Quantum Anomalous Hall Insulator (QAHI) State

• QAHI state is a two-dimensional topological phase characterized by an insulating bulk and quantized Hall conductance from intrinsic magnetism.
• It is underpinned by massive Dirac and lattice Hamiltonians with a nonzero Chern number enforcing dissipationless, unidirectional edge conduction.
• Realized in platforms like magnetically doped topological insulators and oxide heterostructures, QAHIs offer tunability for metrological, spintronic, and quantum computing applications.

The quantum anomalous Hall insulator (QAHI) state is a two-dimensional topological electronic phase characterized by an insulating bulk and quantized Hall conductance at integer multiples of $e^2/h$ in the complete absence of external magnetic field. The QAHI effect originates from spontaneous time-reversal symmetry (TRS) breaking due to intrinsic magnetism, not from Landau quantization by external fields. The topological invariant governing this state is the Chern number $C$, which counts the net number of dissipationless chiral edge modes at a sample's boundary. QAHIs have been theoretically predicted and experimentally realized in diverse platforms, including magnetically doped topological insulator (TI) films, intrinsic magnetic van der Waals materials, oxide heterostructures, graphene and transition-metal dichalcogenide (TMD) moiré superlattices, and nanopatterned 2DEGs, and are central to metrological, spintronic, and quantum information applications (Chang et al., 2022, Wang et al., 2014, Feng et al., 2015, Zhao et al., 2020, Zhang et al., 2011, Baidya et al., 2016, Fang et al., 2013, Zhang et al., 2010).

1. Theoretical Model and Topological Invariant

Minimal theories of the QAHI state employ either massive Dirac models or lattice Hamiltonians incorporating both spin–orbit coupling (SOC) and exchange splitting. In a prototypical 2D TI or Chern insulator thin film, the low-energy physics near a Dirac point is governed by

$$H(\mathbf{k}) = v_F(k_x \sigma_x + k_y \sigma_y) + [M(\mathbf{k}) + \Delta] \sigma_z,$$

where $v_F$ is a Dirac velocity, $M(\mathbf{k})$ may encode hybridization or sublattice asymmetry, and $\Delta$ is the TRS-breaking exchange mass induced by magnetic order (Chang et al., 2022, Wang et al., 2014). In the lattice context, the Haldane model realizes an explicit QAHI via a honeycomb structure with complex next-nearest-neighbor hoppings:

$$H_H(\mathbf{k}) = \sum_{ij} t_{ij} c^\dagger_i c_j + \lambda \sum_{⟨⟨ij⟩⟩} e^{i\phi_{ij}} c^\dagger_i c_j + m \sum_i \xi_i c^\dagger_i c_i.$$

The hallmark of the QAHI phase is a nonzero Chern number

$$C = \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2 k\, \Omega(\mathbf{k}),$$

where the Berry curvature $\Omega(\mathbf{k})$ arises from the Bloch wavefunctions of filled bands and $\mathrm{BZ}$ is the Brillouin zone. For gapped mean-field Hamiltonians of the form $$H = \mathbf{d}(\mathbf{k}) \cdot \boldsymbol{\sigma}$$, the Chern number is computed as

$$C = \frac{1}{4\pi} \int d^2k \, \hat{\mathbf{d}} \cdot (\partial_{k_x} \hat{\mathbf{d}} \times \partial_{k_y} \hat{\mathbf{d}}),$$

with $\hat{\mathbf{d}} = \mathbf{d}/|\mathbf{d}|$ (Feng et al., 2015, Wang et al., 2014). Only integer values are allowed for $C$, securing the quantization of the Hall conductivity: $\sigma_{xy} = C\,e^2/h$.

