Quantum Anomalous Hall Regime

• Quantum Anomalous Hall (QAH) regime is a zero-field topological state characterized by quantized Hall conductance, vanishing longitudinal resistance, and chiral edge states due to broken time-reversal symmetry.
• Microscopic routes to QAH include magnetic doping, gate-tuned transitions, and moiré engineering, all leveraging nontrivial band topology and strong spin-orbit coupling.
• Experimental platforms span thin films, multilayers, and mesoscopic devices, revealing tunable high-Chern-number states and unconventional metallic regimes with quantized Hall conductivity.

The quantum anomalous Hall (QAH) regime denotes the zero-external-field realization of quantized Hall transport in a topological state with broken time-reversal symmetry. In its conventional form, it is the transport fingerprint of a Chern insulator: the Hall response is quantized while longitudinal dissipation vanishes, typically $R_{xy}=h/(Ce^2)$ and $R_{xx}\to 0$, or equivalently $\sigma_{xy}=Ce^2/h$, where $C$ is the Chern number (Zhao et al., 2020, Chang et al., 2022, Liu et al., 2015). The same literature also shows that the phrase “QAH regime” has broadened to include gate-tuned plateau transitions, high-Chern-number realizations, thick-film and multilayer implementations, and unconventional metallic or semimetallic variants in which Hall conductivity remains quantized although longitudinal conduction does not vanish (Zhao et al., 2023, Bai et al., 2023, Wan et al., 29 Dec 2025, Liu et al., 27 May 2026).

1. Topological definition and quantization

The defining theoretical structure of the conventional QAH regime is the Chern-insulating band topology of a two-dimensional system with internally broken time-reversal symmetry. In that setting, the Hall conductance is fixed by the TKNN relation

$\sigma_{xy}=\frac{e^2}{h}C,$

with

$$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$

and

$$\Omega_{xy,s}(\mathbf{k})=i\left(\langle \partial_{k_x}u_s|\partial_{k_y}u_s\rangle-\langle \partial_{k_y}u_s|\partial_{k_x}u_s\rangle\right)$$

as the Berry curvature of occupied Bloch states (Chang et al., 2022). In review form, the same content is expressed as the Brillouin-zone Berry-curvature flux, yielding $\sigma_{xy}=C e^2/h$ as an integer topological invariant rather than a semiclassical anomalous Hall response (Liu et al., 2015).

This topology is reflected in boundary transport. In the conventional QAH regime, a bulk insulating gap coexists with chiral edge states that carry current without dissipation, so the zero-field Hall response is quantized and the longitudinal channel ideally vanishes (Chang et al., 2022). The data repeatedly emphasize that the effect is “anomalous” only in the sense that it occurs without an external magnetic field; the transport signature remains the quantum Hall signature of quantized Hall conductance together with vanishing longitudinal resistance (Liu et al., 2015).

A useful microscopic language is the gapped Dirac-cone decomposition. For simple Dirac sectors, each gapped two-dimensional Dirac cone contributes a half-quantized Hall conductance, $\pm e^2/2h$, and an integer QAH state emerges when these contributions add rather than cancel (Chang et al., 2022). This language is central in magnetic topological-insulator films, where top and bottom surface Dirac cones can each contribute half-quantized Hall responses of the same sign, producing the $C=1$ state (Liu et al., 2015).

2. Microscopic routes to the QAH regime

Across the cited work, two ingredients recur: nontrivial band topology and broken time-reversal symmetry. One review states this in condensed form as “inverted band structure” plus “ferromagnetic insulating behavior,” with spin-orbit coupling supplying the inversion and magnetic order preventing cancellation between time-reversed sectors (Liu et al., 2015). A second review generalizes the same point across material classes: magnetically doped topological-insulator films, intrinsic magnetic topological insulators such as MnBi$R_{xx}\to 0$0Te$R_{xx}\to 0$1, graphene moiré systems, and transition-metal-dichalcogenide moiré systems all realize QAH through distinct microscopic mechanisms but the same topological transport phenomenology (Chang et al., 2022).

