Insulating Altermagnets: Spin-Split Insulators

• Insulating altermagnets are magnetic insulators with zero net magnetization that exhibit momentum-dependent spin splitting via symmetry-governed operations.
• They enable pure magnonic and electronic spin currents and large anomalous and thermal Hall effects through unconventional band topology.
• Their unique spin-resolved tunneling and filtering capabilities support high-performance spintronic and multiferroic device architectures.

Insulating altermagnets constitute a distinct class of magnetic insulators that combine zero net magnetization with spontaneous, nonrelativistic, symmetry-governed spin splitting in their electronic or magnon band structures. Diverging fundamentally from both conventional Néel antiferromagnets (which are spin degenerate owing to translation or inversion symmetry) and traditional ferromagnets (which possess a net magnetization), insulating altermagnets exhibit uncompensated, momentum-dependent spin polarization without invoking spin–orbit coupling. This unique band topology enables phenomena such as pure magnonic or electronic spin currents, large anomalous and thermal Hall effects, cross-coupling with lattice distortions, strong optical response, and robust functionalities for next-generation spintronic, caloritronic, and multiferroic devices.

1. Symmetry Principles and Electronic Structure

The key defining attribute of insulating altermagnets is the realization of momentum-dependent spin splitting in the absence of net magnetization. Unlike Néel-type antiferromagnets with ordering wave vector ${\bm q} \neq 0$, which preserve a combined time-reversal/translation operation and yield spin degenerate bands, altermagnets feature staggered spin ordering with ${\bm q} = 0$ across crystallographically nonequivalent but symmetry-related sublattices (e.g., in perovskite, rutile, and oxychalcogenide structures) (Naka et al., 17 Nov 2024, Wei et al., 18 Oct 2024, Goswami, 3 Sep 2025). This leads to the breaking of macroscopic time-reversal symmetry, while local (microscopic) compensation of magnetic moment is retained.

Symmetry operations such as a combination of twofold rotation and half-lattice translation (C$_2$T) or antiferroelectricity-mediated mappings preclude a simple translation operation from connecting up and down spin sublattices. Consequently, the Hamiltonian acquires spin splitting terms that change sign under discrete lattice operations—for example, a $d_{x^2-y^2}$ form factor given by $S_{\bm k}^{(o)} \propto k_x^2 - k_y^2$ in a dx$^2$–y$^2$-type altermagnet (Wei et al., 18 Oct 2024).

In the minimal tight-binding description, the single-particle energy spectrum can display: $$E_{{\bm k}, \sigma} = \epsilon_0({\bm k}) + \sigma\, \Delta({\bm k})$$ where $\epsilon_0({\bm k})$ is symmetric in momentum and $\Delta({\bm k})$ encodes a momentum-dependent splitting (e.g., $\Delta({\bm k}) \propto \cos k_x - \cos k_y$ or $k_x^2 - k_y^2$). The combined action of lattice symmetry breaking, collinear compensated order, and crystal distortions leads to d-, g-, or i-wave nodal structures in the spin-resolved bands (Jungwirth et al., 16 Sep 2024).

2. Spin-Split Magnon Bands and Caloritronic Responses

In insulating altermagnets, the symmetry-breaking exchange interactions not only split electronic bands but also engineer nondegenerate magnon branches (spin wave excitations) (Cui et al., 2023, Sarkar et al., 6 Jul 2025). For a Hamiltonian with anisotropic intra-sublattice exchange couplings $J_2 \neq J_2'$, the magnon spectrum reads: $$\begin{aligned} \omega_{\bm k}^\alpha &= \frac{S}{\hbar} \big[ (J_2 - J_2')(\gamma_1 - \gamma_2) + \frac{4 J_1 \gamma_3 \Delta}{\gamma_e} \big], \ \omega_{\bm k}^\beta &= \frac{S}{\hbar} \big[ (J_2' - J_2)(\gamma_1 - \gamma_2) + \frac{4 J_1 \gamma_3 \Delta}{\gamma_e} \big], \end{aligned}$$ yielding a frequency splitting: $$\Delta \omega = \frac{2S}{\hbar} |J_2 - J_2'|\,|\gamma_1 - \gamma_2|,$$ which locks opposite spin angular momenta (or chiralities) to the respective modes (Cui et al., 2023). The resulting magnonic structure directly supports both longitudinal and transverse spin currents when subjected to thermal gradients (spin Seebeck and Nernst effects), even in the absence of Berry curvature or spin–orbit coupling: $[j_m^z] = [\sigma_{mn}](-\partial_n T),$ where $\sigma_{mn}$ encodes the magnonic spin conductivity tensor, with off-diagonal components activated by anisotropic magnon velocities (Sarkar et al., 6 Jul 2025).

