EQAH: Extended Quantum Anomalous Hall Effect

• EQAH is a phenomenon where integer-quantized Hall conductance arises over extended carrier densities via interplay of electron correlations, symmetry breaking, and edge effects.
• It involves mechanisms such as the parity anomaly, crystalline order, multilayer stacking, and metallic conduction, yielding diversified quantized plateau behaviors.
• Experimental realizations in moiré superlattices, topological insulator stacks, and ferromagnetic metals challenge traditional QAH paradigms with variable temperature and bias transitions.

The Extended Quantum Anomalous Hall Effect (EQAH)—also referred to as the Extended Integer Quantum Anomalous Hall (EIQAH) effect and its crystal variants—denotes the emergence of robust, integer-quantized Hall conductance beyond the conventional settings of clean, fully filled topological bands at zero external magnetic field. Instead, the quantized Hall plateau persists over a broad range of carrier densities or moiré fillings, in systems where electron correlations, translation symmetry breaking, multi-layer stacking, external fields, metallic conduction, or edge-driven entropy all play decisive roles. Originally grounded in the parity anomaly of 2D Dirac-like systems, EQAH states have now been observed and theoretically characterized across moiré superlattices, van der Waals stacks, ferromagnetic metals, multilayer topological insulators, and even quasi-3D systems, raising fundamental questions about the interplay of topology, symmetry-breaking, and electron interactions in condensed-matter platforms.

1. Definition and Taxonomy of EQAH States

The EQAH effect generalizes the conventional quantum anomalous Hall (QAH) phase in two critical directions: (i) the quantized Hall plateau $\sigma_{xy}=Ce^2/h$ (typically $C=\pm 1$) occurs over an extended filling range, not only at commensurate integer band fillings as in standard Chern insulators, and (ii) quantization persists under competing bulk conduction, disorder, finite temperature, and external fields (Lu et al., 2024, Patri et al., 2024, Wan et al., 29 Dec 2025). The integer-quantized plateau often spans $v\in[0.5,1.3]$ in rhombohedral graphene/hBN multilayers and is seen at $C=\pm N$ in stacked topological-insulator multilayers (Jiang et al., 2018), as well as up to $C=\pm 5$ in spin–orbit proximitized pentalayer graphene (Han et al., 2023).

EQAH states can be classified according to:

Type	Description	Prototypical System
Crystalline EQAH	Integer QAH phase stabilized by density-wave crystal order, breaking translation symmetry	Moiré superlattices, multilayer graphene (Patri et al., 2024)
Multilayer/high-Chern	Multiple decoupled QAH layers, each $\mathcal{C}=1$, giving total $C=N$	MBE-grown (Bi,Sb)$_2$Te$_3$/CdSe stacks (Jiang et al., 2018)
Extended metallic EQAH	Coexistence of quantized $\sigma_{xy}$ with finite $\sigma_{xx}$, chiral edges + bulk metallicity	Ferromagnetic metal bands with band inversion (Wan et al., 29 Dec 2025)
Edge-driven/EQAH-FCI	Plateau upheld by fast edge mode entropy, giving way to FCI (fractional Chern insulator) as edge entropy exceeds bulk order	Rhombohedral graphene pentalayer devices (Wei et al., 2024)
Parity-anomaly (orbital field)	Robust QAH plateau surviving orbital fields, violating Onsager relation	BHZ/HgMnTe, Dirac-like models (Böttcher et al., 2019, 2002.01443)

2. Microscopic Mechanisms and Model Hamiltonians

The microscopic origins of EQAH states span band structure engineering, spontaneous symmetry breaking, and orbital/spin textures:

• Parity anomaly and Dirac mass reversal: 2D Dirac-based QAH models exhibit a parity anomaly, where the Chern–Simons term induces quantized Hall conductance $\sigma_{xy} = C\,e^2/h$ that survives even in orbital magnetic fields, up to a critical field $H_{crit}$ where the Dirac mass closes (Böttcher et al., 2019, 2002.01443). The low-energy Hamiltonian generically reads $$H(\mathbf{k}) = [M - B\,\mathbf{k}^2]\sigma_z - D\,\mathbf{k}^2\sigma_0 + \mathbf{A} \cdot \bm{\sigma}$$.
• Crystalline order in moiré bands: In correlated moiré superlattices, proximity to integer filling prompts either a crystalline arrangement of electrons ("anomalous Hall crystal"—AHC$_\rho$) or holes on an IQAH background, breaking translational symmetry to stabilize the quantized plateau across a filling continuum (Patri et al., 2024). The Landau–Ginzburg free energy

$$F[u] = \int d^2 r\, [ C_L(\nabla \cdot u)^2 + C_T(\nabla \times u)^2 - V_0 \sum_G \cos(G \cdot (r - u(r)))]$$

captures the elastic and pinning aspects of the electronic crystal, with $u(r)$ the displacement field.

