Half-Quantized Anomalous Hall Conductance

• Half-quantized anomalous Hall conductance is a phenomenon where a single Dirac sector produces a Hall response of ±e²/2h without an external magnetic field.
• It arises in semimagnetic and asymmetric topological insulators by breaking parity symmetry, with one surface exchange gapped and the other remaining metallic.
• Experimental findings in asymmetric magnetic TI trilayers show half-quantized plateaus alongside finite longitudinal conductance, highlighting a hybrid transport regime.

Searching arXiv for papers on half-quantized anomalous Hall conductance, semimagnetic topological insulators, and related transport interpretations. Half-quantized anomalous Hall conductance denotes a Hall response of magnitude $\pm e^2/2h$ in zero external magnetic field. In the most stringent usage, it refers to a parity-anomaly-related Hall response of an effectively single Hall-active Dirac sector, typically realized in semimagnetic or asymmetric magnetic topological-insulator structures in which one surface is exchange gapped while another remains gapless or only weakly perturbed (Wang et al., 2023, Sattar et al., 12 Apr 2026). The same numerical value, however, also appears in a separate literature on superconductor–quantum anomalous Hall insulator hybrids, where it is often a two-terminal hybrid-device conductance rather than a Hall conductance proper; multi-terminal analyses now treat these as distinct phenomena (Uday et al., 2024).

1. Terminology and scope

The literature uses closely related phrases for several nonequivalent responses. The distinction is essential because the same value, $e^2/2h$, can arise from different observables, different geometries, and different microscopic mechanisms.

Context	Half-quantized quantity	Characteristic interpretation
Semimagnetic or asymmetric magnetic TI	$\sigma_{xy}=\pm e^2/2h$	Parity-anomaly Hall response of a single effective Dirac sector
SC–QAHI hybrid transport	$\sigma_{\mathrm{2T}}\approx e^2/2h$	Geometry-dependent two-terminal conductance plateau
Mirror-symmetric TI film	$\sigma_{xy}^{\chi}=\chi e^2/2h$ per mirror sector	Mirror-resolved anomaly; total electric Hall conductance cancels
In-plane-field SOC 2D system	Nearly $\pm e^2/2h$	Berry-curvature response of a Zeeman-shifted avoided crossing

In semimagnetic TI and TI/ferromagnet work, the half-quantized value is explicitly discussed as a Hall-sector response, usually with finite $\sigma_{xx}$ and without the requirement of a globally insulating Chern phase (Zhou et al., 2022, Bi et al., 27 Aug 2025). In the mirror-symmetry setting, each mirror sector carries $\pm e^2/2h$, but the net electric Hall conductance vanishes and the quantized response survives only in the mirror channel (Fu et al., 2024). In contrast, the SC–QAHI literature increasingly emphasizes that a measured $e^2/2h$ plateau should not be called a half-quantized anomalous Hall conductance unless the quantity is genuinely a Hall response rather than a Landauer–Büttiker conductance of a hybrid terminal geometry (Uday et al., 2024).

2. Parity anomaly and single-Dirac-cone formulations

The conceptual foundation is the parity anomaly of a single $(2+1)$-dimensional Dirac fermion. In topological-insulator language, a magnetically gapped surface Dirac cone acquires a local Hall response

$e^2/2h$0

with the sign fixed by the sign of the exchange-induced Dirac mass (Sattar et al., 12 Apr 2026). This is closely tied to the topological magnetoelectric description of a strong TI with $e^2/2h$1,

$e^2/2h$2

for which a single gapped surface is the boundary manifestation of the bulk axion response (Sattar et al., 12 Apr 2026).

