Switchable chiral antiferromagnetism through nonlinear magnetic susceptibility

Technical Summary: Switchable Chiral Antiferromagnetism through Nonlinear Magnetic Susceptibility

Problem Statement
Antiferromagnets (AFM) have recently attracted significant attention due to their time-reversal symmetry (TRS) breaking properties, which enable phenomena such as the anomalous Hall effect and spin-polarized currents. However, a major technological hurdle remains: the deterministic selection of a single AFM domain. In most AFM materials, the two TR partners of the ground state are energetically degenerate in the absence of external perturbations. While external magnetic fields can bias these states in materials with non-zero net magnetization (e.g., weak ferromagnets), there is currently no general strategy to deterministically switch TR partners in fully compensated AFM materials (where net magnetization $M=0$). Existing approaches, such as current-induced spin-orbit torques, are limited to specific metallic systems. Furthermore, while nonlinear magnetic susceptibility ($\chi^{(1)}$) has been discussed in scattered literature since the 1970s, primarily for noncollinear Ising spin systems, a comprehensive first-principles framework applicable to general AFM materials has been lacking.

Methodology
The authors develop a unified theoretical framework to calculate and understand the lowest-order nonlinear magnetic susceptibility, $\chi^{(1)} = \partial^2 M / \partial B^2$, in AFM materials. The methodology proceeds in three stages:

1. Symmetry Analysis: The paper establishes that for AFM states with strictly zero net magnetization, a non-zero $\chi^{(1)}$ (or generally any odd-order susceptibility $\chi^{(2n+1)}$) allows TR partners to be switched by a uniform magnetic field. This requires the absence of specific combined symmetries (e.g., $T$, $TI$, $C_{2x,y}$, etc.).
2. First-Principles Formalism: The authors derive analytic formulas for $\chi^{(1)}$ within the spin density functional theory (DFT) framework. They decompose the susceptibility into:
   • Mean-field (Independent-electron) contribution: Calculated via Fermi-surface and Fermi-sea integrals involving band energies and spin matrix elements.
   • Self-consistent-field (SCF) correction: Recognizing that the independent-electron approximation often misses dominant effects in AFM, the authors propose a numerical scheme to extract the SCF contribution. This involves performing self-consistent DFT calculations under small, symmetric Zeeman fields ($B^+$ and $B^-$) and fitting the resulting net magnetization $M(B)$ to a quadratic form to extract $\chi^{(1)}$.
3. Modeling: To gain physical intuition, the authors employ a minimal 3-sublattice Heisenberg spin model with nearest-neighbor exchange ($J$) and easy-axis anisotropy ($K$). This model is used to analyze the temperature dependence of $\chi^{(1)}$ and the competition between longitudinal and transverse spin responses.

Key Results
The formalism is applied to the noncollinear AFM family $Mn_3XN$ ($X = Ni, Ag, Ga, Zn, Sn$) in the $\Gamma_{5g}$ phase, which possesses a triangular spin structure with zero net magnetization.

• Dominance of SCF Corrections: For the $Mn_3XN$ family, the SCF-corrected $\chi^{(1)}$ values are found to be at least two orders of magnitude larger than the bare mean-field values. This indicates that the deformation of the ordered spin configuration under a finite field is the dominant mechanism for $\chi^{(1)}$ in these materials.
• Material Specifics: Among the calculated compounds, $Mn_3AgN$ exhibits the largest nonlinear susceptibility, reaching $\sim -6 \times 10^{-3} \, \mu_B T^{-2}$ per unit cell. At moderate fields (1 T), this results in a difference in net magnetization between TR partners of $\sim 10^{-2} \, \mu_B$, comparable to weak ferromagnets.
• Temperature Dependence and Sign Change: The 3-sublattice toy model predicts a non-trivial temperature dependence for $\chi^{(1)}$. At low temperatures, $\chi^{(1)}$ is positive and driven by the transverse canting of spins (scaling as $1/n$, where $n$ is the ordered spin magnitude). As temperature approaches the Néel temperature ($T_c$), the longitudinal response of the spins (which has a negative $\chi^{(1)}$ due to the concave nature of the Langevin function) becomes dominant. Consequently, the model predicts that $\chi^{(1)}$ undergoes a sign change between $T=0$ and $T_c$, a feature verified by self-consistent mean-field calculations.
• Role of Anisotropy: The magnitude of $\chi^{(1)}$ is shown to scale with magnetic anisotropy. Since anisotropy scales quadratically with spin-orbit coupling, materials containing heavier elements (e.g., Ag, Sn in Row 5 of the periodic table) exhibit significantly larger $\chi^{(1)}$ than those with lighter elements (Ni, Ga, Zn in Row 4).

Significance and Claims
The paper claims to establish a general protocol for accessing and switching TR partners in fully compensated AFM materials using standard magnetic-field techniques, bypassing the need for net magnetization. By generalizing the mechanism previously observed in the kagome spin ice $HoAgGe$, the authors demonstrate that nonlinear susceptibility serves as a universal order parameter for chiral AFM switching.

The work highlights that geometrically frustrated AFMs with strong anisotropy are promising candidates for "field-switchable chiral antiferromagnetism." The authors modestly note that while their DFT calculations provide a theoretical baseline, experimental values may differ due to factors such as orbital contributions, strain, and correlation effects not fully captured in the current model. However, the identification of the SCF correction as the dominant mechanism provides a clear physical picture: the switching capability arises from the self-consistent deformation of the spin texture under an external field, a phenomenon that can be quantified and predicted for a broad class of materials.