Superconducting Altermagnets

• Superconducting altermagnets are systems where superconductivity coexists with altermagnetic order that produces momentum-dependent spin splitting and finite-momentum pairing.
• They impose unique pairing constraints that favor unconventional and mixed-parity superconducting states, influencing phenomena like nonreciprocal transport and Josephson effects.
• These systems offer potential for novel device functionalities such as superconducting diodes and quantum circuits, while intrinsic material realization remains an open challenge.

Superconducting altermagnets are systems in which superconductivity coexists with, or is induced in the presence of, altermagnetic order: a magnetic state that breaks Kramers spin degeneracy, retains zero net magnetization, and produces momentum-dependent spin splitting with symmetry-enforced sign changes in the Brillouin zone. In current usage, the term includes both intrinsic superconductivity in altermagnetic metals and proximity-induced superconductivity in altermagnet–superconductor heterostructures. Across these settings, the defining themes are constrained pairing symmetries, broad stabilization of finite-momentum pairing, nonreciprocal supercurrent transport, and symmetry-shaped nodal or topological boundary phenomena; however, no intrinsically superconducting altermagnet has yet been discovered (Chakraborty et al., 2024, Heinsdorf et al., 3 Sep 2025).

1. Definition and conceptual scope

Altermagnetism is distinct from both conventional ferromagnetism and conventional antiferromagnetism. Like a ferromagnet, it breaks Kramers spin degeneracy and yields spin-split bands. Unlike a ferromagnet, it has zero net magnetization. Like an antiferromagnet, opposite spins occupy different substructures, but the two spin sublattices are not related by a symmetry such as translation or inversion that preserves spin degeneracy, so the splitting is strongly anisotropic in momentum space (Chakraborty et al., 2024). In representative square-lattice models, the splitting has a $d$-wave form factor such as $\cos k_x-\cos k_y$, vanishes on symmetry lines $|k_x|=|k_y|$, and changes sign across the Brillouin zone (Chakraborty et al., 2024).

Within superconductivity, two broad meanings coexist. One is intrinsic coexistence, where the altermagnetic metal itself develops superconducting order. The other is proximity-induced or hybrid superconductivity, where a conventional superconductor, Josephson junction, or superconducting thin film acquires altermagnet-specific behavior through coupling to an altermagnet (Wei et al., 2023, Giil et al., 2023). This distinction is not terminological only. It separates theories of pairing instability inside an altermagnetic band structure from theories of inverse proximity, interfacial symmetry conversion, and superconducting transport through an altermagnetic region.

The literature now spans two-dimensional minimal models, three-dimensional nodal superconductors, Josephson weak links, thin-film proximity structures, and superconducting circuits. A plausible implication is that “superconducting altermagnets” has become a unifying label for a family of symmetry-controlled superconducting phenomena whose common microscopic input is momentum-dependent, magnetization-free spin splitting rather than a single canonical material realization.

2. Symmetry, band structure, and pairing constraints

A central result of the current theory is that spin-sublattice locking imposes severe constraints on admissible pairing channels. In the square-lattice $d_{x^2-y^2}$-wave altermagnet analyzed in a low-energy $(A,\uparrow)$, $(B,\downarrow)$ description, uniform on-site spin-singlet $s$-wave pairing is not allowed. The natural even-frequency spin-singlet channels are instead nearest-neighbor $d_{x^2-y^2}$ and extended $s$-wave pairing, while the simplest equal-spin-triplet nearest-neighbor $p$-wave state is not allowed and mixed-spin-triplet $\cos k_x-\cos k_y$0-wave is allowed (Chakraborty et al., 2024). The same work further shows that these restrictions persist for finite-momentum pairing and that odd-frequency pairing reopens channels forbidden in the even-frequency problem, including odd-frequency equal-spin-triplet onsite $\cos k_x-\cos k_y$1-wave structure.

These constraints reorganize the usual intuition imported from conventional superconductors. In ordinary single-orbital systems, the rule $\cos k_x-\cos k_y$2 is often sufficient for classifying pairing. In altermagnets, however, the low-energy Hilbert space already carries sublattice content tied to spin. As a result, a pairing channel can satisfy the formal spin-parity rule and still be microscopically inaccessible because the requisite same-site or same-sublattice opposite-spin states are absent (Chakraborty et al., 2024). This is why even proximity to a conventional $\cos k_x-\cos k_y$3-wave superconductor is nontrivial in an altermagnet.

