Altermagnetic Second-Order Topological Insulators

• Altermagnetic second-order topological insulators are higher-order phases defined by anisotropic, momentum-dependent spin splitting that gaps first-order boundary modes to create in-gap corner or hinge states.
• Theoretical models, including proximity-induced, first-principles, and non-Hermitian studies, demonstrate how altermagnetism acts as a directional mass-engineering principle distinguishing adjacent edges or surfaces.
• Symmetry protection via rotation, inversion, and time-reversal constraints ensures quantized fractional boundary charges and stability of spin-selective excitations in both 2D and 3D realizations.

Searching arXiv for papers on altermagnetic second-order topological insulators and adjacent altermagnetic higher-order topology. Altermagnetic second-order topological insulators are higher-order topological phases in which altermagnetic order—characterized by zero net magnetization together with momentum-dependent spin splitting—gaps first-order boundary modes and leaves lower-dimensional boundary states such as corner modes in two dimensions or hinge modes in three dimensions. Across recent model, heterostructure, first-principles, and non-Hermitian studies, the recurrent mechanism is that altermagnetism acts as a symmetry-selective mass term: it preserves a bulk insulating gap while changing the topology of edges or surfaces, thereby converting helical edge or surface states into gapped boundaries that bind corner or hinge excitations (Li et al., 2024, Li et al., 2024, Ji et al., 30 Dec 2025).

1. Core physical mechanism

The basic construction begins from a first-order topological insulator and introduces altermagnetic order as an anisotropic, momentum-dependent perturbation. In a 2D topological-insulator heterostructure, the altermagnetic term is written as

$$H_{\rm AM}(k)=2J_0(\cos k_x-\cos k_y)\,\hat{\mathbf s}\cdot \hat{\mathbf n},$$

and the boundary Dirac mass depends on edge orientation $\alpha$ and Néel-vector direction $(\theta,\varphi)$ through

$$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$

Adjacent edges can therefore acquire opposite mass signs, and the corner states arise as Jackiw–Rebbi-type zero modes at mass-domain walls (Li et al., 2024).

An analogous edge-gapping mechanism appears in the Kane–Mele setting with a $d$-wave altermagnetic term. There the pristine $\mathbb{Z}_2$ topological insulator has gapless helical edge states, while altermagnetism breaks $\mathcal T$ without immediately closing the bulk gap, gaps the edge states, and drives the system into a second-order topological-insulator phase with corner-localized in-gap states at nanoflakes (Li et al., 2024). The same logic is realized in a non-Hermitian 2D topological insulator proximitized by an altermagnet, where finite altermagnetic strength $J$ opens a gap in the helical edge states and produces four in-gap corner states, marking a transition from a first-order to a second-order phase (Ji et al., 30 Dec 2025).

In three dimensions, proximity to $d$-wave altermagnets plays the same role for surface Dirac fermions. A 3D topological insulator coupled to $d_{x^2-y^2}$- and $\alpha$0-type altermagnetic exchange fields develops surface masses whose relative signs determine whether the system is a hybrid-order phase or a pure second-order topological insulator. Hinge modes then appear where two gapped surfaces meet and the induced masses differ in sign (Subhadarshini et al., 3 Dec 2025).

Taken together, these constructions indicate that altermagnetism is not merely an additional symmetry-breaking field. A plausible implication is that it serves as a directional mass-engineering principle: because the altermagnetic splitting is anisotropic in momentum space, it can distinguish adjacent edges or surfaces in a way that conventional uniform exchange fields do not.

2. Symmetry protection and topological diagnostics

The altermagnetic SOTI literature does not rely on a single universal invariant. Instead, different symmetry settings admit different but compatible bulk or boundary diagnostics.

For 2D altermagnets with spin-corner locking, the decisive protecting symmetry is $\alpha$1, not mirror symmetry alone. In that setting the fractional corner charge is diagnosed by

$\alpha$2

and, in the spin-resolved analysis discussed for the altermagnetic model, each spin channel yields

$\alpha$3

When uniaxial strain breaks $\alpha$4 but preserves $\alpha$5, the second-order topology survives and becomes corner-polarized (Yang et al., 15 Oct 2025).

