Topological piezomagnetic effect in two-dimensional Dirac quadrupole altermagnets

Altermagnets provide a natural platform for studying and exploiting piezomagnetism. In this paper, we introduce a class of insulating altermagnets in two dimensions (2D) referred to as Dirac quadrupole altermagnets, and show based on microscopic minimal models that the orbital piezomagnetic polarizability of such altermagnets has a topological contribution described by topological response theory. The essential low-energy electronic structure of Dirac quadrupole altermagnets can be understood from a gapless parent phase (i.e., the Dirac quadrupole semimetal), which has important implications for their response to external fields. Focusing on the strain-induced response, here we demonstrate that the topological piezomagnetic effect is a consequence of the way in which strain affects the Dirac points forming a quadrupole. We consider two microscopic models: a spinless two-band model describing a band inversion of $s$ and $d$ states, and a Lieb lattice model with collinear Néel order. The latter is a prototypical minimal model for altermagnetism in 2D and is realized in a number of recently proposed material compounds, which are discussed.

💡 Research Summary

The authors investigate a novel class of two‑dimensional (2D) insulating altermagnets that inherit their low‑energy electronic structure from a gapless “Dirac quadrupole” semimetal. In such a parent phase four Dirac nodes form a quadrupolar arrangement in momentum space. By introducing a symmetry‑breaking mass term the Dirac points are gapped, yielding an insulating altermagnet whose net spin magnetization vanishes but whose orbital magnetization can respond to mechanical strain.

Two minimal microscopic models are studied. The first is a spinless two‑band tight‑binding model on a square lattice with s‑ and d_xy‑orbitals. Its Bloch Hamiltonian H₀(k)=ε_k + t_z(k)τ_z + t_x(k)τ_x + Δτ_y reproduces the Dirac quadrupole when Δ=0 and the insulating altermagnet when Δ≠0. Expanding around the four Dirac points gives low‑energy Dirac Hamiltonians H_K=±(v₁q_yτ_z+v₂q_xτ_x)+Δτ_y, where the velocities v₁ and v₂ are functions of the hopping parameters and the band‑inversion strength δ.

Strain is incorporated by allowing the nearest‑neighbor hopping t₀ to acquire opposite fractional changes along x and y, which is encoded in a term W(k)=χ(k)I with χ(k)=−2t₀(cos k_x−cos k_y). This deformation shifts the Dirac nodes asymmetrically, effectively turning the quadrupole into a “Dirac dipole”. The linear piezomagnetic response is defined by M_i=Λ_{ijk}ε_{jk}; symmetry of the square lattice leaves only Λ_{zxx}=−Λ_{zyy}=Λ non‑zero. Using linear‑response theory the authors derive a compact expression for Λ as an integral over the Berry curvature of the occupied valence band:

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