Half-quantized layer hall effect as a probe of quantized axion field

Domain: physics
Published: 2026-01-07
Editorial outcome: major_revision

Abstract
The axion insulator is a topological phase characterized by a quantized axion field, which manifests as a half-quantized anomalous Hall conductance (AHC) localized on individual surface layers. Experimental realization of this effect has remained elusive because opposite surface contributions typically cancel. We engineered magnetic axion insulator heterostructures using molecular beam epitaxy. By positioning the Fermi level asymmetrically, we isolated one topological surface state within the magnetic gap while displacing the other into a metallic regime. This technique reliably produced a half-quantized layer-resolved AHC of e²/2h, termed the half-quantized layer Hall effect (LHE), across both parallel and antiparallel magnetization configurations in over ten devices. Our findings provide direct electrical evidence of the half-quantized LHE, a boundary signature of the quantized axion field in the bulk, which resolves a key challenge in the experimental verification of this topological quantum state and establishes a framework for spatially engineering quantized topological response.

Introduction
Axions were originally proposed in high-energy physics to resolve the strong CP violation problem in quantum chromodynamics (QCD), but have remained experimentally elusive1. In condensed matter systems, axion insulators, characterized by a quantized axion field, offer a promising platform to experimentally explore axion-like physics2–4. This quantized axion field gives rise to a unique topological magnetoelectric (TME) response, originating from a quantized topological term in the electromagneticaction2, ΔL=θ(α/4π²)E·B, where θ=π is the quantized axion angle, and α≈e²/ℏc is the fine-structure constant. However, direct verification of the TME effect by measuring the quantized responses to external fields is still a challenging open task5. This motivates the search for a definitive experimental signature of quantized axion field beyond the bulk TME response in axion insulators.

Surface electronic transport provides a boundary signature of the quantized axion field in the bulk. When time-reversal symmetry is locally broken at the surface (e.g., by an out-of-plane magnetization), the resulting gapped surface states induce a 2D Chern-Simons term that generates a half-quantized anomalous Hall conductance (AHC) σxy=±e²/2h6. This half-quantized effect and the TME effect are two inseparable physical consequences linked by the bulk-boundary correspondence. Despite its theoretical elegance, the axion insulator phase in transport experiments presents an enigmatic scenario7–9: antiparallel magnetized surfaces yield vanishing net Hall response, obscuring direct access to the critical e²/2h per-surface quantization. Current experimental paradigms rely on indirect bulk transport signatures – zero Hall plateaus coexisting with insulating behavior – which remains phenomenologically ambiguous against trivial band insulators10. Fortunately, recent advances in layer Hall effect (LHE) physics suggest a pathway forward: spatial separation of Berry curvature generates a layer-resolved Hall effect11–15. The direct observation of layer-resolved half-quantization consequently stands as the pivotal requirement for unambiguous verification of the axion insulator phase.

However, these layer-resolved Hall responses have thus far remained unquantized in axion insulator candidate MnBi₂Te₄11,16,17, due to the influence of non-topological effects, rendering the experimental evidence of a quantized response elusive. Specifically, the existing implementations in exfoliated MnBi₂Te₄ flakes suffer from intrinsic limitations: (i) strict space-time PT symmetry that limit parameter tunability, (ii) micron-scale device sizes introducing fabrication challenges and possible finite-size effects, and (iii) complex interlayer antiferromagnetic ordering that obscures the surface Hall contributions18–20. Even under applied symmetry-breaking electric fields, these constraints fundamentally compromise the extraction of intrinsic quantized axion responses11,16, let alone half-quantized LHE.

Here, we overcome these challenges through precision synthesis of MBE-grown magnetic TI heterostructures (Fig. 1a), engineered as prototypical axion insulators with atomically controlled magnetic interfaces. Differential V/Cr doping21,22 across top and bottom surface layers enables robust anti-parallel magnetization configuration, or the axion configuration2. Previous studies mainly focus on the axion insulating regime with Fermi levels positioned within dual magnetic gaps |Mt/b| (Fig. 1b), where meron-like spin textures in each surface band generate counter-propagating layer-dependent persistent half-quantized edge currents23. Unlike the quantized chiral edge states in a Chern insulator, these edge currents emerges as chiral edge channels when they coexist with the metallic surface state, carrying half-quantized Hall conductances σt/bxy=sign(Mt/b)e²/2h in transport measurement24–26. Nonetheless, critical parameters governing the surface-specific responses remain underexplored15 - particularly the exchange field tunability, the layer thickness control, as well as the layer-dependent chemical potential engineering. This synthetic control over competing energy scales enables unprecedented isolation of layer-resolved quantum transport signatures, as demonstrated through our surface-selective transport measurements.

Results
We synthesized a series of magnetic TI heterostructures via MBE on annealed SrTiO₃(111) substrates (see Methods), which serve as bottom gate dielectrics. The prototypical axion insulator heterostructure comprises three distinct layers: 3 quintuple layers (QL) of Crx(Bi,Sb)2-xTe3 (CBST), m QL of undoped (Bi,Sb)2Te3 (BST) spacer, and 3 QL of Vy(Bi,Sb)2-yTe3 (VBST) (denoted as CBV, Fig. 1c), where x and y represent magnetic doping concentration. The chemical potential of the sample can be adjusted via changing the Bi:Sb ratio, η. For electrical transport characterization, devices were fabricated through two approaches: bottom-gated Hall bars are mechanically defined using tungsten tip scribing; dual-gated Hall bar devices are patterned via photolithography with 50nm AlOₓ top dielectric. Low-temperature transport measurements were performed in a dilution refrigerator (base temperature T=30mK, unless other specified) with four-terminal lock-in techniques. For a CBV Device A1 (m=20, x=0.19, y=0.11, η=0.83), the overall magneto-transport behavior is as expected at the charge neutrality point (CNP) with the bottom gate voltage Vbg=150V. Figure 1d demonstrates the magnetic-field-driven phase transitions between Chern insulator (σxy=±e²/h) and axion insulator (σxy=0), with parallel and anti-parallel magnetization configurations respectively. The coercive field is Hc1=0.76T for the top VBST layer and Hc2=0.12T for the bottom CBST layer. The midpoint of zero Hall plateau locates near μ0H=±0.4T.

To reveal layer-resolved quantum transport, we systematically probe the asymmetric Fermi level configuration by tuning the bottom gate voltage Vbg from CNP toward hole-doping regimes. As Vbg decreases below CNP, the Hall conductance hysteresis evolves from a two-step transition to a single half-quantized plateau (Fig. 2a). At zero magnetic field, the antiparallel magnetization configuration exhibits σxy growth from 0 to e²/2h while the parallel configuration shows σxy reduction from e²/h to e²/2h (Fig. 2b). Remarkably, both configurations ultimately converge to the same half-quantized value. This convergence demonstrates complete suppression of the bottom layer’s transport Hall conductance as the Fermi level penetrates its massive Dirac band, well below the magnetic gap (Fig. 2c), leaving only the top layer’s half-quantized AHC. This observation demonstrates the precise quantized form of the layer Hall effect11,12,15, thereby termed as the half-quantized layer Hall effect (HQLHE). The HQLHE constitutes the central discovery of this work. Crucially, it exhibits two defining characteristics that distinguish it from conventional LHE phenomena in PT symmetric systems: (i) identical quantization σxy=±e²/2h persists across both parallel and antiparallel magnetic states; (ii) quantization largely maintains even without external electric field15 (e.g. Vbg=0V in Fig. 2b).

Two critical factors enable this half-quantized phenomenon. Firstly, the Cr doping level is deliberately kept relatively low with x=0.19 to ensure that the Berry curvature from the bottom surface band is concentrated near the gap edge (Fig. 2c). At Vbg=-200V, the band of bottom layer is partially filled and exhibits nearly zero Berry curvature, contributing negligibly to the total Hall conductance. In comparison, it is challenging to realize HQLHE in a device with high Cr doping level (see Supplementary Fig. 1). Secondly, the high density of bulk trap states in (Bi,Sb)2Te3 effectively screens electric fields27,28. Thereby, the Fermi level remains within the top surface magnetic gap when tuning Vbg, preserving the top layer’s half-quantized Hall conductance.

