Many-body effects on superconductivity mediated by double-magnon processes in altermagnets

Altermagnets exhibit a large electron spin splitting which can be understood as a result of strong coupling between itinerant electrons and localized spins. We consider superconductivity due to electron-magnon scattering, using strong-coupling Eliashberg theory to capture many-body effects that are not covered by a weak-coupling approach. The characteristic band structure of altermagnets puts significant constraints on the spin structure of electron scattering on the Fermi surface. We emphasize the role of spin-preserving, double-magnon scattering processes compared to conventional spin-flip processes involving a single magnon. Then, we derive the Eliashberg equations for a situation where double-magnon scattering mediates spin-polarized Cooper pairs, while both double-magnon and single-magnon scatterings contribute to many-body effects. These many-body effects impact superconducting properties in a way that differs significantly from systems where conventional spin-flip processes mediate superconductivity. To highlight the role of $d$-wave magnetism on superconductivity in altermagnets, we compare our results to those found in ferromagnetic half-metals and conventional antiferromagnetic metals.

Figure 1

(a) Feynman diagram for single-magnon electron-magnon coupling (EMC). Solid lines are electrons, and dashed lines are magnons. When arising from a colinear magnetic state, the magnon flips the electron's spin. An effective electron-electron interaction is derived by connecting two such diagrams. Note that the two incoming electrons must have opposite spin. (b) Feynman diagram for double-magnon EMC, where two magnon lines connect to the same interaction vertex. In this case, the electron's spin is not flipped, permitting effective electron-electron interactions that can generate spin-polarized Cooper pairs. Here, spin up is chosen as an example. Note the free momentum $\mathit{q}$ in the double-magnon-mediated electron-electron interaction.

Figure 2

Sketches of Fermi surfaces (FSs) in metallic ferromagnets (FMs), antiferromagnets (AFMs) and altermagnets (AMs). Here, $\mathit{\Gamma }=(0,0)$ is the center of the first Brillouin zone (1BZ). We illustrate possible zero-momentum Cooper pairs by filled circles, where orange (blue) corresponds to electrons with spin up (down). In a FM metal, the electron bands are spin split. Here, we imagine a situation where only one spin component has an FS. In that case, only spin-polarized Cooper pairs are relevant, which can occur due to double-magnon processes. The crystal symmetry of AFMs prohibits spin splitting. This allows Cooper pairs with opposite spin and opposite momenta, which can be induced by single-magnon processes. The AM has a $d$-wave spin splitting of the electron bands. Both spins have FSs, but like the FM, only spin-polarized Cooper pairs carry zero net momentum.

Figure 3

Feynman diagrams illustrate the Dyson equation on the form $G=G_0+G_0\mathrm{\Sigma }G$. The self-energy has four contributions, named in the order $\mathrm{\Sigma }={\mathrm{\Sigma }}^{\text{S}1}+{\mathrm{\Sigma }}^{\text{S}2}+{\mathrm{\Sigma }}^{\text{EL}}+{\mathrm{\Sigma }}^{\text{ML}}$. The single-magnon sunset diagram ${\mathrm{\Sigma }}^{\text{S}1}$ involves a spin flip at each interaction vertex. The double-magnon diagrams ${\mathrm{\Sigma }}^{\text{S}2}, {\mathrm{\Sigma }}^{\text{EL}}$, and ${\mathrm{\Sigma }}^{\text{ML}}$ do not involve spin flips. ${\mathrm{\Sigma }}^{\text{S}2}$ comes from a sunset diagram containing two magnons (S2), while ${\mathrm{\Sigma }}^{\text{EL}}$ and ${\mathrm{\Sigma }}^{\text{ML}}$ are tadpolelike with an electron-loop (EL) and a magnon-loop (ML), respectively. Vertex corrections are illustrated in the sunset diagrams and quantified by ${\mathrm{\Gamma }}^{\text{S}1}$ and ${\mathrm{\Gamma }}^{\text{S}2}$ for the single-magnon and double-magnon sunset diagram, respectively. The interaction strengths $g^{\text{S}1}$ and $g^{\text{S}2}$ depend on the momenta and magnon modes at the vertices.

Figure 4

(a) The 2D Lieb lattice for our microscopic model of an AM. The gold sites are nonmagnetic, and the orange (blue) sites have a localized spin with spin up (down). Arrows illustrate the hopping parameters, while wavy lines illustrate the exchange interactions. (b) The magnon bands are plotted along a path in the 1BZ. The high-symmetry points in the 1BZ are illustrated in Fig. 2. (c) The electron bands show a $d$-wave spin splitting resulting in a spin-independent density of states $N_{\sigma }\left(\epsilon \right)$, which is plotted in the right panel. Parameters: $t_2/t=0.04, t_3/t=0.02, \mu /t=0, J_{\text{sd}}S/t=0.4, J_d/J_{\text{AB}}=-0.2, J_{\text{nm}}/J_{\text{AB}}=-0.4$, and $K/J_{\text{AB}}=0.002$.

