Dynamical Effective Hamiltonian Approach to Second-Harmonic Generation in Quantum Magnets: Application to NiI$_2$

TL;DR

The paper addresses the challenge of quantitatively predicting second-harmonic generation in insulating magnets. It introduces a dynamical effective operator framework in which SHG operators are computed from local many-body cluster models using a d+NTO basis, AMF double counting, and exact diagonalization, followed by a mapping to spin operators. Applied to NiI2, the approach identifies dominant nearest-neighbor ring-current pathways and reproduces rotational anisotropy SHG data, yielding a spiral tilt angle consistent with neutron measurements. This work provides a first-principles, quantitative method for nonlinear optical responses in quantum magnets and establishes a path to extend to other nonlinear probes and materials.

Abstract

Although second harmonic generation (SHG) is a promising and widely used method recently for studying 2D magnetic materials, the quantitative analysis of the full SHG tensor is currently challenging. In this letter, we describe a first-principles-based approach towards quantitative analysis of SHG in insulating magnets through formulation in terms of dynamical effective operators. These operators are computed by solving local many-body cluster models. We benchmark this method on NiI$_2$, a multiferroic 2D van der Waals antiferromagnet, demonstrating quantitative analysis of reported Rotational Anisotropy (RA)-SHG data. SHG is demonstrated to probe local ring-current susceptibilities, which provide sensitivity to short-range chiral spin-spin correlations. The described methods may be easily extended to other non-linear optical responses and materials.

Figures (4)

• Figure 1: (a) Unit cell of NiI$_2$ showing Van der Waals stacked structure. (b) Helical magnetic structure showing spiral plane tilted by angle $\alpha$ from $c$-axis. (c,d) Double-sided Feynman diagrams describing SHG process.
• Figure 2: (a) DFT bandstructure of NiI$_2$. (b) One-particle density of states for nearest neighbor cluster model including Ni $d$-orbital Wannier functions and ligand Natural Transition Orbitals (NTOs). (c,d) Total density $\sum |\psi(\vec{r})|^2$ of (c) $d$-orbital WFs and (d) NTOs.
• Figure 3: (a-f) Computed ($X,Y,Z$) components of the SHG operator couplings presented as (a-c) $2(\hbar\omega/e)^3\text{Im}[\mathbf{C}_{ij}^{\mu\xi\nu}(\omega)+\mathbf{C}_{ij}^{\mu\xi\nu\dagger}(-\omega)]$ and (d-f) $2(\hbar\omega/e)^3\text{Abs}[\mathbf{C}_{ij}^{\mu\xi\nu}(\omega)+\mathbf{C}_{ij}^{\mu\xi\nu\dagger}(-\omega)]$ in units of Å$^3$. Polarization indices are given in ($x,y,z$) coordinates. (g,h) Energy ranges for resonant SHG, referring to eq'n (\ref{['eq:O']}). (i-l) Example ring current excitation pathways contributing to SHG operators at $\hbar\omega = 1.25$ eV. Arrows refer to the transfer of holes. The relative signs of $\langle g|\hat{\mathcal{L}}^\nu(\omega)|n\rangle$, $\langle n|\hat{\mathcal{L}}^\xi(\omega)|m\rangle$, and $\langle m|\hat{\mathcal{L}}^\mu(-2\omega)|g\rangle$ matrix elements are indicated. (m) Depiction of LMCT and intersite $d$-$d$ excitations.
• Figure 4: Fitted RA-SHG intensity (lines) with respect to polarization angle $\theta$ for best fit spin spiral angle $\alpha$ for (a) parallel, (b) perpendicular polarization. Dots are experimental data on CVD-grown NiI$_2$ crystals with $\omega \approx 1.25$ eV extracted from song2022evidence. Fitted $\alpha$ values and relative population of domains with in-plane $\mathbf{Q}_1 = (q,0), \mathbf{Q}_2 = (q,-q), \mathbf{Q}_3=(-q,q)$ are indicated.