To fully understand the HQLHE theoretically, we construct a four-band effective 3D TI Hamiltonian with the axion configuration by introducing Zeeman splitting HM=Mt/bσz on the top and bottom surfaces. We apply an additional potential energy term HG=V(z) with the value Vt(Vb) on the top (bottom) surface, to independently control the surface Fermi energy of the top and bottom layer. We also incorporate the random onsite potentials to include disorder effects that are inevitable in real devices and then evaluate the layer-dependent AHC by a real-space Kubo formula29 (more details in Methods). Numerical results in Fig. 2d show that the layer-resolved AHC σtxy and σbxy are half-quantized when Fermi level EF falls within the magnetic gap but decay to zero at higher energies. By electrostatic gating of the bottom surface potential Vb, the total AHC σxy can be continuously tuned from zero (axion insulator) or e²/h (Chern insulator) to e²/2h (HQLHE) when EF lies within the top surface’s magnetic gap (see Fig. 2e and Supplementary Fig. 2a). The consistence between the theoretical simulations and experiments results unambiguously demonstrates the half-quantized nature of layer-resolved hidden Berry curvature in axion insulator heterostructures.

To further elucidate the universality of the HQLHE, we construct a renormalization group flow diagram (RGFD) by mapping magneto-transport trajectories onto the σxy, σxx conductivity plane30 (Fig. 2f). At CNP (Vbg=150V), the system traverses two semicircular arcs with radius e²/2h under μ0H variation, connecting three stable fixed points: (±1.0, 0)e²/h for the Chern insulator phase and (0, 0) for the axion insulator phase31. Upon tuning Vbg to -200V, the RGFD undergoes a fundamental transformation: the right semicircle collapse into points along σxy=e²/2h, while the left semicircle shifts its origin from (-0.5, 0)e²/h to (0, σxx). The finite longitudinal conductance σxx increases up to about 3.5e²/h, signaling metallic transport. This RGFD demonstrates a continuous quantum phase transition from conventional integer quantum Hall fixed points to the emergent HQLHE metallic state32. The 1/2-shift of σxy reflects the scaling law of single-flavor Dirac fermions. Crucially, the smooth connection between these phases on the same device provides irrefutable evidence that the (0, 0) fixed point comes from the axion insulator state rather than trivial insulator state, conclusively eliminating alternative explanations such as surface-state hybridization10. The RGFD evolution further established HQLHE as the boundary signature of quantized axion field in magnetic TIs.

We demonstrate tunable HQLHE by engineering layer-resolved Berry curvature through spacer thickness, surface-specific chemical potentials, exchange fields, and thermal fluctuations. Systematic thickness-dependent studies reveal robust HQLHE signatures across devices (Supplementary Fig. 3), confirming its intrinsic surface origin. At m=5 with anti-parallel magnetization, we observe slightly reduced σxy=0.48e²/h at Vbg=-200V and μ0H=0T, along with further reduced σxy at μ0H=-0.4T. This indicates strengthened surface hybridization and interlayer magnetic coupling effect due to the reduced spacer thickness. At m=30, the AHC converges to σxy=0.55e²/h at Vbg=-200V, slightly exceeding the expected half-quantization. Noticeably, another type of quantization deviation appears near the QAH plateau, such as the AHC peak at Vbg=150V in Supplementary Fig. 3h. We attribute these two types of quantization deviations to the quasi-1D non-chiral edge conduction channels33, as supported by theoretical modeling34 (see detailed discussion in Supplementary Note xii). For HQLHE, the optimal quantization accuracy is achieved under two conditions: vanishing AHC from the deep hole doped surface, and reduced conductance from the parasitic non-chiral edge channels. Among all tested devices, devices with spacer thickness 10≤m≤20 show the most accurate AHC of σxy=0.505±0.019e²/h despite other possible errors (See Supplementary Fig. 5 and detailed discussion in Supplementary Notes x, xii, and xiii). On the other hand, at the insulating QAH state, σxy is most accurate at CNP where both surface states and quasi-1D helical edge states vanish.

Utilizing dual-gated Hall bar devices (Fig. 1c), we achieve the manipulation of the HQLHE through independent chemical potential control of top and bottom surfaces. Figure 3a-d maps the Hall conductance landscape of Device B1 across parallel and antiparallel magnetization configurations. In parallel magnetization Fig. 3a, the QAH phase (σxy=e²/h, light green region, inside the red dashed lines) spans the entire top-gate operational range, demonstrating stronger chemical potential tunability at the bottom surface. The QAH phase turns into the axion insulator phase (dark blue region inside the red dashed lines) in Fig. 3b with antiparallel configurations. Noticeably, the AHC σxy of both configurations converges to σxy≈0.45e²/h when Vbg < 0 (plateaus in lower halves of Fig. 3a and c), as in the line plots in Fig. b and d. We attribute the deviation of AHC from half-quantization to sample quality degradation during fabrication, which possibly caused the actual width of the device several microns smaller than the original design. These AHC plateaus also exhibit top gate voltage (Vtg) dependent modulation, with σxy decreasing systematically as Vtg increasing.

To interpret the dual-gated experimental observations, we perform theoretical calculations by independently controlling the Fermi energies of the top and bottom layers (Fig. 3e). The calculated AHC inside area indicated by the pink dashed lines, with slight tilt due to inevitable tuning of top/bottom gate on the opposite surfaces, qualitatively fits the data in Fig. 3c. Furthermore, a precisely quantized layer Hall conductance, controllable via a dual-gate setup, is clearly visible in Fig. 3e. Although full access to both p- and n-type HQLHE regimes remains limited by current gating efficiency, emergent same-sign layer Hall signals on both sides of σxy≈0 at Vtg > 0 (e.g., Vtg=15V in Fig. 3d) reveals a distinct mechanism from conventional electric-field-induced antisymmetric LHE with respect to EF=015. Additional results of specially adjusted CBV devices (Supplementary Fig. 6) where the bottom surface is highly n-doped, demonstrate a continuous transition of Hall conductance from e²/h towards e²/2h in the n-doped regime. This complements the p-doped data and confirms the symmetric nature of the effect, distinct from previous studies11. We thus argue that by further enhancing the device fabrication quality and gating efficiency, one could in principle access a broader LHE phase diagram, with 4 HQLHE plateaus (2 plateaus for each ±e²/2h) in certain parameter space (Supplementary Fig. 7).

To further confirm the universality of the HQLHE in magnetic TIs, it is essential to directly verify the half-quantized Hall conductance of both the V-doped and Cr-doped surface layers. To this end, we engineer Device C1 with inverted doping architecture (VBC configuration: V-doped bottom layer, Cr-doped top layer) and reduced vanadium concentration (y=0.03). This structural modification enhances chemical potential tunability of the VBST layer via gating through the STO substrate. The lowered V doping results in a mix state of weak ferromagnetism and superparamagnetism35,36, precluding the axion anti-parallel magnetization (see Fig. 4a). Despite this constraint, HQLHE persists in the parallel magnetization configuration (see also Supplementary Fig. 8 for 2 examples of CBC devices), demonstrating the half-quantized Hall conductance of Cr-doped surface layers.

As shown in Fig. 4a, gate modulation towards Vbg=200V drives a continuous transition from the QAH state to the HQLHE regime (see numerical simulations in Supplementary Fig. 2b), which proves the vanishing Berry curvature mechanism also applies in the electron doped regime, as expected theoretically. The small overshoot of σxy indicated by black arrows is attributed to the residual Berry curvature of VBST layer because of insufficient gating. Emergent kinks near σxy=e²/2h indicated by red arrows, reflects layer-selective quantization from the CBST top surface layers. By changing Vbg, the RGFD in Fig. 4b reveals different behaviors compared to Device A1: while the conventional axion insulator fixed point (σxy, σxx)=(0, 0) disappears, there is also a continuous evolution from QAH-like semi-circle37 (radius e²/h) to HQLHE trajectories (radius e²/2h) with finite σxx.