A recurring subtlety is ultraviolet completion. A single isolated Dirac cone cannot occur in a strictly two-dimensional lattice theory without a regulator, and several papers therefore emphasize that the measurable half-quantized response depends on how the low-energy Dirac sector is completed at high energy. In one semimagnetic-TI formulation, the zero-field Hall response in the parity-symmetric regime is

$e^2/2h$3

and this contribution is explicitly traced to occupied states far below the Fermi level rather than to a low-energy Landau-level half step (Wang et al., 2023). The same work shows that finite magnetic field generically restores integer Hall quantization in the insulating regime, so the half-quantized value is best interpreted there as a parity-anomaly Hall response of a metallic single-cone system rather than as a persistent half-integer quantum Hall plateau (Wang et al., 2023).

A broader topological formalization recasts the intrinsic Hall response of a metal in terms of a Fermi-loop Berry phase and proposes a $e^2/2h$4 classification: $e^2/2h$5 In that framework, the half quantization is protected by local unitary or anti-unitary symmetries near the Fermi surface rather than by a global symmetry of the entire Brillouin zone (Fu et al., 2024). This places half-quantized anomalous Hall conductance in a category distinct from both ordinary metallic ferromagnets with nonuniversal intrinsic AHE and fully gapped integer-QAH phases (Fu et al., 2024).

3. Semimagnetic and ferromagnet/TI realizations

The most direct material route is a semimagnetic topological insulator in which one surface is magnetically gapped and the opposite surface remains effectively gapless. A first-principles proposal based on MnBi$e^2/2h$6Te$e^2/2h$7/Sb$e^2/2h$8Te$e^2/2h$9 uses a single septuple layer of MnBi$\sigma_{xy}=\pm e^2/2h$0Te$\sigma_{xy}=\pm e^2/2h$1 on five quintuple layers of Sb$\sigma_{xy}=\pm e^2/2h$2Te$\sigma_{xy}=\pm e^2/2h$3 and predicts a switchable Hall plateau

$\sigma_{xy}=\pm e^2/2h$4

over a broad energy range when the chemical potential is tuned into the top-surface magnetic gap (Muzaffar et al., 5 Jul 2025). In that analysis, the top surface is exchange gapped while the bottom surface remains gapless, and full-band Berry-curvature accounting leads to the specific conclusion that, when the chemical potential lies inside the top-surface gap, the gapped surface bands give no net Hall contribution and the net half-quantized Hall response is carried by the gapless Dirac sector (Muzaffar et al., 5 Jul 2025). The same work estimates operation near $\sigma_{xy}=\pm e^2/2h$5 for MnBi$\sigma_{xy}=\pm e^2/2h$6Te$\sigma_{xy}=\pm e^2/2h$7/Sb$\sigma_{xy}=\pm e^2/2h$8Te$\sigma_{xy}=\pm e^2/2h$9 and up to $\sigma_{\mathrm{2T}}\approx e^2/2h$0 for the Cr-substituted CrBi$\sigma_{\mathrm{2T}}\approx e^2/2h$1Te$\sigma_{\mathrm{2T}}\approx e^2/2h$2/Sb$\sigma_{\mathrm{2T}}\approx e^2/2h$3Te$\sigma_{\mathrm{2T}}\approx e^2/2h$4 variant (Muzaffar et al., 5 Jul 2025).