In chiral altermagnetic three-dimensional systems with $\alpha$6 symmetry, the topology is expressed through rotation eigenvalues. The rotation-based data are packaged as

$\alpha$7

with quantized fractional corner charge

$\alpha$8

For the $\alpha$9 plane of K[Co(HCOO)$(\theta,\varphi)$0], each spin channel carries a fractionally quantized corner charge $(\theta,\varphi)$1, and this is the bulk indicator for 1D hinge states in the 3D crystal (Xie et al., 18 Aug 2025).

A more general model-Hamiltonian framework uses the spin Chern number,

$(\theta,\varphi)$2

with $(\theta,\varphi)$3 or $(\theta,\varphi)$4 symmetry relating the spin blocks. In 3D, the decisive data are the SCNs on the $(\theta,\varphi)$5 and $(\theta,\varphi)$6 planes; their equality or mismatch distinguishes weak- and strong-TI-like phases, while certain weak-TI-like regimes exhibit hinge or corner localization in finite geometry (Gonzalez-Hernandez et al., 31 Jul 2025).

For 3D altermagnetic heterostructures, the first-order component is diagnosed by a dipolar winding number

$(\theta,\varphi)$7

whereas second-order topology is captured by quadrupolar winding numbers such as

$(\theta,\varphi)$8

The phases labelled SOTI$(\theta,\varphi)$9 and SOTI$$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$0 are distinguished by whether $$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$1 or $$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$2 (Subhadarshini et al., 3 Dec 2025).

Non-Hermitian altermagnetic SOTIs require an additional diagnostic. Under cylindrical geometry, the relevant quantity is the spectral winding number

$$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$3

which predicts whether edge states skin-localize under full open boundary conditions (Ji et al., 30 Dec 2025). For bosonic honeycomb altermagnets, higher-order topology is instead diagnosed by parity eigenvalues, bosonic Berry connection, and Wilson-loop/Wannier-center flow, with the symmetric Wannier spectrum signaling the absence of ordinary first-order topology and the presence of a second-order topological magnon insulator (Guo et al., 30 Jul 2025).

3. Two-dimensional electronic realizations and corner-localized phases

In 2D electronic systems, altermagnetism has been used in three distinct ways: proximity engineering of a parent topological insulator, intrinsic modification of a topological lattice model, and first-principles identification of intrinsic altermagnetic materials.

The heterostructure proposal based on a MnF$$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$4/Bi/MnF$$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$5 sandwich is the clearest proximity-engineered example. The 2D topological-insulator layer is buckled square-lattice bismuthene, described as having a large gap of about $$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$6 eV. First-principles calculations show that proximity to MnF$$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$7 induces a $$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$8-wave spin splitting in Bi of about $$M(\alpha,\theta,\varphi)\sim J_0\sin\theta\cos(2\alpha)\cos(\varphi-\alpha).$$9 meV, gaps the helical edge states by about $d$0 meV for in-plane altermagnetism, and produces two in-gap corner states in a finite square sample. Rotating the Néel vector moves these corner states between different corners, while the states remain robust provided the mass-domain-wall criterion is satisfied (Li et al., 2024).

The Kane–Mele route is conceptually complementary. There the altermagnetic term effectively behaves as $d$1, and the resulting phenomenology depends strongly on the Néel-vector orientation. For in-plane altermagnetism, the system becomes a second-order topological insulator with two in-gap corner states localized at the obtuse angles of a rhombic or flake geometry; for out-of-plane altermagnetism, the corner states appear at the acute angles. With increasing altermagnetic strength and intrinsic Rashba spin-orbit coupling, the same model undergoes sequential transitions into QAHE phases with several reported Chern numbers, including $d$2 and $d$3 (Li et al., 2024).

