Formulation of the orbital magnetic moment in multiorbital tight-binding models: Application to the inverse Faraday effect

TL;DR

The paper presents a rigorous framework for the orbital magnetization in multiorbital tight-binding systems, identifying four contributions to the total magnetic moment and highlighting the Wannier-corrected electric dipole term $\bm{M}_{P}$. Using an $s$-$p$ minimal model and Floquet theory, it derives analytical expressions via time-dependent Schrieffer-Wolff transformation and validates them with nonequilibrium Floquet calculations, showing that the orbital part of the IFE can dominate and that the three orbital contributions are of comparable size. The study reveals that orbital degrees of freedom drive a stronger IFE response than spin, with a distinct Fermi-sea character for orbital moments and resonant enhancements near band-crossing energies. These results underscore the importance of incorporating Wannier corrections and multiorbital physics when predicting light-induced magnetization, with potential implications for optospintronics and ultrafast control of magnetism.

Abstract

We establish a theoretical formulation of the orbital magnetic moment in multiorbital tight-binding models, focusing on the role of the electric dipole. We demonstrate that the total magnetic moment can be decomposed into several contributions in multiorbital tight-binding models generally. In particular, we reveal that the electric dipole moment of Wannier orbitals also contributes to the orbital magnetic moment, which is not included in the conventional expression for the orbital magnetic moment in lattice systems. The derived formulation for the magnetic moment is applied to the inverse Faraday effect (IFE), a phenomenon where circularly-polarized light induces a magnetic moment. To account for all possible contributions, we adopt an $s$-$p$ tight-binding system as a minimal model for studying the IFE. Using an analytical approach based on the Schrieffer-Wolff transformation, we clarify the physical origins of these contributions. Additionally, we quantitatively evaluate each contribution on an equal footing through a numerical approach based on the Floquet formalism. Our results reveal that the orbital magnetic moment exhibits a significantly larger response compared to the spin magnetic moment, with all contributions to the orbital magnetic moment being comparable in magnitude. These findings highlight the essential role of orbital degrees of freedom in the IFE.

Figures (8)

• Figure 1: Schematic picture of the inverse Faraday effect. When circularly-polarized light is irradiated into the material, an effective magnetic field is generated, inducing spin angular momentum $\langle S^z \rangle$, orbital angular momentum $\langle L^z \rangle$ and kinetic orbital magnetic moment $\langle M^z_{\text{lat}} \rangle$ and $\langle M^z_{P} \rangle$.
• Figure 2: Schematic picture of the model. The system is defined on a square lattice with $s$, $p_x$, and $p_y$ orbitals. The system is irradiated with circularly-polarized light.
• Figure 3: Band structure without an applied electric field with $\lambda = 0.1 t_s$, $\Delta = 2.0 t_s$. (a) The band structure along the high symmetry line. The dashed line represents the chemical potential $\mu = -2.0 t_s$ used in Figs. \ref{['fig: EOmega_heatmap']} and \ref{['fig: DeltaOmega_heatmap']}. (b) 3D plot of the band structure in the first Brillouin zone.
• Figure 4: Floquet band structure under an applied electric field along the high symmetry line. The parameters are set to $\lambda = 0.5 t_s, \Delta = 2.0 t_s, \Omega = 8.0 t_s, E = 2.0 t_s / a$. (a) The red and blue bands correspond to the spin-up band and spin-down band, respectively. (b-d) The color represents the magnitude of the (b) orbital angular momentum $-L^{z}(\bm{k})$, (c) kinetic contribution $M^z_{\text{lat}}(\bm{k})$, and (d) Wannier polarization contribution $M^z_{P}(\bm{k})$.
• Figure 5: Distribution of (a-f) orbital angular momentum $-L^z(\bm{k})$, (g-l) kinetic contribution $M^z_{\text{lat}}(\bm{k})$, and (m-r) Wannier polarization contribution $M^z_{P}(\bm{k})$ in the first Brillouin zone for each band. The parameters are the same as Fig. \ref{['fig: Floquet band']}. (a,g,m) lower band, up spin; (b,h,n) lower band, down spin; (c,i,o) middle band, up spin; (d,j,p) middle band, down spin; (e,k,q) higher band, up spin; (f,l,r) higher band, down spin. We note that, for the lower band, the color bar range is narrowed to clearly show the distribution near the $\Gamma$ point.