A complementary first-principles study of TI/ferromagnet van der Waals heterostructures treats 6QL Bi$\sigma_{\mathrm{2T}}\approx e^2/2h$5Se$\sigma_{\mathrm{2T}}\approx e^2/2h$6 interfaced on one side with Cr$\sigma_{\mathrm{2T}}\approx e^2/2h$7Ge$\sigma_{\mathrm{2T}}\approx e^2/2h$8Te$\sigma_{\mathrm{2T}}\approx e^2/2h$9, CrI$\sigma_{xy}^{\chi}=\chi e^2/2h$0, or MnBi$\sigma_{xy}^{\chi}=\chi e^2/2h$1Se$\sigma_{xy}^{\chi}=\chi e^2/2h$2 (Sattar et al., 12 Apr 2026). There the magnetized top surface is the dominant Hall-active component, with the local Chern contribution spatially concentrated near the top interface. In the slab Berry-curvature calculation, CGT/Bi$\sigma_{xy}^{\chi}=\chi e^2/2h$3Se$\sigma_{xy}^{\chi}=\chi e^2/2h$4 yields $\sigma_{xy}^{\chi}=\chi e^2/2h$5 with $\sigma_{xy}^{\chi}=\chi e^2/2h$6, while CrI$\sigma_{xy}^{\chi}=\chi e^2/2h$7/Bi$\sigma_{xy}^{\chi}=\chi e^2/2h$8Se$\sigma_{xy}^{\chi}=\chi e^2/2h$9 yields approximately $\pm e^2/2h$0 with $\pm e^2/2h$1 (Sattar et al., 12 Apr 2026). This treatment emphasizes a different microscopic decomposition from the MnBi$\pm e^2/2h$2Te$\pm e^2/2h$3/Sb$\pm e^2/2h$4Te$\pm e^2/2h$5 analysis: the Hall response is localized near the magnetized surface, whereas the ungapped opposite surface mainly supplies a parallel dissipative channel (Sattar et al., 12 Apr 2026). Taken together, these papers indicate that the existence of a half-quantized Hall plateau is more robust than any single microscopic partition of its band-resolved origin.

Transport-focused semimagnetic-TI theory further shows that the half-quantized Hall response need not imply a Hall insulator. In a model with a gapped top surface and gapless bottom and side surfaces, the Hall conductance approaches

$\pm e^2/2h$6

under strong dephasing, while $\pm e^2/2h$7 remains finite because the system is metallic (Zhou et al., 2022). This is precisely the regime later described as a “half-quantized Hall metal,” where weak disorder preserves the half plateau and strong disorder drives a crossover first to a marginal metal and then to an Anderson insulator (Bi et al., 27 Aug 2025).

The most explicit experimental realization in the supplied literature is an MBE-grown asymmetric magnetic TI trilayer composed of 3 QL V-doped $\pm e^2/2h$8/6 QL undoped $\pm e^2/2h$9/3 QL Cr-doped $\sigma_{xx}$0 (Zhuo et al., 19 Sep 2025). In that structure, the differing anisotropy fields of the top and bottom magnetic layers allow an in-plane-field window in which the top surface remains gapped while the bottom becomes gapless. At $\sigma_{xx}$1 and $\sigma_{xx}$2, the reported parity-anomaly regime shows

$\sigma_{xx}$3

which is interpreted as a robust $\sigma_{xx}$4 state rather than an incipient integer-QAH plateau (Zhuo et al., 19 Sep 2025).

4. Metallic transport, edge current, and sidewall structure

A defining property of half-quantized anomalous Hall conductance in semimagnetic TI platforms is its coexistence with finite longitudinal transport. This differs from the conventional QAH paradigm, in which quantization is tied to an insulating bulk and a dissipationless integer chiral edge state. In the semimagnetic case, the opposite surface and the sidewalls remain active, so the measured state is often metallic even when $\sigma_{xx}$5 is pinned near $\sigma_{xx}$6 (Zhou et al., 2022, Sattar et al., 12 Apr 2026).

One transport theory attributes the half plateau to a robust half-chiral current in a strongly dephasing metal. In that picture, the local transmission imbalance along the relevant edge approaches

$\sigma_{xx}$7

while normal metallic channels contribute a separate longitudinal component; in the large-width limit this gives

$\sigma_{xx}$8

so Hall quantization and finite dissipation coexist naturally (Zhou et al., 2022). A disorder study arrives at a related conclusion from a different direction: when the Fermi level intersects only the single gapless bottom-surface Dirac cone, finite-size scaling gives $\sigma_{xx}$9 in the thermodynamic limit, while the same phase retains weak antilocalization and finite $\pm e^2/2h$0 (Bi et al., 27 Aug 2025).

