Minimal Models for Altermagnets

• Minimal models for altermagnets are symmetry-driven Hamiltonian frameworks that distill key microscopic ingredients like odd-parity hopping, sublattice exchange, and spin–orbit coupling.
• They reveal how momentum-dependent terms and crystal symmetries produce robust spin splitting, nodal degeneracies, and nontrivial Berry curvature in electronic bands.
• These models provide tractable insights into topological responses and multipolar orders across diverse lattice architectures, guiding both theoretical research and material design.

Altermagnets are a distinct class of collinear, compensated magnetic systems that break time-reversal symmetry while exhibiting zero net magnetization. Their defining characteristic is a momentum-dependent, symmetry-enforced spin splitting of the electronic bands—typically with $d$-, $g$-, or higher-wave symmetry—that is directly protected by the magnetic space group and crystal symmetry. Minimal models for altermagnets distill these essential physical ingredients to provide a transparent, symmetry-driven Hamiltonian framework capturing the emergence, band structure, and topological responses of the phase. Such models have become central to theoretical and material proposals, offering both microscopic understanding and analytic tractability.

1. Core Hamiltonian Structure and Microscopic Ingredients

The archetype of a minimal model for altermagnets is the two-sublattice tight-binding Hamiltonian on a bipartite lattice, with Pauli matrices $\tau_i$ (sublattice) and $\sigma_i$ (spin) (Lee et al., 1 Dec 2025, Roig et al., 2024):

$$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$

• $\varepsilon_0(\mathbf{k})$: even under inversion and $\mathbf{k}\to-\mathbf{k}$, capturing spin-independent hopping and chemical potential.
• $t_x(\mathbf{k})$: inter-sublattice, IR trivial under point group.
• $t_z(\mathbf{k})$: intra-sublattice, transforms as the nontrivial "altermagnetic" 1D IR $\Gamma_N$ (e.g., $g$0-, $g$1-, $g$2-wave).
• $g$3: collinear Néel order parameter, opposite on the two sublattices.
• $g$4: spin–orbit coupling (SOC), even-inversion, odd-$g$5, dictates anisotropy, topology, and Hall response.

In real space, microscopic origins of each term for prototypical models such as MnF$g$6 involve alternating on-site exchange (Hund's coupling) and sublattice-odd kinetic hopping created by crystallographically inequivalent ligand environments (e.g., cage distortions), generating the symmetry-required $g$7 (Lee et al., 1 Dec 2025).

Symmetry and Group-Theoretical Generalization

For any crystal point group $g$8, altermagnetic order is tied to a 1D IR, $g$9, under which the two sublattices and their moments change sign under certain operations (rotations, mirrors) but remain invariant under translations and inversion. The form of $\tau_i$0 encodes the essential harmonic character:

• Square/Tetragonal ($\tau_i$1): $\tau_i$2: $\tau_i$3 (B$\tau_i$4), $\tau_i$5: $\tau_i$6 (B$\tau_i$7)
• Hexagonal ($\tau_i$8): $\tau_i$9-wave: $\sigma_i$0 (A$\sigma_i$1)
• Cubic ($\sigma_i$2): $\sigma_i$3-wave: $\sigma_i$4 (A$\sigma_i$5) (Roig et al., 2024, Sorn et al., 21 May 2025).

2. Minimal Two-Sublattice Model: Band Structure and Spin Splittings

In the canonical two-sublattice, spin-1/2 model (Lee et al., 1 Dec 2025, Roig et al., 2024, Sorn et al., 21 May 2025), the Bloch Hamiltonian is

$\sigma_i$6

• $\sigma_i$7: sublattice-mixing hopping, typically $\sigma_i$8
• $\sigma_i$9: sublattice-odd hopping, e.g., $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$0
• $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$1: on-site exchange splitting, alternates sign on A/B

Spin splitting $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$2 for bands $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$3 is

$$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$4

Maximal splitting arises when both $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$5 and $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$6 are nonzero; the nodes ($$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$7) are symmetry-protected by the underlying $$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$8 or mirror symmetry. First-principles comparison in MnF$$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\tau_0 + t_x(\mathbf{k})\,\tau_x + t_z(\mathbf{k})\,\tau_z + \tau_z(\mathbf{J}\cdot\boldsymbol{\sigma}) + \tau_y(\boldsymbol{\lambda}_{\mathbf{k}}\cdot\boldsymbol{\sigma}).$$9 shows the model accurately reproduces DFT-predicted nodal lines and $\varepsilon_0(\mathbf{k})$0-dependent spin splittings (magnitudes $\varepsilon_0(\mathbf{k})$1–$\varepsilon_0(\mathbf{k})$2 eV) (Lee et al., 1 Dec 2025).

