Nonlocal Hall Conductivity in Quantum Regimes

• Nonlocal Hall conductivity is a wave-vector-dependent Hall response emerging in systems with spatial dispersion, capturing quantum geometry and topological invariants.
• It quantifies finite-q corrections in Bloch bands, where quadratic terms reveal band-projected electric quadrupoles and establish links with Hall viscosity.
• It underlies device-level phenomena in multiterminal setups, where nonlocal resistance measurements expose spin, valley, and anomalous Hall effects.

Nonlocal Hall conductivity denotes Hall response beyond the strictly local, homogeneous limit. In translationally invariant settings it is the wave-vector-dependent Hall conductivity $\sigma_{xy}(\mathbf q,\omega)$, typically expanded at small $\mathbf q$; in slowly inhomogeneous backgrounds it can denote an averaged Hall response written in phase space; and in multiterminal transport it is often accessed indirectly through nonlocal resistance rather than through a distinct conductivity tensor. Across these usages, the common theme is that Hall transport depends on spatial variation, internal geometry, or spatially separated conversion processes rather than only on the local $q=0$ conductivity (Shu et al., 26 May 2026, Fialkovsky et al., 2019, Tuan et al., 2016).

1. Meanings and scope

The term is not used uniformly across the literature. In lattice-band theory, nonlocal Hall conductivity is the static Hall response to a spatially nonuniform electric field with small wave vector $\mathbf q$, with the isotropic expansion

$$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$

In this usage, the $q^2$ term is a genuine transport coefficient tied to band geometry rather than merely a formal gradient correction (Shu et al., 26 May 2026).

In continuum quantum Hall and Dirac-Landau problems, the same phrase refers to the finite-$q$ Hall conductivity obtained from current-current linear response. For gapped graphene, for example, the valley-resolved quantity is defined by

$$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$

and is then expanded through order $q^2$ (Sherafati et al., 2018).

A different usage appears in inhomogeneous magnetic backgrounds. There the Hall response is formulated with the Wigner-transformed two-point Green function $G_W(R,p)$ and the Moyal star product, and the final object emphasized is an area-averaged Hall conductivity,

$\mathbf q$0

with $\mathbf q$1 a phase-space topological invariant (Zhang et al., 2019, Fialkovsky et al., 2019).

In multiterminal experiments and device theory, by contrast, the directly measured quantity is usually a nonlocal resistance. Several works are explicit that a measured nonlocal voltage is not, by itself, a nonlocal Hall conductivity. In Au-decorated graphene, the bulk quantities computed are the spin Hall conductivity and spin Hall angle, whereas the experimental observable is

$\mathbf q$2

which can contain large non-spin-Hall backgrounds (Tuan et al., 2016).

Setting	Central quantity	Meaning of nonlocality
Bloch-band response	$\mathbf q$3 or small-$\mathbf q$4 expansion	Spatial dispersion in a homogeneous crystal
Inhomogeneous-field topology	$\mathbf q$5	Phase-space dependence on position and momentum
Multiterminal transport	$\mathbf q$6	Remote voltage generated by Hall-mediated neutral or spin currents
Interfacial Hall effects	Layer-resolved Hall current	Spatial separation of magnetization, spin transport, and spin-orbit conversion

This plurality of meanings is consequential. It implies that statements about “nonlocal Hall response” must specify whether the subject is a constitutive tensor, an averaged topological response, or a device-level transresistance.

2. Finite-wave-vector Hall response in lattice bands

For an isolated filled Chern band in two dimensions, the finite-$\mathbf q$7 Hall response was developed as a lattice-band theory in which the quadratic wave-vector correction is controlled by quantum geometry. The zeroth-order Hall conductivity remains the quantized value

$\mathbf q$8

for Chern number $\mathbf q$9, but the $q=0$0 coefficient is governed by the Brillouin-zone average of the product $q=0$1, where $q=0$2 is the quantum metric and $q=0$3 is the Berry curvature (Shu et al., 26 May 2026).

A central result is that, in a Chern band, nonlocal Hall conductivity is not just a higher-gradient correction to $q=0$4. Its microscopic origin is a band-projected electric quadrupole determined by quantum geometry. In isotropic notation the quadratic term is the coefficient of $q=0$5; in the general lattice setting it is tensorial. This places the finite-wave-vector Hall response on the same footing as other geometric response coefficients, rather than treating it as a purely electrodynamic afterthought (Shu et al., 26 May 2026).

