Orbital piezomagnetic polarizability of pure insulating altermagnets in two dimensions

TL;DR

This work develops a general theory of orbital piezomagnetism in two-dimensional pure altermagnets, linking strain-induced orbital magnetization to the Berry curvature of occupied bands. By formulating a minimal tight-binding framework with spin-sector decoupling and applying it to three tetragonal models (Lieb, 2D rutile, and octahedral-rotation), it derives simple microscopic expressions for the orbital magnetization $M_z$ and the linear piezomagnetic polarizability $\Lambda$, and reveals how topology (Dirac points and their masses) governs the response, including a topological discontinuity in the linear response for the Lieb model. For the $g$-wave octahedral-rotation model, the linear effect is forbidden by symmetry, and a nonlinear piezomagnetic response appears as $M_z = \Lambda^{(2)} \phi_1 \phi_2$, with a maximum in $\Lambda^{(2)}$ at modest $N_z$. Overall, the results establish a framework that connects orbital magnetization under strain to Berry curvature in 2D altermagnets and highlights the role of band topology and symmetry in determining the magnitude and form of the piezomagnetic response, with implications for materials in the rutile family and related altermagnets.

Abstract

The distinctive symmetry properties of pure altermagnets make them natural candidates for piezomagnetism. Previous work motivated by the piezomagnetic properties of altermagnets has primarily focused on the spin magnetization response to applied strain. In this paper we study orbital piezomagnetic effects--the orbital magnetization response to applied strain--in minimal lattice models of pure insulating altermagnets in two dimensions. We obtain general microscopic expressions for the orbital magnetization in the presence of strain, as well as the orbital piezomagnetic polarizability, i.e., the defining response characteristic of pure altermagnets. We apply these expressions to three specific tetragonal lattice models, two corresponding to $d$-wave altermagnets and one describing a $g$-wave altermagnet. Whereas the $d$-wave altermagnets are associated with a linear piezomagnetic polarizability, the $g$-wave altermagnet exhibits a nonlinear piezomagnetic effect. Our analysis reveals how the polarizabilities are related to and determined by the Berry curvature of the occupied bands. Connections to materials of current interest are discussed.

Figures (9)

• Figure 1: Lieb lattice model. (a) Schematic of the two-dimensional Lieb lattice. The magnetic $A$ and $B$ sites are shown in blue and red, and the nonmagnetic site is shown in white. The ordered moments of the collinear altermagnetic Néel state point out of the plane, as indicated. (b) Energy spectrum with and without strain (red/blue and solid/dashed black, respectively) of the Lieb lattice model specified by Eqs. \ref{['eq:eps_k-t_xk']}--\ref{['eq:Lieb-strain']} with parameters $(t_0,t_1,\lambda,N_z) = (0.25t_d,0.5t_d,0.25t_d,2.0t_d)$ and $\phi=0.1$ for the case with strain. Blue and red bands correspond to $\sigma=\uparrow$ and $\sigma=\downarrow$, respectively.
• Figure 2: 2D rutile lattice model. (a) Schematic of the 2D rutile lattice with layer group $P4/mbm$. The magnetic sites (red and blue) have an anisotropic crystallographic environment. (b) Energy spectrum with and without strain (red/blue and solid/dashed black, respectively) of the model defined by Eq. \ref{['eq:eps_k-t_xk']} and Eqs. \ref{['eq:rutile']}--\ref{['eq:rutile-strain']} with parameters $(t_0,t_1,\lambda,N_z) = (0.25t_d,0.5t_d,0.25t_d,2.0t_d)$ and $\phi=0.1$ for the strained case. As in Fig. \ref{['fig:Lieb']}(b), blue and red bands correspond to $\sigma=\uparrow$ and $\sigma=\downarrow$, respectively.
• Figure 3: Octahedral rotation lattice. (a) A lattice of magnetic sites (shown in blue and red) surrounded by rotated square cages of nonmagnetic sites (shown in white). (b) Momentum-space resolved plot of the energy difference $(E^\uparrow_{{\bf k}1} -E^\downarrow_{{\bf k}1})/8t_g$ between the valence bands of opposite spin in the absence of strain. The $g$-wave structure of the spin splitting is evident. Energies are calculated for the model defined in Eqs. \ref{['eq:eps_k-t_xk']} and \ref{['eq:octahedral']} with parameters $(t_0,t_1,\lambda_2,N_z) = (0.25t_g,0.5t_g,0.5t_g,2.0t_g)$.
• Figure 4: Orbital magnetization. (a) Orbital magnetization $M_z$ as a function of the dimensionless strain field $\phi$, calculated for the Lieb lattice model (see inset) using Eq. \ref{['eq:Mz-altermagnet']}. We have used the parameters $(t_0,t_d,\lambda) = (0.5t_1,2t_1,0.5t_1)$ and different curves correspond to $N_z/t_1= 1.0,2.0,3.0,4.0,5.0$ (from blue to magenta). Here $a$ is a length scale on the order of the lattice spacing defined via $\hbar^2/2m_e t_1 \equiv a^2$. Note further that the value of the $M_z$ has been multiplied by a factor $10$. (b) Same as in (a) but for the 2D rutile lattice model (see inset). For the rutile lattice model we have used the same model parameters $(t_0,t_d,\lambda)$ but different curves correspond to $N_z/t_1= 2.0,\ldots,6.0$ (from blue to magenta).
• Figure 5: Berry curvature distribution. (Left) Plot of the Berry curvature of the Lieb lattice model without strain. The upper and lower panels correspond to the $\sigma=\uparrow$ and $\sigma=\downarrow$ valence bands, respectively. The Berry curvature is concentrated around the gapped Dirac crossings, which exist on the $k_y=\pi$ and $k_x=\pi$ lines (respectively) and are indicated by black markers. Here we have used $( t_d,\lambda,N_z) = (2t_1,0.3t_1,4.0t_1)$. (Right) Plot of the Berry curvature of the (unstrained) 2D rutile lattice model, with upper and lower panels corresponding to $\sigma=\uparrow$ and $\sigma=\downarrow$, respectively. For the rutile model we have used $(t_d,\lambda,N_z) = (2t_1,0.5t_1,2.0t_1)$. The location of the Dirac points on the body diagonal, obtained as described in Appendix \ref{['app:2D-rutile']}, is indicated by black dots.