Another strand of theory emphasizes that the associated current distribution is not that of a conventional integer-QAH edge mode. In a class of “parity anomalous semimetals,” the Hall response is realized by extended massless Dirac states, while the nontrivial Berry-curvature structure is supplied by a massive Dirac sector (Zou et al., 2022). The resulting edge current density decays from the boundary with a power law rather than exponentially,

$\pm e^2/2h$1

and the integrated response approaches $\pm e^2/2h$2 in the thermodynamic limit (Zou et al., 2022). This provides a bulk-edge correspondence distinct from that of integer QH or integer QAHE.

Realistic finite samples also contain sidewall states. In TI/ferromagnet van der Waals heterostructures, nanoribbon calculations show that the top-sidewall states near the magnetized surface are spin-polarized and chiral, yet not “proper” topological chiral edge states in the integer-QAHE Chern-insulator sense (Sattar et al., 12 Apr 2026). The current experimental interpretation of the asymmetric magnetic TI trilayer is broadly consistent with this non-idealized picture: nonlocal and nonreciprocal transport are strongly enhanced in the $\pm e^2/2h$3 regime and are taken as evidence for a half-quantized chiral edge current localized at the boundary of the top gapped surface, but this boundary current hybridizes with dissipative channels on the gapless surfaces (Zhuo et al., 19 Sep 2025).

5. Distinction from SC–QAHI half-plateau transport

The literature on superconductor–QAHI devices created prolonged ambiguity because the number $\pm e^2/2h$4 also emerged there, initially as a proposed signature of a single chiral Majorana edge mode. In the simplest version of that proposal, a QAH chiral fermion entering a proximitized strip in the $\pm e^2/2h$5 topological-superconductor phase yields a two-terminal conductance plateau

$\pm e^2/2h$6

and this was defended in follow-up commentary as distinct from more trivial near-$\pm e^2/2h$7 signals (Zhang et al., 2019).

Subsequent work showed that the same plateau is not uniquely Majorana-related. In a clean NSN strip, orbital magnetic motion can favor a single charged finite-$\pm e^2/2h$8 fermionic mode and generate the same two-terminal value $\pm e^2/2h$9 without any Majorana mode (Osca et al., 2018). In disordered QAHI–SC–QAHI junctions, percolation and dephasing can also produce a nearly identical half plateau, so the electrical conductance alone does not establish an $e^2/2h$0 chiral topological superconductor (Huang et al., 2017). Domain-wall scenarios were proposed to reconcile half-quantized conductance plateaus with nonideal Hall-bar transport, again in terms of two-terminal hybrid conductance rather than a true half Hall conductivity (Chen et al., 2016).

The decisive conceptual clarification is now multi-terminal. In a superconductor–QAHI Hall bar, ordinary QAH transport away from the strip remains

$e^2/2h$1

while the measured half value appears in a distinct Landauer–Büttiker relation for terminal conductance (Uday et al., 2024). In that analysis, the superconducting electrode equilibrates the potentials of incoming chiral edge states, and the observed $e^2/2h$2 follows from edge-potential equilibration at the SC contact rather than from a chiral Majorana mode (Uday et al., 2024). Experimental studies of transparent QAH–Nb interfaces reach a similar conclusion: strong Andreev reflection, transparent coupling, and contact equilibration can lock the two-terminal conductance to $e^2/2h$3 throughout the aligned-QAH regime, even persisting after the Nb strip loses superconductivity (Kayyalha et al., 2019).

For the subject of half-quantized anomalous Hall conductance, the consequence is straightforward. A hybrid-device plateau at $e^2/2h$4 should not be identified with a Hall coefficient $e^2/2h$5 unless the measured observable is genuinely the Hall response of the Hall sector itself (Uday et al., 2024).