A third line of work develops intrinsic 2D altermagnetic SOTIs with spin-selective corners. In a square-lattice two-sublattice model, the altermagnetic symmetry constraint

$d$4

encodes anisotropic spin splitting and supports a conventional SOTI phase with four corner-localized in-gap states and fractional corner charge $d$5. When uniaxial strain breaks $d$6 but leaves $d$7 intact, the phase becomes a corner-polarized second-order topological insulator (CPSOTI), in which the corner states are spin-polarized and energetically separated. The reported spin-resolved corner-state splitting reaches about $d$8 meV in the effective model and about $d$9 meV in the CrO calculations. First-principles analysis identifies monolayer CrO and Cr$\mathbb{Z}_2$0Se$\mathbb{Z}_2$1O as candidates; the CrO phase diagram additionally contains an altermagnetic Weyl semimetal region, a spin-corner-locked SOTI along $\mathbb{Z}_2$2, and CPSOTI and QAHI phases when $\mathbb{Z}_2$3 (Yang et al., 15 Oct 2025).

These 2D results collectively establish several non-equivalent meanings of “altermagnetic second-order topology”: movable corner states induced by proximity, corner states emerging from a topological-insulator parent under $\mathbb{Z}_2$4-wave altermagnetic perturbation, and intrinsically spin-resolved corner modes in 2D altermagnetic crystals.

4. Three-dimensional and chiral hinge phases

In three dimensions, the second-order boundary manifestation is typically the hinge state rather than the corner state. Two strands are especially prominent: chiral altermagnetic materials and altermagnet/topological-insulator heterostructures.

K[Co(HCOO)$\mathbb{Z}_2$5] has been identified as a chiral altermagnetic second-order topological insulator in the chiral space group $\mathbb{Z}_2$6 (No. 182). It combines a chiral crystal structure with left- and right-handed enantiomers, altermagnetism with $\mathbb{Z}_2$7-wave spin-split bands, and second-order topology with 1D hinge states on hexagonal nanotubes. The hinge modes form an alternating triangular pattern on three of the six hinges, with spin-up and spin-down channels occupying complementary hinge sets. Chirality inversion switches the hinge-state spin polarization, and the two enantiomers exhibit the same magnitude but opposite sign of $\mathbb{Z}_2$8; the optical Hall conductivity, Kerr rotation, and Faraday rotation likewise reverse sign between enantiomers, with reported maxima of approximately $\mathbb{Z}_2$9 for Kerr rotation and $\mathcal T$0 deg/cm for Faraday rotation (Xie et al., 18 Aug 2025).

A heterostructure route realizes 3D altermagnetic SOTIs by coupling a 3D topological insulator to $\mathcal T$1-type and $\mathcal T$2-type altermagnetic exchange fields. With only $\mathcal T$3, the system is a hybrid-order topological phase: the $\mathcal T$4 surface remains gapless, the $\mathcal T$5 and $\mathcal T$6 surfaces are gapped with opposite masses, and numerical rod geometry shows eight hinge modes total, four propagating along $\mathcal T$7 and four along $\mathcal T$8. When $\mathcal T$9 is added, the previously gapless surface also gaps out, and two distinct SOTI phases emerge. For $J$0, SOTI$J$1 has hinge modes propagating along $J$2 and $J$3 and is characterized by $J$4; for $J$5, SOTI$J$6 has hinge modes along $J$7 and $J$8 and is characterized by $J$9. The two phases are further distinguished by two-terminal differential conductance, which is quantized at $d$0 for the hinge contribution in the relevant hybrid-order regime and vanishes in the chosen $d$1-transport geometry for SOTI$d$2 (Subhadarshini et al., 3 Dec 2025).

A more abstract model-Hamiltonian perspective reaches similar conclusions. Tight-binding altermagnetic topological insulators protected by $d$3 or $d$4 support surface states in strong-TI regimes and hinge or corner localization in weak-TI-like regimes, with the boundary structure controlled by magnetic symmetry and local moments (Gonzalez-Hernandez et al., 31 Jul 2025). This suggests that three-dimensional altermagnetic SOTIs need not be restricted to one crystallographic setting; the hinge phase can arise whenever rotation-time-reversal symmetry and the altermagnetic spin splitting cooperate to gap surfaces in a sign-alternating pattern.

