Spin Chern Marker Overview

• Spin Chern Marker is a real-space topological index that decomposes spin contributions to reveal local quantum spin Hall behavior.
• It is computed by projecting Berry curvature onto spin-resolved subspaces, ensuring quantization in the bulk and signaling phase transitions.
• The framework applies to diverse systems, including disordered lattices, altermagnets, and fractal geometries, providing insights into spin-resolved topological phases.

The spin Chern marker (SCM) is a real-space-resolved topological index that generalizes the global spin Chern number to systems lacking translational symmetry, spatially inhomogeneous environments, or conditions where spin is not a perfect quantum number. It is foundational in the study and classification of quantum spin Hall insulators, altermagnets, and related topological phases in both crystalline and amorphous materials. The SCM is constructed by projecting the conventional Chern marker framework onto spin-resolved (or spin-projected) subspaces, yielding a marker that is locally quantized in the bulk of topologically nontrivial phases and encodes both topological phase boundaries and local responses, including in strongly disordered, interacting, or fractal geometries (Júnior et al., 2024, Lage et al., 4 Mar 2025, Gonzalez-Hernandez et al., 2024, Gebert et al., 2019, Qin et al., 31 Dec 2025).

1. Mathematical Definition and Construction

The SCM is formulated via the local decomposition of the spin Chern number. In the translationally invariant limit, the spin Chern number is given as the difference of Berry-curvature Chern numbers for two spin sectors: $$C_s = \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2k \ \left[ \Omega_{xy}^{\uparrow}(\mathbf{k}) - \Omega_{xy}^{\downarrow}(\mathbf{k}) \right]$$ where $\Omega_{xy}^{\sigma}$ is the Berry curvature for spin $\sigma$ (Gonzalez-Hernandez et al., 2024, Chen, 2022, Lage et al., 4 Mar 2025).

In real space, for a tight-binding basis $\{\ket{\mathbf{r}}\}$ (site and spin), the local spin Chern marker at position $\mathbf{r}$ is

$$\mathfrak{C}_s(\mathbf{r}) = \frac{\mathfrak{C}_+(\mathbf{r}) - \mathfrak{C}_-(\mathbf{r})}{2}$$

where $\mathfrak{C}_{\pm}(\mathbf{r})$ are local Chern markers for the positive/negative branches of the spectrum of the valence-band-projected spin operator $$M^{s_z}_{\rm v.s.} = \mathcal{P} \hat{s}_z \mathcal{P}$$, with $\mathcal{P}$ the projector onto occupied bands. Each branch marker is (Júnior et al., 2024): $$\mathfrak{C}_\sigma(\mathbf{r}) = 2\pi\,\mathrm{Im}\left\langle \mathbf{r}\,|\, \left( \mathcal{Q}_\sigma \hat{X} \mathcal{P}_\sigma \hat{Y} \mathcal{Q}_\sigma - \mathcal{P}_\sigma \hat{X} \mathcal{Q}_\sigma \hat{Y} \mathcal{P}_\sigma \right) |\,\mathbf{r} \right\rangle$$ with projectors $\Omega_{xy}^{\sigma}$0 ($\Omega_{xy}^{\sigma}$1) onto the subspaces of positive and negative eigenvalues of $\Omega_{xy}^{\sigma}$2.

In the special case of perfect spin conservation, the SCM reduces to a difference of local Chern markers for spin-up/down sectors as in

$\Omega_{xy}^{\sigma}$3

with $\Omega_{xy}^{\sigma}$4 the conventional Bianco–Resta local Chern marker built from spin-$\Omega_{xy}^{\sigma}$5 projectors (Gebert et al., 2019).

An alternative fully symmetric, regularized definition for Dirac systems employs unitary, strictly periodic position operators, and leads to (Qin et al., 31 Dec 2025): $\Omega_{xy}^{\sigma}$6 where $\Omega_{xy}^{\sigma}$7 projects onto filled states, $\Omega_{xy}^{\sigma}$8 is a generalized spin operator, and $\Omega_{xy}^{\sigma}$9 are regularized position operators.

2. Physical Interpretation and Spectral Gap Protection

The SCM provides a local indicator of topological order and the existence of bulk spin currents associated with quantum spin Hall phases or other spin-resolved topological phases. The quantization of $\sigma$0 to integer values (typically $\sigma$1 or $\sigma$2 in $\sigma$3 or mod $\sigma$4 in $\sigma$5 systems) in the interior of a system is protected by two gaps:

• The energy gap $\sigma$6 in the single-particle spectrum, ensuring the system remains an insulator.
• The projected spin gap $\sigma$7, the minimal difference between positive and negative eigenvalues of $\sigma$8, ensuring that the decomposition into spin branches is unambiguous (Júnior et al., 2024, Lage et al., 4 Mar 2025).

If either gap closes (e.g., at a phase transition), the SCM loses its quantization and physical meaning, signaling a change in topological phase (Júnior et al., 2024). In systems with strong spin mixing, these gaps are not symmetry protected but are generically nonzero except at critical points.

