Turov's Invariants in Collinear Antiferromagnets

• Turov’s invariants are group-theoretical quantities that specify when a uniform magnetization arises from Néel order in collinear two-sublattice antiferromagnets.
• They are defined through symmetry-allowed linear coupling terms in the Landau free energy, linking the magnetization and Néel order parameter.
• Applications to square-lattice models illustrate distinct cases for genuine antiferromagnets, ferrimagnets, and weak ferromagnets, directly impacting the anomalous Hall effect.

Turov's invariants are group-theoretical quantities that characterize the existence of a uniform magnetization in collinear two-sublattice antiferromagnets, described as the lowest-order linear coupling terms in the Landau free energy that are simultaneously linear in the magnetization $\mathbf{M}$ and odd in the Néel order parameter $\mathbf{L}$, yet transform as true scalars under the full magnetic (Néel)-space-group of the underlying crystal. These invariants provide a necessary and sufficient condition for the induction of weak ferromagnetism and, consequently, the appearance of an intrinsic anomalous Hall effect in metallic collinear antiferromagnets by identifying the precise symmetry-allowed directions in which $\mathbf{M}$ can be generated from $\mathbf{L}$ (Golubinskii et al., 4 Nov 2025).

1. Formal Definition and Symmetry Criteria

Turov's invariants are defined by considering the transformation properties of the net magnetization $\mathbf{M} = \mathbf{M}_1 + \mathbf{M}_2$ and the Néel vector $\mathbf{L} = \mathbf{M}_1 - \mathbf{M}_2$ under spatial rotations/reflections $R$, time reversal $\Theta$, and operations $g$ that may exchange magnetic sublattices. For the general interaction term

$$F_\mathrm{int} = \sum_{\alpha, \beta} K_{\alpha \beta} M_\alpha L_\beta,$$

one seeks the set of tensors $K_{\alpha\beta}$ for which $F_\mathrm{int}$ is invariant under every $g$ in the magnetic space group $G$:

$$\det(R_g) s_g R_g^{\alpha\gamma} R_g^{\beta\delta} K_{\gamma\delta} = K_{\alpha\beta},$$

where $s_g = -1$ if $g$ exchanges sublattices and $+1$ otherwise. Nontrivial solutions ($K_{\alpha\beta} \neq 0$) define the Turov invariants. If, for all $\alpha, \beta$, no such $K_{\alpha\beta}$ is allowed by symmetry, then $\mathbf{M}\equiv 0$ in all directions of $\mathbf{L}$, forbidding weak ferromagnetism and the anomalous Hall effect.

2. Application to Square-Lattice Antiferromagnetic Models

For two-sublattice, collinear systems on a square lattice in the $x$–$y$ plane, both $\mathbf{M}$ and $\mathbf{L}$ are axial vectors. Crystal symmetry and magnetic-group analysis reveals:

• Genuine antiferromagnets: The symmetry group includes a translation $T$ that exchanges the sublattices, combined with time reversal. Under $g=T\Theta$, no $M_\alpha L_\beta$ term remains invariant, implying $\mathbf{M} = 0$ everywhere and prohibiting the anomalous Hall effect.
• Ferrimagnets: The sublattices are inequivalent, so all symmetries leave each sublattice distinct. For point-group $C_{4v}$ (four-fold rotation $C_4$ and mirrors $\sigma_x$, $\sigma_y$), two independent invariants exist:
  1. $F_\mathrm{int} \supset \lambda_z M_z L_z$
  2. $$F_\mathrm{int} \supset \lambda_{x-y} M_z (L_x - L_y)$$

These allow weak ferromagnetism and anomalous Hall effect for appropriately oriented $\mathbf{L}$.

• Weak (Dzyaloshinskii) ferromagnets: The relevant symmetry combines mirror reflection in the $x$–$z$ plane ($y\to -y$) with time reversal. Only the invariant $F_\mathrm{int} \supset \lambda_y M_z L_y$ survives, so weak ferromagnetism and anomalous Hall effect are allowed only when $\mathbf{L} \parallel y$.

3. Microscopic Models and Explicit Calculations

The presence or absence of Turov's invariants is corroborated by explicit tight-binding Hamiltonians and Berry curvature calculations.

