Quantum Coherence in Hall Effect

• The Coherence Hall Effect is a family of phenomena where quantum coherence, rather than classical Lorentz forces, drives transverse Hall responses.
• It includes engineered phase coherence in quantum Hall interferometers, coherence recovery from electron injection, and interaction-induced interference periodicity.
• The effect also serves as a precise diagnostic tool that reveals hidden coherence in layered, correlated, and disordered systems, offering insights into quantum transport.

The term “Coherence Hall Effect” is plausibly understood as an umbrella designation for phenomena in which a Hall response, a Hall-like transverse current, or a Hall measurement is governed by quantum coherence rather than by a purely semiclassical carrier picture. Across the works considered here, coherence enters in several distinct ways: as tunable phase coherence of integer quantum Hall edge states, as interaction-induced transfer of interference periodicity between adjacent Hall channels, as coherent accumulation of tunneling or Andreev phases that generates transverse transport, as interlayer quantum hybridization that produces an intrinsic planar Hall effect in atomically thin systems, and as a transverse probe that reveals coherence scales that are obscure in longitudinal transport (Huynh et al., 2012, Sivan et al., 2017, Zeng, 1 Mar 2026, Zheng et al., 2024, Zhang et al., 10 Jun 2026, Yi et al., 2016).

1. Conceptual scope

The papers suggest two broad meanings for the expression. In one class, coherence is the mechanism of the Hall response itself. This includes a tunneling Hall effect produced by coherent multiple reflections and a light-induced phase in a semi-Dirac normal-metal/normal-metal/superconductor junction, a tunneling valley Hall effect generated by coherent geometric phase accumulation in an $\alpha$-$\mathcal{T}_3$ double barrier, a non-adiabatic Hall effect tied to field-induced electron-hole coherence near a Berry-curvature hot spot, and an intrinsic planar Hall effect in bilayer and trilayer van der Waals systems that disappears when layer coherence is absent (Zeng, 1 Mar 2026, Zeng, 2024, Tu et al., 2020, Zheng et al., 2024).

In the second class, the Hall response is primarily a diagnostic of coherence. In the integer quantum Hall regime, interferometric visibility measures the coherence length of an edge channel and shows that electrostatic control of a neighboring edge state can increase coherence length by nearly a factor of two (Huynh et al., 2012). In the doped Hubbard model, the Hall conductivity reveals a crossover from semi-classical to quantum-coherent transport while the resistivity remains robustly $T$-linear (Zhang et al., 10 Jun 2026). In $\beta$-YbAlB$_4$, the Hall coefficient exposes two distinct coherence or Kondo scales, and in organic semiconductors the Hall effect distinguishes coherent band-like carriers from hopping carriers that partially compensate the Hall voltage (O'Farrell et al., 2012, Yi et al., 2016).

A recurring implication is that “Hall” here should not be restricted to the textbook Lorentz-force geometry. Several of the cited mechanisms operate without relying on ordinary magnetic deflection, and some produce no net transverse charge current at all, but rather a transverse valley current or a planar Hall conductivity induced by coherence between internal quantum sectors (Zeng, 2024, Zheng et al., 2024).

2. Integer quantum Hall edge-state coherence engineering and recovery

In the integer quantum Hall regime at filling factor $\nu=2$, a Mach-Zehnder interferometer can be used as a direct coherence probe for chiral edge transport. The relevant physical setting contains two co-propagating one-dimensional edge states: the outer edge state (OES), which interferes, and the inner edge state (IES), which acts as an effective environment. The visibility of the Aharonov-Bohm signal is described by

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$

and the decoupling gate $DG1$ reshapes the IES into a small loop of length about $8\,\mu\text{m}$, with energy spacing

$$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$

The coupling is monitored through

$\mathcal{T}_3$0

Reducing the coupling lowers the dephasing rate $\mathcal{T}_3$1, and the temperature dependence of the coherence length can be varied by about a factor of two (Huynh et al., 2012).

