Dynamical response of noncollinear spin systems at constrained magnetic moments

TL;DR

This work introduces a general, first-principles framework to compute the dynamical response of noncollinear spin systems by constraining local magnetic moments via a penalty-functional and linking different magnetic functionals through Legendre transforms. The approach stabilizes the linear-response problem, removes problematic magnon resonances at low energies, and allows exact post-processing recovery of physically meaningful relaxed-spin responses. By formulating both static and finite-frequency spin-phonon-electron couplings within dynamical DFPT, the authors derive frequency-dependent dielectric, magnetic, and magnetoelectric susceptibilities and demonstrate the method on ferromagnetic CrI$_3$ and antiferromagnetic Cr$_2$O$_3$, revealing electromagnons and large lattice-mediated ME effects. The results show that including magnon masses (SOA) and spin-phonon Berry-curvature contributions yields significantly improved agreement with the underlying physics, enabling efficient, accurate predictions of coupled spin-lattice dynamics in magnetic insulators with potential for THz-optical control of magnetism.

Abstract

Noncollinear magnets are notoriously difficult to describe within first-principles approaches based on density-functional theory (DFT) because of the presence of low-lying spin excitations. At the level of ground-state calculations, several methods exist to constrain the magnetic moments to a predetermined configuration, and thereby accelerate convergence towards self-consistency. Their use in a perturbative context, however, remains very limited. Here we present a general methodological framework to achieve parametric control over the local spin moments at the linear-response level. Our strategy builds on the concept of Legendre transform to switch between various flavors of magnetic functionals, and to relate their second derivatives via simple linear-algebra operations. Thereby, we can address an arbitrary response function at the time-dependent DFT level of theory with optimal accuracy and minimal computational effort. In the low frequency limit, we identify the leading correction to the existing adiabatic formulation of the problem [S. Ren \emph{et al.}, Phys. Rev. X {\bf 14}, 011041 (2024)], consisting in a renormalization of the phonon and magnon masses due to electron inertia. As a demonstration, we apply our methodology to the THz optical response of bulk CrI$_3$ and Cr$_2$O$_3$, where we identify contributions from hybrid (electro)magnons with mixed spin-lattice character.

Figures (13)

• Figure 1: (a) Side view and (b) top view of a three-layer section of the CrI$_3$ structure. The dashed lines in (a) outline the primitive cell used in the numerical calculations, with the three real-space primitive vectors labeled as ${\bf a}_i$. Thick black arrows indicate the magnetic moments, aligned along the $z$-axis. In panel (b), only the Cr sites are shown to clearly illustrate the ABC hexagonal stacking pattern.
• Figure 2: Side view of the Cr$_2$O$_3$ rhombohedral cell. The dashed lines outline the three real-space primitive vectors, labeled as ${\bf a}_i$. Thick black arrows indicate the magnetic moments, aligned along the $z$-axis.
• Figure 3: Convergence rate of first-order local magnetic moments versus number of SCF iterations within standard noncollinear DFPT (top panels) and within our constrained-${\bf B}$ method (bottom panels). The following types of perturbations are considered: static uniform Zeeman field (a,b), static displacement of Cr1 (c,d), static electric field (e,f) and dynamic electric field at $\hbar\omega=32.7$ meV (g,h). In the latter case, only the real part of the magnetic moment is shown. The first-order magnetic moments are expressed in Cartesian coordinates in panels (a,b). In (c,d), we show their projection on a direction that is either parallel (l) or perpendicular (t) to the atomic displacement in the $xy$ plane. In (e,f,g,h) the antiferromagnetic component of the induced moments is projected on the applied electric field direction (l) or its normal (t).
• Figure 4: (a) Squared residual of the SCF potential and (b) longitudinal in-plane first-order atomic magnetic moments as a function of the SCF iterations for the linear-response calculation to an electric field of $\hbar\omega=32.7$ meV. Colored curves illustrate the results obtained with different values of the magnetic penalty stiffness ($\alpha$).
• Figure 5: (a) Frequency dependence of the optical-magnon coefficients of Eq. \ref{['bfu']}, as obtained from polynomial frequency interpolation at different degrees: $n=4$ (dark-solid), $n=2$ (turquoise-dashed) and $n=1$ (orange-dashed). Panel (b) shows an amplified view of the abscissas axis around the intersection of the curves shown in (a).