3. Computational Methodologies

Evaluation of the SCM proceeds as follows:

1. Diagonalize the Hamiltonian $\sigma$9 on the relevant geometry (finite flake, fractal, or lattice).
2. Construct the valence-band projector $\{\ket{\mathbf{r}}\}$0 onto all occupied states.
3. Form and diagonalize the projected spin matrix $\{\ket{\mathbf{r}}\}$1 for the chosen spin component ($\{\ket{\mathbf{r}}\}$2).
4. Split the valence subspace into two branches by the sign of corresponding eigenvalues $\{\ket{\mathbf{r}}\}$3 of $\{\ket{\mathbf{r}}\}$4.
5. Construct branch projectors $\{\ket{\mathbf{r}}\}$5 and their complements.
6. Compute the local marker $\{\ket{\mathbf{r}}\}$6 for each site using the formulas above.
7. To obtain a global invariant, sum or average $\{\ket{\mathbf{r}}\}$7 over the bulk region (Júnior et al., 2024, Lage et al., 4 Mar 2025, Qin et al., 31 Dec 2025).

For random/disordered systems, averaging the SCM over sites yields the global spin Chern number, while its variance across sites detects topological phase boundaries and disorder-induced transitions (Qin et al., 31 Dec 2025).

4. Applications: Model Hamiltonians and Geometries

The SCM framework is now standard in the study of time-reversal-invariant insulators, Rashba-coupled Kane–Mele models, Sierpinski fractals, and collinear altermagnets:

• Kane–Mele–Rashba Models: The SCM captures the topology in honeycomb lattices with various spin–orbit couplings, including strong Rashba interaction ($\{\ket{\mathbf{r}}\}$8) and sublattice-dependent or in-plane spin Hall phases. It accurately maps the real-space phase diagrams and is robust even when spin is only approximately conserved (Júnior et al., 2024).
• Fractal and Disordered Systems: On Sierpinski carpets, the SCM remains quantized in the bulk regardless of internal edge proliferation, provided the projected spin gap stays open (Lage et al., 4 Mar 2025). Open boundaries or disorder merely induce local anomalies at edges or voids.
• Altermagnets and d-wave Materials: For $\{\ket{\mathbf{r}}\}$9-symmetric altermagnets, the spin Chern marker defines the unique $\mathbf{r}$0 bulk topological invariant in 2D and directly signals Weyl points in 3D, serving as a robust order parameter for phases without net anomalous Hall effect (Gonzalez-Hernandez et al., 2024).

5. Connection to Experiment and Local Probes

Recent advances allow direct experimental access to SCMs or closely related markers:

• Circular Dichroism and ARPES: Spin-resolved photoemission and optical absorption can measure the spatial profile or frequency dependence of the spin Berry curvature, enabling extraction of local markers and their critical behavior near phase transitions (Chen, 2022).
• Driven–Dissipative Protocols in Bosonic Systems: In magnonic and photonic platforms, local topological markers (Chern or spin Chern) can be mapped by preparing band-projected localized states, applying circular drives, and monitoring excitation rates—yielding single-site resolution of topological quantization (Bermond et al., 24 Apr 2025).
• Quantum-Gas Microscopy: In ultracold atomic realizations of topological Hamiltonians, the single-particle density matrix can be reconstructed from local correlation measurements, allowing direct computation of local Chern and spin Chern markers (Gebert et al., 2019).

6. Extensions and Theoretical Significance

The SCM not only generalizes the bulk spin Chern number to real space, but also connects to other topological invariants:

• In Dirac systems, the regularized SCM is equivalent to the spin Bott index in classes AII and DIII, thus unifying multiple topological indices within a singular computational framework (Qin et al., 31 Dec 2025).
• The SCM is sensitive to topological phase transitions: its variance peaks at critical disorder strengths, and its spatial correlators (spin Chern correlator) encode the divergence of topological correlation lengths at quantum criticality (Chen, 2022).
• The marker is adaptable to scenarios with approximate spin conservation or strong spin–orbit scattering, and applies equally to finite, disordered, or amorphous systems.

7. Summary Table: Key Formulas and Physical Context

Quantity	Mathematical Definition (Example)	Physical Context/Significance
Spin Chern number (global)	$\mathbf{r}$1	Bulk topological index, QSH phase
Local spin Chern marker $\mathbf{r}$2	$\mathbf{r}$3	Real-space resolved local topological order
Projected spin gap $\mathbf{r}$4	$\mathbf{r}$5 (from $\mathbf{r}$6 eigenvalues)	Ensures branch decomposition, SCM quantization
Regularized SCM (Dirac/AII)	$\mathbf{r}$7	Equivalence to spin Bott index, universal in bulk

The SCM provides a flexible, physically meaningful, and computationally feasible tool for the diagnosis and spatial imaging of topological character in a broad array of quantum materials, particularly where traditional momentum-space invariants are not applicable (Júnior et al., 2024, Lage et al., 4 Mar 2025, Gonzalez-Hernandez et al., 2024, Qin et al., 31 Dec 2025, Gebert et al., 2019).