This experiment establishes a concrete sense in which coherence is engineered rather than merely measured. The strengthening of phase coherence at finite temperature arises from a reduction of the coupling between co-propagating edge states, not from temperature itself enhancing coherence. The same work also separates two contributions to finite-bias visibility: a Gaussian envelope,

$\mathcal{T}_3$2

which is surprisingly insensitive to the coupling, and a beating component,

$\mathcal{T}_3$3

which is strongly affected by inter-edge coupling. The theory of Levkivskyi and Sukhorukov is invoked to interpret this beating in terms of fast charged and slow neutral collective modes (Huynh et al., 2012).

A related but distinct issue is coherence recovery after injection of a single electron into a $\mathcal{T}_3$4 edge. In the geometry analyzed by Tewari et al., an electron emitted from a quantum dot with energy $\mathcal{T}_3$5 fractionalizes into charge and neutral modes with velocities

$\mathcal{T}_3$6

For a linear plasmon spectrum and a symmetric interferometer, the interference current is the difference of two quasiparticle contributions, and these cancel exactly at large $\mathcal{T}_3$7, so visibility vanishes in the minimal model. The key point is that visibility loss comes from destructive interference between two internally coherent fractional excitations, not from their individual decoherence (Goremykina et al., 2017).

The same analysis shows that coherence recovery occurs when that exact balance is broken. Dissipation or dispersion before the interferometer lowers the effective energy density of the wave packet, introducing a scale $\mathcal{T}_3$8 and a threshold

$\mathcal{T}_3$9

estimated as

$T$0

Experimentally, the visibility drops and then saturates to a plateau, with a recovery of about $T$1 at higher energies. A periodic bias between the interferometer arms is proposed as a way to reveal directly that the charge and neutral modes remain coherent and only cancel in the unperturbed symmetric setup (Goremykina et al., 2017).

3. Interaction-induced interference periodicity in Hall interferometers

A more stringent form of coherence control appears in screened Fabry-Perot interferometers in the integer quantum Hall effect regime. At bulk filling $T$2, the outermost edge channel exhibits conventional single-electron interference with flux period $T$3 in the usual Aharonov-Bohm condition $T$4. At $T$5, however, the oscillation period doubles, corresponding to $T$6, and shot-noise measurements show an effective interfering charge $T$7. The Aharonov-Bohm phase is written as

$T$8

and the dominant frequency at $T$9 corresponds to

$\beta$0

The measured charge tracks visibility: when visibility is high, the extracted $\beta$1 approaches $\beta$2; as visibility is reduced, it continuously approaches $\beta$3 (Sivan et al., 2017).

The central observation is that the coherence and interference periodicity of the interfering outermost channel are solely determined by the coherence and enclosed flux of the adjacent first-inner channel. At $\beta$4, grounding the inner channel does not affect outer-channel interference. At $\beta$5, directing the first-inner channel to the grounded center contact quenches the interference of the outermost channel completely, even though the average current carried by the outermost channel is unchanged. This shows that the coherence of the outermost channel is not self-contained; it depends on the adjacent channel’s coherence (Sivan et al., 2017).

A second set of experiments uses an interior island and two internal quantum point contacts to tune which channel encloses the relevant flux. At $\beta$6, the transition between Aharonov-Bohm periods occurs when the first-inner channel is fully reflected around the island, while the outermost channel can still pass through the internal constriction. The interference frequency of the outermost channel is therefore set by the flux enclosed by the adjacent inner channel, not just by the geometric loop of the outermost path (Sivan et al., 2017).

In a combined Mach-Zehnder/Fabry-Perot geometry, the naive single-particle picture predicts oscillatory terms such as

$\beta$7

but at $\beta$8 the dominant Fabry-Perot contribution is doubled and additional oscillatory components arise from the influence of the confined first-inner channel. The proposed interpretation is that interactions between the two outer edge channels generate a neutral chiral excitation central to the pairing phenomenon. A narrow constriction can preserve coherence of this neutral object, whereas a wide constriction disrupts recombination and causes dephasing. This suggests that the integer quantum Hall regime hosts interaction-induced composite interfering objects, channel-to-channel coherence transfer, and effective charge doubling even in a nominally simple topological phase (Sivan et al., 2017).