6. Symmetry-protected extensions and alternative routes

The half-quantized Hall response has also been generalized beyond the one-gapped-surface semimagnetic-TI setting. A symmetry-based classification proposes that a metallic or semimetallic phase with intrinsic Hall response

$e^2/2h$6

is characterized by a $e^2/2h$7 invariant extracted from the line integral of the Berry connection around symmetry-preserving Fermi loops (Fu et al., 2024). In that construction, local anti-unitary symmetries such as $e^2/2h$8, $e^2/2h$9, or $(2+1)$0, and local unitary symmetries such as $(2+1)$1 or $(2+1)$2, quantize the Fermi-loop Berry phase to $(2+1)$3 while global preservation of the same symmetry would force $(2+1)$4 (Fu et al., 2024). The proposal was explicitly applied to semimagnetic Bi$(2+1)$5Se$(2+1)$6 and Bi$(2+1)$7Te$(2+1)$8 films with $(2+1)$9 and to SnTe films with $e^2/2h$00 or $e^2/2h$01 (Fu et al., 2024).

A conceptually distinct extension is the “half quantum mirror Hall effect.” In a mirror-symmetric strong-TI film, each mirror sector hosts an effective single Dirac cone and carries

$e^2/2h$02

while the total electric Hall conductance remains zero by time-reversal symmetry and the quantized response survives only as a mirror Hall conductance

$e^2/2h$03

This is therefore a symmetry-resolved analogue of half-quantized Hall physics rather than an ordinary net anomalous Hall response (Fu et al., 2024).

A third route invokes strong SOC and broken $e^2/2h$04 symmetry in nominally nonmagnetic 2D systems. For a $e^2/2h$05 2DEG with Rashba SOC, cubic warping, and in-plane Zeeman coupling, the field shifts a band crossing to finite momentum and opens an avoided crossing whose Berry curvature produces a nearly half-quantized intrinsic Hall conductance. In the ideal $e^2/2h$06 limit at $e^2/2h$07,

$e^2/2h$08

while the material-specific analysis of Sb$e^2/2h$09Te$e^2/2h$10 thin films requires roughly $e^2/2h$11 and $e^2/2h$12 and explicitly describes the result as nearly, not exactly, half quantized (Sun et al., 2022).

7. Experimental status and unresolved issues

The current experimental picture is that half-quantized anomalous Hall conductance is most credible in asymmetric or semimagnetic topological-insulator geometries that isolate one Hall-active Dirac surface while leaving other channels metallic. The strongest direct evidence in the supplied literature is the asymmetric V/undoped/Cr magnetic TI trilayer, where the field-tuned plateau near $e^2/2h$13 is accompanied by enhanced nonlocal and nonreciprocal transport consistent with a boundary current tied to the top gapped surface (Zhuo et al., 19 Sep 2025). Earlier semimagnetic-TI studies are interpreted by theory as parity-anomaly Hall responses that can cross over to integer quantum Hall values under finite magnetic field and disorder, rather than as ordinary half-integer Hall plateaus of an insulating Chern phase (Wang et al., 2023).

At the same time, several issues remain structurally important. One is the metallic background itself: finite $e^2/2h$14 is not an experimental imperfection extraneous to the phenomenon, but in many models an intrinsic consequence of the gapless opposite surface and sidewalls (Zhou et al., 2022, Sattar et al., 12 Apr 2026). Another is the microscopic bookkeeping of the Hall response. Full-band analyses do not yet speak with a single voice on whether the measured half plateau should be assigned primarily to the gapped magnetized surface sector or, after lattice regularization, to the remaining gapless sector (Muzaffar et al., 5 Jul 2025, Sattar et al., 12 Apr 2026). A third is disorder: weak disorder can stabilize a half-quantized Hall metal, but stronger disorder can generate a marginal metallic phase with nonuniversal Hall response before eventual Anderson localization (Bi et al., 27 Aug 2025).

The most stable consensus is therefore negative as much as positive. Positively, a Hall response near $e^2/2h$15 can occur in zero field as a parity-anomaly manifestation of a single effective Dirac sector in semimagnetic or asymmetric magnetic topological-insulator structures (Zhuo et al., 19 Sep 2025, Wang et al., 2023). Negatively, the same number is not by itself diagnostic across all platforms, especially not in SC–QAHI hybrids where multi-terminal transport shows that $e^2/2h$16 often belongs to a geometry-dependent two-terminal conductance rather than to the Hall response of the anomalous Hall sector (Uday et al., 2024).