Interplay of Altermagnetism and Coupled Quasi-Altermagnetic states in Sliding Two-dimensional Square Lattice

Abstract: The emergence of non-relativistic spin splitting (NRSS) in altermagnetic systems has introduced a new paradigm in antiferromagnets with vanishing net magnetization. Although sliding-induced valley-polarized phases have recently been demonstrated in two-dimensional altermagnets, the observed valley-polarized state represents only a partial manifestation of altermagnetism, and a comprehensive classification based on spinsplitting characteristics remains lacking. Here, using first-principles calculations, general stacking theory, and spin-Laue symmetry analysis, we propose a coupled quasialtermagnetic state representing a distinct subclass of altermagnetism, in which reversible type-IV NRSS is controlled through interlayer sliding. Accordingly, the sliding-induced phases are classified into two categories: altermagnetic and quasi-altermagnetic states. We establish a direct correspondence between reciprocal-space spin splitting and real-space switching between the two quasi-altermagnetic states. Importantly, the spin-polarized bands in these states remain spin split at Γ point even in the absence of spin-orbit coupling (SOC), distinguishing them within the proposed classification framework. To demonstrate the interplay between altermagnetic and quasi-altermagnetic states, we investigate the two-dimensional Lieb-lattice material Mn2WS4 and its Janus derivative Mn2WS2Se2, analysing how changes in the local environment influence the different magnetic phases. Importantly, the underlying mechanism is broadly applicable to a wide class of twodimensional square-lattice systems. We further investigate the effects of SOC, focusing on spin texture and transport signatures in coupled quasi-altermagnetic states.

• The paper develops a symmetry-based framework using spin-Laue group analysis to classify sliding-induced altermagnetic and quasi-altermagnetic states.
• It employs a minimal Lieb lattice model and DFT+U calculations to validate spin splitting in materials like Mn₂WS₄ and Mn₂WS₂Se₂.
• The study reveals that interlayer sliding enables reversible control of spin degeneracy and valley/spin transport, suggesting new switchable spintronic device concepts.

Interplay of Altermagnetism and Coupled Quasi-Altermagnetic States in Sliding Two-Dimensional Square Lattices

Introduction and Motivation

This work formalizes the classification and mechanisms of altermagnetism and introduces the concept of coupled quasi-altermagnetic states in 2D square-lattice systems with interlayer sliding. Altermagnets are systems with zero net magnetization but exhibit robust, SOC-independent spin splitting throughout the Brillouin zone, enabled by specific spin-group symmetries; they bridge key characteristics between AFMs and FMs. For 2D systems, stacking and sliding-induced symmetry modifications—particularly in square lattices—enable a set of nontrivial magnetic phases beyond conventional altermagnetism.

Symmetry Framework for Sliding-Induced Magnetic Phases

The paper develops a rigorous symmetry-based framework using spin-Laue group analysis to classify sliding-induced bilayer phases:

• Altermagnetic states: Defined by intact symmetry connectors between opposite-spin sublattices, maintaining $\Gamma$-point spin degeneracy even in the absence of SOC, e.g., AA and AB stacking.
• Coupled quasi-altermagnetic states: Realized when specific sliding operations (e.g., AC$_1$ and AC$_2$ geometry) break the relevant connectors, resulting in lifted spin degeneracy at $\Gamma$ despite vanishing net magnetization and preserved overall antiferromagnetic arrangement.

The spin splitting in these quasi-altermagnetic phases is reversible via real-space operations, producing pairs of coupled states with reciprocal valley-spin character. The symmetry analysis, including exclusion of certain connectors (like $$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$), enables precise delineation of when true altermagnetism is lost.

Figure 1: Square-lattice stacking configurations and their symmetry connectors, with sliding directionality and broken/preserved degeneracy at $\Gamma$ highlighted.

Two-Dimensional Lieb Lattice Model and Material Realization

The authors construct a minimal Lieb lattice model as the canonical platform for 2D altermagnetism. In particular, Mn$_2$WS$_4$ and its Janus derivative Mn$_2$WS$_2$Se$_1$0 are chosen as representative materials. The explicit site symmetry and Wyckoff positioning in the Lieb lattice enables:

• d-wave altermagnetic states (quadratic spin winding at high-symmetry points).
• Strong spin splitting along non-centrosymmetric paths (maximum splitting at $_1$1 as a function of next-nearest neighbor hopping anisotropy).
• Retention of zero net moment due to compensated sublattices.

  Figure 2: Lieb lattice model, Mn$_1$2WS$_1$3 structure, and calculated band structure demonstrating symmetry-driven altermagnetic spin splitting.

Further, the authors perform DFT+U calculations (U = 3.0 eV for Mn) to validate electronic structure, spin polarization, and DOS in both monolayer and bilayer variants.

Bilayer Stacking, Sliding and Classification of Magnetic Phases

The paper systematically explores all symmetry-distinct bilayer stacking geometries:

• AA/AB stacking: Preserves altermagnetic phase, maintains $_1$4-point degeneracy and symmetric spin DOS/band edges.
• AC$_1$5/AC$_1$6 stacking: Induces coupled quasi-altermagnetic states. Here, $_1$7-point degeneracy is lifted, band edges shift oppositely for spin channels, and the valley-spin correspondence reverses upon real-space symmetry transformation.