Limiting regimes:

• $\varepsilon_0(\mathbf{k})$3: bands split mainly by $\varepsilon_0(\mathbf{k})$4; A (B) sublattice preferentially hosts majority (minority) spin.
• $\varepsilon_0(\mathbf{k})$5: small spin splitting with maximal $\varepsilon_0(\mathbf{k})$6-dependence. Nodal degeneracies are enforced for all $\varepsilon_0(\mathbf{k})$7 along symmetry lines $\varepsilon_0(\mathbf{k})$8 or $\varepsilon_0(\mathbf{k})$9.

3. Multipolar and Cluster-Based Minimal Models

Beyond the two-sublattice paradigm, minimal models can be constructed via spin clusters or multi-site unit cells, as in the Lieb lattice, 1/5-depleted square nets, or checkerboard motifs (Tani et al., 24 Jul 2025, Huo et al., 19 Dec 2025, Zhu et al., 12 Apr 2025). These frameworks yield higher-order magnetic multipole structures:

• Three-site (A: nonmagnetic; $\mathbf{k}\to-\mathbf{k}$0, $\mathbf{k}\to-\mathbf{k}$1): effective $\mathbf{k}\to-\mathbf{k}$2-octupole ($\mathbf{k}\to-\mathbf{k}$3), with zero net magnetization; leading $\mathbf{k}\to-\mathbf{k}$4 spin splitting (Tani et al., 24 Jul 2025).
• Quartet, octet, or hexad clusters: $\mathbf{k}\to-\mathbf{k}$5-, $\mathbf{k}\to-\mathbf{k}$6-, and $\mathbf{k}\to-\mathbf{k}$7-wave orderings, defined by applying $\mathbf{k}\to-\mathbf{k}$8 operations for symmetry-imposed vanishing net moment but nonzero $\mathbf{k}\to-\mathbf{k}$9-odd spin-multipole. Coexisting ferro- and antiferro- exchange within and between clusters are essential, with stability analyzed via mean-field theory and Hubbard interaction thresholds (Zhu et al., 12 Apr 2025).
• Such multipolar order uniquely appears at minimal order compatible with the crystal symmetry and magnetic space group, e.g., octupole for square ($t_x(\mathbf{k})$0), hexadecapole for FeSe monolayer (Tani et al., 24 Jul 2025).

Cluster constructions enable systematic enumeration and classification of minimal parent altermagnetic models across different lattice geometries.

4. Symmetry, Susceptibility, and Mean-Field Instability

Altermagnetic order emerges as the leading instability when the system’s susceptibility for sublattice-staggered, symmetry-odd ($t_x(\mathbf{k})$1 IR) order exceeds that of uniform ferromagnetism. The critical $t_x(\mathbf{k})$2 for a Hubbard-type interaction is lowered in the presence of non-symmorphic band degeneracies and symmetry-protected nodal planes.

Formally, the susceptibility matrix reflects the symmetry channel:

$t_x(\mathbf{k})$3

The robust stabilization of altermagnetism in wide parameter regimes—especially away from half-filling or in the presence of strongly interacting electrons on cluster lattices—has been confirmed both analytically and numerically (Roig et al., 2024, Zhu et al., 12 Apr 2025). The minimal form factor $t_x(\mathbf{k})$4 is determined by projective symmetry analysis of the crystal space group.