In magnetic topological-insulator thin films, exchange fields gap topological surface states. In the thin-film limit, this can be cast as a phase-boundary problem controlled by out-of-plane and in-plane exchange fields, structural inversion asymmetry, and orbital-dependent spin coupling. For symmetric coupling, the QAH criterion becomes

$R_{xx}\to 0$2

so an in-plane field can substantially deform phase boundaries and generically induce the QAH effect (Hattori, 2015). This makes the QAH regime a matter not only of large out-of-plane magnetization, but also of thin-film hybridization and electrostatic tuning.

The same topological logic appears in lattice and correlated-electron proposals. The review of the Haldane model presents the first explicit zero-field Chern insulator, where complex next-nearest-neighbor hopping gaps honeycomb-lattice Dirac cones without net magnetic flux (Liu et al., 2015). In bilayer graphene, one proposed QAH state spontaneously breaks discrete time-reversal symmetry but no continuous symmetry; within a hidden SU(4) structure, zero-point fluctuations favor that QAH state at low temperature (Nandkishore et al., 2010). In the organic Mn-DCA Kagome lattice, ferromagnetism, spin polarization, and SOC open a topological gap in Kagome bands, producing $R_{xx}\to 0$3, $R_{xx}\to 0$4, and gapless chiral edge states in an intrinsic QAH phase (Wang et al., 2017). In single-quintuple-layer GdBiTe$R_{xx}\to 0$5, ab initio calculations likewise predict a stoichiometric QAH insulator in which strong SOC, intrinsic ferromagnetism from Gd, and a thin-film band structure near a topological inversion point yield one chiral edge mode and a nonzero Chern number (Zhang et al., 2011).

Correlated graphene-based platforms add another route. In rhombohedral pentalayer graphene/monolayer WS$R_{xx}\to 0$6, the reported $R_{xx}\to 0$7 QAH states arise from the synergy of electron correlation in flat bands, displacement-field tuning, and proximity-induced Ising SOC (Han et al., 2023). The mechanism is described as Haldane-like: valley masses acquire opposite signs, Berry curvatures add constructively, and a nonzero net Chern number results.

3. Transport phenomenology, edge channels, and local current structure

The conventional transport fingerprint remains the starting point: a zero-field Hall plateau with suppressed or vanishing longitudinal channel. Yet the cited work shows that the microscopic realization of that transport is more varied than the simplest edge-only picture suggests. The most direct statement comes from magnetic imaging of a QAH insulator, which visualizes transport current through the QAH plateau and identifies a regime in which the sample transports current primarily in the bulk rather than along the edges; combined magnetic imaging and gating suggest that incompressible regions carry the current and that self-consistent electrostatics plays a central role in determining the current distribution (Ferguson et al., 2021). This does not negate chiral-edge transport as the conventional description, but it shows that the local current distribution within the QAH regime need not be trivially edge-confined.

Magnetization reversal reveals another distinctive transport structure. In a Cr-doped $R_{xx}\to 0$8 QAH insulator, the transition between the $R_{xx}\to 0$9 plateaus proceeds through a well-defined zero Hall plateau near the coercive field, with $\sigma_{xy}=Ce^2/h$0 over a finite field range and $\sigma_{xy}=Ce^2/h$1 at coercivity (Feng et al., 2015). The proposed microscopic picture is a dense network of chiral edge states located at magnetic domain boundaries. Because this network forms during magnetization reversal, the plateau transition is related to but not identical with the conventional quantum Hall network model.

A later disorder study sharpens this distinction by separating two zero-field ground states: the low-disorder QAH insulator and the higher-disorder anomalous Hall insulator (Liu et al., 2021). In the former, $\sigma_{xy}=Ce^2/h$2 and $\sigma_{xy}=Ce^2/h$3; in the latter, the Hall response can remain finite or nearly quantized while the longitudinal channel becomes insulating. In the low-disorder limit, a universal quantized longitudinal resistance $\sigma_{xy}=Ce^2/h$4 is observed at coercive field. The proposed control parameter is the transmission between chiral edge states at magnetic domain boundaries, tuned by magnetic disorder and magnetic field rather than by ordinary potential disorder.

These observations underlie a common misconception addressed repeatedly in the literature: the QAH regime is not simply the zero-field limit of the integer quantum Hall effect. The cited papers emphasize that magnetic domain formation, reversal dynamics, and spatially random magnetization create transport phenomena and critical behavior unavailable in the ordinary Landau-level problem (Feng et al., 2015, Liu et al., 2021).