4. Coherent tunneling, Andreev, valley, and non-adiabatic Hall responses

One major class of coherence Hall phenomena is built from phase-coherent scattering amplitudes rather than from conventional semiclassical orbits. In a semi-Dirac normal-metal/normal-metal/superconductor junction, off-resonant circularly polarized light applied to the central normal region generates an effective static Hamiltonian

$\beta$9

with

$_4$0

Multiple reflections between the left boundary and the NS interface make the Andreev amplitude

$_4$1

so the total phase

$_4$2

becomes physically effective only through coherence across repeated scattering events. The paper identifies the phase decomposition

$_4$3

whose last term is odd in $_4$4. This yields asymmetric Andreev reflection,

$_4$5

and a transverse conductance $_4$6 that is odd in $_4$7,

$_4$8

Right- and left-handed circularly polarized light therefore reverse the tunneling Hall current, whereas the longitudinal conductance is insensitive to light handedness and acquires only a finite phase shift with varying intensity (Zeng, 1 Mar 2026).

A parallel mechanism appears in an $_4$9-$\nu=2$0 double-barrier junction. There, the backreflected electron at each barrier acquires a valley-dependent geometric phase

$\nu=2$1

with

$\nu=2$2

For two barriers, the total transmission takes the Fabry-Pérot form

$\nu=2$3

with

$\nu=2$4

Because $\nu=2$5 is valley odd and angle asymmetric, the coherent denominator produces skew tunneling,

$\nu=2$6

Time-reversal symmetry enforces

$\nu=2$7

but a charge-neutral transverse valley current survives,

$\nu=2$8

The effect is electrically tunable by the gate voltages $\nu=2$9 and $\mathcal{V}(T)\propto e^{-T/T_\varphi},$0, and it vanishes for equal barrier heights $\mathcal{V}(T)\propto e^{-T/T_\varphi},$1 (Zeng, 2024).

A third coherence-based route is the non-adiabatic Hall effect at a Berry-curvature hot spot. In this framework, an electric field induces valence-conduction coherence described by

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$2

In the adiabatic limit, the anomalous Hall velocity reduces to

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$3

but beyond that limit the coherent evolution produces an ac Hall velocity that retains memory of the history of the state. When environmental noise is added, the system relaxes into field-dressed eigenstates with dressed gap

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$4

and the Hall velocity becomes

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$5

The dc nonlinear Hall response then contains only odd powers of the electric field. In this setting, the Hall current remains controlled by Berry curvature, but only after it has been renormalized by nonperturbative field-induced electron-hole coherence (Tu et al., 2020).

Taken together, these mechanisms suggest a coherent-transmission paradigm in which Hall-like transport emerges from phase accumulation in scattering amplitudes, not from a conventional Lorentz-force picture. The relevant coherence may reside in Andreev phases, valley-dependent geometric phases, or field-dressed interband amplitudes, but in each case the transverse response disappears or changes character when the corresponding coherent phase structure is lost (Zeng, 1 Mar 2026, Zeng, 2024, Tu et al., 2020).

5. Layer-coherent planar, multipole, and valley Hall transport

A distinct formulation of coherence Hall physics is provided by the intrinsic planar Hall effect of layer-coherent electrons in bilayer and trilayer van der Waals materials. The starting point is that a few-layer electron carries an out-of-plane charge dipole. For a bilayer,

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$6

and for general $\mathcal{V}(T)\propto e^{-T/T_\varphi},$7-layer systems,

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$8

Lateral motion produces an in-plane magnetic dipole moment,

$\mathcal{V}(T)\propto e^{-T/T_\varphi},$9

which couples directly to an in-plane magnetic field through

$DG1$0

This coupling can also be written as a magneto-Stark effect. The Hall conductivity is then expressed in Berry-curvature form as

$DG1$1

and to linear order in the in-plane field,

$DG1$2

The crucial statement is that the effect disappears if the layer degree of freedom is conserved, so the intrinsic planar Hall effect is specifically a consequence of hybridized layer wave functions (Zheng et al., 2024).