• Ferrimagnet: The Hamiltonian,

$$\hat{H}_\mathrm{ferri}(k) = \begin{pmatrix} \vec{m}\cdot\vec{\sigma} - t_k & \xi_k + i\vec{\gamma}_k\cdot\vec{\sigma} \ \xi_k - i\vec{\gamma}_k\cdot\vec{\sigma}^* & -\vec{m}\cdot\vec{\sigma} + t_k \end{pmatrix}$$

leads to a Berry curvature

$$\omega_{k;xy} = 8 t m^2 [1 - \cos^2 k_x \cos^2 k_y]\left[\gamma^2 m_z + \gamma_z\gamma(m_y - m_x)\right]$$

showing nonzero $\sigma_{xy}$ only when $M_z L_z$ or $M_z(L_x - L_y)$ is symmetry-allowed.

• Weak ferromagnet: The reduced Hamiltonian generates a Berry curvature that is nonzero only if $m_y \neq 0$ (i.e., $\mathbf{L}\parallel y$), exactly reflecting the symmetry criterion from Turov’s invariants.

The anomalous Hall conductivity is given by

$$\sigma_{xy} = \frac{e^2}{\hbar} \int_\mathrm{BZ} \frac{d^2k}{(2\pi)^2} \sum_n \Omega_{k;xy}^{(n)} f(\epsilon_k^{(n)}).$$

4. Physical Consequences: Weak Ferromagnetism and Anomalous Hall Effect

The physical content of Turov’s invariants is precise: whenever a symmetry-allowed linear $M_\alpha L_\beta$ coupling exists, the development of Néel order $\mathbf{L}$ induces a small uniform moment $\mathbf{M}$, giving rise to weak ferromagnetism. This is essential for the intrinsic anomalous Hall effect (AHE) in metallic collinear antiferromagnets, which otherwise would be forbidden by symmetry. The occurrence of the AHE is thus strictly tied to the existence and orientation-dependence of these invariants. For the square-lattice models:

• Genuine antiferromagnets: No AHE in any direction.

• Ferrimagnets: AHE present if $\mathbf{L} \parallel z$ or $\mathbf{L} \parallel [1,-1]$.
• Weak ferromagnets: AHE present if $\mathbf{L}\parallel y$ and forbidden otherwise.

5. Group-Theoretical Evaluation and Procedure

To determine Turov’s invariants for a given crystal and magnetic structure, one:

1. Identifies the full magnetic (Néel)-space-group $G$ of the ordered phase.
2. Enumerates its generators, classifying those that exchange sublattices and spatial symmetries.
3. Applies the constraint

$$\det(R_g) s_g R_g^{\alpha\gamma} R_g^{\beta\delta} K_{\gamma\delta} = K_{\alpha\beta}$$

to every generator $g$.

1. Solves for the set of $K_{\alpha\beta}$ allowed by the group, thereby identifying the nonzero $M_\alpha L_\beta$ couplings that constitute Turov’s invariants.

This group-theoretical criterion fully classifies when symmetry permits a collinear two-sublattice system to display weak ferromagnetism and, hence, the AHE.

6. Context and Connections in Magnetism

Turov’s invariants provide a rigorous symmetry-based framework for understanding weak ferromagnetism in systems lacking net macroscopic magnetization by virtue of their antiferromagnetic order, yet where certain broken symmetries and spin-orbit coupling make weak moments symmetry-allowed. These results elucidate the fundamental constraints on magnetoelectric and transport phenomena in metallic antiferromagnets, offering necessary and sufficient criteria for the appearance of effects tied to net magnetization, such as the AHE. The approach generalizes to other multicomponent magnetic orderings on lattices with complex symmetry.

7. Summary Table: Turov’s Invariants on the Square Lattice

Magnetic ordering	Symmetry-allowed Turov invariant(s)	Weak ferromagnetism/AHE condition
Genuine AF	None	Forbidden in all directions
Ferrimagnet	$M_z L_z$, $M_z(L_x-L_y)$	If $\mathbf{L}\parallel z$ or $[1,-1]$
Weak Dzyaloshinskii FM	$M_z L_y$	Only if $\mathbf{L}\parallel y$

The presence or absence of Turov’s invariants, determined by full group-theoretical analysis, provides an unambiguous symmetry diagnostic for the emergence of weak ferromagnetism and associated anomalous Hall effects in collinear metallic antiferromagnets (Golubinskii et al., 4 Nov 2025).