4. Realized platforms and experimentally accessed regimes

The reported experimental realizations span multiple material classes and dimensionalities. A review identifies four realized classes of QAH insulators: thin films of magnetically doped $\sigma_{xy}=Ce^2/h$5, thin films of the intrinsic magnetic topological insulator MnBi$\sigma_{xy}=Ce^2/h$6Te$\sigma_{xy}=Ce^2/h$7, moiré materials formed from graphene, and moiré materials formed from transition metal dichalcogenides (Chang et al., 2022). The data block further documents multilayers, thick trilayers, and non-moiré proximitized graphene as additional experimentally relevant architectures.

Platform	Reported regime	Representative signature
Magnetic topological-insulator multilayers	Tunable high-Chern-number QAH	$\sigma_{xy}=Ce^2/h$8 tuned up to 5 with $\sigma_{xy}=Ce^2/h$9 at zero field (Zhao et al., 2020)
MnBi$C$0Te$C$1/Bi$C$2Te$C$3 superlattice	Berry-curvature-driven “Q-window”	$C$4 plateau when $C$5 is tuned into a large surface gap and the bulk Hall contribution is essentially zero (Deng et al., 2020)
Rhombohedral pentalayer graphene/monolayer WS$C$6	Large-Chern-number QAH at charge neutrality	$C$7, $C$8, $C$9, up to about 1.5 K (Han et al., 2023)
Magnetic topological-insulator trilayers of hundred-nanometer thickness	Three-dimensional QAH effect	well-quantized $\sigma_{xy}=\frac{e^2}{h}C,$0 Hall resistance and vanishing longitudinal resistance at zero field (Zhao et al., 2023)
CVBST/CdSe superlattice	QAH multilayer behaving as $\sigma_{xy}=\frac{e^2}{h}C,$1	$\sigma_{xy}=\frac{e^2}{h}C,$2 for $\sigma_{xy}=\frac{e^2}{h}C,$3 (Jiang et al., 2018)

High-Chern-number engineering is a recurring theme. Magnetic topological-insulator multilayers fabricated by molecular beam epitaxy realize the QAH effect with tunable Chern number up to $\sigma_{xy}=\frac{e^2}{h}C,$4, controlled by magnetic doping concentration or the thickness of interior magnetic TI layers (Zhao et al., 2020). A distinct multilayer strategy stacks decoupled QAH layers separated by CdSe, producing $\sigma_{xy}=\frac{e^2}{h}C,$5 and a superlattice that behaves as a high-Chern-number QAH insulator with $\sigma_{xy}=\frac{e^2}{h}C,$6 (Jiang et al., 2018). In proximitized rhombohedral graphene, the observed $\sigma_{xy}=\frac{e^2}{h}C,$7 state shows that high-Chern-number QAH need not rely on magnetic dopants or a moiré superlattice (Han et al., 2023).

The MnBi$\sigma_{xy}=\frac{e^2}{h}C,$8Te$\sigma_{xy}=\frac{e^2}{h}C,$9/Bi$$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$0Te$$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$1 superlattice extends the QAH regime in a different direction. There, robust ferromagnetism opens a large surface gap, and an above-Kelvin “Q-window” is identified in which the anomalous Hall conductance from the bulk is to a high precision zero, allowing an $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$2 plateau even though the bulk need not be fully insulating (Deng et al., 2020). In hundred-nanometer magnetic topological-insulator trilayers, the QAH effect persists in the three-dimensional regime, implying that nonchiral side-surface states are gapped and do not spoil quantization (Zhao et al., 2023).

5. Geometry, finite size, and mesoscopic transport

The QAH regime is highly sensitive to geometry, edge-state overlap, and mesoscale disorder. In Mn-doped HgTe quantum wells, a finite-width analysis identifies four transport regimes—quantum spin Hall, QAH, edge conducting, and normal insulator—all with insulating bulk but distinct edge spectra (Fu et al., 2014). Because edge-state coupling is spin dependent and depends on both ribbon width and doping concentration, one spin sector can gap out before the other. For $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$3 and $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$4, the QAH regime occurs roughly for $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$5 (Fu et al., 2014). This suggests that in nanostructures the QAH regime can be selected geometrically rather than only chemically.

