--- title: Altermagnetic Superconductors url: https://www.emergentmind.com/topics/altermagnetic-superconductors type: topic --- # Altermagnetic Superconductors 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 [2509.03774], [2509.03247], [2306.09413]. ## 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 [2510.18145]. 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 [2306.09413], [2509.03774]. 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 \(\mathcal T\mathcal M_{xy}\) can protect degeneracies at selected momenta while allowing spin splitting at generic \(\mathbf{k}\) [2509.03247], [2409.01008]. ## 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 \(s\)-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 \(s\)-wave leads [2509.03774], [2403.10456], [2510.18145]. 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 \(s\)-wave and spin-triplet \(p\)-wave pairings, and the altermagnetism is beneficial to the triplet channel [2305.10479]. A different intrinsic route appears when \(d\)-wave altermagnetism coexists with spin-singlet \(d\)-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 [2309.14427]. 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 [2411.02489]. ## 3. Pairing structure, gap anisotropy, and spectral signatures Microscopic proximity theory for thin altermagnetic films yields an induced pairing matrix \[ \Sigma_{\mathbf k}= \begin{pmatrix} s_{\mathbf k} & p_{\mathbf k}\\ p_{\mathbf k}^* & s_{\mathbf k} \end{pmatrix}, \] showing directly that conventional \(s\)-wave proximity can generate a mixed singlet/triplet state inside the altermagnet. In the minimal \(d\)-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 [2509.03774]. 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 \(d\)-wave gap structures there [2509.03247]. Thermodynamic calculations in AM/S bilayers reveal a distinct contrast with ferromagnetic bilayers. The superconducting transition remains second-order over the investigated \(T\)-\(H_{AM}\) plane, and the angle-averaged density of states remains spin-degenerate despite local spin splitting. The same DOS develops logarithmic peaks at \(E=\pm|\Delta+H_{AM}|\), shoulder-like features at \(E=\pm|\Delta-H_{AM}|\), and a gapless regime once \(H_{AM}>\Delta\) [2403.10456]. 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 [2404.10038]. 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 [2409.01008]. ## 4. Interfaces, Andreev states, and Josephson structures At altermagnet/superconductor interfaces, Andreev reflection depends qualitatively on Fermi-surface orientation. In the continuum model \[ H_{AM}=t_0(k_x^2+k_y^2)+\big[t_{J1}(k_y^2-k_x^2)+2t_{J2}k_xk_y\big]\sigma_z-\mu, \] the \(t_{J1}\)-oriented case behaves ferromagnet-like because a range of transverse momenta lacks matching Andreev-reflected holes, whereas the \(t_{J2}\)-oriented case remains close to a nonmagnetic interface. Strong-barrier geometries inherit the same distinction: the \(t_{J1}\) orientation supports pronounced subgap resonances and stronger disorder sensitivity, while the \(t_{J2}\) orientation does not [2305.03856]. 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, \(d\)- and \(g\)-wave altermagnets can split, curve, or flatten surface Andreev states of \(d\)-wave and chiral \(d\)- and \(p\)-wave superconductors, and the resulting subgap bands support a large spin conductance at zero net magnetization [2508.03364]. Finite-width Josephson junctions make the symmetry dependence especially explicit. In altermagnet-based short junctions, \(d_{x^2-y^2}\) order produces spin-split Andreev bound states already in a single transverse mode, whereas \(d_{xy}\) 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 \(\pi\)-periodic form, \[ U_J(\phi)\approx -E_J^{(2)}\cos(2\phi), \] which underlies a magnetic-field-free, parity-protected qubit architecture called the altermon [2510.18145]. ## 5. Topological and higher-order superconductivity Altermagnetism supplies a route to topological superconductivity without uniform magnetization. In a Rashba nanowire proximitized by an \(s\)-wave superconductor and an altermagnet, the BdG Hamiltonian \[ h(k)=\left[\epsilon(k)+\lambda_R\sin k\,\sigma^2+J_A\cos k\,\sigma^3\right]\tau^3+\Delta\,\tau^2\sigma^2 \] becomes topological when \[ \sqrt{\Delta^2+(t-\mu)^2}