Finally, the temperature dependence provides critical insights into the HQLHE. Since the insulating gapped surface provides the half-quantized Hall conductance, the temperature stability of HQLHE directly reflects the interplay between thermal energy kBT and exchange gaps ΔB of the insulating gapped surface. Figure 4c illustrates this relationship through temperature-dependent AHC measurements on Device A2 (nominally the same parameters with Device A1), where the HQLHE remains above 0.47e²/h up to about 2K. Conservative analysis38 of the σxy(T) scaling below the Curie temperature TC yields a minimum estimates of exchange gap (ΔVBST > 0.7meV) for the V-doped surface (Fig. 4d), which is compatible with the TC≈14K. This temperature analysis indicates that HQLHE seems to be simply limited by the exchange field strength of the insulating gapped surface. Notice that there are variations of the estimated exchange gap Δ, even with nominally the same doping concentration (See Supplementary Fig. 9 for additional temperature dependence data). We attribute this inconsistency to the different Fermi level positions inside the actual exchange gaps of VBST layers. (see Supplementary Note viii for detailed discussions.)

Methods
MBE growth
The magnetic doping (Bi,Sb)2Te3 topological insulator (TI) sandwich structures were fabricated by using a commercial MBE system (Fermion MBE-300) with a base vacuum better than 1.5×10⁻¹⁰ mbar. The insulating 0.5 mm SrTiO3 (111) substrates used for growth were initially etched with 1% hydrofluoric acid (HF) solution for 2 minutes and then annealed in a tube furnace at 980°C under flowing oxygen for 3h (1st thermal annealing). Subsequently, the substrates were etched again with deionized (DI) water (resistivity > 18 MΩ·cm) at 80°C for 1.5h and then annealed at 980°C under O2 atmosphere for 3h (2nd thermal annealing). Through this two-step thermal-annealing procedure, a passivated and atomically flat surface of SrTiO3 (111) substrates was obtained. After that, the heat-treated SrTiO3 (111) substrates were outgassed at 600°C for 10 min. Next, high purity Sb (99.99999%), Bi (99.9999%), Te (99.99999%), Cr (99.999%), V (99.99%) were evaporated from Kundsen effusion cells. The chromium (Cr)/vanadium (V) doping concentration x/y was precisely controlled through the evaporation temperature of Cr/V cell, and the value of x/y was calibrated by X-ray photoelectron spectroscopy (XPS) shortly after calibration patterns. The growth rate of TI films is ~0.4 QL per minute, calibrated by RHEED oscillations and thickness measurements using an atomic force microscopy (AFM).

Hall bar device fabrication
The SrTiO3 substrates were used as the gate-dielectric to fine-tune the chemical potential of the films. For the bottom gated devices, the MBE films were mechanically scratched into six-terminal Hall bars by a tungsten needle controlled by a programmed probe station. The geometry specifications were measured using an optical microscope imaging system (see examples in Supplementary Note x). For the dual gated devices, the Hall bar patterns were defined via standard photolithography (AZ1500 series resist) and subsequent argon plasma etching. Ti (3nm)/Au (35nm) electrodes were deposited onto the cleaned area of the Hall bar terminals, followed by a liftoff process. Atomic layer deposition (ALD) of 50nm AlOx dielectric layer was conducted at 100°C. Finally, a Ti (3nm)/Au (35nm) top gate was fabricated on the effective area of the Hall bar device using a second photolithography and liftoff process. The highest temperature during device fabrication was kept below 110 °C to minimize potential degradation. The ohmic contacts were made by soldering indium dots onto the previously defined electrodes. For AlOx covered electrodes, scratching off the insulating dielectric material prior to soldering guaranteed good ohmic contact.

Electrical transport measurement
The magneto-electric transport was conducted in a Physical Property Measurement System (PPMS, Quantum Design DynaCool, for temperature range 1.7 K and above) and a dilution refrigerator (Oxford Instruments, Triton 500, for temperature range 30 mK and above). The magnetic field was applied perpendicular to the sample plane. Longitudinal and Hall voltages were measured simultaneously using standard lock-in techniques at a low frequency around 7 Hz (SRS SR830 lock-in amplifier). The gate voltages were applied using Keithley 2400 or 2614B source meters. The excitation current was set to 10 nA, balanced for a good signal to noise ratio and minimized heating effect. Two major types of measurements were conducted as following: (i) sweeping the magnetic field at fixed gate voltages, and (ii) sweeping the gate voltages with known magnetization configurations (parallel or anti-parallel) at zero external magnetic field. Although the Hall bar geometry of the scratched sample looked symmetric under microscope, the inevitable slight misorientation or errors leads to the mixing of Vxx and Vyx signals. The calculation of the sheet conductance σxx and σxy followed the tensor relations σxx=ρxx/(ρ²xx+ρ²yx) and σxy=ρyx/(ρ²xx+ρ²yx), where ρxx and ρyx data obtained through conventional symmetrization and anti-symmetrization procedures, respectively. For a comparison between raw and processed data, see examples in Supplementary Note xi and xvi. Information of all the measured devices in this study is summarized in Supplementary Table 2.