The same work separates the quadratic response into a uniform topological part and lattice-specific fluctuation corrections. The uniform part is what survives in the Landau-level-like limit of uniform geometry, while the correction is a covariance term built from metric and curvature fluctuations over the Brillouin zone. This shows explicitly why a strict continuum relation between Hall viscosity and nonlocal Hall conductivity fails on a lattice: once Galilean invariance is lost, velocity is no longer locked to momentum, and the electromagnetic response acquires lattice-geometric corrections (Shu et al., 26 May 2026).

This formulation also identifies an electrical diagnostic of band quality. The small-$q=0$6 Hall response is presented as a transport probe of “geometric idealness,” meaning proximity to the uniform-geometry structure of a Landau level. A plausible implication is that finite-$q=0$7 Hall measurements can distinguish topologically equivalent Chern bands that differ strongly in their internal quantum geometry.

3. Quantum geometry, Hall viscosity, and ideal Chern bands

The geometric interpretation proceeds through band projection. After projection to an isolated band, the coordinates satisfy the noncommutative algebra

$q=0$8

so Berry curvature sets the projected-coordinate algebra. The primitive electric quadrupole is then built from the projected coordinates, and its expectation value in a semiclassical wave packet equals the quantum metric. The data summarize this point as follows: Berry curvature controls the algebra of projected coordinates, while the quantum metric controls the intrinsic quadrupole carried by the projected wave packet; together they are the imaginary and real parts of the quantum geometric tensor (Shu et al., 26 May 2026).

Because Hall viscosity is determined by expectation values of the projected quadrupole, and the same quadrupole controls the $q=0$9 Hall response in a nonuniform electric field, the paper establishes a lattice viscosity-conductivity relation. Its key claim is not that Hall viscosity alone fixes $\mathbf q$0, but that both arise from the same projected quadrupole structure. In generic lattice bands the relation contains covariance corrections; in ideal bands the structure simplifies sharply (Shu et al., 26 May 2026).

Ideal bands are defined by the trace condition

$\mathbf q$1

For a sign-definite ideal Chern band satisfying this condition, the remaining deviation from the Landau-level form is quantified solely by Berry-curvature fluctuations. The paper introduces a dimensionless nonlocal Hall ratio

$\mathbf q$2

and shows that in a sign-definite ideal band this reduces to

$\mathbf q$3

while in the uniform-curvature Landau-level limit $\mathbf q$4 (Shu et al., 26 May 2026).

Three representative models were compared with this prediction. The Kapit–Mueller model gives $\mathbf q$5 close to $\mathbf q$6, matching the nearly uniform-curvature ideal limit. Chiral twisted bilayer graphene shows a moderate enhancement of $\mathbf q$7, attributed to finite Berry-curvature fluctuations in an otherwise ideal-like moiré Chern band. The Qi–Wu–Zhang model shows a substantially larger $\mathbf q$8, signaling stronger nonideal lattice geometry. Within this framework, nonlocal Hall conductivity becomes an electrical signature of Hall viscosity and a transport diagnostic of geometric idealness (Shu et al., 26 May 2026).

The same analysis emphasizes the geometric inequality

$\mathbf q$9

Upon integration, this yields a lower bound on the isotropic Hall-viscosity density, $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$0. This suggests that lower bounds on dissipationless stress response and bounds on nonlocal Hall transport are both constrained by local quantum geometry rather than solely by topology.

4. Landau levels and graphene Dirac systems

In strong perpendicular magnetic field, the continuum Dirac theory of gapped graphene provides an explicit finite-$$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$1 Hall-conductivity problem distinct from the lattice-band geometric setting. The system is a non-topological insulator with two valleys and a Semenoff mass, treated at zero temperature within linear response. The nonlocal Hall conductivity is computed from Landau-level matrix elements and expanded through order $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$2 at all frequencies (Sherafati et al., 2018).

One of the paper’s sharpest conclusions is that both Hall viscosity and Hall conductivity vanish when the Fermi level lies in the gap separating the lowest Landau level in the conduction band from the highest Landau level in the valence band. In that regime,

$$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$3

for the total two-valley system. Outside the gap, the familiar gapless-graphene formulas are recovered. In the dc limit and for $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$4, the total conductivity through order $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$5 is

$$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$6

A striking feature is that the total dc $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$7 coefficient is independent of the mass gap, even though the valley-resolved conductivities depend strongly on it (Sherafati et al., 2018).