Theoretical analyses indicate that the spin Seebeck conductivity in altermagnetic insulators is as high as those in antiferromagnets requiring external fields to lift mode degeneracy, while spin Nernst conductivities are up to two orders of magnitude higher than in monolayers relying on relativistic Berry curvature (Cui et al., 2023, Sarkar et al., 6 Jul 2025). The pure transverse magnon spin current can exert spin-splitter torques suitable for magnetization switching in adjacent ferromagnetic layers (with torque fields of order mT) (Sarkar et al., 6 Jul 2025).

3. Spin-Filtering, Tunneling, and Magnetoresistance

The momentum-resolved and symmetry-protected spin splitting in insulating altermagnets enables efficient spin filtering and high-performance tunneling magnetoresistance (TMR) in electronic junctions (Samanta et al., 30 Aug 2024, Chi et al., 2023). In spin-filter magnetic tunnel junctions (SF-MTJs) where the barrier is an altermagnetic insulator (AMI) of rutile-type MF$_2$ (M = Fe, Co, Ni), the spin-dependent decay rates of evanescent states $K_+(k_\perp)$ and $K_1(k_\perp)$ (extracted from the complex band structure) yield a strong $k_\perp$-dependent spin polarization: $$p(k_\perp) = \tanh\left([K_+(k_\perp) - K_1(k_\perp)] d\right),$$ where $d$ is the barrier thickness. For parallel and antiparallel Néel vector configurations in a double-barrier setup, this gives rise to a TMR ratio,

$\text{TMR} = \frac{T_P - T_{AP}}{T_{AP}},$

with predicted values in the range $150$–$170\%$ for CoF$_2$ or NiF$_2$ SF-MTJs when the Fermi energy aligns near the valence band maximum (Samanta et al., 30 Aug 2024).

In RuO$_2$-based altermagnetic tunnel junctions (ATMTJs), first-principles calculations show ultrahigh TMR ($\sim6100\%$) for transport along the [110] axis, with near-zero TMR along [001] due to the symmetry of spin-resolved conduction channels (Chi et al., 2023). This behavior permits current polarization and readout of the Néel vector orientation, robust operation against stray fields, and facilitates dense device integration for spintronic memory.

4. Nontrivial Hall and Optical Phenomena

Altermagnetic symmetry further enables spontaneous thermal Hall effects in insulating magnets without requiring external fields or net magnetization (Hoyer et al., 8 May 2024). In rutile-structure-based models (MnF$_2$, CoF$_2$, NiF$_2$), the Dzyaloshinskii–Moriya interaction (DMI) provides the necessary time-reversal-asymmetric ingredient, resulting in finite Berry curvature in both magnon bands and a nonvanishing thermal Hall conductivity: $$\kappa_H = -\frac{k_B^2 T}{\hbar V} \sum_{\bm k} \Omega_{\bm k} \left[c_2(\rho(E_{\bm k,\alpha})) + c_2(\rho(E_{\bm k,\beta}))\right],$$ where $\Omega_{\bm k}$ is the Berry curvature and $c_2(x)$ is a statistical weight function. The sign and magnitude of $\vec\kappa_H$ are tunable by the Néel vector orientation or by applying strain, breaking further symmetries to activate previously forbidden components (Hoyer et al., 8 May 2024). A complementary spin Nernst response also emerges.

In the optical regime, light-induced magnetization in insulating altermagnets (e.g., CoF$_2$) displays canted, polarization-sensitive orbital and spin responses, expandable in tensor series over the Néel vector direction: $$\delta \mathcal{O}_i = \varphi_{ijk}^{(3a)} E_j E_k^* + \varphi_{ijkl}^{(4p)} E_j E_k^* N_l + \cdots.$$ Including spin–orbit coupling enhances the effect by over an order of magnitude and introduces a strong polarization anisotropy, allowing for optical switching and probing of the altermagnetic order (Adamantopoulos et al., 15 Mar 2024).

5. Disorder, Correlation, and Topological Responses

Insulating behavior in altermagnets commonly originates from strong electronic correlations and structural distortions (e.g., GdFeO$_3$-type rotations in perovskite ABX$_3$ compounds), which promote collinear AFM ordering on non-equivalent sublattices and anisotropic hopping (Naka et al., 17 Nov 2024). The resulting multiband Hubbard models display a sequence Metal $\rightarrow$ AFM Metal $\rightarrow$ AFM Insulator as $U$ or lattice distortion is increased, with the insulating phase preserving a $d$-wave spin-splitting texture.