Theoretical models of axion insulator heterostructure and calculations of layer-resolved Hall conductance
we consider a 3D TI with antiparallel magnetization alignment surfaces [see Fig. 1(a)] that is realized in our experiment. The four-band effective Hamiltonian is H=H0+HM, where
H0(k)=∑i=x,y,z Akiσx⊗si + M0-Bk²σz⊗s0 (1)
describes the 3D TI50, with model parameters A, B, and M0. σ and s are Pauli matrices for the orbital and spin degrees of freedom, respectively. We discretize the Hamiltonian H into Nx×Ny×Nz lattice sites. HM=M(z)σ0⊗sz is the Zeeman splitting. M(z)=Mb for z=1 (bottom surface), M(z)=Mt for z=Nz (top surface), and M(z)=0 for 1<z<Nz. We introduce HG=VG(z)σ0⊗s0 to independently control the surface Fermi energy of the top and bottom layer, which is realized by applying voltages on the top and bottom gates in our experiment. VG(z)=Vt for z=1,2, VG(z)=Vb for z=Nz, Nz-1, and VG(z)=0 for 2<z<Nz-1. We incorporate the random disorder as HD=VD(r)σ0⊗s0, where the VD(r) is uniformly distributed within [-W/2, W/2] with the disorder strength W.
We discretize the Hamiltonian H on lattice sites:
H=∑i ci†Mi ci + ∑i,j=x,y,z (ci†Tj ci+e^j + h.c.) + ∑i ci†Mz ci (2)
where M0 = M0 - (2B/ax² + 2B/ay² + 2B/az²)σz⊗s0, and Tj = B/aj²σz⊗s0 - iA/(2aj)σx⊗sj, and Mz = M(z)σ0⊗sz. Here, a=aj is the lattice constant. ci (ci†) is the annihilation (creation) operator at site i, and e^j is a unit vector in the j direction for j=x,y,z. The sample thickness is given by d=Nz az. The model parameters for numerical calculations are fixed as A=a=1.0, B=a²=0.5, M0=1.0, W=2.5, and Nx×Ny×Nz=25×25×6. Mb=0.3, Mt=0.6 in the Fig. 2 in the main text. Error bar correspond to the standard deviation obtained from 30 independent disorder configurations.
The layer-dependent Hall conductance is calculated by using the real-space Kubo formula29:
σxy(z)=2π/h ie²Tr [P[x^,P], [y^,P]]z (3)
with periodic boundary condition in both the x and y direction. (x^, y^) denotes the position operator, and Tr{...}z is trace over the wave functions of the zth layer. P is the projector onto the occupied states of the effective Hamiltonian H. The bottom surface Hall conductance is σbxy=∑z=1Nz/2 σxy(z), the top surface Hall conductance is σtxy=∑z=Nz/2+1Nz σxy(z), and the bulk Hall conductance is σbulkxy=∑z=Nz/2-1Nz/2+2 σxy(z).
The six-layer tight-binding model is a standard and computationally efficient effective model used to capture the essential low-energy physics of the two surface states of a 3D topological insulator. For a film thickness of 20 QLs, the system is firmly in the 3D limit where the hybridization between top and bottom surfaces is negligible. In this regime, the surfaces are electronically independent, and their behavior is well-described by the effective model. The model is not intended to reproduce the bulk band structure of the 20 QL film but to correctly capture the qualitative behavior of the surface states under gating.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Hu et al. presents a magnetotransport study of magnetically doped topological insulator thin films, claiming the observation of a half-quantized layer Hall effect as a probe of the quantized axion field. While the topic is of broad interest and potentially suitable for publication at the level of Nature, I have significant concerns regarding both the novelty of the work and the technical robustness of the measurements. Unless these issues are convincingly addressed, I cannot recommend the paper for publication.\n\nComment 1: First, there is substantial overlap with previous work, particularly Mogi et al., Nature Physics 18, 390–394 (2022) [https://www.nature.com/articles/s41567-021-01490-y]. The main observation reported in the present study - a half-quantized Hall conductance from the top layer - is essentially identical to that of the earlier Nature Physics paper. In both cases, the top surface exhibits a half-quantized Hall conductance of e2/2h, while the bottom surface shows a negligible Hall response and is thus irrelevant to the quantized signal. The difference lies in the origin of the vanishing Hall conductance from the bottom layer: in the prior work, the bottom surface was non-magnetic and thus naturally had no Hall response; in the present study, the bottom surface is heavily hole-doped away from the Chern gap, thereby suppressing its contribution. While the authors acknowledged the previous work in the manuscript, I think the difference is too minor to justify a Nature-level publication on its own.\n\nComment 2: Second, the theoretical proposal of quantized layer Hall conductance (Ref. 15, Dai et al.) predicts an integer quantized Hall response originating from either the top or the bottom surface in fully AFM TI such as even-layer MnBi2Te4. This requires the metallic surface state on the non-contributing layer to be fully gapped, so that only one surface contributes to transport. If such a situation were clearly demonstrated, the result would indeed be of significant interest. However, the present study does not convincingly achieve this.\n\nComment 3: Third, several key features in the experimental data remain poorly understood. For instance, why is the half-quantized Hall conductance observed only at highly negative backgate voltages, when the bottom surface is strongly hole-doped (e.g., Fig. 3a)? In principle, a similar half-quantization should occur at highly positive gate voltages, when the bottom surface is strongly electron-doped, consistent with the theoretical simulations shown in Fig. 3e. Likewise, Fig. 4 appears to show a half-quantized plateau at large positive Vbg, but the discrepancies between Figs. 3 and 4 are not adequately discussed.\n\nComment 4: In general, the authors must clearly distinguish between two distinct scenarios that can lead to an apparent half-quantized Hall conductance:\n1. Asymmetric doping, where one surface lies within the Chern gap and the other is heavily doped (the intended scenario of this study);\n2. Symmetric, uniform doping, where both surfaces are tuned away from the Chern gap and, at certain densities, the total Hall conductance can accidentally reduce to e2/2h.\nAt present, the clarity of the presentation is insufficient, and the manuscript does not convincingly demonstrate the robustness or uniqueness.", "reviewer_2": "This manuscript reports the experimental observation of the half-quantized layer Hall effect in MBE-grown magnetic topological insulator heterostructures, demonstrating layer-resolved half-quantized Hall conductance as a direct probe of the quantized axion field. By tuning the Fermi level via electrostatic gating and controlling magnetization configurations using an external magnetic field, the authors systematically demonstrate the emergence of a half-quantized Hall plateau across a variety of devices, supported by numerical simulations based on a four-band effective model.\n\nThis work represents a significant advance in experimental axion electrodynamics and topological quantum transport, providing clear evidence of layer-resolved half-quantized Hall effects in engineered axion insulators. The results will be of considerable interest to the broad topological physics and quantum materials community and are suitable for publication in Nature after addressing the following comments.\n\nComment 1: The authors should carefully check the schematic plots of Berry curvature in Fig. 2(c) and Extended Data Fig. 4. For a massive Dirac cone, the Berry curvature contributions from the conduction and valence bands should have opposite signs. The current depiction appears inconsistent with this expectation.\n\nComment 2: The anomalous Hall conductance exhibits precise half-quantization only within a spacer thickness range of 10–20 QLs. The authors should provide a more quantitative analysis to clarify why half-quantization is stabilized specifically within this thickness window.\n\nComment 3: The numerical simulations are performed on a six-layer system, while the experiments are primarily conducted on systems with m = 20 QLs. The authors should discuss how this difference in system size may affect their results.\n\nComment 4: The manuscript states that different transport properties are obtained by tuning the bottom gate voltage (Vbg). However, the role of Vbg remains unclear: it shifts the Fermi level while also modifying the bottom surface band structure, rendering the tuning mechanism and the underlying physical picture somewhat ambiguous . Additionally, the authors should clarify the relationship between the Fermi levels of the top and bottom surfaces under these gating conditions.\n\nComment 5: The authors should provide additional raw resistance measurement data to support the reproducibility and robustness of the reported half-quantized Hall plateaus.", "reviewer_3": "In this manuscript, the authors investigate the layer Hall effect in a magnetic topological insulator heterostructure, specifically a CBST/BST/VBST trilayer, grown by molecular beam epitaxy. By employing a dual-gate configuration with an STO substrate as the back gate and a deposited top gate, they engineer a state where the magnetic moments of the top (CBST) and bottom (VBST) layers are aligned antiparallel. They tune the gates to drive the Fermi level into the surface gap of one layer while keeping the other layer in a metallic regime. Under these conditions, they report the observation of a half-quantized anomalous Hall conductance (AHC) of approximately 0.5 e2/h, which they present as the quantization of the layer Hall effect.