictions. Overall, our results establish LNHE as a universal phenomenon in a wide range of layered quantum materials and provide a robust route toward electrically tunable nonlinear transport.

T

-linear across parameter sets, the Hall response is highly sensitive to particle-hole asymmetry, Fermi surface topology, and many-body correlation effects. Specifically, the combination of these effects determine a crossover scale in which the system becomes quantum-coherent, and is reflected in the Hall conductivity. Our results demonstrate that while the

T

-linearity in resistivity appears universal, the Hall response reveals a crossover from semi-classical to quantum-coherent transport otherwise masked in the longitudinal channel.