3D Quantum Hall Effect: Antichiral Modes

• 3D Quantum Hall Effect is defined by the emergence of robust antichiral hinge, surface, or edge modes in engineered 3D lattices using synthetic dimensions and gauge fields.
• Model Hamiltonians, derived from stacked 2D Chern insulators or synthetic extensions, enable precise tuning of mode directionality and group velocity via dispersion tilts and flux engineering.
• Experimental systems such as photonic arrays, stacked heterostructures, and metamaterials demonstrate quantized, lossless transport and reveal non-Hermitian skin effects in topological settings.

The 3D Quantum Hall Effect, in the context of contemporary condensed matter and synthetic systems literature, refers to the emergence and transmission of higher-order topological modes—specifically antichiral edge, surface, or hinge channels—in three-dimensional lattices under the interplay of artificial gauge fields, non-Hermiticity, and topological band engineering. While conventional quantum Hall effects in 3D magnetic materials yield quantized Hall conductance on 2D surfaces, in modern artificial matter platforms the effect is architected through dimensional stacking or synthetic-dimensional augmentation of lower-dimensional antichiral systems, leading to robust, co-propagating one-way channels localized on crystalline hinges or surfaces. This phenomenon generalizes the well-known chiral edge currents of 2D quantum Hall insulators and is distinguished by the unique antichiral property: all hinge modes propagate in the same direction despite spatial separation, drawing a direct link to both higher-order topology and non-Hermitian skin effects.

1. Theoretical Foundations and Model Hamiltonians

Construction of 3D antichiral topological matter typically proceeds from stacking or synthetic extension of 2D Chern/antichiral insulators with suitably engineered interlayer or synthetic dimension couplings. A fundamental approach is realized in models based on the generalized Haldane architecture or related Qi–Wu–Zhang lattices, extended in a third direction using either physical stacking or synthetic dimensions.

A prototypical Hamiltonian is:

$$H_{\mathrm{3D}}(k) = \sum_{j=1}^4 h_j(k)\,\Gamma_j + H_u(k_z)$$

where the $h_j(k)$ encode hopping and mass terms across all lattice and synthetic degrees of freedom, and $H_u(k_z)$ provides a tunable dispersion tilt:

$$H_u(k_z) = \sum_{s} u_s\,\sin k_z\,|s\rangle\langle s|$$

with $s$ labeling sublattice/polarization indices. The $\Gamma_j$ form a Dirac-matrix basis ($4\times4$ Pauli algebra) acting on the internal degrees of freedom, and $k = (k_x, k_y, k_z)$ samples the full 3D Brillouin zone. For synthetic platforms, $k_y$ and $k_z$ can represent indices such as photonic orbital angular momentum ($\ell$) and optical frequency ($n$), respectively (Wei et al., 21 Jun 2025, Cheng et al., 2021). The realization of simultaneous fluxes or synthetic gauge fields in multiple directions is essential for generating quantized boundary currents.

In the case of non-Hermitian ladders, the 3D extension is achieved by locally non-uniform modulation imbuing triangular (or higher) plaquettes of the synthetic lattice with additional fluxes, allowing for the manifestation of higher winding numbers and antichiral surface transport (Ye et al., 2023).

2. Band Topology, Antichiral Hinge Modes, and Bulk–Boundary Correspondence

The hallmark of the 3D quantum Hall antichiral regime is the appearance of hinge-localized modes connecting two surface or bulk Dirac points, each mode exhibiting unidirectional propagation set by the sign of an engineered tilt. For a broad range of parameter sets, the systems show robust, symmetry-protected boundary states whose group velocity is controlled through tunneling asymmetry or modulation phase engineering.

The emergent antichiral hinge state at a given hinge (labelled by sublattice index $s$) follows an exact dispersion:

$E_{\mathrm{hinge}}(k_z) = u_s\,\sin k_z$

with

$$v_s(k_z) = \frac{dE_{\mathrm{hinge}}}{dk_z} = u_s\,\cos k_z$$

The sign and magnitude of $u_s$ can be individually programmed for each hinge, yielding full control over mode directionality and group velocity (Wei et al., 21 Jun 2025). For proper design of model parameters ($|\gamma + J\cos k_z| < |\lambda|$, $|\gamma'+J'\cos k_z| < |\lambda'|$), hinge modes are guaranteed to exist and span a topological spectral gap. These modes are topologically protected by higher-order invariants; notably, in stacking constructions, nonzero slab Chern numbers $C^z_{\mathrm{slab}} = \pm1$ force the existence of copropagating hinge channels (Cheng et al., 2021).

In non-Hermitian 3D synthetic lattices constructed from 2D Hall ladders, the spectral winding number $w$ achieves higher integer values (e.g., $|w| = 2$), resulting in “bipolar” skin effect with doubled amplitude for ACEC surface currents, directly confirmed in tight-binding and Floquet numerics (Ye et al., 2023).