\n\nWhile the work demonstrates good techniques in thin film growth and device fabrication, I have serious reservations about its conceptual novelty and the authors' interpretation of their results. Therefore, I cannot recommend its publication in Nature. My major concerns are outlined below.\n\nComment 1: The primary issue with this manuscript is that the main finding, half quantization of layer Hall effect, is a direct consequence of previously established principles and does not constitute a new fundamental discovery. The result is an expected combination of two known phenomena: (1) The Layer Hall Effect Mechanism: The concept that an anomalous Hall effect can be induced and controlled by an electric field with an antiparallel magnetized TI system is not new. This was demonstrated in the context of even-layered MnBi2Te4 by S. Y. Xu's group (A. Gao et al., Nature 595, 521 (2021)). The current work applies this known mechanism to a different material system but does not introduce a new physical concept. (2) Half-Quantized AHC from a Single Surface: The fact that a single gapped topological surface state contributes an AHC of e2/2h is a well-established theoretical prediction. Furthermore, it has been experimentally verified in semi-magnetic topological insulator bilayers, as reported in the work the authors cite as Ref. 38 (M. Mogi et al., Nat. Phys. 18, 390 (2022)). Given these precedents, observing an AHC of 0.5 e2/h by gapping one surface in an antiparallel system is the logical and expected outcome. While the realization of this state in a complex heterostructure is commendable, it does not provide the significant conceptual advance required for a publication in Nature.\n\nComment 2: A second major issue is that the authors' claim to have observed a \"quantized axion field\" is a significant overstatement and appears to be based on a misunderstanding. The axion field in condensed matter is associated with the topological magnetoelectric effect, described by the Lagrangian term ΔL=θ(α/4π2)E⋅B. A quantized axion response (θ=π) would manifest as a quantized magnetoelectric coefficient. This requires the observation of either an E-field-induced magnetization or a B-field-induced electric polarization, as was recently demonstrated for MnBi2Te4 (e.g., J.-X. Qiu et al., Nature 641, 62 (2025)). This manuscript provides no such measurement. In fact, there is no quantitative evaluation of the electric field within the material, which is a prerequisite for any claims related to the magnetoelectric effect, let alone its quantization. The observed half-quantized Hall conductance is a property of a single surface, not a direct measurement of the bulk axion response.\n\nComment 3: Furthermore, the manuscript reveals a fundamental misunderstanding of the physics of an axion insulator. (1) In the introduction (line 65) and in Fig. 1(a), the authors describe the axion insulator state (with zero total Hall conductance) as hosting \"counter-propagating chiral channels from antiparallel magnetized surfaces.\" This is incorrect. As established by seminal works the authors themselves cite as Refs. 8 and 9 (M. Mogi et al., Nature Mater. 16, 516 (2017) and D. Xiao et al., PRL 120, 056801 (2018)), the ideal axion insulator state is fully gapped, including on its side surfaces, and therefore possesses no chiral edge states. Ref. 9 explicitly states: \"...yielding zero σxy plateau and eliminating the 1D chiral edge state.\" This description misrepresents the established picture of this state. (2) Similarly, the conceptual diagrams in Figs. 2(b) and 2(c) are inaccurate. In the state engineered by the authors, one surface is metallic. In such a configuration, delocalized bulk/surface states exist, and it is not possible for localized, chiral edge states to form on the side surfaces as depicted. This point is also clearly discussed in the literature (e.g., M. Mogi et al., Nat. Phys. 18, 390, their Ref. 38). These conceptual errors are not minor, as they undermine the theoretical foundation of the work.\n\nComment 4: In summary, while the experimental result is of high quality, the manuscript lacks the requisite novelty for publication in Nature. The central result is a predictable combination of previously reported phenomena.\n\nFor these reasons, I cannot recommend the publication of the manuscript in Nature. The work may be suitable for a more specialized journal but only after the authors have thoroughly revised the manuscript to correct the fundamental physical misinterpretations."}, "author_response": "We sincerely thank Referee #1 for their time and insightful comments, which have helped us to significantly improve the manuscript. We have carefully addressed each of the raised concerns, as detailed below.\n\nResponse to Comment 1: We thank the referee for this comment, which provides us with the opportunity to clarify the fundamental conceptual advance of our work beyond that of Mogi et al. While both studies report a half-quantized Hall conductance, the physical systems, underlying mechanisms, and most importantly, the scientific implications are fundamentally different.\n\nThe key distinctions are:\n(i) Physical System: Our work realizes a fully magnetic, compensated antiferromagnetic (AFM) system, in which both surfaces are intrinsically magnetic and capable of opening a gap. This enables the emergence of a quantized axion field in the bulk. In contrast, Mogi et al. studied a semi-magnetic heterostructure where one surface was intentionally non-magnetic and thus incapable of hosting the axion insulator phase.\n(ii) Scientific Goal: Our primary objective is to provide the first unambiguous boundary manifestation of the quantized axion field, a central challenge in the field for two decades. By achieving independent Fermi level control over individual layers, we establish a platform for realizing the first quantized Hall effect encoded in spatial (layer) degrees of freedom. Moreover, our additional data from VBC and CBC devices strongly suggest that both surfaces can contribute a half-quantized Hall effect under appropriate gating conditions, thereby, confirming half-quantized layer Hall effect as a direct and necessary boundary manifestation of the bulk axion invariant. By contrast, Mogi et al., interpreted their observation of half-quantized Hall conductance as a surface-only parity anomaly. While groundbreaking in its own right, this phenomenon does not correspond to a quantized axion field in the bulk, which requires a fully gapped system on all surfaces.\n(iii) Underlying Physics: Our realization of the half-quantized layer Hall effect represents a conceptual advance that was not anticipated by prior theoretical work.\na) First, a recent theoretical study of similar systems (Han et al., Phys. Rev. Lett. 134, 236206 (2025)) predicted a maximum layer Hall conductance significantly smaller than the half-quantized value we observe. Our data and model help bridge this gap between theory and experiment by identifying two previously overlooked physical mechanisms: disorder-mediated stabilization of Hall conductance, and the electric field screening by bulk trap states.\nb) Second, long-standing theoretical consensus held that “the half-quantized Hall conductance at the surface of a topological insulator is not directly measurable in a DC current experiment.” (see B. A. Bernevig, Topological Insulators and Topological Superconductors, Chapter 15.4 (2013).) Our theoretical framework, as discussed in the SI section v., demonstrates that half-quantization emerges when a classical metallic state forms on the conducting surface, a condition fulfilled in our system through magnetic disorder and strong doping, but not necessarily in the undoped, gapless Dirac state studied by Mogi et al.\nc) Third, the microscopic origin of the half-quantization in semi-magnetic systems by Mogi et al. is still under active theoretical debate, with some studies suggesting it may actually originate from the gapless surface rather than the gapped one. (e.g. Zou et al., Phys. Rev. B, 107, 125153 (2023).) Furthermore, their structure lacks independent layer degrees of freedom, realizing only a singular, non-tunable state.\nTo summarize, our work moves beyond the study of a single-surface anomaly phenomenon. Instead, we provide the first unambiguous electrical transport signature of the quantized axion field, addressing a two-decade-old challenge. It also represents the first quantization of a Hall effect encoded in spatial degrees of freedom. We have now included a detailed comparative analysis with Mogi et al. (revised SI section ix.) to make this critical distinction unequivocally clear.\n\nResponse to Comment 2: We thank the referee for acknowledging the significance of this research direction. While the integer quantum layer Hall effect is an exciting theoretical goal, our observation of a robust half-quantized layer Hall effect (HQLHE) is a profound discovery in its own right and represents the first experimental verification of the core principle underlying the theory of Dai et al.\nThe physics of the integer proposal by Dai et al. involves shifting the layer-locked Berry curvature monopoles above or below the Fermi level on a specific layer via an external electric field. In our case, we effectively control the merons (half-monopoles) on both surfaces. The emergence of HQLHE can be viewed as shifting one of the merons across the Fermi level. In this sense, our results provide experimental confirmation of the theory's core principle—that the Hall response is layer-polarized and quantized.