5. Berry Curvature, Orbital Magnetism, and Nontrivial Responses

Minimal models reveal that in altermagnets, Berry curvature, and hence anomalous Hall effects (AHE), can be generated even with negligible net ferromagnetic moment, by leveraging symmetry-allowed SOC components (Roig et al., 2024, Sorn et al., 21 May 2025). The Berry curvature in minimal two-sublattice models is linear in the relevant SOC (Roig et al., 2024):

$t_x(\mathbf{k})$5

Depending on symmetry, SOC can either contribute linearly to the AHE and FM moment, or (if quasi-symmetries are present) shift the latter to higher order in SOC strength.

Strain-induced orbital piezomagnetism is captured via similar models, where the piezomagnetic polarizability tensor is dictated by the Berry curvature hotspots associated with Dirac points or nodal lines. Linear (for $t_x(\mathbf{k})$6-wave) or nonlinear (for $t_x(\mathbf{k})$7-wave) responses are found depending on the model symmetry, with topological signatures linked to underlying Dirac quadrupoles (Bell et al., 10 Feb 2026, Radhakrishnan et al., 5 Feb 2026).

Excitation of chiral magnons via domain walls or by coherent driving in pure altermagnets, as captured in minimal tight-binding models, can give rise to magnon-driven AHE and emergent orbital magnetization even when equilibrium AHE is forbidden by symmetry (Sorn et al., 21 May 2025, Liu et al., 20 Mar 2026).

6. Generalizations: sAM and Synthetic Altermagnets

Minimal model concepts extend to generalized, fully gapped, spin-compensated "extended s-wave altermagnets" (sAMs) (Dürrnagel et al., 27 Aug 2025). Here, emergent valley-exchange symmetries enforce isotropic spin splitting between related Fermi pockets and result in unusual spin-selective transport and pair-density wave superconductivity. The minimal sAM model illustrates that zero net magnetization can be enforced by generalized ("hidden-symmetry") combinations of momentum translation and spin flip, beyond standard crystallographic symmetry.

Similarly, synthetic altermagnets comprising bilayers of anisotropic ferromagnets with engineered coupling serve as minimal models that mimic $t_x(\mathbf{k})$8-wave altermagnetic features, including the momentum-dependent splitting, Berry curvature, and spin-current phenomena (Asgharpour et al., 2024).

7. Higher-Order Topology and Unconventional Phases

Minimal models constructed on geometries such as the Lieb lattice demonstrate that altermagnetic patterns can induce higher-order topological phases, e.g., corner modes in the presence of SOC and in-plane exchange fields, by reconstructing mirror or crystalline Chern band topology (Huo et al., 19 Dec 2025). Cluster or multi-orbital minimal models also provide solvable settings for studying altermagnetic spin liquids, encompassing chiral and d- or g-wave orbital altermagnetic phases realized with Majorana representations (Sobral et al., 30 Dec 2025).

Summary Table: Key Ingredients in Prototypical Minimal Models

Ingredient	Typical Term / Parameter	Physical Role
Sublattice structure	$t_x(\mathbf{k})$9 or $t_z(\mathbf{k})$0	Encodes two-site magnetic unit cell
Exchange (Néel) field	$t_z(\mathbf{k})$1	Momentum-odd band splitting, zero net moment
Odd parity hopping	$t_z(\mathbf{k})$2	Sets $t_z(\mathbf{k})$3-, $t_z(\mathbf{k})$4-, $t_z(\mathbf{k})$5-wave form factors; symmetry channel
SOC	$t_z(\mathbf{k})$6	Induces AHE, determines moment anisotropy
Cluster/multisite term	$t_z(\mathbf{k})$7	Higher multipolar splitting, e.g. octupole or hexadecapole
Hubbard $t_z(\mathbf{k})$8	$t_z(\mathbf{k})$9	Drives spontaneous symmetry breaking

Minimal models for altermagnets thus provide a symmetry-driven, physically transparent framework that yields the full spectrum of phenomena characteristic of altermagnetic phases: momentum-dependent spin splitting, strict enforcement of zero net magnetization, topological and orbital responses, and multipolar order. The unifying structure is the interplay of symmetry-imposed sublattice (or cluster) patterning with local exchange and odd-parity hopping, capturing the essence of altermagnetism across a range of materials and theoretical constructs (Lee et al., 1 Dec 2025, Roig et al., 2024, Zhu et al., 12 Apr 2025, Huo et al., 19 Dec 2025).