This mechanism is explicitly distinguished from conventional planar Hall scenarios. In strictly two-dimensional van der Waals layers there is no out-of-plane orbital motion, so the three-dimensional orbital-motion mechanism is quenched. The paper also targets systems in which the spin-orbit-based mechanism is absent or ineffective. The resulting planar Hall response is therefore a spinless, layer-specific, 2D intrinsic Hall mechanism (Zheng et al., 2024).

Symmetry breaking is essential. The largest point group that can support the intrinsic planar Hall effect is

$DG1$3

and strain or interlayer sliding activates the response. In strained twisted bilayer graphene on aligned hBN, $DG1$4 symmetry forbids the effect in the absence of strain, but with $DG1$5 uniaxial strain the intrinsic planar Hall conductivity becomes sizable, with angular dependence

$DG1$6

Around charge neutrality,

$DG1$7

and the planar Hall coefficient can reach

$DG1$8

In twisted trilayer MoTe$DG1$9, interlayer sliding over roughly $8\,\mu\text{m}$0–$8\,\mu\text{m}$1 of the moiré periodicity yields sizable conductivity, with a representative value of about

$8\,\mu\text{m}$2

at low hole doping (Zheng et al., 2024).

The same formalism extends beyond ordinary charge transport. Because current can be layer resolved, the paper introduces a planar multipole Hall effect, including a dipole Hall current defined as the difference between currents on the two outer layers in twisted trilayer MoTe$8\,\mu\text{m}$3. Heterostrain further acts as a layer-dependent vector potential,

$8\,\mu\text{m}$4

which functions like an in-plane pseudo-magnetic field and generates a time-reversal-preserving valley Hall effect even in centrosymmetric materials. The same layer mechanism also provides a route to a quantized Hall response driven by an in-plane field through a topological band inversion in twisted trilayer MoTe$8\,\mu\text{m}$5 (Zheng et al., 2024).

A plausible broader implication is that layer coherence should be regarded as an internal quantum degree of freedom on the same conceptual footing as spin, valley, or edge-channel index. In this formulation, the Hall response is generated by coherence within that sector rather than added on top of it.

6. Hall measurements as probes of coherence formation in correlated and disordered conductors

Several works use the Hall channel not as the response to be generated, but as the best available detector of coherence formation. In the doped Hubbard model with next-nearest-neighbor hopping $8\,\mu\text{m}$6 and magnetic field $8\,\mu\text{m}$7, determinant quantum Monte Carlo shows that the resistivity remains robustly $8\,\mu\text{m}$8-linear across parameter sets, whereas the Hall conductivity $8\,\mu\text{m}$9 and Hall coefficient $$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$0 are highly sensitive to particle-hole asymmetry, Fermi-surface geometry, and many-body spectral rearrangements. The Hall coefficient is

$$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$1

and the high-temperature expansion of the Hall conductivity is

$$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$2

The temperature at which $$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$3 departs from its high-$$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$4 behavior coincides with the minimum in the average double occupancy $$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$5, defining an empirical coherence scale $$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$6. Only a narrow region yields the familiar $$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$7 behavior; more generally,

$$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$8

The conclusion is that the Hall response reveals a crossover from semi-classical to quantum-coherent transport that is mostly hidden in $$E_\delta \sim h v_D/L_\delta \sim 50\,\mu\text{eV}.$$9 (Zhang et al., 10 Jun 2026).

In $\mathcal{T}_3$00-YbAlB$\mathcal{T}_3$01, the Hall effect reveals a two-component coherence structure. The Hall resistivity is decomposed as

$\mathcal{T}_3$02

so that

$\mathcal{T}_3$03

On cooling from $\mathcal{T}_3$04 K, $\mathcal{T}_3$05 changes sign around $\mathcal{T}_3$06 K, reaches a clear minimum near $\mathcal{T}_3$07 K, rises again below $\mathcal{T}_3$08 K, and shows a kink just below $\mathcal{T}_3$09 K. Over roughly one decade in temperature, from about $\mathcal{T}_3$10 K down to $\mathcal{T}_3$11 K,