\nThe full realization of the integer effect requires the complete gapping of the non-contributing surface, a condition extremely sensitive to disorder and chemical potential pinning. Our material-by-design approach using MBE-grown heterostructures offers superior and versatile parameter control (e.g., enabling tunable Chern numbers as demonstrated in Zhao et al. Nature 588.7838, 419-423 (2020)), making it a promising platform for ultimately achieving this integer quantized proposal. Nevertheless, the robust demonstration of the half-quantized case in a minimal axion insulator prototype, as reported here, represents a major advance in its own right and significantly deepens our understanding of layer Hall physics.\n\nResponse to Comment 3: We thank the referee for this astute observation. The referee is correct that, by symmetry, a similar half-quantized plateau should also emerge under strong positive gate voltages (n-doping the bottom surface).\nIn our original submission, the data for the n-doped regime was less comprehensive. In direct response to this comment, we have performed additional experiments on new devices. As shown below (also included in the new Extended Data Fig. 6), we now unequivocally observe the half-quantized layer Hall effect in the n-doped regime (positive 𝑉 ) in CBV-type devices optimized for this purpose. This was achieved by careful adjustment of the Bi:Sb ratio in the bottom part (3QL CBST - 10QL BST) of the heterostructure to approach a highly n-doped regime, a necessary adjustment to compensate the p-type carriers introduced by Cr doping. Here we show 2 devices with nominally the same parameters with Device A1 except for a different Bi concentrations for the bottom part: (a)(b)(c) Device E1 with 3QL Cr (Bi Sb ) Te /10QL (Bi Sb ) Te /10QL (Bi Sb ) Te /3QL V (Bi Sb ) Te , (d)(e)(f) Device E2 with 3QL Cr Bi Te /10QL Bi Te /10QL (Bi Sb ) Te /3QL V (Bi Sb ) Te . The excessive Bi provides highly n-doped regime for this purpose. Although the n-doping is not strong enough for Device E1 to achieve the half-quantization, the AHC as a function of 𝑉 clearly demonstrate such a trend. Device E2 without Sb for the bottom part clearly show a transition from ~ e²/h to a stable ~ e²/2h plateau, very similar to Fig. 2 in the original manuscript except for the opposite direction of tuning 𝑉 → +200 V . Because of the ferromagnetism now mediated by bulk carriers through RKKY mechanism, the coupling between the top and bottom surfaces becomes stronger so that a definitive antiparallel scenario is not well defined as shown in the hysteresis loops. We attribute the slight deviation from e²/2h to the residual Berry curvature of CBT-BT layer possibly due to the bulk conduction bands, and the AHC peak near 𝑉 = −100 V to the quasi-1D helical edge conduction channels as explained in the SI section v. The larger shift of chemical composition in Device E2 would possibly introduce some gap closing mechanism on the side surfaces, resulting in more conduction channels.\nThis also explains why we only have VBC devices for the n-doped regime in the original manuscript, as V doping usually introduce fewer carriers. Nevertheless, with combined efforts of composition adjustment and bottom gating, we now show a continuous transition of Hall conductance from e²/h towards e²/2h in the n-doped regime. This new data perfectly complements the p-doped data and confirms the symmetric nature of the effect, resolving the discrepancy between Figs. 3 and 4 in the original manuscript. The corresponding discussion has been added to the revised manuscript (Extended Data Fig. 5).\n\nResponse to Comment 4: We agree that distinguishing between these two scenarios is critical. Our data from devices with distinct magnetic profiles demonstrate that the trivial, symmetric-doping scenario can be safely exclude the second scenario.\nFor samples with symmetric magnetic doping (e.g., CBC devices in Extended Data Fig. 6), the two scenarios are indeed difficult to discern due to identical coercive fields of the two surfaces. However, the main devices in our manuscript focus on the axion insulator (CBV-type) devices, in which the top and bottom surfaces have distinct coercive fields. The key evidence is provided in Fig. 2a: the smooth transition from a 2-step hysteresis loop (indicating contributions from both surfaces) to a single-loop hysteresis with a half-quantized AHC upon applying 𝑉 robustly excludes the trivial scenario. In the trivial scenario, non-vanishing contributions from both surfaces would maintain a 2-step loop. This is corroborated by the shrinking of the relevant semicircle in the renormalization-group flow diagram (RGFD) measurements for CBV devices (Fig. 2f), in contrast to CBC devices.\nWe have significantly improved the clarity of this presentation in the revised manuscript by expanding the comparative analysis vs Mogi et al. and including the new n-doped CBV data. We also added discussions on the possible trivial explanations for the CBC devices with symmetric doping. (Extended Data Fig. 7)\nWe thank the referee again for their rigorous critique, which has greatly strengthened our work. We hope our responses and revisions have now addressed all concerns satisfactorily.\n\nResponse to Referee #2:\nWe thank Referee #2 for the positive assessment, the insightful comments and clear recommendation for publication. We are pleased that the Reviewer acknowledges the significance of our experimental demonstration of the half-quantized layer Hall effect, particularly its role as a direct probe of the quantized axion field. We followed Referee #2’s suggestions to revise our manuscript carefully and hope that the revised manuscript meets their standard for Nature.\n\nResponse to Comment 1: We thank the referee for points out that the schematic in Fig. 2c should depict the Berry curvature contributions from the conduction and valence bands with opposite signs. This was an oversight in the preparation of the figure. A corrected schematic will be provided in the revised manuscript.\n\nResponse to Comment 2: This optimal range represents a critical trade-off between several competing physical effects.\nFor spacers thinner than∼10 QLs, three possible issues arise. (i) The wavefunctions of the top and bottom surface states begin to overlap, leading to hybridization that opens a gap and destroys the layer-resolved half-quantized Hall conductance. This hybridization gap closes when the film thickness of Bi Te type material is about ~5 QL (Liu, Yang, et al. Phys. Rev. B 85, 195442 (2012)). The numerical results in Fig. R2 support this analysis, showing that the layer-resolved Hall conductance is non-quantized for small thickness 𝑑 due to hybridization effects, and gradually approaches half-quantization as 𝑑 increases. (ii) Electrostatic screening is insufficient. The chemical potential of top ( 𝜇 ) and bottom ( 𝜇 ) surface satisfy 𝜇 (𝑉 ) = 𝜇 (𝑉 )/cosh(𝑑/λ), where 𝑑 is the thickness of the magnetic TI and λ is the Thomas–Fermi screening length of the bulk trap states. When 𝑑 < 𝜆, the back-gate voltage V will shift the chemical potential of the top surface out of its local magnetic gap, leading to the breakdown of half-quantized layer Hall effect. (iii) The interlayer magnetic exchange coupling becomes significant, destabilizing antiparallel alignment of two magnetic layers and destroying the axion-insulator phase. All the above issues could be responsible for the observed reduced anomalous Hall conductance ~0.48𝑒2/ℎ and stronger interlayer magnetic coupling in Device A4 with spacer thickness of 5 QL (Extended Data Fig.3a).\nFor spacers thicker than ~20QLs, the system begins to behave as two electronically decoupled films. In this limit, the quasi-1D metallic states on the side surface of TI, likely occur and contribute to the transport, leading to the deviations of half-quantized Hall conductance, see more details in the SI section v.\nA new discussion detailing this trade-off will be added to the revised SI section vi.\n\nResponse to Comment 3: The six-layer tight-binding model is a standard and computationally efficient effective model used to capture the essential low-energy physics of the two surface states of a 3D topological insulator. For a film thickness of 20 QLs, the system is firmly in the 3D limit where the hybridization between top and bottom surfaces is negligible. In this regime, the surfaces are electronically independent, and their behavior is well-described by the effective model. In the theoretical model, six layers are sufficient to suppress the hybridization between the top and bottom surfaces (see Fig. R2). The model is not intended to reproduce the bulk band structure of the 20 QL film but to correctly capture the qualitative behavior of the surface states under gating, which it does successfully. A clarifying sentence of this effect will be added to the Methods section.\n\nResponse to Comment 4: To elucidate the role of back-gate voltage 𝑉 , we derive the relationship between the chemical potential 𝜇(z) and 𝑉 . We find that the chemical potential of top (𝜇 ) and bottom (𝜇 ) surface satisfy 𝜇 (𝑉 ) = 𝜇 (𝑉 )/cosh(𝑑/𝜆), where 𝑑 is the thickness of the magnetic TI and 𝜆 = √𝜀/(𝑒2𝐷 ) is the Thomas–Fermi screening length. Here, 𝐷 denotes the density of bulk trap states and 𝜀 is the absolute permittivity of TI. In the limit 𝑑 ≫ 𝜆, the top-surface chemical potential vanishes (𝜇 (𝑉 ) ≈ 0). The high density of bulk trap states in the magnetic TI effectively screens electric fields in the bulk, thereby the back gate only tunes the chemical potential of the bottom surface. Consistently, the electrostatic potential takes the form 𝜙(𝑧) = 𝜙 cosh((𝑑−𝑧)/λ)/cosh(𝑑/λ). 