A distinct but compatible route appears in thin-film proximity structures. A metallic altermagnetic film coupled to a conventional $\cos k_x-\cos k_y$4-wave superconductor acquires superconducting order only when weak interfacial Rashba spin-orbit coupling is present, and the induced state is generically nodal with a mixed singlet/triplet order parameter. In the minimal $\cos k_x-\cos k_y$5-wave altermagnet model, this nodal state has eight Dirac points per Brillouin zone, is predominantly singlet near the Brillouin-zone diagonals where the altermagnetic splitting vanishes, and predominantly equal-spin triplet away from those lines (Heinsdorf et al., 3 Sep 2025). This establishes that even conventional parent pairing is generically reshaped into altermagnet-specific mixed-parity superconductivity rather than simply transferred intact.

A related proximity problem yields a different nodal phenomenology. When a two-dimensional altermagnet with anisotropic spin splitting $\cos k_x-\cos k_y$6 is proximitized by a conventional $\cos k_x-\cos k_y$7-wave superconductor, the anisotropic spin splitting alone can drive the system into a gapless superconducting phase and generate a pair of finite-energy mirage gaps without spin-orbit interaction and without external in-plane Zeeman field (Wei et al., 2023). The resulting state exhibits spin-degenerate segmented Fermi surfaces at $\cos k_x-\cos k_y$8, spin-polarized segmented Fermi surfaces at finite energy, and coexisting spin-singlet and spin-triplet pair amplitudes with $\cos k_x-\cos k_y$9-wave character inherited from the altermagnetic splitting.

3. Intrinsic pairing channels and finite-momentum superconducting phases

Microscopic and Ginzburg–Landau studies converge on the conclusion that altermagnets strongly favor finite-momentum pairing. In a minimal two-band model of a $|k_x|=|k_y|$0-wave altermagnetic metal with nearest-neighbor spin-singlet pairing, the $|k_x|=|k_y|$1–$|k_x|=|k_y|$2 phase diagram at $|k_x|=|k_y|$3, $|k_x|=|k_y|$4, $|k_x|=|k_y|$5 contains a low-field BCS phase, a broad FF phase extending down to $|k_x|=|k_y|$6, a re-entrant field-induced BCS phase, a second finite-momentum phase denoted FF$|k_x|=|k_y|$7, and a normal phase (Chakraborty et al., 2024). The striking feature is that finite-$|k_x|=|k_y|$8 pairing is stabilized over a wide region of phase space even without an external magnetic field, in contrast to ordinary field-induced Fulde–Ferrell states in nonmagnetic systems.

A complementary weak-coupling Ginzburg–Landau analysis classifies five pair-density-wave states in a two-dimensional metallic $|k_x|=|k_y|$9-wave altermagnet: FF, FF$d_{x^2-y^2}$0, UD, BD1, and BD2. In the pure altermagnetic case, $d_{x^2-y^2}$1-wave pairing yields only inversion-preserving UD and BD2 states, whereas $d_{x^2-y^2}$2-wave pairing follows the sequence $d_{x^2-y^2}$3 as $d_{x^2-y^2}$4 increases, with approximately $d_{x^2-y^2}$5 for UD, $d_{x^2-y^2}$6 for BD2, and $d_{x^2-y^2}$7 for FF$d_{x^2-y^2}$8 (Froldi et al., 8 Oct 2025). Only FF and FF$d_{x^2-y^2}$9 break both inversion and time-reversal symmetry in that analysis.

The form of the modulated state is also nonstandard. A microscopic and GL theory of coexistence between superconductivity and altermagnetism finds that genuine multi-sublattice altermagnetic band structure stabilizes a phase-modulated Fulde–Ferrell state at zero external field, whereas conventional Zeeman-driven systems more typically favor amplitude-modulated LO order. In the altermagnetic case, the gradient coefficient $(A,\uparrow)$0 becomes negative while the quartic coefficient $(A,\uparrow)$1 remains positive near the transition, so the stable nonuniform state is the single-$(A,\uparrow)$2, phase-modulated FF state rather than LO (Sumita et al., 27 Jun 2025). This result depends sensitively on the multi-sublattice structure or an effective $(A,\uparrow)$3-type anisotropic Zeeman field; simpler anisotropic Zeeman models do not generically reproduce it.