  Figure 3: Structural and electronic characteristics of AA, AB, AC$_1$8, and AC$_1$9 bilayers, with degenerate/split bands at $_2$0 and symmetry markers.

Orbital analysis (Mn $_2$1-projection) reveals that the spin-split bands in these systems are predominantly derived from $_2$2, $_2$3, and $_2$4 orbitals, with filling-dependent contributions tightly correlated to stacking geometry.

Figure 4: Mn $_2$5-orbital resolved band projections for AA, AB, AC$_2$6, and AC$_2$7 bilayers, distinguishing spin-resolved orbital weights.

Structural distortion analysis reveals that volume difference in Mn tetrahedra (between spin-up and spin-down sublattices) is absent in altermagnetic AA/AB stacks but significant in AC$_2$8/AC$_2$9, directly correlating to lifted spin degeneracy.

Figure 5: Structural comparison of Mn tetrahedra in AA and AC$\Gamma$0 bilayers, highlighting volume mismatch in quasi-altermagnetic phase.

Electronic DOS, Fermi Surface, and Spin Texture Analysis

The spin-resolved DOS profiles unambiguously prove that while AA/AB configurations retain identical spin DOS band edges, AC$\Gamma$1/AC$\Gamma$2 display opposite shifts, producing asymmetric demagnetization but zero net moment.

Figure 6: Spin-resolved DOS for AA, AB, AC$\Gamma$3, and AC$\Gamma$4 bilayers, with schematic band edge shifts and opposite differences in quasi-altermagnetic configurations.

Fermi surface topology further distinguishes the altermagnetic and quasi-altermagnetic phases, showing that while AA has symmetry-related $\Gamma$5/$\Gamma$6 Fermi surfaces, AC$\Gamma$7/AC$\Gamma$8 lack connectors between spin channels but remain related by sliding-induced real-space symmetry.

Figure 7: Spin-resolved Fermi surfaces for AA, AC$\Gamma$9, and AC$$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$0 configurations, encapsulating valley-spin reversal under sliding.

Upon SOC inclusion, spin textures predominantly project onto $$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$1 with negligible in-plane components for AC$$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$2/AC$$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$3 states, and degeneracy protected by $$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$4 symmetry in AA/AB stacks is lifted for quasi-altermagnets.

Figure 8: Spin-resolved band structure and spin texture with SOC for all stacking configurations, revealing dominant out-of-plane spin polarization.

Transport Signatures and Device Implications

Berry curvature in AA/AB is antisymmetric between $$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$5 and $$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$6 valleys, resulting in VHE. Quasi-altermagnetic states (AC$$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$7/AC$$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$8) exhibit both magnitude and sign reversal in Berry curvature, producing an anomalous VHE and tunable AHC with transport directionality that is fully switchable via sliding. Sliding-induced transitions accomplish control of Hall response and spin-current directionality without external fields.

Figure 9: AHC and schematic device architectures leveraging sliding-controlled coupled quasi-altermagnetic states for TMR-like switching.

Proposed device concepts exploit bilayer Mn$$\{\tau, \mathcal{P}, \mathcal{C}_{2z}, \mathcal{M}_{z}\}$$9WS$\Gamma$0 to realize low/high resistance states based on parallel/antiparallel configuration of AC$\Gamma$1/AC$\Gamma$2 electrodes, with AA stack as tunneling barrier, paving the way for contactless (mechanical, sliding-controlled) valley/spintronics memory elements.

Extension to Janus Derivatives and Stacking Engineering

The Janus compound Mn$\Gamma$3WS$\Gamma$4Se$\Gamma$5 enables further symmetry manipulation via S/Se substitution and stacking order. Two distinct phases are realized:

• Type-I (non-Janus): Valley-polarized phase due to breaking of $\Gamma$6 symmetry.
• Type-II (Janus): Altermagnetic states for AA/AB stacking; coupled quasi-altermagnetic states for AC$\Gamma$7/AC$\Gamma$8 stacking; field-induced transition between AFM and AM in antiparallel alignment.

  Figure 10: Diagrammatic summary of Type-I and Type-II Mn$\Gamma$9WS$_2$0Se$_2$1 structural evolution and valley/altermagnetic phase emergence.

Further stacking manipulation (AA vs. AA$_2$2) and control of interlayer spin alignment enable precise toggling between altermagnetic, quasi-altermagnetic, and AFM states, with corresponding band structure mapping.

Figure 11: Spin-polarized band structures for Type-II AA Mn$_2$3WS$_2$4Se$_2$5 in various stacking and magnetic configurations.

Figure 12: Spin-polarized band structures for Type-II AA$_2$6 Mn$_2$7WS$_2$8Se$_2$9, showing stacking-induced quasi-altermagnetic phenomena.

Conclusion

The study provides a symmetry-driven, first-principles validated classification of sliding-induced altermagnetic and coupled quasi-altermagnetic phases in 2D square-lattice systems. The work rigorously establishes the correspondence between real-space stacking, local atomic environment, and reciprocal-space spin splitting, with practical implications for nonvolatile and mechanically switchable spintronic and valleytronic devices. The methodology and theoretical framework are widely extensible to other 2D square-lattice materials. The findings demonstrate that interlayer sliding can induce, reverse, and control spin degeneracy and transport signatures at the $_4$0 point, without requiring external fields or intrinsic polarization. This opens a pathway toward symmetry-engineered spin-electronic phase control and device architectures in next-generation 2D materials.