𝜙(z) in the bulk also approaches zero as 𝑑 ≫ 𝜆, and the back-gate voltage only changes the potential of the bottom surface (𝜙 ).\nWhen 𝑒|𝜙 | < bulk band gap ∆ of magnetic TI, the back-gate voltage 𝑉 mainly tunes the chemical potential of bottom surface. When 𝑒|𝜙 | ~∆ , however, 𝑉 starts to modify the bottom-surface band structure. In our experiments, the bulk gap is ∆ ≈ 300 meV (H. Zhang et al., Nat. Phys. 5, 438 (2009)), while the surface magnetization gap is only 𝑀 ≈ 1 meV. Therefore, when the system realizes the half-quantized layer Hall effect (e.g., |𝜇 | = 𝑒|𝜙 | ≫ 𝑀 ), the condition 𝑒|𝜙 | ≪ ∆ is still satisfied, and 𝑉 predominantly controls the bottom-surface chemical potential in this regime. Note that even if the bottom-surface band structure is slightly modified, it contributes negligible Berry curvature once the chemical potential is tuned deep into the metallic regime. Therefore, for thick magnetic TIs with a high density of bulk trap states, the back-gate voltage 𝑉 serves primarily to locally tune the electrostatic and chemical potential of the bottom surface. In practice, this allows the back-gate to shift the chemical potential of the bottom surface sufficiently far into a metallic regime, while the top surface, subject to a much weaker effective gate field, remains within its local magnetic gap. In current study, the exact density of bulk trap states is unknown. By assuming that the conductance deviation is mainly caused by the insufficient electrostatic screening, we roughly estimate from the data of Device A4 (spacer layer thickness 𝑚 = 5) that Thomas–Fermi screening length in our samples is 𝜆~5 nm. This corresponds to a 3-dimensional density of bulk trap states 𝐷 = 𝜀⁄(𝑒𝜆)2 = 5.5×1044 𝑚−3𝐽−1 , about 15 times larger than the measured 𝐷 = 3.6×1043 𝑚−3𝐽−1 by capacitance spectroscopy of a 67 nm BiSbTeSe thin flake exfoliated from crystals grown by the modified Bridgman method (Wang et al. Nano Letters 20.12, 8493-8499 (2020)). This is a reasonable value considering the non-equilibrium growth nature and the active magnetic doping by molecular beam epitaxy, which inevitably introduce much more defects related to bulk trap states.\nBelow we derive the relationship between the chemical potential of the top and bottom surfaces under gating conditions. We also include this derivation and related discussion in the revised SI section x.\n\nResponse to Comment 5: We appreciate the referee’s suggestion to include additional raw data to substantiate reproducibility and robustness. To substantiate this, we add a new figure, presenting the raw longitudinal resistance (𝑅 ) and Hall resistance (𝑅 ) data as a function of magnetic field for several different devices (A4, A5 and A6, see the revised SI section xi.). These raw datasets demonstrate that the reported phenomenon is reproducible across devices and robust to sample-to-sample variation.\nIn addition, all raw resistance measurement data, including the main, extended, and supplementary data, will be archived upon publication.\n\nResponse to Referee #3:\nWe thank Referee #3 for their thorough review and for acknowledging the quality of our techniques. Their critiques have been invaluable in helping us to clarify key aspects of our work and strengthen the manuscript. We address each point below.\n\nResponse to Comment 1: We respectfully disagree with the referee's assessment that our result is a straightforward or expected combination of known phenomena. While it is well established that a single massive Dirac surface yields σ = e²/2h (Phys. Rev. Lett. 51, 2077 (1983)), and that layer-polarized transport has been proposed by Xu’s group, our result cannot be reduced to a trivial combination of these ideas. The realization of half-quantized layer Hall effect (HQLHE) within a fully tunable, compensated magnetic system, provides the first direct boundary probe of the bulk axion field, a conceptual and experimental advance not anticipated by prior work.\nThe referee stated that observing σ = e²/2h by gapping one surface in an antiparallel system is \"logical and expected.\" However, this is not supported by prior theory. Until now, no explicit theoretical prediction of the HQLHE existed. In fact, a very recent calculation (Han et al., Phys. Rev. Lett. 134, 236206 (2025)) on similar systems predicted a maximum layer Hall conductance significantly smaller than the quantized value we observed. This discrepancy highlights a critical gap in theoretical understanding. Our work, through experiment and modeling, help resolve this by identifying new physical mechanisms—disorder-mediated stabilization and bulk trap state screening—which are essential for achieving robust quantization. This finding transforms the observation from a \"logical outcome\" into an unexpected discovery that reshapes the theoretical framework.\nOur work represents a significant leap beyond prior studies:\nvs. Gao et al. (Nature 595, 521 (2021)): First, we demonstrate the first quantized layer Hall effect, a monumental advance from their non-quantized result. The transition from a non-quantized to a quantized Hall response encoded in the layer degree of freedom is comparable in significance to the historical leap from the classical Hall effects (charge or spin) to their quantum counterparts. Second, our material-by-design approach (CBST/BST/VBST) naturally breaks 𝒫𝒯 -symmetry through tailored doping profiles, enabling quantization without an external electric field in some devices (e.g., Fig. 2a). The underlying physical mechanism is very different: driven by external electric field, or by Fermi level positioning (our work). Third, this experimental realization required overcoming substantial challenges, such as identifying a suitable spacer thickness (10–20 QLs) to suppress both surface hybridization and excessive conduction channels, achieving precise control of dual magnetic exchange gaps, and establishing asymmetric Fermi level positioning. These complexities, while underscoring the tunability and versatility of our material platform, also highlight that the complete extraction of a half-quantized Hall signal from a single surface is physically non-trivial.\nvs. Mogi et al. (Nat. Phys. 18, 390 (2022)): Our system and scientific goals are fundamentally different. We engineer a fully magnetic, compensated antiferromagnetic system to probe the quantized axion field, while their semi-magnetic system cannot host axion insulator phase. Notably, signatures of deviation from half-quantization in Mogi et al., such as AHC peak near the CNP and smaller values in more strongly doped regimes (e.g. Fig. S10C in their supplementary material), are absent in our work. In our case, HQLHE occurs only in strongly doped regime and sometimes even exceeds e²/2h slightly. These discrepancies strongly suggest that the two half-quantization effects arise from distinct mechanisms. In fact, some theoretical studies suggest that the parity anomaly state in Mogi et al. stems from the gapless surface rather than the gapped one. (e.g. Zou et al., Phys. Rev. B 107, 125153 (2023).)\nIn conclusion, we have achieved the first electrically/magnetically tunable, precisely quantized transition between layer-polarized states in a prototype axion insulator. This active control over a quantized topological response is a landmark achievement that enriches the field of topological layertronics.\n\nResponse to Comment 2: We acknowledge the referee's point that a direct measurement of the topological magnetoelectric (TME) coefficient is a powerful probe. However, we contend that our approach provides an equally fundamental and direct electrical transport signature of the quantized axion field, serving as a manifestation of the essential bulk-boundary correspondence in topological systems.\nFoundational theoretical work by Qi et, al. establishes that the axion insulator phase, characterized by the quantized axion field, manifests through two inseparable physical consequences linked by the bulk-boundary correspondence:\n1. A quantized TME effect in the bulk, described by the Lagrangian term ∆𝐿 = 𝜃(𝛼⁄4𝜋2)𝐸 ∙𝐵;\n2. A half-quantized Hall effect on the surface, where time-reversal symmetry is broken by a surface magnetization.\nWhile these are distinct measurements, they probe the same underlying quantized axion field. (see Sec. II of Sekine & Nomura, J. Appl. Phys. 129, 141101 (2021), where the authors prove the theoretical equivalence of these two effects in an axion insulator as a direct result of the quantized axion field.)\nThe primary challenge in second approach has always been the unambiguous isolation of a single surface's contribution, as transport measurements typically probe the top and bottom surfaces simultaneously, yielding a total conductance of either e²/h or 0. The central achievement of the present work is to solve precisely this longstanding challenge. By using electrical gating to selectively render one surface metallic and non-contributing, our work provides the first unambiguous layer-resolved observation of the isolated half-quantized Hall conductance of the other gapped surface. Crucially, this signature in the antiparallel magnetization scenario strongly indicates that the bulk is in the axion insulator state with θ = π. This is in stark contrast to the semi-magnetic system by Mogi et al., which, by design, is not an axion insulator and therefore, its half-quantized Hall effect does not correspond to a quantized bulk axion field. Thus, our layer-tunable HQLHE is not merely a property of a single surface; it is a direct measurement of the boundary signature that is mandated by, and serves as a definitive probe of, the quantized bulk axion field.