3. Physical Realizations and Experimental Platforms

A diverse set of platforms realizes 3D antichiral quantum Hall phenomena:

• Photonic cavity arrays with synthetic dimensions: These exploit frequency and angular momentum as synthetic axes, with electro-optic modulators configuring inter-mode tunneling and dispersion tilts. Transversal and longitudinal EOMs, and q-plates structure the synthetic lattice, allowing hinge states to be directly visualized through frequency-selective transmission spectroscopy (Wei et al., 21 Jun 2025).
• Stacked Haldane bilayer heterostructures: Here two 2D quantum-anomalous-Hall (QAH) sheets of opposite Chern number are joined and stacked. Sublattice-dependent interlayer coupling patterns localize the antichiral hinge channels at specific corners or edges of the 3D sample (Cheng et al., 2021).
• 3D synthetic ladders in photonic fiber-ring resonators: Discrete synthetic dimensions created by ring resonator modes modulated at specific frequencies can map directly onto 3D lattice models with tunable flux and nonreciprocal hopping, yielding higher-order ACECs (Ye et al., 2023).
• Acoustic and electric lattice systems: Both higher-order 3D versions of circuit-based modified Haldane models and photonic metamaterials have demonstrated copropagating hinge channels (Cheng et al., 2021, Wei et al., 21 Jun 2025).

4. Transmission Properties, Disorder Robustness, and Skin Effects

Robust, quantized, and unidirectional hinge transmission is numerically and experimentally observed in clean samples, with the transmission coefficient for hinge-to-hinge propagation quantized to unity provided the topological bulk gap (or winding number) remains open (Wei et al., 21 Jun 2025, Cheng et al., 2021). Disorder resilience is a central feature: copropagating hinge channels remain sharply localized and perfectly transmitting up to moderate disorder strengths $U \lesssim 2t$, beyond which extended bulk states can hybridize with boundary modes and lead to localization only if the topological gap closes (Cheng et al., 2021).

In the non-Hermitian setting, 3D antichiral currents serve as a direct probe of point-gap winding and skin effects, with the measured edge or surface currents closely matching the theoretical winding number $w$. The amplitude of the skin effect, as seen in ACEC, is doubled for $|w|=2$ compared to $|w|=1$ (Ye et al., 2023).

5. Topological Protection, Tunability, and Applications

The protection of 3D antichiral hinge or surface channels is ensured by nontrivial higher-order invariants: either layer/slab Chern numbers, quadrupole moments in synthetic-dimensional implementations, or point-gap spectral winding in non-Hermitian realizations (Cheng et al., 2021, Wei et al., 21 Jun 2025, Ye et al., 2023). Unlike ordinary quantum Hall edge channels, all antichiral hinge modes can be programmed to propagate in the same direction, even if spatially separated. Furthermore, the directionality, group velocity, and localization of each hinge can be precisely tuned via external parameters—for example, the amplitude and phase of the EOM drive in synthetic photonic lattices (Wei et al., 21 Jun 2025).

Notable applications include:

• Directional, lossless waveguides for photonics or acoustics immune to backscattering,
• Programmable delay lines and switching devices,
• Topological sensors leveraging higher-order mode localization,
• Photonic or electronic devices utilizing robust antichiral transport on programmable hinges,
• Experimental testbeds for novel higher-order non-Hermitian skin effects and spectral topology.

6. Outlook, Limitations, and Open Questions

The 3D quantum Hall effect via antichiral hinge or surface modes represents a substantial generalization of the original 2D quantum Hall paradigm, enabled by advances in synthetic gauge fields, engineered non-Hermiticity, and higher-order band topology. The current experimental and theoretical understanding demonstrates robustness to moderate disorder and flexibility for device engineering (Wei et al., 21 Jun 2025, Ye et al., 2023, Cheng et al., 2021). Nonetheless, several open issues remain:

• For gapless or pseudo-topological systems (antichiral semimetals), coexistence with bulk states can compromise topological protection in the presence of strong disorder unless design modifications reopen the gap or decouple bulk and hinge spectra (Mannaï et al., 2023).
• Systematic paper of nonlinear, interacting effects (as in Bose–Hubbard ladders) in 3D antichiral platforms is at an early stage, with self-trapping and “trap-skin” phenomena indicating rich dynamical regimes that modify transmission characteristics and skin localization (Chen et al., 29 Oct 2024, Ye et al., 2023).

Table: Key Features of 3D Quantum Hall Antichiral Regimes

Platform	Protected Quantity	Robustness Threshold
Photonic synthetic-dimension arrays	Winding number $w$, quadrupole moments	$U \lesssim$ bulk gap
Stacked Haldane bilayers	Slab/local Chern number	$U \lesssim 2t_1$
Non-Hermitian synthetic lattices	Spectral winding $w$	$|γ|,\;|φ|$ set skin amplitude; loss/gain tune robustness

“Hinge” and “surface” channels exhibiting copropagating antichiral currents represent a robust, programmable, and higher-dimensional realization of Hall physics. These modes provide both fundamental platforms for testing topological quantum phenomena beyond the conventional Hall effect, as well as concrete design principles for future technology in optics, acoustics, and quantum simulation (Wei et al., 21 Jun 2025, Ye et al., 2023, Cheng et al., 2021).