Separate microscopic pairing mechanisms strengthen the conclusion that altermagnets can host intrinsic unconventional superconductivity even outside finite-$(A,\uparrow)$4 singlet scenarios. In a minimal three-site two-dimensional altermagnet, nondegenerate magnons generated by the same crystal-magnetic structure that produces $(A,\uparrow)$5-wave-like spin splitting mediate a dominant spin-polarized $(A,\uparrow)$6-wave superconducting instability, and the corresponding critical temperature can be significantly enhanced by tuning the chemical potential (Brekke et al., 2023). In a minimal Lieb-lattice altermagnet, a weak-coupling phonon calculation finds that the leading instability is odd in momentum and even in spin with fully spin-polarized Cooper pairs, showing that a spinless boson can favor zero-momentum unconventional superconductivity once the pairing problem is projected onto altermagnetic spin-split Fermi surfaces (Leraand et al., 12 Feb 2025).

4. Nonreciprocal transport, Josephson effects, and superconducting spin transport

The superconducting diode effect is one of the most developed consequences of finite-momentum superconductivity in altermagnets. In a $(A,\uparrow)$7-wave altermagnet with unconventional spin-singlet pairing, the diode efficiency

$(A,\uparrow)$8

vanishes in BCS and normal phases, is typically negative deep in the FF phase with reverse-diode efficiencies up to about $(A,\uparrow)$9, becomes positive near BCS boundaries with forward efficiencies up to about $(B,\downarrow)$0, and reaches perfect diode efficiency, $(B,\downarrow)$1, in part of the high-field FF$(B,\downarrow)$2 regime (Chakraborty et al., 2024). The largest response is attributed not merely to finite-$(B,\downarrow)$3 pairing itself but to competition between nearby finite-$(B,\downarrow)$4 and zero-momentum BCS states, sharpened near a field-driven topological nodal-to-nodeless transition of the spin-split normal-state Fermi surfaces.

A Ginzburg–Landau study of pair-density-wave altermagnetic states reaches the same general conclusion at smaller efficiency scales. In that framework only FF and FF$(B,\downarrow)$5 states exhibit an intrinsic diode effect, the field-free spontaneous FF$(B,\downarrow)$6 state reaches $(B,\downarrow)$7, and the largest response occurs in a field-induced $(B,\downarrow)$8-wave FF state with Rashba spin-orbit coupling and in-plane field, where $(B,\downarrow)$9 around $s$0 (Froldi et al., 8 Oct 2025). This suggests that explicit breaking of inversion and time-reversal symmetry can optimize nonreciprocal transport beyond what spontaneous symmetry breaking alone provides.

A proximity platform reaches a still different regime. A conventional $s$1-wave superconducting thin film coupled to a $s$2-wave altermagnet develops a non-collinear superconducting diode effect already in the nominal BCS state, with the critical-current anisotropy showing a fourfold $s$3 symmetry. At larger splitting the system enters an FF state, the anisotropy evolves continuously into an effectively unidirectional $s$4 pattern, the BCS-to-FF transition occurs continuously at approximately $s$5, and the FF-to-normal transition becomes first-order around $s$6 (Yang et al., 7 Jul 2025). The defining transport feature is that the optimal FF momentum locks to discrete crystal diagonals, eliminating the rotational Goldstone mode typical of isotropic FFLO models.

Josephson transport through altermagnets shows equally distinctive signatures. In a two-terminal $s$7 junction on a square lattice, altermagnets induce $s$8-$s$9 oscillations despite zero net magnetization, and both the oscillation period and decay profile differ qualitatively from ferromagnetic junctions. The response is strongly crystallographic: for $d_{x^2-y^2}$0, the first $d_{x^2-y^2}$1-$d_{x^2-y^2}$2 transition occurs at $d_{x^2-y^2}$3 for a straight junction but only at $d_{x^2-y^2}$4 for a diagonal junction (Ouassou et al., 2023). This crystallographic anisotropy is a direct consequence of the sign-changing momentum dependence of the altermagnetic spin splitting.