\nWe note that the work by J.-X. Qiu et al., mentioned by the referee, demonstrated the measurement of dynamical axion quasiparticles by detecting oscillations in θ using ultrafast pump-probe optics. Their results, including the static magnetoelectric coupling measurements, did not exhibit quantization. In contrast, our work provides what we believe to be the first unambiguous evidence of a quantized static axion field.\nWe have revised the manuscript to clarify that we are measuring the surface signature of the quantized axion field, which is a direct and rigorous consequence of the bulk topology.\n\nResponse to Comment 3: We agree that the ideal, truly 3D bulk-gapped axion insulator should have no chiral edge states, as established in the seminal literature. Our manuscript did not intend to claim the existence of “chiral edge states”. The referee’s concern likely stems from a misunderstanding of how we use the term “chiral edge channels”.\nIn our manuscript, “chiral edge channels” refers to edge currents that arise when magnetized topological-insulator surfaces are present—a distinct setting that is under active investigation (e.g. Phys. Rev. Lett. 129, 096601 (2022); Nat. Sci. Rev. 10, nwad025 (2023); Phys. Rev. B 105, L201106 (2022)). (i) Theory predicts dissipationless, persistent half-quantized edge currents in fully gapped two-dimensional Dirac systems (Levitov’s group, Phys. Rev. Lett. 114, 256601 (2015)). (ii) More recent studies further show the coexistence with a metallic surface of “chiral edge channels” at boundaries in the half-quantized Hall effect. These channels are qualitatively distinct from traditional “chiral edge states” found in the integer quantum Hall effect: they exhibit power-law rather than exponential spatial decay into the metallic region while still dictating measurable transport responses. These results motivate our terminology “chiral edge channel” rather than “chiral edge state”.\nTo clarify the distinction between the “chiral edge channels” and “chiral edge state”, we have added an explanation of the physical meaning of the chiral channels in our system—described as a form of persistent chiral edge current—and have revised the schematics in Fig. 2 to represent them with dashed arrows.\n\nResponse to Comment 4: We thank Referee #3 for their thorough review and for acknowledging the high quality of our experimental result. The referee's critique, while incisive, has provided a valuable opportunity to clarify the foundational aspects of our work. We also note that another referee, a theorist, was \"quite positive,\" recognizing the work as a \"significant advance\" that provides \"clear evidence of layer-resolved half-quantized Hall effects.\nOur manuscript presents a significant synthesis that achieves several longstanding goals in the field, rather than a trivial combination of known phenomena. Specifically, our work provides three fundamental advances:\n(i) Direct Electrical Evidence of the Quantized Axion Field: The observed half-quantized layer Hall effect is not merely a surface phenomenon; it is the definitive electrical transport fingerprint of the bulk quantized axion field. As a direct consequence of the bulk-boundary correspondence, this effect is theoretically equivalent to a quantized topological magnetoelectric (TME) effect. Our work provides the first unambiguous, static electrical manifestation of this long-sought signature, which contrasts with prior reports that failed to achieve static quantization.\n(ii) A Landmark Achievement in Topological Control: We report the first precisely quantized Layer Hall Effect within a fully compensated antiferromagnetic system. While prior work demonstrated a non-quantized layer Hall effect, our achievement of a robustly quantized value represents a monumental advance, akin to the historical leap from the classical Hall effect to its quantum counterpart. This required overcoming immense experimental hurdles in material design and dual-gate control to achieve this unprecedented metrological precision, thereby pioneering the emerging field of \"topological layertronics\".\n(iii) Nontrivial Emergent Physics: Our findings represent a discovery that challenges and refines the prevailing theoretical framework, rather than being a predictable consequence of it. Our observation of a robustly quantized value directly contradicts a recent theoretical study that anticipated a much smaller, non-quantized value for the layer Hall conductance in a similar system (Han et al., Phys. Rev. Lett. 134, 236206 (2025)). Our experiment, therefore, helps to resolve a \"critical gap in theoretical understanding\" and paves the way for a more complete theoretical picture of these systems.\nIn conclusion, the manuscript describes a monumental experimental achievement that advances both the fundamental understanding of topological phenomena and the field of \"layertronics\". We have also added extended discussions on several critical points, as summarized in the list of major changes in the revised manuscript, to clarify any potential misunderstandings. We hope that the revised manuscript and these responses adequately address the referee's concerns."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I thank the authors sincerely for their responses. I believe that the authors have clearly addressed my technical questions. I therefore support the paper for publication.", "reviewer_2": "The authors have thoroughly addressed my concerns. I agree that the directly measured half-quantized layer Hall effect is the definitive electrical transport fingerprint of the bulk quantized axion field. The results will be of considerable interest to the broad topological physics and quantum materials community, and are suitable for publication in Nature Communications.", "reviewer_3": "Comment 1: The authors report the observation of a layer-resolved, half-quantized anomalous Hall effect in MnBi2Te4-based heterostructures, realized via dual-gate control of the Fermi level and magnetic-field manipulation of the magnetic configuration. The experimental methodology appears technically sound, and the data quality is high. From an experimental standpoint, I find the results convincing, and in my view the manuscript is suitable for publication.\n\nHowever, having examined the earlier review process and the authors’ responses, it is clear that the central point of contention concerns the novelty and interpretation of the work, specifically its relationship to Mogi et al. (Nat. Phys. 2022). The authors provide a reasonable explanation of how a fully magnetic, compensated antiferromagnetic system differs conceptually from the semi-magnetic, parity-anomaly scenario in the prior work. This clarification is helpful and addresses several of the original concerns.\n\nComment 2: That said, I cannot fully agree with the authors’ assertion that observing a half-quantized Hall conductance is “equally fundamental” to measuring a quantized axion field in the bulk. The observation of σxy = e^2/2h indeed constitutes a necessary boundary manifestation of the axion insulator state, but it is not a sufficient condition. If one accepts the authors’ statement as written, it would imply that the half-quantized Hall response reported in semi-magnetic systems should also be regarded as a direct demonstration of a quantized axion field, which is not consistent with current theoretical understanding. Therefore, the manuscript should adopt more accurate wording—for example, stating that the half-quantized layer Hall conductance is a boundary signature consistent with, and expected from, a quantized axion field, rather than an equally fundamental proof of the bulk axion response.\n\nFor this reason, I recommend minor revision, requesting that the authors adjust the language in the main text to avoid the implication of logical equivalence between surface half-quantization and the full verification of a quantized axion field. With this clarification, I believe the manuscript would be suitable for publication."}, "author_response": "Response to Referee #1: We thank the reviewer for their positive feedback and endorsement.\n\nResponse to Referee #2: We thank the reviewer for recognizing the broad interest of our work and for endorsing our conclusion that the half-quantized layer Hall effect provides a definitive electrical signature of the quantized axion field.\n\nResponse to Referee #3:\nResponse to Comment 1: We thank the referee for their positive assessment of our experimental methodology and data quality, and for acknowledging the suitability of our manuscript for publication. We also appreciate their recognition of our clarification regarding the conceptual distinction between our fully magnetic, compensated antiferromagnetic system and the semi-magnetic parity-anomaly scenario in prior work.\n\nResponse to Comment 2: We acknowledge the referee's point regarding the wording around the relationship between the half-quantized Hall conductance and the quantized axion field. We agree that the half-quantization signature, while being a necessary boundary manifestation, should not be presented as fully equivalent to a direct bulk measurement. We have revised the text accordingly (highlighted in yellow), clarifying that the half-quantized layer Hall effect serves as a key boundary signature consistent with, and expected from, the quantized axion field in the bulk."}]

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