Altermagnetic Superconductors

• Altermagnetic superconductors are materials with momentum-dependent, sign-changing exchange fields that yield zero net magnetization and enable mixed singlet/triplet pairing.
• They feature anisotropic spin splitting and finite-momentum pairing, resulting in unconventional gap structures and potential routes to topological superconductivity.
• Engineered heterostructures and intrinsic altermagnetic metals demonstrate distinctive spectral signatures and spintronic functionalities, informing novel quantum device designs.

Altermagnetic superconductors are superconducting states or superconducting hybrids in which the essential time-reversal-breaking ingredient is the momentum-dependent, sign-changing exchange field of altermagnetism: bands are spin split, yet the net magnetization vanishes. In current usage, the term covers both superconductivity developing inside altermagnetic metals and proximity-induced superconductivity in altermagnet–superconductor heterostructures. Across these realizations, the recurring features are anisotropic spin splitting, mixed singlet/triplet pairing, finite-momentum pairing tendencies, unconventional gap structures, and access to topological superconductivity without uniform magnetization (Heinsdorf et al., 3 Sep 2025, Rasmussen et al., 3 Sep 2025, Ghorashi et al., 2023).

1. Magnetic symmetry and normal-state electronic structure

Altermagnets are collinear magnets with zero net magnetization but momentum-dependent spin splitting. In continuum $d$-wave models, the exchange field can be written as

$$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$

with pure $d_{xy}$ and $d_{x^2-y^2}$ altermagnetism realized by $t_1\neq 0,t_2=0$ and $t_2\neq 0,t_1=0$, respectively. The resulting splitting changes sign under point-group operations while averaging to zero over momentum space, so time-reversal symmetry is broken without producing macroscopic magnetization (Vosoughi-nia et al., 20 Oct 2025).

On lattices, representative altermagnetic exchange fields include $J_A(\mathbf{k})\propto(\cos k_x-\cos k_y)\sigma^3$ in 1D and 2D topological-superconductivity models and the even-in-momentum splitting relation

$$\varepsilon_{\uparrow}(\mathbf{k})-\varepsilon_{\downarrow}(\mathbf{k}) = \varepsilon_{\uparrow}(-\mathbf{k})-\varepsilon_{\downarrow}(-\mathbf{k})$$

in proximitized thin-film models. These forms distinguish altermagnets from ferromagnets, where the exchange field is essentially momentum independent, and from conventional collinear antiferromagnets, where bands often remain spin-degenerate in the primitive Brillouin zone (Ghorashi et al., 2023, Heinsdorf et al., 3 Sep 2025).

The same physics acquires additional structure in multi-sublattice formulations. Minimal two-sublattice models make the altermagnetic mechanism explicit through Hamiltonians of the form

$$\mathcal H(\mathbf{k}) = \varepsilon_{0,\mathbf{k}}\tau_0+t_{x,\mathbf{k}}\tau_x+t_{z,\mathbf{k}}\tau_z+\tau_z\mathbf N\cdot\boldsymbol\sigma,$$

where the combined effect of sublattice structure and Néel order yields strongly anisotropic spin-split Fermi surfaces. In square-lattice models, antiunitary symmetries such as $\mathcal T\mathcal C_{4z}$ and $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$0 can protect degeneracies at selected momenta while allowing spin splitting at generic $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$1 (Rasmussen et al., 3 Sep 2025, Maiani et al., 2024).

2. Routes to superconductivity in altermagnetic settings

A persistent misconception is that an altermagnetic superconductor must be an intrinsically superconducting altermagnet. Much of the literature instead studies proximity-induced realizations. A thin metallic altermagnet proximitized by a conventional $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$2-wave superconductor acquires superconducting order through interface tunneling, and this route is emphasized precisely because no intrinsically superconducting altermagnets have yet been discovered. Other basic geometries include AM/S bilayers in the thin-film quasiclassical limit and Josephson junctions in which a semiconducting weak link is proximitized by an altermagnet and by $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$3-wave leads (Heinsdorf et al., 3 Sep 2025, Chourasia et al., 2024, Vosoughi-nia et al., 20 Oct 2025).

Intrinsic superconductivity has nevertheless been modeled in altermagnetic metals. In a 2D altermagnetic metal with Rashba spin-orbit coupling and an extended attractive Hubbard interaction, self-consistent mean-field calculations favor a mixture of spin-singlet $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$4-wave and spin-triplet $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$5-wave pairings, and the altermagnetism is beneficial to the triplet channel (Zhu et al., 2023).

A different intrinsic route appears when $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$6-wave altermagnetism coexists with spin-singlet $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$7-wave superconductivity. In that setting, zero-field finite-momentum superconductivity emerges when the superconducting nodes coincide with the altermagnetic nodes, and the same model supports field-induced superconductivity from a parent zero-field normal state (Chakraborty et al., 2023).