2. Microscopic Mechanisms and Material Realizations

The QAHI state can arise in both noninteracting band-structure models and interaction-driven regimes:

• Magnetically Doped TIs: In Cr- or V-doped (Bi,Sb)$_2$Te$_3$ thin films and MnBi$_2$Te$_4$, localized moments couple to Dirac surface states, opening an exchange gap ($\Delta$) that allows a single chiral edge channel on each boundary when the chemical potential lies in the gap (Chang et al., 2022, Wang et al., 2014, Zhang et al., 2011).
• Intrinsic Magnetic TIs and Crystalline Insulators: Systems such as intrinsic MnBi$_2$Te$_4$ or thin-film topological crystalline insulators with engineered band inversion and magnetic order exhibit QAHI phases including $|C| > 1$ regimes—by tuning subband inversion via thickness, doping, or strain (Fang et al., 2013, Zhao et al., 2020).
• Correlated Oxide Heterostructures: In ultrathin 3d–5d double perovskite films (e.g., Ba$_2$FeReO$_6$), a combination of strong SOC, non-relativistic orbital-Rashba effect, and robust room-temperature magnetism can produce a C=1 QAHI with large bandgaps ($\sim 100$ meV) (Baidya et al., 2016).
• Moiré Systems and 2DEGs: Flat bands in twisted bilayer/trilayer graphene and TMD superlattices with spontaneous valley polarization, as well as patterned 2DEGs with combined Rashba and Dresselhaus SOC and weak in-plane Zeeman terms, have been proposed or observed to support QAHI in selected fillings or modulation regimes (Chang et al., 2022, Zhang et al., 2010).
• Interaction-Driven QAHI: Strong electronic interactions—e.g., nearest- and next-nearest-neighbor repulsion in checkerboard-lattice semimetals—can stabilize a QAHI with $C=1$ via spontaneous loop-current (TRS-breaking) ordering, even when noninteracting bands are semimetallic (Sur et al., 2018, Lu et al., 2021).

3. Edge-State Physics, Bulk–Boundary Correspondence, and Domain Walls

The most robust transport signature of the QAHI is dissipationless, unidirectional chiral edge conduction, enforced by the bulk–boundary correspondence. Each unit of Chern number $C$ guarantees the existence of one chiral edge channel per boundary (Wu et al., 2014, Chang et al., 2022). The edge state inherits topological spin textures that wind around sample boundaries, as established in models with chiral-like symmetry:

• Chiral edge spin texture is a direct real-space reflection of the nontrivial bulk topology, corresponding to a quantized Berry phase and a 1D winding number linked to $C$ (Wu et al., 2014).
• Edge-state chirality (direction of current propagation) is set by the sign of the local magnetization, which may be controlled via field, current, or exchange bias (Yuan et al., 2022, Zhang et al., 2022).
• During domain reversal or near coercive fields, multi-domain configurations yield a network of chiral channels at domain boundaries, leading to zero-Hall plateaus or distinct scaling phenomena (Feng et al., 2015, Liu et al., 2021).

4. Phase Diagrams, Tunability, and Chern Number Switching

The QAHI phase can be accessed and tuned in diverse parameter regimes:

• Exchange mass, hybridization, and subband inversion: The competition between magnetic exchange and hybridization in TI thin films, or structural distortions in crystalline insulators, controls topological phase boundaries. Tuning film thickness, doping concentration, or perpendicular electric fields can drive transitions between trivial, QAHI, and axion-insulator regimes (Wang et al., 2014, Fang et al., 2013, Lu et al., 4 Sep 2025).
• Chern number tuning: Multilayer magnetic TI stacks with repeated QAH/normal sequences support $|C|$ up to five, limited by number of magnetically doped bilayers and their internal structure. Each additional QAH layer adds an independent chiral channel, observable as a reduction in Hall resistance plateau $R_{xy}=h/(C e^2)$ and proportional increase in breakdown current and bandwidth (Zhao et al., 2020).
• Magnetoelectric and electrical switching: Edge-state chirality can be electrically reversed by applying spin–orbit torque (SOT) current pulses or by exchange-bias field training at a QAHI/antiferromagnetic interface, offering a route to programmable chiral interconnects and logic devices (Yuan et al., 2022, Zhang et al., 2022, Lu et al., 4 Sep 2025).

