Topological piezomagnetic effect in two-dimensional Dirac quadrupole altermagnets

Abstract

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.

Figures (4)

• Figure 1: (a) Schematic depiction of the piezomagnetism in 2D altermagnets: an orbital magnetization $M_z$ develops in response to applied strain. (b) Configuration of Dirac points in a Dirac quadrupole semimetal, with positive and negative helicities color coded red and blue, respectively.
• Figure 2: (a) Energy bands of the orbital altermagnet model defined in Eq. \ref{['eq:Hk-orb']} and schematically depicted in the inset (upper right corner). Here we have set $(t_0,t_d,\delta) = (0.25t_1,0.5t_1,0.8t_1)$; the orange and thin black curves correspond to $\Delta=0.3t_1$ and $\Delta=0.0$, respectively. The Dirac points and their valley labels are indicated. (b) Orbital piezomagnetic polarizability $\Lambda$ of the orbital altermagnet model, computed using Eq. \ref{['eq:Lambda-2band']} and shown as a function of $\Delta$ (in units of $t_1$). Here we have used $(t_0,t_d) = (0.5t_1,0.5t_1)$. The red dashed lines correspond to $\pm (1/\pi) (t_0 \delta/t^2_1)$. (c) Without strain ($\phi=0$) all Dirac points occur at the same energy. (d) When strain is applied ($\phi \neq0$) nodes of opposite topological charge are shifted in opposite directions.
• Figure 3: (a) Lieb lattice altermagnet model. The non-magnetic sites are shown in grey and the magnetic sites are shown in red and blue ($A$ and $B$ sublattice, respectively), indicating opposite alignment of spin moments along the $\hat{z}$ direction. (b) Location of the Dirac points on the BZ boundary in the topological regime ($|N_z|<4t_d$). Valleys labeled red and blue correspond to the $\sigma=\uparrow$ and $\sigma=\downarrow$ sectors, respectively (assuming $N_z>0$, see Appendix \ref{['app:Lieb']}). (c) Effect of strain on the (tilted) Dirac points. The Dirac points of the valleys $\mathbf{K}_\pm$ and $\mathbf{K}'_\pm$ are shifted in energy in opposite directions. (d) Orbital piezomagnetic polarizability as a function of $\lambda$ for different $N_z$ (both in units of $t_1$), calculated using Eq. \ref{['eq:Lambda-Lieb']}. Here we have set $(t_0,t_d) = (0.5t_1,1.5t_1)$. (e) Same as in (d), showing contributions from the geometric and interband terms in Eq. \ref{['eq:Lambda-Lieb']} (for $N_z=2.0t_1$).
• Figure 4: (a) Piezomagnetic polarizability $\Lambda$ of the Lieb lattice model as a function of $N_z$ (in units of $N_{z,c}\equiv 4t_d$) for different $t_0$ (in units of $t_1$). We have set $(t_d,\lambda) = (3t_1,0.2t_1)$. (b) Same as in (a), showing separate contributions from the geometric and interband terms in Eq. \ref{['eq:Lambda-Lieb']} (for $t_0=0.4t_1$).