• Chiral edge states plus metallic bulk: In ferromagnetic metals, band inversion and spin–orbit coupling produce chiral edge modes with $C=\pm 1$ even as the bulk remains metallic (without a gap), and strong dephasing enforces parallel edge and bulk transport channels such that $\sigma_{xy}$ remains quantized but $\sigma_{xx} > 0$ (Wan et al., 29 Dec 2025), a significant extension beyond insulating QAH paradigms.
• Multilayer stacking: Stacking $N$ decoupled QAH layers yields net $C=N$ and a Hall resistance $R_H = h/(N e^2)$, as each layer provides an independent chiral edge channel (Jiang et al., 2018). Block-diagonalizing the Hamiltonian for weakly coupled layers gives the sum rule for total Chern number.

3. Edge Physics, Disorder, and Transport Signatures

EQAH phases are characterized by the central role of gapless, chiral edge modes, domain walls, and edge-driven phase transitions:

• Edge entropy and phase competition: In both crystalline EQAH and fractional Chern insulator (FCI) regimes, the entropy associated with edge modes is crucial. The FCI edge, with lower velocity $v_{FCI}$, generates higher thermal entropy $S_{edge} \sim (\pi T L)/(6 v_{edge})$, enabling thermal and current-driven transitions from EQAH to FCI phases at critical temperature and bias (Wei et al., 2024).
• Domain walls and edge reconstruction: Disorder nucleates domains with opposite quantized Hall conductance, hosting networks of co-propagating edge states. Edge reconstruction, inter-channel interaction $w$, and disorder further reduce edge velocities, boosting edge entropy and advancing the EQAH$\to$FCI transition (Wei et al., 2024).
• Transport plateaus and breakdown: Experimentally, the EQAH manifests as $R_{xy}=h/e^2$, $R_{xx} \approx 0$ over extended fillings, with sharp breakdown (depinning) at critical current $I_c$ or temperature $T_c$. At generic fillings, the depinned phase is metallic; at Jain fractions, an equilibrium transition to FQAH states occurs, driven by the polarizability difference $\chi_{Jain} - \chi_{cryst}$ in the presence of applied field (Patri et al., 2024).

4. Generalization to High Chern Number, Three Dimensions, and Metallic Regimes

EQAH phenomena exhibit robust quantized conductance well beyond the unit Chern insulator case and conventional dimensions:

• High-Chern-number EQAH: In MBE-superlattices [(Bi,Sb)$_2$Te$_3$/CdSe], stacked up to $N=4$, $R_H$ scales as $h/(N e^2)$, confirming dissipationless conduction by $N$ parallel chiral channels (Jiang et al., 2018). In pentalayer graphene/WS$_2$, Chern numbers $C=\pm 5$ are observed with Hall resistance $R_{xy}=h/(5e^2)$ at base temperature up to 1.5 K (Han et al., 2023).
• Three-dimensional QAHE: In Rashba-gapped Weyl semimetals, a bulk Chern invariant $\mathcal{C}_{xy}=1$ yields fully three-dimensional anomalous Hall response—chiral surface and hinge states, with Hall resistance quantized to $0$, $h/e^2$, or $\pm h/e^2$ dependent on transport and Fermi energy (Zhang et al., 2 Jan 2025).
• Non-insulating (metallic) EQAH: Ferromagnetic metals with topological band inversion display robust $\sigma_{xy}=e^2/h$ plateaus even as bulk conduction remains finite ($\sigma_{xx}>0$), enabled by strong dephasing (Büttiker probes) that force diffusive transport (Wan et al., 29 Dec 2025).