Disorder introduces an additional dimension to the physics: Two-dimensional altermagnets can display a disorder-driven Kosterlitz–Thouless-type transition from a marginal metallic state (with robust spin anisotropy) to an insulator (where spin splitting is suppressed) (Li et al., 14 Jul 2025). The localization length diverges exponentially near the transition, and spin-resolved spectral features fade, consistent with experimental ARPES and TMR observations in real samples.

Topologically nontrivial phases also appear: Model Hamiltonians for insulating altermagnets including DMI, exchange, and d-wave/g-wave terms demonstrate quantum anomalous Hall regimes with nonzero Chern number, even in zero net magnetization systems (Goswami, 3 Sep 2025). Non-Hermitian extensions (introducing imaginary potentials to simulate dissipation/gain) yield exceptional points and nontrivial effects in the Berry curvature and quantum geometric tensor.

6. Multiferroicity, Magnetoelectric Coupling, and Electrical Control

A major avenue for functionalizing insulating altermagnets lies in electrically tunable magnetoelectric multiferroics. In antiferroelectric altermagnets (AFEAM), the coexistence of antiferroelectricity and altermagnetic order allows the toggling between spin-split (altermagnetic) and spin-degenerate (standard AFM) insulating states by the application of weak electric fields (Duan et al., 8 Oct 2024). The universal design principle is symmetry-based: Antiferroelectric ordering enforces C$_2$T symmetry (enabling spin splitting, $P_S \neq 0$), whereas ferroelectric alignment restores pure translation symmetry (forbidding spin splitting).

Additionally, spin-driven exchange striction mechanisms in low-Z altermagnetic multiferroics (LiMnO$_2$, NaMnO$_2$, strained RuF$_4$) generate robust polarizations ($\sim1$–$1.8~\mu$C/cm$^2$), overtaking SOC-based multiferroics by 1–2 orders of magnitude (Cao et al., 29 Dec 2024). The magnetoelectric coupling in these systems is correspondingly strong, and the phase diagrams reveal regimes where electric or magnetic fields enable selective switching between altermagnetic and Kramers-degenerate antiferromagnetic insulating phases.

7. Material Platforms and Prospects

A growing number of insulating altermagnet candidates have been theoretically and experimentally identified in diverse families:

Material Class	Representative Compounds	Remarks
Rutile-type fluorides	MnF$_2$, CoF$_2$, NiF$_2$	Prototypical insulating altermagnets, thermal/rutile Hall effects, spin filtering, optical response
Layered oxychalcogenides	La$_2$O$_3$Mn$_2$Se$_2$	d$_{x^2-y^2}$-wave altermagnetism, robust correlation, tunable spin splitting (Wei et al., 18 Oct 2024)
Chromium oxytellurides	Cr$_2$Te$_2$O, Cr$_2$Se$_2$O	Efficient pure magnon spin transport (Cui et al., 2023)
Supercell altermagnets	MnSe$_2$, RbCoBr$_3$, BaMnO$_3$	Energy-efficient control of order parameter orientation (Jaeschke-Ubiergo et al., 2023)
Perovskite oxides	BiCrO$_3$, LaTiO$_3$, LaFeO$_3$	Multiferroic functionalities, strain engineering (Naka et al., 17 Nov 2024, Duan et al., 8 Oct 2024)
Van der Waals materials	CuMoP$_2$S$_6$, CuWP$_2$S$_6$	2D AFE-altermagnets, electrical tunability (Duan et al., 8 Oct 2024)

Recent innovations include light-induced dynamical engineering of odd-parity altermagnetism (p-wave order) using circularly polarized fields to drive topological transitions to Chern insulator and Weyl semimetal phases (Liu et al., 25 Aug 2025), as well as the use of multi-terminal Coulomb drag devices that exploit the unique anisotropic spin splitting of insulating altermagnets for device probing and functionality (Lin et al., 18 Dec 2024).

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Insulating altermagnets thus offer a robust and highly tunable platform for exploiting pure spin transport, high-performance filtering, Hall responses, ultrafast magnetoelectric switching, and topologically protected functionalities without the drawbacks of net moment or reliance on heavy elements and strong SOC. Their realization and exploitation is at the forefront of spintronics, magnonics, and multiferroics, offering both fertile ground for fundamental paper and routes toward energy-efficient device architectures.