Mirror-Spin Chern Number in 2D Altermagnets

• Mirror-spin Chern number is a topological invariant defined as the difference in Chern numbers between two mirror-symmetry enforced spin channels, ensuring quantized spin transport.
• It is determined from independent Chern subsystems in 2D altermagnets where mirror-spin coupling robustly separates spin-up and spin-down states even in the presence of strong spin-orbit coupling.
• This invariant underpins quantized spin-Hall conductivity and nearly 100% spin-polarized edge states, making it a pivotal concept for designing robust spintronic devices.

A mirror-spin Chern number is a symmetry-protected topological invariant that arises in systems where mirror symmetry locks the spin degree of freedom to definite mirror eigenvalues. This invariant is defined as the difference of Chern numbers between two orthogonal mirror-spin channels, and is quantized as long as mirror symmetry is preserved. In the context of two-dimensional altermagnets with a horizontal mirror plane and out-of-plane Néel vector, the mirror-spin Chern number is central to predicting and realizing a quantized spin-Hall conductivity, as recently demonstrated in Fe$_2$Te$_2$O (Zhang et al., 11 Mar 2025). In such systems, mirror-spin coupling (MSC) enforces a robust separation between spin-up and spin-down sectors, making the quantized spin transport resilient even in the presence of strong spin-orbit coupling (SOC).

1. Mirror-Spin Coupling: Symmetry Principle and Mechanism

Mirror-spin coupling (MSC) is a symmetry-induced phenomenon in crystalline materials with a well-defined horizontal mirror plane, implemented via a mirror operator $\mathcal{M}_z$ with eigenvalues $m_z = \pm i$. In the presence of MSC, spin-up and spin-down states are locked to distinct mirror eigenvalues: e.g., spin-up with $m_z = +i$ and spin-down with $m_z = -i$. This structure is guaranteed whenever there exists an additional operator $\mathcal{O}$ (such as an in-plane two-fold rotation or its composition with time reversal) that anticommutes with $\mathcal{M}_z$, i.e.,

$\{\mathcal{M}_z, \mathcal{O}\} = 0.$

Under this algebra, the action of $\mathcal{O}$ flips both the spin and the mirror eigenvalue, so eigenstates can be labeled by definite $(m_z, s)$ combinations. As a result, when the electronic states are expressed as $\mathcal{M}_z$ eigenstates, spin hybridization between the two mirror channels is strictly forbidden by symmetry, even with finite SOC. This decoupling enables each channel to be treated as an independent Chern subsystem, allowing for the definition of a mirror-spin Chern number.

2. Quantized Spin-Hall Conductivity from Mirror-Spin Topology

In monolayer Fe$_2$Te$_2$O, ab initio calculations and model analysis show that, in the absence of SOC, the system exhibits altermagnetic order and hosts Weyl points, acting as an altermagnetic Weyl semimetal. Each Weyl node exists in both mirror-spin channels and is protected by the MSC-induced decoupling. When SOC is switched on, these Weyl points are gapped, and the bulk spectrum opens a gap of $\sim$0.17 eV. Crucially, unlike conventional quantum spin Hall (QSH) insulators where SOC generically hybridizes spin channels, in Fe$_2$Te$_2$O the MSC suppresses such hybridization. This ensures that both bulk and edge states maintain nearly perfect spin polarization.

Each mirror-spin sector (say, $m_z = +i$ for spin-up) then acquires its own Chern number, here $C_{+} = +1$ (spin-up, $m_z=+i$) and $C_{-} = -1$ (spin-down, $m_z = -i$). The mirror Chern number is

$\mathcal{C}_m = \frac{C_{+} - C_{-}}{2} = 1.$

This non-trivial value underlies the quantized spin-Hall conductivity $\sigma_{xy}^z = e/(4\pi)$ that appears in the bulk band gap, a haLLMark of robust spin-polarized edge transport.

3. Model Hamiltonians and Topological Analysis

At low energy near Weyl points (in the semimetallic phase), the effective Hamiltonian for a given mirror-spin sector (for example, spin-up) takes the form

$$H_X = (m + t_x k_x^2 + t_y k_y^2) \sigma_z + r k_x k_y \sigma_x.$$

SOC introduces a mass term $\Delta \sigma_y$ that gaps these Weyl points: $$H_{\pm}^{\uparrow} = \pm \sqrt{|m/t_y|} (2 t_y k_y \sigma_z + r k_x \sigma_x ) + \Delta \sigma_y,$$ where $\Delta \approx 0.103$ eV in Fe$_2$Te$_2$O. Each gapped Weyl point contributes a half-integer to the Chern number of its mirror-spin channel; summing both gives $C_{+} = +1$ and $C_{-} = -1$. Because mirror-spin mixing remains negligible, the edge states at boundaries of the sample remain almost perfectly spin-polarized even after including SOC, guaranteeing quantized spin Hall transport.

4. Phase Evolution: From Altermagnetic Weyl Semimetal to Magnetic Mirror Chern Insulator

The phase evolution in Fe$_2$Te$_2$O upon including SOC is as follows:

• No SOC: The system is an altermagnetic Weyl semimetal; Weyl points in both $m_z = +i$ and $-i$ (spin-polarized) channels are symmetry-protected by MSC.
• With SOC: Weyl points are gapped, the spectrum is insulating, and the system becomes a magnetic mirror Chern insulator (MCI) with $\mathcal{C}_m = 1$.
• Edge States and Spin Protection: Owing to the persistent MSC, the topological edge states are counterpropagating and retain nearly $100\%$ spin polarization along the edge—an essential feature for robust spin-Hall conductivity.

This mechanism sharply contrasts with standard QSH insulators, where SOC typically causes significant spin hybridization and may suppress sharp quantization in realistic materials.

5. Theoretical and Material Implications

The symmetry-protected decoupling mechanism provided by MSC in Fe$_2$Te$_2$O circumvents fundamental limitations of traditional QSH systems by allowing strong SOC while suppressing spin-mixing. The quantized spin-Hall conductivity, realized here in a system not limited by weak SOC, is experimentally accessible due to a sizable bulk gap (0.17 eV), which should be robust even at room temperature.

By demonstrating mirror-spin topological protection in an altermagnetic system, this work broadens the class of materials predicted to host robust quantized spin transport. Monolayer Fe$_2$Te$_2$O thus becomes the prototype for a magnetic mirror Chern insulator displaying mirror-spin-coupling-protected quantized spin-Hall conductivity. This suggests that other 2D altermagnets with a horizontal mirror plane and an appropriate Néel vector orientation may be candidates for similar symmetry-protected spin-Hall phenomena.

6. Broader Context and Future Directions

The identification of mirror-spin Chern numbers in magnetic systems with MSC provides a framework for the classification and design of new topological phases. By generalizing the mirror-spin coupling mechanism and its consequences—quantized spin transport and persistent spin-polarized edge states—researchers can systematically search for or engineer further magnetic MCIs. These developments expand the pool of viable materials for next-generation spintronic devices, offering promising pathways to achieve robust spin manipulation at elevated temperatures. Moreover, the topological invariants defined via MSC make it possible to identify and distinguish phases in magnetic materials that would otherwise be considered topologically trivial under the conventional $\mathbb{Z}_2$ paradigm.