High Spin Chern Number Phase

• High spin Chern number phases are topological states defined by invariants |C| > 1 or |𝒞ₛ| > 1, leading to multiple robust edge channels and quantized Hall responses.
• They are realized through lattice engineering, multilayer structures, and strong electronic correlations in systems like TI heterostructures, quantum spin liquids, and cold-atom platforms.
• These phases enable enhanced spin and charge transport, offer potential for spintronics and quantum computation, and challenge conventional topological classifications.

A high spin Chern number phase is a topological phase characterized by a spin Chern number (𝒞ₛ) or, more broadly, a Chern number (C), whose absolute value exceeds one. These phases generalize the concept of quantum Hall and quantum spin Hall states to systems with Chern numbers |C| > 1 and spin Chern numbers |𝒞ₛ| > 1, resulting in enriched edge-state structures, novel bulk–edge correspondences, and new fractionalized or interacting topological phases. High spin Chern number phases can occur in a wide variety of systems including engineered lattice models, multilayer and superlattice topological insulators, quantum spin Hall materials, strongly interacting electron systems, quantum spin liquids, and engineered cold-atom or magnetic systems.

1. Definition and Topological Characterization

A system exhibits a high spin Chern number phase when its topological invariant — the (spin) Chern number — satisfies |C| > 1 or |𝒞ₛ| > 1. The Chern number for a Bloch band is defined by integrating the Berry curvature Ω(k) over the Brillouin zone: $$C = \frac{1}{2\pi} \int_{\text{BZ}} d^2 k\, \Omega(k)$$ For spinful systems with approximate spin conservation, the spin Chern number is

$$\mathcal{C}_S = \frac{1}{2}\left(C_\uparrow - C_\downarrow\right)$$

where C_↑ and C_↓ are the Chern numbers of the spin-up and spin-down sectors. Nontrivial high spin Chern number phases yield multiple helical or chiral edge channels and quantized spin (or charge) Hall responses proportional to the value of the (spin) Chern number. In interacting systems or magnetic materials, high Chern numbers can also appear in magnon bands, Bogoliubov quasiparticle bands, or Majorana bands, and are evaluated via the Berry curvature of the appropriate quasiparticle eigenstates.

2. Mechanisms and Physical Realizations

Multiple microscopic scenarios lead to high (spin) Chern number phases:

Lattice Engineering and Multilayers

• TI multilayer structures: Alternating stacks of magnetic-doped and undoped topological insulator layers give rise to high-Chern number phases through a sequence of band inversions. When multiple subbands invert at symmetry points (e.g., Γ), the Chern number can jump by more than one, leading to C ≫ 1 (Wang et al., 2021, Wang et al., 2022). The resulting Chern number is a sum over the contributions of all inverted Dirac cones, with the total given by C = N_+ – N_–, counting subbands with positive and negative mass inversions.
• Bilayer/multilayer type-II QSHI: Stacking monolayers of type-II QSHI (each with 𝒞ₛ = 1) leads to bilayers with 𝒞ₛ = 2; further stacking produces phases with even larger spin Chern numbers. Unlike type-I QSHI stacking, which yields trivial phases, type-II stacking systematically builds high spin Chern number QSHIs, exemplified in materials such as Nb₂SeTeO (Tan et al., 7 Aug 2025).
• Synthetic lattice models with SO symmetry: Tight-binding models on various lattices (e.g., pyrochlore, kagome, triangular) engineered to have higher Chern number bands via spin–orbit coupling and multiorbital or multilayer structures can stabilize fractional Chern insulators at unusual filling fractions and high Chern numbers (Liu et al., 2012, Alase et al., 2021).

Strong Correlations and Exotic Phases

• Fractional Chern insulators (FCIs): In flat bands with high Chern number, strongly correlated electron interactions at fractional filling stabilize Abelian or non-Abelian fractionalized states that generalize Laughlin physics to high Chern number contexts; e.g., in C = N > 1 bands, stable insulating FCIs for fermions arise at ν = 1/(2N+1) and for bosons at ν = 1/(N+1) (Liu et al., 2012).
• Kitaev magnets and quantum spin liquids: Application of Zeeman field and the inclusion of non-Kitaev interactions (such as Dzyaloshinskii–Moriya and off-diagonal exchange) in the Kitaev honeycomb or star lattice drive transitions to gapped spin liquid phases with Chern number |C| = 2, 3, or higher. These transitions can often be tuned continuously via electric field or pressure (Kwon et al., 29 May 2025, Zou et al., 3 Apr 2025).
• Magnonic Chern insulators: In spin systems such as the Kitaev-Heisenberg ferromagnet with further-neighbor interactions, magnon bands with nontrivial topology exhibit Chern numbers up to |C| = 4, associated with multiple chiral magnon edge modes and quantized thermal Hall effects (Deb et al., 2019).
• Topological superfluids: Attractive fermionic systems with synthetic SO coupling and Zeeman fields can enter a topological superfluid phase with Chern number C = 2. In these phases, two branches of Majorana edge modes exist and each vortex traps two Majorana zero modes (Yi-Xiang et al., 2016).

3. Edge States, Bulk–Edge Correspondence, and Transport

High Chern and spin Chern number phases fundamentally alter edge and transport properties:

• Multiple edge channels: The bulk–edge correspondence ensures that |C| (or |𝒞ₛ|) chiral (or helical for spin) edge modes exist per edge. For 𝒞ₛ = 2, two pairs of helical edge modes cross the gap, leading to doubled edge conductance compared to conventional QSHIs (Bai et al., 2022, Cook et al., 2023).
• Quantized Hall and spin Hall conductances: The Hall response is quantized as

$$\sigma_{xy} = C\frac{e^2}{h},\qquad \sigma_{xy}^z = \mathcal{C}_S\frac{e}{2\pi}$$

and in magnon or thermal systems the thermal Hall conductivity obeys

$$\kappa_{xy}/T = C \left(\frac{\pi}{12}\frac{k_B^2}{\hbar}\right)$$

Edge state multiplicity is a robust haLLMark even when symmetry (e.g., U(1) spin rotation) is weakly broken, due to topological protection (Tan et al., 7 Aug 2025, Song et al., 2016).