$\mathcal{T}_3$12

indicating resonant skew scattering from incoherent local moments down to about $\mathcal{T}_3$13 K. The magnetic part of the resistivity peaks at

$\mathcal{T}_3$14

whereas the Hall coefficient has its minimum only at

$\mathcal{T}_3$15

The field dependence is modeled by

$\mathcal{T}_3$16

and the growth of $\mathcal{T}_3$17 below $\mathcal{T}_3$18 K with saturation near $\mathcal{T}_3$19 K is interpreted as emergence of a second coherent transport channel. The Hall effect thus exposes a two-stage evolution of $\mathcal{T}_3$20-$\mathcal{T}_3$21 hybridization that the longitudinal channel alone does not resolve (O'Farrell et al., 2012).

Organic semiconductors provide an analogous but materially different example. In organic field-effect transistors, coexisting band-like carriers $\mathcal{T}_3$22 and hopping carriers $\mathcal{T}_3$23 give a longitudinal conductivity

$\mathcal{T}_3$24

Only the band carriers feel the magnetic Lorentz force directly, but the hopping carriers respond to the Hall electric field. The Hall field becomes

$\mathcal{T}_3$25

and the Hall mobility is

$\mathcal{T}_3$26

The carrier coherence factor is

$\mathcal{T}_3$27

This framework explains the underdeveloped Hall effect characterized by

$\mathcal{T}_3$28

and also clarifies that near agreement between Hall and FET measurements does not necessarily imply purely band transport. Here again, the Hall response measures the balance between coherent and incoherent carriers more sharply than a single drift mobility can (Yi et al., 2016).

7. Unifying principles and interpretive boundaries

Across these examples, the common principle is that the transverse response is unusually sensitive to coherent closed-loop, phase-accumulating, or sector-hybridizing processes. In integer quantum Hall interferometers, the relevant coherence resides in coupled chiral edge channels or in the charge and neutral modes formed by fractionalization (Huynh et al., 2012, Goremykina et al., 2017, Sivan et al., 2017). In semi-Dirac, $\mathcal{T}_3$29-$\mathcal{T}_3$30, and Berry-hot-spot settings, it resides in coherent multiple reflections, geometric phase accumulation, or field-induced electron-hole mixing (Zeng, 1 Mar 2026, Zeng, 2024, Tu et al., 2020). In layertronics, it is the interlayer coherence of the Bloch state itself (Zheng et al., 2024). In heavy-fermion, Hubbard, and organic systems, the Hall response functions as a selective readout of when coherence is formed, transferred, or only partially developed (Zhang et al., 10 Jun 2026, O'Farrell et al., 2012, Yi et al., 2016).

A frequent misconception is that these results can all be subsumed under ordinary Hall transport with modified parameters. The cited works indicate otherwise. Some effects produce transverse responses without relying on conventional orbital motion in a magnetic field; some persist with zero net transverse charge current but finite transverse valley current; some manifest as changes in interference periodicity or visibility rather than as a simple Hall coefficient; and some show that longitudinal resistivity can remain nearly featureless while the Hall channel changes qualitatively (Zeng, 2024, Zheng et al., 2024, Zhang et al., 10 Jun 2026).

Another important boundary concerns terminology. A plausible synthesis is that “Coherence Hall Effect” denotes not a single universal mechanism but a family of phenomena in which Hall observables are controlled by quantum coherence. This suggests a classification into at least three forms: engineered coherence Hall transport, where coherence is deliberately tuned; coherence-mediated Hall transport, where the transverse response is created by coherent phases or hybridization; and Hall-detected coherence, where the Hall channel reveals coherence scales obscured elsewhere. That synthesis is interpretive rather than formal, but it captures the structure common to the cited literature.

In that inferred sense, the coherence Hall effect marks a shift in emphasis from carrier deflection alone to the quantum organization of the states that carry, screen, tunnel, interfere, or hybridize. The Hall response then becomes either a direct output of coherence or a particularly sensitive lens on how coherence emerges.