The valley structure is essential. For a single valley, the nonlocal Hall conductivity remains finite and gap dependent, whereas the single-valley Hall viscosity diverges with opposite signs in the two valleys and only becomes finite after summation. The paper therefore concludes that the Hoyos–Son formula relating Hall viscosity to the $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$8 coefficient of $$\sigma_H(\mathbf q)=\sigma_H^{(0)}+\sigma_H^{(2)}q^2+O(q^4).$$9 cannot be applied valley by valley in gapped graphene, but only to the full two-valley system (Sherafati et al., 2018).

This example clarifies an important conceptual point. Finite-$q^2$0 Hall conductivity can remain a well-defined valley-resolved observable even when the corresponding stress response is ill-defined at the same level of resolution. The problem of defining a meaningful “valley Hall viscosity” is stated to remain open (Sherafati et al., 2018).

5. Multiterminal nonlocal transport and Hall-mediated resistance

In device physics, “nonlocal Hall” phenomena are often operationalized through remote voltages. In Au-decorated graphene, numerically exact Kubo and Landauer–Büttiker calculations were used to obtain the spin Hall conductivity, the spin Hall angle, and the six-terminal nonlocal resistance, but the study is explicit that it does not compute a distinct nonlocal Hall conductivity. Its central decomposition is

$q^2$1

with positive spin-Hall and Ohmic terms, a negative quasiballistic term, and a positive pseudodiffusive contribution. The authors also show that $q^2$2 even when $q^2$3, so a nonlocal voltage can survive with spin-orbit coupling fully switched off (Tuan et al., 2016).

This has direct interpretive consequences. A large nonlocal resistance signal is not, in itself, evidence for a nonlocal Hall conductivity, because multiterminal scattering backgrounds can be comparable to or larger than the spin-Hall contribution. The same work also reports strong suppression of the spin Hall angle at room temperature and proposes a modified geometry with adatoms removed from the connecting channel in order to suppress pseudodiffusive and quasiballistic contamination (Tuan et al., 2016).

A related but experimentally earlier graphene literature measured a giant nonlocal response near the Dirac point in Hall-bar geometry. There the observable was again a nonlocal resistance $q^2$4, not a conductivity tensor, and the signal was interpreted in terms of long-range neutral flavor currents generated by spin Hall and possibly valley Hall effects. The classical Ohmic estimate,

$q^2$5

is exponentially small for $q^2$6, yet the measured finite-field response near neutrality is orders of magnitude larger. The supplement presents the neutral-current diffusion form

$q^2$7

which connects remote voltage to Hall-mediated flavor conversion and diffusion length $q^2$8 (Abanin et al., 2011).

A nonequilibrium Green-function theory of six-terminal graphene bars reformulated the same phenomenon in terms of direct and inverse Zeeman-splitting-driven spin Hall effects. In that framework the nonlocal signal is

$q^2$9

while the generating coefficient in a four-terminal bridge is the spin Hall conductance

$q$0

The theory finds that momentum-relaxing dephasing reduces the spin Hall conductance and nonlocal voltage at the Dirac point by two orders of magnitude and washes out side features away from the Dirac point (Chen et al., 2011).

In valley Hall systems the same transduction logic reappears. For gated bilayer graphene and related two-dimensional semiconductors, the local charge Hall conductivity vanishes by time-reversal symmetry, while the local valley Hall conductivity $q$1 is nonzero. The measurable nonlocal response is then a sequence of valley generation, diffusion, and inverse conversion, with the large-distance good-metal scaling

$q$2

The paper emphasizes that the source term for valley accumulation is nonzero only where $q$3 changes spatially, effectively at the sample boundary, so the measured nonlocal response depends crucially on edge pumping and on enhanced edge longitudinal conductivity rather than on bulk $q$4 alone (Sekine et al., 2020).

6. Interfacial and multilayer nonlocal anomalous Hall effects

A distinct line of work uses “nonlocal” to denote spatial separation of magnetization and spin-orbit conversion. In heavy-metal/ferromagnetic-insulator bilayers such as Pt/YIG, the nonlocal anomalous Hall effect was formulated as the combined action of bulk spin Hall physics and spin-dependent scattering at the magnetic interface, even when no induced magnetization exists in the metal itself. The transport theory is written in terms of integral conductivity operators $q$5, $q$6, $q$7, and $q$8, with the Hall current density

$q$9

The first term, $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$0, is the nonlocal anomalous Hall current and is first order in the spin Hall angle $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$1, in contrast to the older double-spin-Hall mechanism proportional to $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$2 (Zhang et al., 2015).