A third route dispenses with a pre-existing altermagnet altogether. A square adatom superlattice on an unconventional superconducting substrate can stabilize an orbital-altermagnetic superconductor with loop currents and zero net orbital moment; when spin-orbit coupling is included, the Bogoliubov bands develop altermagnetic spin splitting and non-trivial spin textures in the superlattice unit cell (Pupim et al., 2024).

3. Pairing structure, gap anisotropy, and spectral signatures

Microscopic proximity theory for thin altermagnetic films yields an induced pairing matrix

$$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$8

showing directly that conventional $$h_z(\mathbf{k})=\frac{\hbar^2}{m}\left[t_1 k_x k_y+t_2(k_y^2-k_x^2)\right],$$9-wave proximity can generate a mixed singlet/triplet state inside the altermagnet. In the minimal $d_{xy}$0-wave altermagnet on a square lattice, the resulting superconductor is generically nodal, with eight Dirac nodes per Brillouin zone, and the low-energy gap is separated into singlet-dominated and triplet-dominated sectors of the Fermi surface (Heinsdorf et al., 3 Sep 2025).

Realistic multi-sublattice models show that anisotropy is not merely a consequence of choosing anisotropic pairing interactions. In nonsymmorphic altermagnets, a momentum-independent bare attraction channel produces a superconducting gap that must vanish on Brillouin-zone edges if the Fermi surface crosses them. By contrast, nearest-neighbor pairing is allowed on the Brillouin-zone edges and can favor $d_{xy}$1-wave gap structures there (Rasmussen et al., 3 Sep 2025).

Thermodynamic calculations in AM/S bilayers reveal a distinct contrast with ferromagnetic bilayers. The superconducting transition remains second-order over the investigated $d_{xy}$2-$d_{xy}$3 plane, and the angle-averaged density of states remains spin-degenerate despite local spin splitting. The same DOS develops logarithmic peaks at $d_{xy}$4, shoulder-like features at $d_{xy}$5, and a gapless regime once $d_{xy}$6 (Chourasia et al., 2024).

Inverse proximity adds further structure. Even a normal-metal/superconductor bilayer develops a minigap in the superconducting layer together with a four-peak DOS, and coupling to an altermagnet converts the induced splitting into a momentum-dependent one. In that setting, the minigap can be closed while the integrated DOS remains spin-degenerate, reflecting the same distinction between angle-resolved spin splitting and zero net magnetization (Sukhachov et al., 2024).

Local probes sharpen this picture. In altermagnetic superconductors, non-magnetic impurities can induce spin-polarized subgap states whose spatial extension reflects the magnetic order of the host. If the impurity preserves the bulk generalized time-reversal symmetries, these states form spin-degenerate doublets; if it breaks them, or if a Zeeman field parallel to the Néel vector is applied, the doublets split (Maiani et al., 2024).

4. Interfaces, Andreev states, and Josephson structures

At altermagnet/superconductor interfaces, Andreev reflection depends qualitatively on Fermi-surface orientation. In the continuum model

$d_{xy}$7

the $d_{xy}$8-oriented case behaves ferromagnet-like because a range of transverse momenta lacks matching Andreev-reflected holes, whereas the $d_{xy}$9-oriented case remains close to a nonmagnetic interface. Strong-barrier geometries inherit the same distinction: the $d_{x^2-y^2}$0 orientation supports pronounced subgap resonances and stronger disorder sensitivity, while the $d_{x^2-y^2}$1 orientation does not (Papaj, 2023).

Altermagnetic fields acting on unconventional superconductors further enrich the subgap spectrum. When the symmetry of altermagnetism aligns with that of the superconducting order parameter, bulk zero-energy flat bands can emerge and generate a zero-bias conductance peak. More generally, $d_{x^2-y^2}$2- and $d_{x^2-y^2}$3-wave altermagnets can split, curve, or flatten surface Andreev states of $d_{x^2-y^2}$4-wave and chiral $d_{x^2-y^2}$5- and $d_{x^2-y^2}$6-wave superconductors, and the resulting subgap bands support a large spin conductance at zero net magnetization (Lu et al., 5 Aug 2025).

Finite-width Josephson junctions make the symmetry dependence especially explicit. In altermagnet-based short junctions, $d_{x^2-y^2}$7 order produces spin-split Andreev bound states already in a single transverse mode, whereas $d_{x^2-y^2}$8 order preserves spin degeneracy and instead produces orbital splitting by intermode hybridization. Summing the negative BdG eigenenergies yields a Josephson potential that can be electrically tuned into an approximately $d_{x^2-y^2}$9-periodic form,

$t_1\neq 0,t_2=0$0

which underlies a magnetic-field-free, parity-protected qubit architecture called the altermon (Vosoughi-nia et al., 20 Oct 2025).