Figure 5

(a) The dimensionless couplings ${\lambda }_1^{\text{S}1}\left(i{\omega }_{\nu }\right),{\lambda }_1\left(i{\omega }_{\nu }\right),$ and ${\lambda }_2\left(i{\omega }_{\nu }\right)$ as functions of bosonic Matsubara frequencies up to ${\omega }_{\nu }\approx 2M$. This includes 534 discrete Matsubara frequencies at the chosen temperature, so the curves look smooth, and markers are omitted. (b) $Z$ and (c) $\varphi /{\varphi }_{\text{max}}$ as a function of fermionic Matsubara frequencies up to ${\omega }_n\approx M$. All functions in this plot are real and even in frequency. At larger frequencies $Z\to 1$ and ${\lambda }_1^{\text{S}1},{\lambda }_1,{\lambda }_2,\varphi \to 0$. We plot a normalized version of $\varphi ={\varphi }_{\sigma }$ since its amplitude is arbitrary when solving linearized equations. Parameters: $t_2/t=0.04, t_3/t=0.02, \mu /t=0.39, J_{\text{sd}}S/t=0.4, S=3/2, J_{\text{AB}}S/t=0.01, J_d/J_{\text{AB}}=-0.2, J_{\text{nm}}=-0.4, K/J_{\text{AB}}=0.002, T/t=1.8\times {10}^{-5}\approx T_c^E/t, M/t=0.03, N={80}^2$ points in sum over $\mathit{q}$, and 54 points on the FS for each spin. ${\psi }_{\sigma }\left(\mathit{k}\right)$ is set to the normalized solution of the linearized BCS gap equation, shown in Fig. 6.

Figure 6

(b) Critical temperature from BCS theory $T_c^{\text{BCS}}$ (divided by 500, blue line, diamonds), Allen-Dynes estimate $T_c^{\text{AD}}$ (orange line, crosses), and from solving the linearized Eliashberg equations $T_c^E$ (green line, circles at calculated points). Parameters: $t_2/t=0.04, t_3/t=0.02, J_{\text{sd}}S/t=0.4, S=3/2, J_{\text{AB}}S/t=0.01, J_d/J_{\text{AB}}=-0.2, J_{\text{nm}}/J_{\text{AB}}=-0.4, K_z/J_{\text{AB}}=0.002, M/t=0.03, N={80}^2$ points in sum over $\mathit{q}$, and 54 or 58 points on the FS for each spin, depending on the chemical potential. ${\psi }_{\sigma }\left(\mathit{k}\right)$ is set to the normalized solution of the linearized BCS gap equation, shown for $\mu /t=0.35$ in (a) and $\mu /t=0.39$ in (c). The black vertical line in (b) shows the transition from nodeless $p_y$- to nodal $p_x$-wave symmetry for the spin-up gap. The magnon gap is ${\omega }_0/t\approx 0.0018$, the maximum magnon frequency is ${\omega }_c/t\approx 0.064$, and, if $t=1$ eV, then $T_c^{\text{BCS}}\approx 88$ K and $T_c^E\approx 0.21$ K at $\mu /t=0.39$.

Figure 7

The normalized solutions to the linearized BCS gap equation, ${\psi }_{\sigma }\left(\mathit{k}\right)$, are shown on the FS for (a) $\mu /t=0.381$ and (b) $\mu /t=0.382$. The transition from nodeless $p_y$- to nodal $p_x$-wave symmetry for the spin-up gap occurs without any dramatic change in the shape of the FS. Here, there are 110 points on the FS for each spin and otherwise the parameters are the same as in Fig. 6.

Figure 8

(a) The Allen-Dynes estimate $T_c^{\text{AD}}$ of the critical temperature, (b) ${\omega }_0$ and ${\omega }_{\text{log}}$, and (c) the dimensionless couplings at zero frequency, all plotted as a function of the easy-axis anisotropy $K$. Parameters: $t_2/t=0.04, t_3/t=0.02, J_{\text{sd}}S/t=0.4, \mu /t=0.39, S=3/2, J_{\text{AB}}S/t=0.01, J_d/J_{\text{AB}}=-0.2$, and $J_{\text{nm}}/J_{\text{AB}}=-0.4$. ${\psi }_{\sigma }\left(\mathit{k}\right)$ is set to the normalized solution of the linearized BCS gap equation. To get accurate results, we used 110 points on the FS and $N={150}^2$ points in the sum over $\mathit{q}$ at the lowest magnon gap.

Figure 9

(a) The electron bands in a metallic FM for a case where only spin up crosses the FS. The bands are plotted along $k_x$, with $k_y=0$, giving an approximately circular FS around the $\mathit{\Gamma }$ point. (b) A normalized $p_x$-wave gap for spin up, ${\mathrm{\Delta }}_{\mathit{k},\uparrow }$, shown on the FS. With parameters chosen as $t=1$ eV, $S=1, \mu /t=-3.8, J_{\text{sd}}S/t=0.4, JS/t=0.005, J_x/J=0.9999$, and $J_y/J=0.3$, we get ${\omega }_c/t\approx 0.064, {\omega }_0/t\approx 0.0003, \lambda \approx 0.138$, and $T_c^{\text{BCS}}\approx 0.6$ K.