5. Non-Hermitian and bosonic extensions

The notion of altermagnetic second-order topology is not confined to Hermitian fermionic electrons. Two recent generalizations extend it to non-Hermitian boundary physics and to magnonic BdG systems.

In the non-Hermitian setting, a 2D non-Hermitian topological insulator proximitized by an altermagnet realizes second-order topology together with skin effects. The Hamiltonian combines a time-reversal-invariant topological-insulator sector, a nonreciprocal hopping term,

$d$5

and an altermagnetic term

$d$6

For $d$7, the system hosts gapless helical edge states; finite $d$8 gaps them and yields four in-gap corner states. With nonreciprocity, bulk states exhibit the non-Hermitian skin effect, while the edge and corner states undergo a hybrid skin-topological effect. The edge-state spectral winding under cylindrical geometry determines which corner is favored. For $d$9, $d_{x^2-y^2}$0 and both edge and corner states accumulate at the lower-left corner; for $d_{x^2-y^2}$1, the winding reverses to $d_{x^2-y^2}$2 and the favored corner becomes the upper-right; at the critical point $d_{x^2-y^2}$3, the real edge dispersion becomes flat, $d_{x^2-y^2}$4, and the skin-topological effect is suppressed (Ji et al., 30 Dec 2025).

In bosonic systems, a honeycomb altermagnet with bond dimerization and, in 3D, AA-type stacking with antiferromagnetic interlayer coupling realizes higher-order topological magnons. The magnon bands are chirality split, with chirality index

$d_{x^2-y^2}$5

and the bosonic topology is formulated in a para-unitary BdG framework. In 2D, open boundaries yield in-gap corner magnons; in 3D, the stacked system supports hinge magnons protected by mirror symmetry together with anisotropic surface states. The phase is diagnosed through inversion parity, bosonic Berry connection, and Wannier-center flow, and the corner or hinge modes remain robust against small perturbations and local defects as long as the relevant spatial symmetries are maintained (Guo et al., 30 Jul 2025).

These extensions indicate that altermagnetic higher-order topology is a structural idea rather than a narrowly electronic one: the essential ingredient is the anisotropic, symmetry-controlled mass pattern generated by altermagnetic order.

6. Scope, adjacent phases, and recurrent misconceptions

A persistent source of confusion is that not every altermagnetic topological phase with boundary states is second-order. Several closely related systems are explicitly not SOTIs.

Cr$d_{x^2-y^2}$6BAl, a 2D MBene monolayer, is presented as an altermagnetic topological crystalline insulator with Dirac edge states away from $d_{x^2-y^2}$7. Its [100] edge-projected band structure shows Dirac dispersions for the Cr-B terminated edge and their absence for the Cr-Al terminated edge, but the work does not claim second-order topology and does not report corner or hinge states (Sattigeri et al., 12 Jun 2025).

Likewise, a 3DTI thin film interfaced with altermagnetic order realizes helical edge states without time-reversal symmetry through the interplay of the Wilson mass and an anisotropic altermagnetic mass. The resulting boundary modes are ordinary first-order edge states associated with opposite high-symmetry-point Chern numbers such as $d_{x^2-y^2}$8 and $d_{x^2-y^2}$9, not hinge or corner states (Wan et al., 4 Sep 2025).

Even within model-Hamiltonian studies that do display corner or hinge localization, the classification is not always “pure HOTI” throughout the full nontrivial phase diagram. In the $\alpha$00 and $\alpha$01 models, the principal bulk invariant is the spin Chern number, and the higher-order character is most explicit in particular finite-geometry regimes, especially the weak-TI-like 3D phases (Gonzalez-Hernandez et al., 31 Jul 2025).

The resulting taxonomy is therefore more precise than the shorthand phrase might suggest. “Altermagnetic second-order topological insulator” refers specifically to phases with gapped edges or surfaces and corner or hinge boundary modes created, stabilized, or directionally controlled by altermagnetic order. By contrast, altermagnetic topological crystalline insulators and altermagnet-induced first-order helical edge phases remain adjacent but distinct members of the broader landscape of altermagnetic topological matter.