5. Effects of Disorder, Domain Structure, and Critical Phenomena

QAHI systems exhibit rich physics at the intersection of topology, magnetism, and disorder:

• Disorder and Magnetic Domains: Random magnetic doping leads to inhomogeneous exchange mass $M(\mathbf{r})$, which, depending on disorder strength, results in either a globally topological QAH insulator (quantized $\rho_{yx}$, vanishing $\rho_{xx}$) or an anomalous Hall (AH) insulator (finite, nonquantized $\rho_{yx}$, divergent $\rho_{xx}$) (Liu et al., 2021). Domain wall networks at coercivity are responsible for unique scaling behavior and zero Hall plateau phases absent in standard quantum Hall systems (Feng et al., 2015).
• Universal Scaling: At quantum phase transitions between QAHI, AH-insulator, and trivial insulator, observables such as $\rho_{xx}(B,T)$ collapse onto scaling functions with exponents that are distinct from conventional integer quantum Hall transitions—a direct fingerprint of magnetic-domain-driven criticality (Liu et al., 2021, Feng et al., 2015).

6. Proximity Effects and Chiral Majorana Modes

QAHI edges are a promising platform for realizing topological superconductivity upon interfacing with conventional $s$-wave superconductors:

• Chiral Majorana Physics: Superconducting proximity effect in QAHIs can induce chiral Majorana edge modes—observable via quantized half-integer conductance plateaus, negative nonlocal conductance due to crossed Andreev reflection, and zero-bias peak signatures in narrow QAHI/S superconduting nanoribbons (Uday et al., 2023, Atanov et al., 2023).
• Majorana Zero Modes in Quasi-1D Devices: Nanoribbon devices etched from QAHI films and proximitized by Nb exhibit evolution from multi-gap to single zero-bias conductance peaks in increasing perpendicular magnetic field, consistent with the appearance of localized Majorana zero modes in the single-channel regime (Atanov et al., 2023).

7. Material Platforms, Experimental Signatures, and Applications

A wide variety of systems have demonstrated or are predicted to host QAHI states:

Material Platform	Mechanism	Critical Temperature/Bandgap
Cr- or V-doped (Bi,Sb)$_2$Te$_3$ TIs	Exchange gap via FM doping	$T_c \sim 10$–$30$ K, $\Delta \sim 10$ meV (Chang et al., 2022)
MnBi$_2$Te$_4$ (intrinsic magnetic TI)	Layered AF/FM order	$T_N \sim 25$ K, robust $C = 1$ (Chang et al., 2022)
Ultrathin Ba$_2$FeReO$_6$ (double perovskite oxide)	SOC + orbital Rashba + FM	$T_c \sim 300$ K, $\Delta \sim 100$ meV (Baidya et al., 2016)
Topological crystalline insulator films (TCI)	Zeeman + surface Dirac cones	$C$ tunable by $d$, doping, strain (Fang et al., 2013)
Moiré graphene/TMD superlattices	Interaction-driven QAHI	$T_c \lesssim 8$ K ($\nu=3$ TBG) (Chang et al., 2022)
Patterned 2DEG (honeycomb + SOC + $B_y$)	Synthetic gauge fields	$\Delta \sim 0.1$–0.5 meV ($<$6 K) (Zhang et al., 2010)

Experimentally, QAHI phases are identified via:

• Vanishing longitudinal resistance $R_{xx} \rightarrow 0$.
• Quantized Hall plateau $R_{xy} = h/(C e^2)$ to better than $10^{-6}$ precision at sub-Kelvin temperatures (Feng et al., 2015, Zhao et al., 2020).
• Quantized plateau stability under size and disorder, via topological protection (Lu et al., 4 Sep 2025).
• Hysteretic switching, gate or SOT control of chirality, nonlocal edge current mapping.
• Chiral edge-state detection via scanning probes or microwave measurements (Wu et al., 2014).

QAHI-based devices provide metrologically stable resistance standards, ultra-low-dissipation interconnects, electrically reconfigurable spintronic elements, and a platform for topological quantum computing utilizing chiral Majorana physics (Chang et al., 2022, Uday et al., 2023). Their tunable Chern number and the robustness of edge modes to disorder and geometry underpin their technological promise.