Hybrid structures convert these transport effects into device functionality. In AM–SC bilayers and AM–SC–AM trilayers, rotating the Néel vector controls the superconducting transition temperature $d_{x^2-y^2}$5; in the trilayer, switching between effective parallel and antiparallel altermagnetic configurations modulates $d_{x^2-y^2}$6 strongly enough to realize superconducting on/off switching and, operationally, “infinite magnetoresistance” near the transition (Giil et al., 2023). More broadly, intrinsic equal-spin superconducting altermagnets can support both spin-polarized electrical supercurrent and pure spin supercurrent. In the non-relativistic limit the two spin-resolved condensates are decoupled, allowing arbitrary spin polarization of the superflow; with spin-orbit interactions or magnetic disorder present, the spin current develops spatial oscillations but no dissipation or decay (Monkman et al., 29 Jul 2025). Quasiclassical analyses of superconductor–altermagnet hybrids further predict a controllable supercurrent-induced edge magnetization, a Cooper-pair spin-splitter effect, and spin-selective tunneling of triplet Cooper pairs (Giil et al., 2024).

5. Nodal, gapless, and topological structures

The nodal structure of superconducting altermagnets is unusually rich because it is controlled jointly by superconducting symmetry and altermagnetic spin splitting. In proximitized two-dimensional altermagnets with anisotropic spin splitting $d_{x^2-y^2}$7, the main superconducting gap closes directionally when $d_{x^2-y^2}$8, creating a gapless superconducting phase with symmetry-protected gapped sectors around the diagonals and zero-energy quasiparticles elsewhere (Wei et al., 2023). At the same time, finite-energy mirage gaps appear at

$d_{x^2-y^2}$9

with width $s$0, so the quasiparticle spectrum becomes simultaneously gapless at zero energy and gapped at finite energies.

In thin-film heterostructures with weak interfacial Rashba spin-orbit coupling, the induced superconducting state in a metallic altermagnet is generically nodal and mixed-parity, with eight Dirac points in the minimal $s$1-wave model (Heinsdorf et al., 3 Sep 2025). The nodal positions satisfy

$s$2

so they lie on the nonmagnetic Fermi surface and separate singlet-dominated sectors from triplet-dominated sectors. The same state supports a spin-current dynamo effect: a charge supercurrent imposed by a phase gradient in the substrate induces a pure spin supercurrent in the altermagnetic layer.

Three-dimensional models extend these ideas into a topological boundary setting. In $s$3- and $s$4-wave altermagnets with chiral $s$5-wave superconductivity, the interplay of nodal chiral pairing and altermagnetic symmetry yields crossed zero-energy surface flat bands on the [001] surface, Bogoliubov–Fermi surfaces in the bulk, and surface arcs on the [100] surface (Fukaya et al., 16 Oct 2025). The [001] crossed flat bands have four corners for $s$6 and $s$7 altermagnets, with the latter rotated by $s$8, while the $s$9 altermagnet produces eight corners. Their protection is formulated through particle-hole symmetry plus pseudo-magnetic mirror symmetry, which induces chiral symmetry on magnetic mirror planes and allows a one-dimensional winding number. The Bogoliubov–Fermi surfaces interrupt this protection in momentum regions where the bulk is already gapless.

Noncentrosymmetric metallic altermagnets with strong Rashba spin-orbit coupling introduce yet another modulated phase. Numerical Eilenberger analysis finds a stripe superconducting phase with multiple center-of-mass momenta at low temperatures, separated from a single-$p$0 helical phase by an instability determined from a linearized gap equation (Mukasa et al., 2 Nov 2025). The stripe phase is reentrant as a function of altermagnetic splitting $p$1, and in the large-$p$2 regime it becomes approximately LO-like with $p$3. The mechanism is specific to altermagnets: anisotropic $p$4-wave deformation of the Rashba-split Fermi surfaces allows the outer sheet to participate in negative-momentum pairing at large $p$5, producing a nonmonotonic stripe instability absent in simpler Rashba–Zeeman systems.

