Spin-biased quantum spin Hall effect in altermagnetic Lieb lattice

Abstract: Altermagnetic (AM) order, a recently discovered magnetic state, has attracted intense research interest for its potential applications in spintronic and quantum technologies. Here, we theoretically investigate the AM state in the Lieb lattice, a prototypical two-dimensional lattice, using the Hubbard model. We show that AM order emerges with only moderate electronic correlations. Strikingly, spin-orbit coupling drives the system into a topological phase exhibiting a new quantum spin Hall effect (QSHE) with spin-biased topological edge states in one-dimensional nanoribbons. These edge states possess different localizations and velocities, and hence may produce spin and charge currents, fundamentally distinct from that in conventional topological insulators with spin degeneracy. This novel spin-biased QSHE in the AM Lieb lattice unveils exciting opportunities for both fundamental studies and innovative device concepts, motivating immediate experimental exploration.

• The paper demonstrates that SOC transforms an altermagnetic Lieb lattice into a topologically nontrivial insulator with a quantized spin Hall effect.
• It employs the extended Hubbard model to map phase transitions between nonmagnetic metal, altermagnetic metal, and insulator regimes by tuning U and onsite potential differences.
• Spin-biased edge state analysis reveals spatially separated, nondegenerate spin channels that point to new possibilities for spintronic device applications.

Spin-Biased Quantum Spin Hall Effect in the Altermagnetic Lieb Lattice

Introduction

This work provides a theoretical analysis of altermagnetic (AM) order in the two-dimensional Lieb lattice, employing the extended Hubbard model as the core framework. It explores how introducing spin-orbit coupling (SOC) drives the Lieb lattice into a nontrivial topological phase and realizes a novel quantum spin Hall effect (QSHE), distinguished by spin-biased, nondegenerate edge states. The results systematically clarify the interplay among electronic correlation, AM order, SOC, and topological electronic structure in a canonical two-dimensional platform, setting the stage for both further theoretical investigation and experimental realization.

Altermagnetism in the Lieb Lattice

Altermagnetism represents a recently classified magnetic symmetry distinct from conventional AFM and FM orders, featuring compensated sublattice magnetizations connected by crystal rotation or mirror symmetries. The Lieb lattice, with its C$_{4v}$ and mirror symmetries, provides a minimal model for 2D AM order, where opposite spin polarization naturally emerges on the B and C sublattices. Unlike honeycomb or kagome lattices, the Lieb lattice enables AM order regardless of strong electronic correlations and possesses a topology that admits inherent Berry-phase phenomena.

The paper demonstrates that using moderate Hubbard interaction strengths $U$ ($0.7t \lesssim U \lesssim 5t$), the Lieb lattice transitions between three distinct phases as a function of the onsite potential difference $\Delta$ and $U$: a nonmagnetic (NM) metal, an AM metal, and an AM insulator. The emergence of the AM regime is associated with spontaneous symmetry reduction in each spin channel, leading to pronounced momentum-dependent spin-splitting bands without net magnetization.

Topological Effects of Spin-Orbit Coupling

Incorporating SOC—in the symmetry-allowed Kane-Mele-type form—qualitatively alters the electronic structure of the AM metallic state. The key results are:

• SOC opens a global band gap in the AM metal, transforming it into an insulating phase whose gap scales linearly with SOC strength.
• The SOC-driven AM insulating phase is fundamentally distinct from a trivial Mott/Slater insulator, being characterized by topologically nontrivial bulk and edge state properties.

Detailed calculations of the Berry curvature and spin-resolved Chern numbers reveal that the SOC-induced gap enables the realization of a quantized spin Hall conductance $\sigma_{xy}^s = e^2/h$, with $C_s = 1$, while the total Chern number $C$ vanishes due to spin compensation. The AM Lieb lattice thus realizes a QSHE without relying on time-reversal symmetry (TRS), in sharp contrast with conventional topological insulators (TIs) whose QSHE is protected by TRS. This represents a symmetry-unprotected, interaction- and magnetism-assisted topological insulator phase.

Spin-Biased Edge State Phenomenology

A defining feature of the QSHE in the AM Lieb lattice is its spin-biased edge state structure, which is concretely illustrated by nanoribbon modeling:

• Spin-up and spin-down edge states are spatially separated, energetically nondegenerate, and possess opposite edge localization and distinct group velocities.
• The nondegeneracy leads to both spin and charge current contributions along the edge, differing fundamentally from the Kramers-degenerate, purely spin-polarized edge currents of conventional QSHE/TI systems.
• These features stem from the reduction of symmetry from C$_{4v}$ to C$_{2v}$ in each spin channel and are intimately tied to the band topology imparted by the lattice's crystalline symmetries and magnetic configuration.

This form of spin-biased QSHE aligns with theoretical predictions for AM materials [e.g., (Chang et al., 6 Aug 2025, Feng et al., 17 Mar 2025)] and is distinct from quantum spin Hall phases in honeycomb-lattice-based altermagnets, which retain Kramers degeneracy.

Robustness and Phase Transitions

The work traces the evolution of the phase diagram under variation of $U$0, $U$1, and SOC strength. The AM QSHE is shown to be robust to moderate symmetry-breaking perturbations (e.g., Rashba SOC, uniaxial strain), as long as the global band gap is preserved. Close to AM metal-insulator phase boundaries, SOC-driven band inversions provide signatures of topological transition points, shifting the system from trivial to nontrivial topological order as SOC is increased. The study clarifies the conditions for gap closing and reopening, relevant both for tuning device properties and for fundamental understanding of correlated topological matter.

Implications and Outlook

These findings have both immediate and longer-term implications:

• The theoretically predicted AM QSHE provides a new route to quantized spin transport in the absence of global magnetization and without requiring TRS, opening opportunities for low-dissipation, high-coherence spintronic devices.
• The spin-biased edge states enable charge-spin conversion mechanisms not accessible in standard TIs, potentially allowing nonreciprocal device architectures.
• While direct realization of freestanding AM Lieb lattices is experimentally challenging, the model system applies to known quasi-2D materials hosting Lieb-type motifs, and the phenomenology is expected to be realizable in designer heterostructures, ultracold-atom solid-state simulators, photonic lattices, and 2D van der Waals platforms.
• The work highlights the essential role of lattice symmetry in the emergence of and protection for AM order and topological phenomena, reinforcing the ongoing drive to engineer designer band topology for quantum matter research.

Conclusion

This paper establishes the 2D Lieb lattice as a minimal but versatile platform for realizing AM order and a spin-biased QSHE driven by moderate correlations and SOC. The synergy of orbital, spin, and lattice symmetries produces topological phenomena distinct from both AFM and FM systems, suggesting a broad new class of interaction-enabled, symmetry-protected quantum phases. Beyond theoretical impact, the results provide fundamental guidance for experimental realization and control of AM topological insulators with unique edge states and transport properties, with significant implications for the future of spintronics and quantum technology.