• Double-helical Luttinger liquids: In high spin Chern insulator edges (e.g., 𝒞ₛ = 2), the coupled edge channels realize multi-channel (double-helical) Luttinger liquids, which can support a rich array of collective instabilities under electron–electron interactions, such as π-spin density wave (π–SDW) and superconducting order (Hung et al., 11 Dec 2024). Domain walls between distinct edge phases can host Majorana Kramers pairs.

4. Distinctions from Conventional (Low-Chern) Topology

High spin Chern number phases display features distinct from familiar C = 1 or 𝒞ₛ = 1 topological states:

• Band inversion at non-high-symmetry points: In α-antimonene and α-bismuthene, high 𝒞ₛ phases are generated by band inversion at generic k-points, making them “hidden” to symmetry indicator and topological quantum chemistry methods (Bai et al., 2022, Wang et al., 2022).
• Deviation from Z₂ topology: The Z₂ invariant fails to fully capture these phases; even Z₂-trivial states (Z₂ = 0) can feature robust edge states and quantized spin Hall conductivity, provided the spin Chern number is nonzero and even (Wang et al., 2022).
• Entanglement and multilayer uniqueness: In flat bands with Chern number C > 1, the many-body ground states display intrinsic “interlayer” entanglement not present in multilayer Landau level analogs. Even fractional states at “anomalous” filling fractions, not permitted in continuum quantum Hall settings, are stabilized (Liu et al., 2012).
• Flatness and interaction effects: Topologically nontrivial flat bands (particularly in Lieb or decorated honeycomb lattices) can support high Chern number phases with little or no dispersion, potentially enhancing the role of correlations and enabling exotic fractional phases (Karnaukhov et al., 2015, Zou et al., 3 Apr 2025).

5. Tunability and Experimental Platforms

High spin Chern number phases are highly tunable and can be realized in a variety of physical platforms:

• Strain engineering: In α‑Sb monolayers, moderate uniaxial or biaxial strain tunes band inversions at valleys and at Γ, enabling switching among 𝒞ₛ = 0, 1, 2, and 3 phases. The number and nature of edge channels can thus be engineered on demand (Cook et al., 2023).
• Stacking of 2D layers: Controlled stacking of type-II QSHI layers results in stepwise increases of the spin Chern number, opening pathways to highly quantized spin Hall conductance in engineered superlattices (Tan et al., 7 Aug 2025).
• Electric and magnetic fields: Application of electric field or hydrostatic pressure tunes non-Kitaev interactions in quantum spin liquids, driving continuous transitions between topological phases of different Chern numbers (Kwon et al., 29 May 2025). Zeeman fields are vital for achieving and tuning anomalous Hall (Chern insulator) phases in correlated materials (e.g., BaFe₂(PO₄)₂ or TI heterostructures) (Song et al., 2016, Yi-Xiang et al., 2016).
• Curvature and geometric control: Periodic geometric curvature in nanowires with spin–orbit coupling can induce transitions to high Chern number states and control the quantized charge pumping via modulated gate potentials and external magnetization (Siu et al., 2021).
• Cold atom simulators: Recent proposals detail platforms using ultracold atoms in optical lattices (e.g., triangular lattices with designed SO coupling) where high-Chern number Hamiltonians can be engineered, and direct detection of the Chern number is possible using Zeeman spectroscopy and Bloch oscillations (Alase et al., 2021, Hung et al., 11 Dec 2024).

6. Interacting Edge Phenomena and Majorana Physics

The presence of multiple edge channels enables new boundary physics in high spin Chern number systems:

• Domain wall Majorana Kramers pairs: Competing edge instabilities in double-helical (𝒞ₛ = 2) systems support transitions between superconducting and π–spin-density wave phases. The interface between such phases realizes a topologically protected Majorana Kramers pair, which constitutes a robust non-Abelian defect even in systems with time-reversal symmetry (Hung et al., 11 Dec 2024).
• Multi-channel instabilities: The coupled channels can host multi-channel helical Luttinger liquids, with controllable scaling exponents for various ordered states. Cold atom platforms afford unprecedented spatial and interaction control, allowing for the direct engineering of boundary transitions and their zero-energy excitations.

7. Future Directions and Implications

The development and exploration of high spin Chern number phases open several avenues:

• Spintronics and quantum computation: Systems supporting multiple spin-polarized edge channels or chiral Majorana edge states may be exploited in low-dissipation spin current devices, topological quantum computing schemes, or robust thermal management applications.
• Fractionalization and non-Abelian states: Fractional Chern insulator physics in high Chern number bands hints at non-Abelian statistics and new routes to both Abelian and non-Abelian anyonic matter.
• Materials and heterostructures: The identification of α‑bismuthene, α‑antimonene, bilayer Nb₂SeTeO, and multilayer TI structures as high spin Chern number candidates points to the feasibility of room-temperature, highly quantized spin Hall conductances and practical device implementations.
• Beyond standard topological classification: These phases challenge the completeness of band-predicted (TQC/SI) invariants, motivating the development of improved topological diagnostics sensitive to band inversions at generic k-points and to interaction-driven topology.

High spin Chern number phases embody a rapidly expanding class of topological quantum matter, with their diagnosis, manipulation, and application at the forefront of condensed matter, materials science, magnetism, and quantum information research.