5. The Role of External Fields and Parity Anomaly

The EQAH regime is deeply linked to the parity anomaly in Dirac systems and its consequences under external fields:

• Parity anomaly origin: In Dirac-based QAH systems, the requirement of gauge invariance forces a half-integer Hall response that is regularized to integer by momentum-dependent mass terms [$B\,k^2$]. In orbital magnetic fields, the quantized plateau survives up to the critical field $H_{crit}$, at which the renormalized mass reverses and the Chern–Simons contribution vanishes (Böttcher et al., 2019, 2002.01443).
• Onsager relation violation: Unlike the quantum Hall (QH) effect, the Hall conductivity in the EQAH regime can become even in magnetic field, $\sigma_{xy}(-H)=\sigma_{xy}(H)$, violating Onsager reciprocity—directly reflecting parity anomaly and spectral asymmetry mechanisms (Böttcher et al., 2019, 2002.01443).

6. Finite Temperature, Current-Driven, and Phase Transition Behavior

The stability of EQAH states is contingent on competitive entropy, polarizability, and transport energy scales:

• Thermal transitions: The entropy advantage of slow edge modes (FCI) overtakes the lower bulk energy density of crystalline EQAH, resulting in a transition at $T_c$, typically $T_c \sim 100$ mK in rhombohedral graphene/hBN systems near Jain fractions (Patri et al., 2024, Wei et al., 2024).
• Current-driven transitions: Increasing DC bias progressively depins the crystal phase, with sharp jumps and hysteretic transport curves; at Jain fillings, the transition can be pre-empted by an equilibrium switch to FQAH states with matching $R_{xy}=h/(p/(2p+1)e^2)$ (Patri et al., 2024).
• Field-tuned behavior: Displacement fields and gate voltages can induce transitions between EQAH, FQAH, composite Fermi liquids, and conventional insulators, depending on filling, temperature, and device architecture (Lu et al., 2024).

7. Experimental Systems and Realization

EQAH states have been realized or proposed in:

• Rhombohedral graphene multilayers (graphene/hBN moiré superlattices): EQAH plateaus observed over broad moiré fillings at base temperature, with correlated breakdown behavior; transition to FQAH and FCI states at higher temperatures or currents (Lu et al., 2024, Wei et al., 2024, Patri et al., 2024).
• MBE-grown TI multilayers: Stacked (Bi,Sb)$_2$Te$_3$/CdSe devices display high-Chern-number EQAH plateaus (Jiang et al., 2018).
• Ferromagnetic metals: Quantized Hall conductivity ($e^2/h$) coexists with metallic bulk conduction when dephasing dominates (Wan et al., 29 Dec 2025).
• Dirac and BHZ models in orbital fields: EQAH signature quantized plateaus and counterpropagating edge mode coexistence evident in HgMnTe, Bi-based paramagnet systems (Böttcher et al., 2019, 2002.01443).
• Spin-orbit proximitized graphene: Pentalayer graphene/WS$_2$ heterostructures manifest large QAHE with $C=\pm 5$, Hall resistance $h/5e^2$ up to 1.5 K (Han et al., 2023).

8. Conceptual Implications and Open Problems

The existence of EQAH phases challenges the conventional wisdom restricting quantized anomalous Hall effect to insulators, integer fillings, and strict translation invariance. EQAH phases reveal new roles for crystalline symmetry breaking, edge entropy, disorder-induced domain wall physics, metallic conductance, and parity anomaly mechanisms in topologically protected transport.

A plausible implication is that the stability and observability of EQAH states fundamentally depend on the balance between interaction-driven gap opening, disorder pinning, edge mode velocity, and entropy competition. These factors set both the operational regime and the breakdown thresholds for quantized anomalous Hall plateaus in real devices.

Ongoing research focuses on:

• Extending the temperature robustness of EQAH in metallic systems (Wan et al., 29 Dec 2025)
• Realizing chiral Majorana edge states via engineered high-Chern EQAH platforms (Han et al., 2023)
• Quantitative assessment of edge entropy and disorder effects in phase transitions (Wei et al., 2024, Patri et al., 2024)
• Generalizing EQAH behavior to 3D topological platforms (Zhang et al., 2 Jan 2025)
• Exploring device applications in low-dissipation interconnects and in-memory computing (Jiang et al., 2018, Zhang et al., 2 Jan 2025)

The EQAH effect thus stands as a central theme in the broader landscape of correlated, symmetry-broken, and topologically nontrivial quantum matter.