5. Topological and higher-order superconductivity

Altermagnetism supplies a route to topological superconductivity without uniform magnetization. In a Rashba nanowire proximitized by an $t_1\neq 0,t_2=0$1-wave superconductor and an altermagnet, the BdG Hamiltonian

$t_1\neq 0,t_2=0$2

becomes topological when

$t_1\neq 0,t_2=0$3

yielding Majorana zero modes at the wire ends despite vanishing net magnetization. The same work identifies 2D weak and chiral topological superconductors in altermagnet/superconductor heterostructures and higher-order phases with corner Majorana modes in 2D TI/altermagnet/superconductor systems (Ghorashi et al., 2023).

Self-consistent studies of intrinsic 2D altermagnetic metals reach a parallel conclusion. With Rashba spin-orbit coupling and extended attraction, the superconducting state is either mixed $t_1\neq 0,t_2=0$4chiral $t_1\neq 0,t_2=0$5 or mixed $t_1\neq 0,t_2=0$6helical $t_1\neq 0,t_2=0$7. When the $t_1\neq 0,t_2=0$8-wave component dominates, the chiral phase is a first-order TSC with even Chern number, while the helical phase can realize a $t_1\neq 0,t_2=0$9-enforced second-order TSC with Majorana corner modes (Zhu et al., 2023).

Proximity-induced thin-film altermagnets offer a complementary nodal route. There the mixed singlet/triplet state is generically nodal, and the Dirac points imply, by the same bulk-boundary logic used for nodal superconductors, flat band edge modes on appropriately oriented boundaries (Heinsdorf et al., 3 Sep 2025).

6. Response functions, devices, and engineered platforms

Quasiclassical theory predicts three superconducting spintronic functionalities for altermagnets: a controllable supercurrent-induced edge magnetization, a Cooper pair spin-splitter, and spin filtering of Cooper pairs. In S–F–AM–S and related structures, the altermagnet acts as a symmetry-controlled spin splitter and spin filter while retaining zero net magnetization and no macroscopic stray fields (Giil et al., 2024).

Planar AM–SC–AM junctions likewise exhibit thermoelectric response without external or stray magnetic fields. Inverse proximity induces a momentum-dependent spin splitting in the superconducting electrode, directional tunneling through the second altermagnet breaks particle-hole symmetry in transport, and the resulting Seebeck coefficient and figure of merit are comparable to those of ferromagnet–superconductor junctions, with a nonmonotonic dependence of $t_2\neq 0,t_1=0$0 on the altermagnetic splitting (Sukhachov et al., 2024).

Mn$t_2\neq 0,t_1=0$1Pt provides a concrete non-collinear platform. A theory of Mn$t_2\neq 0,t_1=0$2Pt–superconductor heterostructures on the breathing kagome lattice shows that altermagnetic spin textures can remain magnetization-free even in the presence of spin-orbit coupling and can produce a superconducting diode effect after proximity coupling. The angular dependence of the critical current, including symmetry-enforced zeros of the diode efficiency, directly probes the underlying magnetic order (Schrade et al., 6 Jan 2026).

Altermagnetic superconductivity can also be engineered directly in the condensate. A square adatom superlattice on an unconventional superconductor stabilizes an orbital-altermagnetic superconducting phase with loop-current patterns, zero net orbital moment, and a non-zero Berry curvature quadrupole moment; with spin-orbit coupling, the Bogoliubov bands acquire altermagnetic spin splitting and non-trivial spin textures in the superlattice unit cell, again with zero net spin moment (Pupim et al., 2024).

Candidate materials and heterostructures mentioned across this literature include RuO$t_2\neq 0,t_1=0$3, La$t_2\neq 0,t_1=0$4O$t_2\neq 0,t_1=0$5Mn$t_2\neq 0,t_1=0$6Se$t_2\neq 0,t_1=0$7, Mn$t_2\neq 0,t_1=0$8Pt, Rb$t_2\neq 0,t_1=0$9V$J_A(\mathbf{k})\propto(\cos k_x-\cos k_y)\sigma^3$0Te$J_A(\mathbf{k})\propto(\cos k_x-\cos k_y)\sigma^3$1O, FeSb$J_A(\mathbf{k})\propto(\cos k_x-\cos k_y)\sigma^3$2, OsO$J_A(\mathbf{k})\propto(\cos k_x-\cos k_y)\sigma^3$3, and conventional superconductors such as Al, Pb, and Nb (Vosoughi-nia et al., 20 Oct 2025, Heinsdorf et al., 3 Sep 2025).

Taken together, these works define altermagnetic superconductors as a family of intrinsic states and engineered hybrids in which a sign-changing exchange field restructures pairing, quasiparticle spectra, and transport without uniform magnetization. The recurrent outcomes are symmetry-controlled spin splitting, mixed-parity or finite-momentum pairing, unconventional gap structures, and zero-field routes to Josephson, topological, thermoelectric, and spintronic functionality.