Emergence of chiral $p$-wave and $d$-wave states in $g$-wave altermagnets

Abstract

Altermagnets emerge as a novel platform for realizing unconventional superconductivity through their exotic momentum-dependent spin-splitting of electronic band structures. Recent experiments have uncovered a novel form of altermagnetism with distinctive $g$-wave symmetry in CrSb. However, the potential for unconventional superconductivity arising from $g$-wave altermagnetism in such systems remains largely unexplored. In this study, we discover the emergence of chiral superconducting states in three-dimensional $g$-wave altermagnetic metals. Through systematic self-consistent mean-field analysis on the extended attractive Hubbard model combined with $g$-wave altermagnetic exchange fields in a three-dimensional hexagonal lattice, as observed in CrSb, we find that the altermagnetic spin splitting of Fermi surfaces favors chiral $p$-wave states as the dominant pairing channel under strong altermagnetic fields and high electron densities, while chiral $d$-wave states become predominant under weak altermagnetic fields and intermediate electron densities. Conversely, at weak altermagnetic fields and typical electron densities, non-chiral $s$-, extended $s$-, or $f$-wave states become stabilized. We also showcase the possible experimental detection using the quasiparticle energy dispersions and the density of states to distinguish different pairing symmetries. These findings underscore the potential of $g$-wave altermagnets to host sought-after chiral and gapless superconductivity.

Figures (10)

• Figure 1: Altermagnetic spin splitting in electronic band structures. (a) Illustration of a three-dimensional hexagonal lattice (left) and its Brillouin zone (BZ; right). In the left panel, blue and red spheres represent the A and B sublattice layers, respectively. Red and blue arrows indicate magnetic moments oriented in opposite directions on these layers. In the right panel, high-symmetry points $\Gamma$, $\Gamma'$, $M_{1}$, $M_{1}'$, and $A$ are marked. (b) Electronic band structures $E_{\alpha\beta}(\bm{k})$ along high-symmetry lines. The four bands correspond to combinations of band index $\alpha=\pm$ and spin index $\beta=\pm$. In the left panel, for each $\alpha$, the spin-up ($\beta=+$) and spin-down ($\beta=-$) bands exhibit $g$-wave altermagnetic spin splitting along the high-symmetry line $-M_{1}'$–$\Gamma$–$M_{1}'$. Conversely, in the right panel, no spin splitting is observed along $-M_{1}$–$\Gamma$–$M_{1}$. (c) Spin-split Fermi surfaces derived from (left) the upper band ($\alpha=+$) and (right) the lower band ($\alpha=-$). In each panel, green lines highlight characteristic nodal planes. (d) Two-dimensional Fermi surface projections onto the $k_x$-$k_y$ plane at $k_z=\pi/2$.
• Figure 2: Form factors of superconducting states. Each panel depicts: (a) $g_{es}(\bm{k})$, (b) $|g_{d+id}(\bm{k})|$, (c) $\textrm{Re}[g_{d+id}(\bm{k})]$, (d) $\textrm{Im}[g_{d+id}(\bm{k})]$, (e) $g_{f}(\bm{k})$, (f) $|g_{p+ip}(\bm{k})|$, (g) $\textrm{Re}[g_{p+ip}(\bm{k})]$, and (h) $\textrm{Im}[g_{p+ip}(\bm{k})]$. The functions $g_{es}(\bm{k})$, $g_{d+id}(\bm{k})$, $g_{f}(\bm{k})$, and $g_{p+ip}(\bm{k})$ are the form factors associated with the $es$-wave, chiral $d$-wave, $f$-wave, and chiral $p$-wave states, respectively. In each panel, the white solid lines indicate the boundary of the first BZ. In panel (a), K1, K2, M1, M2, and M3 mark the high-symmetry points. In each panel, the $x$-axis represents the $k_x$ value, while the $y$-axis indicates the $k_y$ value.
• Figure 3: Superconducting phase diagrams. (a)–(c) Zero temperature phase diagrams as a function of $J$ and $V_1$ for three different chemical potential $\mu$ values: (a) $\mu=-2$, (b) $\mu=0$, and (c) $\mu=2$. (d)–(f) Zero temperature phase diagrams as a function of $J$ and $\mu$ for three different interaction strength $V_1$ values: (d) $V_1=1$, (e) $V_1=2$, and (f) $V_1=3$. In each panel, the annotations indicate the following: $s$ denotes the $s$-wave phase; $(s, es)$ represents a mixed phase of $s$-wave and $es$-wave pairings; $d+id$ refers to the chiral $d$-wave phase (specifically with $d+id$ pairing); $f$ signifies the $f$-wave phase; $p+ip$ indicates the chiral $p$-wave phase (with $p+ip$ pairing); and $(f, p+ip)$ denotes a mixed phase of $f$-wave and $p+ip$-wave pairings. These phase diagrams reveal the emergence of chiral $d$-wave—or the chiral $p$-wave phase ($p+ip$-wave pairing). The white region corresponds to the normal phase ($N$), where all the pairings are vanishing. In panels (c)–(f), the red boxes around $J \approx 0.6$ and $\mu \approx 2.0$ indicate the relevant parameter space for the candidate material $\mathrm{CrSb}$Reimers2024.
• Figure 4: Distribution of pairing amplitude components. Light colored markers represent three pairing amplitude components for a spin singlet state as a function of $J$: $s$-wave $(\Delta_{aa;s}^{\uparrow\downarrow})$ (yellow), $es$-wave ($\Delta_{aa;es}^{\uparrow\downarrow}$) (orange), chiral $d$-wave ($\Delta_{aa;d+id}^{\uparrow\downarrow}$) (red) pairing amplitudes on the A sublattice. Blue markers represent four pairing amplitude components for a spin triplet state: $f$-wave ($\Delta_{aa;f}^{\uparrow\uparrow}$ and $\Delta_{bb;f}^{\uparrow\uparrow}$) and chiral $p$-wave ($\Delta_{aa;p+ip}^{\uparrow\uparrow}$ and $\Delta_{bb;p+ip}^{\uparrow\uparrow}$) pairing amplitudes on the A and B sublattices. At each $J$ point, the pairing amplitude components are derived from their respective spin singlet and spin triplet energy-minimized configurations. The actual ground state is determined by selecting the configuration with the lower condensation energy. Each panel annotates the stabilized ground state phases within their respective regions separated by dashed lines. Parameter values for $\mu$ and $V_1$ are annotated in each panel.
• Figure 5: Formation of Bogoliubov Fermi surfaces (BFSs). In each panel, the colored areas represent the $\bm{k}$-space volume where BFSs are present. The $\bm{k}$-points are projected onto the $k_x$–$k_y$ plane, with the $k_z$ value indicated by the color scale. Each panel corresponds to an $(s, es)$-wave or chiral $d$-wave state, with specific parameter values annotated. The left and right subpanels in each panel represent the third and fourth lowest energy bands, respectively. The $x$-axis indicates the $k_x$ values, while the $y$-axis represents the $k_y$ values in all panels.