The same paper makes the nonlocality explicit through kernels $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$3 and $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$4, where $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$5 is purely interfacial and decays with $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$6, showing that the conversion is generated at the magnetic boundary and communicated into the bulk over a mean-free-path scale. Roughness is essential; for a perfectly flat interface one has $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$7, so no interfacial spin-charge conversion occurs. Using parameters $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$8, $$\sigma^{\tau}_{xy}(\mathbf q,\omega)=-\frac{1}{\omega}\operatorname{Im}\left[\chi^{\tau}_{xy}(\mathbf q,\omega)\right],$$9, $q^2$0, $q^2$1, and $q^2$2 for Pt(7 nm)/YIG, the estimated anomalous Hall angle is

$q^2$3

stated to be in good agreement with experiment (Zhang et al., 2015).

Metallic multilayers provide a closely related realization. In Pt/$q^2$4 stacks, the measured anomalous Hall effect can be dominated by spin-polarized current leaking from the ferromagnet into the adjacent heavy metal and then being converted there into a transverse charge current. The extracted spin Hall angle for Pt is

$q^2$5

opposite in sign to the effective Hall conversion of $q^2$6, leading to an AHE inversion for thin magnetic multilayers with $q^2$7. The paper attributes the inversion to spin-current leakage and anisotropic electron scattering at Pt/(Co,Ni) interfaces, and states that magnetic proximity effects cannot explain the observations (Dang et al., 2019).

This multilayer perspective generalizes the notion of nonlocal Hall conductivity. The measured Hall response is not a local property of a single magnetic layer; it is a distributed interfacial transport property in which one layer supplies spin polarization and another layer supplies much of the transverse conversion. A plausible implication is that layer-resolved current profiles, interface specularity, and spin-memory loss are intrinsic parts of Hall-coefficient interpretation in spin-orbit-coupled heterostructures.

7. Phase-space topological formulations in inhomogeneous fields

For slowly varying non-uniform magnetic fields, Hall conductivity has also been formulated as a phase-space topological invariant rather than as a finite-$q^2$8 kernel. In a $q^2$9-dimensional tight-binding system, the current density is written in terms of the Wigner-transformed Green function $G_W(R,p)$0 and the Weyl symbol $G_W(R,p)$1 of the inverse propagator,

$G_W(R,p)$2

and the area-averaged Hall conductivity is then identified as

$G_W(R,p)$3

with $G_W(R,p)$4 a phase-space topological invariant built from $G_W(R,p)$5, $G_W(R,p)$6, and the Moyal star product (Zhang et al., 2019).

The significance of this approach is twofold. First, it addresses Hall response in backgrounds that are spatially inhomogeneous but slowly varying, so that response depends simultaneously on position and momentum. Second, it extends perturbatively to interactions: one paper proves that, within the analytic perturbative regime for Coulomb interactions, the same topological formula remains valid if the noninteracting Green function is replaced by the full interacting two-point Green function (Zhang et al., 2019). This is a different sense of nonlocality from finite-$G_W(R,p)$7 transport, because the final emphasized observable is an averaged Hall coefficient rather than a complete conductivity kernel $G_W(R,p)$8 or $G_W(R,p)$9.

A closely related formulation generalizes the Hall invariant to non-uniform tight-binding models in phase space and states explicitly that the local current density is not topological, whereas the current averaged over space and imaginary time is. In $\mathbf q$00 dimensions the averaged Hall conductivity again takes the form $\mathbf q$01, now with $\mathbf q$02 written as a phase-space invariant of the Weyl symbols of the Green function (Fialkovsky et al., 2019).

For constant magnetic field in uniform systems, an alternative magnetic-Brillouin-zone formulation replaces the cumbersome Wigner–Moyal phase-space expression by a Harper-representation formula involving finite $\mathbf q$03 matrices. In that setting the intrinsic dc Hall conductivity is

$\mathbf q$04

where $\mathbf q$05 is a Green-function topological invariant over the magnetic Brillouin zone. This framework is presented for noninteracting systems, while extension to interacting systems and fractional quantum Hall physics is proposed as a hypothesis rather than proved (Suleymanov et al., 2021).

These phase-space and magnetic-Brillouin-zone formulations delimit an important conceptual boundary. They are highly relevant to Hall response in non-uniform fields, but they are not the conventional finite-wave-vector nonlocal Hall conductivity $\mathbf q$06. The literature therefore supports at least three technically distinct categories under the same label: small-$\mathbf q$07 Hall conductivity in homogeneous media, averaged topological Hall response in slowly inhomogeneous backgrounds, and multiterminal Hall-mediated nonlocal resistance in devices.