Sub-micron transport experiments on magnetic topological-insulator devices show that the main QAH features are still preserved for widths down to 600 nm, but miniaturization introduces a back-scattering channel through percolative hopping between two-dimensional compressible puddles (Qiu et al., 2021). The same work reports large resistance fluctuations near coercive field, attributed to collective interference between intersecting paths along domain walls when the device geometry is smaller than the phase coherence length $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$6. Breakdown-current measurements indicate that the chiral edge states are confined at the physical boundary with width on the order of the Fermi wavelength (Qiu et al., 2021). Together with the direct-current imaging results, this indicates that the QAH regime at mesoscopic scales is governed by a nontrivial combination of boundary confinement, bulk puddles, and domain-wall transport.

Thin-film phase-boundary engineering gives another geometric control knob. In magnetic TI thin films with structural inversion asymmetry, an in-plane exchange field strongly deforms the QAH phase boundary and can induce QAH even for extremely weak out-of-plane and in-plane exchange fields when SIA is tuned by gate bias (Hattori, 2015). For antisymmetric coupling, by contrast, the QAH phase is absent without a strong enough in-plane field (Hattori, 2015). This establishes that the “QAH regime” in thin films is not a single fixed phase point, but a tunable region in an extended parameter space of thickness, hybridization, inversion asymmetry, and exchange-field orientation.

6. Unconventional QAH regimes and conceptual boundaries

Recent work expands the meaning of the QAH regime beyond the conventional bulk-insulating Chern insulator. One proposal identifies a metallic quantized anomalous Hall effect in a topological-insulator film with magnetic sandwich heterostructure, where Hall conductance is quantized to $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$7 but longitudinal conductance remains finite (Bai et al., 2023). The effect is attributed to a pair of massless Dirac cones, each contributing half of the Hall conductance due to quantum anomaly, and is stated to be neither characterized by a Chern number nor associated with any chiral edge states (Bai et al., 2023). In this case, Hall conductivity is quantized but Hall resistivity is not, because $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$8.

A distinct metallic-QAH proposal in ferromagnetic metals reaches a different conclusion about edge structure. There, chiral edge channels coexist with isotropic bulk conduction channels without a bulk energy gap, and a six-terminal Hall bar exhibits quantized Hall conductivity together with nonzero longitudinal conductivity once dephasing drives the system from a quantum-ballistic regime to a classical-diffusive regime (Wan et al., 29 Dec 2025). Again, Hall resistivity itself never becomes quantized. The observable hallmark is therefore quantized Hall conductivity with finite $$C=\frac{1}{2\pi}\int_{\rm BZ} d^2k \sum_{s\in {\rm occ}} \Omega_{xy,s}(\mathbf{k}),$$9, not the conventional $$\Omega_{xy,s}(\mathbf{k})=i\left(\langle \partial_{k_x}u_s|\partial_{k_y}u_s\rangle-\langle \partial_{k_y}u_s|\partial_{k_x}u_s\rangle\right)$$0, $$\Omega_{xy,s}(\mathbf{k})=i\left(\langle \partial_{k_x}u_s|\partial_{k_y}u_s\rangle-\langle \partial_{k_y}u_s|\partial_{k_x}u_s\rangle\right)$$1 pair.

An intermediate semimetallic generalization is reported for chiral semimetals. There, conduction and valence bands touch at zero energy, yet the Hall conductivity can remain quantized because the Berry-curvature integral over occupied states remains quantized; finite dephasing helps recover the multiterminal Hall plateau, while longitudinal conductivity remains finite and semimetallic (Liu et al., 27 May 2026). A plausible implication is that the QAH regime is increasingly being defined operationally by topologically quantized anomalous Hall conductivity rather than exclusively by a fully gapped bulk.

These generalizations sharpen an important conceptual distinction. In the conventional usage codified by reviews, the QAH effect is the zero-field quantum Hall effect of a Chern insulator, with insulating bulk and chiral edge states (Chang et al., 2022, Liu et al., 2015). More recent work extends the term to metallic and semimetallic states with quantized Hall conductivity but finite longitudinal transport (Bai et al., 2023, Wan et al., 29 Dec 2025, Liu et al., 27 May 2026). This suggests that current usage contains both a strict definition and a broader transport-oriented usage, and that careful distinction between quantized Hall resistance and quantized Hall conductivity has become essential in discussing the QAH regime.