6. Materials, devices, and unresolved directions

The material landscape is still exploratory. Two-dimensional metallic $p$6-wave altermagnets such as KV$p$7Se$p$8O and Rb$p$9V$\cos k_x-\cos k_y$00Te$\cos k_x-\cos k_y$01O are presented as candidate platforms for finite-momentum superconductivity and superconducting diode physics (Froldi et al., 8 Oct 2025). Josephson and memory proposals emphasize metallic altermagnets such as RuO$\cos k_x-\cos k_y$02 and Mn$\cos k_x-\cos k_y$03Si$\cos k_x-\cos k_y$04, while broader proximitized and spin-transport studies also discuss MnTe, CrSb, and RuO$\cos k_x-\cos k_y$05 (Ouassou et al., 2023, Monkman et al., 29 Jul 2025). Three-dimensional topological modeling is motivated in part by Sr$\cos k_x-\cos k_y$06RuO$\cos k_x-\cos k_y$07, for which both altermagnetic order and three-dimensional chiral $\cos k_x-\cos k_y$08-wave superconductivity have been proposed (Fukaya et al., 16 Oct 2025).

Because intrinsic superconducting altermagnets have not yet been observed, heterostructure design remains central. Proposed low-lattice-mismatch platforms for inducing superconductivity in thin altermagnetic films include Al / Rb$\cos k_x-\cos k_y$09V$\cos k_x-\cos k_y$10Te$\cos k_x-\cos k_y$11O with mismatch $\cos k_x-\cos k_y$12, NbS / FeSb$\cos k_x-\cos k_y$13 with mismatch $\cos k_x-\cos k_y$14 and $\cos k_x-\cos k_y$15 along the orthorhombic axes, Nb$\cos k_x-\cos k_y$16Se$\cos k_x-\cos k_y$17 / FeBr$\cos k_x-\cos k_y$18 with mismatch $\cos k_x-\cos k_y$19, CaKFe$\cos k_x-\cos k_y$20As$\cos k_x-\cos k_y$21 / KV$\cos k_x-\cos k_y$22Se$\cos k_x-\cos k_y$23O with mismatch $\cos k_x-\cos k_y$24, and Pb / OsO$\cos k_x-\cos k_y$25 with mismatch $\cos k_x-\cos k_y$26 (Heinsdorf et al., 3 Sep 2025). These proposals are notable because they target precisely the regime in which conventional $\cos k_x-\cos k_y$27-wave order, interfacial Rashba coupling, and altermagnetic spin splitting cooperate to produce mixed-parity nodal superconductivity and spin-polarized persistent current.

Circuit applications are already being discussed at the level of microscopic Josephson modeling. When altermagnetic Josephson junctions are inserted into transmons, flux qubits, or fluxonium, the junction energy can be written as

$\cos k_x-\cos k_y$28

with the first two harmonics often dominant. In the transmon design, regimes near $\cos k_x-\cos k_y$29-$\cos k_x-\cos k_y$30 transitions and in a $\cos k_x-\cos k_y$31-state simultaneously enhance anharmonicity and suppress decoherence, although they also slow charge-drive gates; strain is proposed as a tuning knob to move the qubit out of the protected regime for fast operations and then return it to the protected point (Tjernshaugen et al., 1 Jun 2026). This is not a bulk-superconductivity result, but it shows how altermagnetic Josephson physics can reshape superconducting-circuit design.

The unresolved questions are therefore sharply defined. The first is material: whether any altermagnet can host intrinsic superconductivity rather than only proximity-induced order. The second is microscopic: how robust the present pairing classifications and phase diagrams remain in multiorbital, material-specific, and disordered settings. The third is diagnostic: which combination of angular critical-current measurements, Josephson interferometry, tunneling spectroscopy, Andreev reflection, and spin-sensitive probes will most cleanly distinguish intrinsic superconducting altermagnets from hybrid proximity structures. Taken together, the literature suggests a coherent picture: altermagnetism is not an incidental magnetic background for superconductivity, but a symmetry structure that reorganizes the superconducting problem at the levels of pairing, transport, and topology.