Superconducting States and Intertwined Orders in Metallic Altermagnets

Abstract

Altermagnets are a newly identified class of magnets with nodal spin-split band structures, providing a fertile platform for studying unconventional superconductivity and intertwined orders. Here we investigate multicomponent superconductivity and fluctuation-induced intertwined orders in an interacting $d$-wave metallic altermagnet that is invariant under a combination of a fourfold rotation $C_4$ and time-reversal symmetry $T$. Within mean-field theory, the superconducting ground-state manifold is described in terms of two equal-spin two-component $p$-wave gap functions $(Δ_A^x,Δ_B^y)$ and $(Δ_A^y,Δ_B^x)$, where $A$ and $B$ refer to the two spin-polarized Fermi surfaces related by $C_4T$ symmetry. Because these two sets of gap functions condense at different temperatures, a rich phase diagram with multiple superconducting phase transitions emerges. Distinct fluctuations of sub-leading normal-state instabilities that compete with altermagnetism lift the degeneracy of the multicomponent pairing state in different ways. While nematic fluctuations enhance competition between distinct superconducting components and stabilize nematic superconducting phases, spin current-loop fluctuations promote coexistence and select a pair of chiral states. Our results uncover the pairing structure and elucidate how intertwined sub-leading fluctuations shape superconducting order in altermagnetic metals, suggesting a route toward realizing nematic and topological superconductivity.

Figures (16)

• Figure 1: (a) Lieb lattice model of the $d$-wave altermagnet antonenko2024. Blue and red dots denote sublattices 1 and 2, respectively, while black squares indicate the crystalline environment. $t_1$, $t_{2a}$, and $t_{2b}$ represent the nearest- and next-nearest-neighbor hoppings, with corresponding interactions $V_1$, $V_{2a}$, and $V_{2b}$. (b) Fermi surface and (c) band structure for $t_1=0.1$, $t_{2a}=1.0$, $t_{2b}=0.5$, and $\mu=-2.1$. The labels $A$ and $B$ denote the bands (eigenstates), and up (down) arrows indicate spin-up (spin-down) states.
• Figure 2: Superconducting susceptibility in the (a) singlet and (b) triplet channels as a function of pair momentum $Q$. In the singlet channel, $\chi_{\text{SC}}(Q)$ exhibits only a finite-momentum peak, reflecting the lack of perfect nesting. In contrast, the triplet $p$-wave susceptibility diverges at $Q=0$, identifying equal-spin $p$-wave pairing as the leading instability. See text for details.
• Figure 3: Typical superconducting phase diagram as a function of the dimensionless temperature $T/t_2$ and the nematic susceptibility $\lambda^2 \chi_{\text{nem}}$ for $\phi = 0.05$, where $\lambda$ denotes the average of $\lambda_1$ and $\lambda_2$. Black solid lines indicate continuous transitions, while the white dashed line marks a first-order transition. For clarity we focus on specific a range of values of the nematic susceptibility.
• Figure 4: Schematic pairing structures corresponding to superconducting phases I–V in Fig. \ref{['nem']}, shown separately for the $A$ and $B$ bands. Each band hosts a $(p_x, p_y)$ order parameter, with the black contour indicating the underlying Fermi surface. Light, faded regions indicate that the corresponding component is absent in that phase. Phases IV and V satisfy $|\Delta_A^x|=|\Delta_B^y|$ and $|\Delta_A^y|=|\Delta_B^x|$. In contrast, phases I-III are characterized by $|\Delta_A^x|\neq|\Delta_B^y|$ and $|\Delta_A^y|\neq|\Delta_B^x|$, reflecting the imbalance between competing superconducting components induced by strong nematic fluctuations.
• Figure 5: Superconducting phase diagrams including spin current-loop fluctuations for $\phi=0.05$, where $\gamma$ denotes the average of $\gamma_1$ and $\gamma_2$. (a) Same parameter set as in Fig. \ref{['nem']}: $\mu=-2.1$, $V_2=2.25$. (b) Alternative parameter set: $\mu=-2.8$, $V_2=3.75$. In (a) [(b)], spin current-loop fluctuations suppress (enhance) the superconducting transition temperature $T_c$. At low temperatures, all four superconducting components coexist; upon increasing $T$, the $(\Delta_A^x,\Delta_B^y)$ components first vanish.