Finite-momentum superconductivity with singlet-triplet mixing in an altermagnetic metal: A pairing instability analysis

Abstract

We analyze the pairing instability of an altermagnetic metal on a square lattice driven by an attractive nearest-neighbor interaction. This interaction enables multiple pairing channels, including even-parity extended $s$-wave and $d$-wave states, as well as two odd-parity $p$-wave channels. We verify that altermagnetic spin-splitting in the single-particle dispersion gives rise to finite-momentum pairing between electrons with unlike spins, in agreement with earlier predictions. Quite unexpectedly, this pairing typically emerges across multiple channels with mixed parity. Consequently, the resulting finite-momentum Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) superconducting phase is expected to exhibit a multi-component order parameter featuring singlet-triplet mixing. We examine several forms of altermagnetism, specifically $d_{xy}$-wave and $d_{x^{2}-y^{2}}$-wave altermagnetic couplings, and present the corresponding phase diagrams. Additionally, we investigate the triplet pairing between electrons with identical spins, and find that it consistently occurs at zero center-of-mass momentum and is unfavorable at weak altermagnetic coupling and low electron filling. The influence of on-site attractive interactions on mixed-parity pairing is also explored.

Figures (15)

• Figure 1: Diagrammatic representation of the two-particle vertex function $\Gamma(\mathbf{k},\mathbf{k}';\mathbf{Q},\omega)$, within the standard ladder approximation, for a general non-separable inter-particle interaction $V_{\uparrow\downarrow}(\mathbf{k},\mathbf{k}')$.
• Figure 2: The largest eigenvalue of the inverse vertex function matrix, $\gamma_{1}(\mathbf{Q}=0)$, is plotted as a function of the lattice filling factor $\nu$, in the case without altermagnetism (i.e., $\lambda=0$). The curves represent the results for pairing between electrons with unlike spins, while the red crosses indicate the triplet pairing results for spin-up electrons. Here and in the following figures, we adopt the interaction parameters $U=-0.01t$ and $V=V_{\sigma}=-1.5t$, and temperature $T=0.01t$, unless otherwise stated.
• Figure 3: (a) The curves represent the four eigenvalues of the inverse vertex function matrix, $\gamma_{n}(\mathbf{Q})$, at the filling factor $\nu=0.2$, as a function of the center-of-mass momentum $\mathbf{Q}$ when it lies along the $x$-axis. The crosses show $\gamma_{1}(\mathbf{Q})$ for $\mathbf{Q}$ directed along the diagonal ($Q_{x}=Q_{y}$). In both orientations, introducing a $d_{xy}$-wave altermagnetic coupling with strength $\lambda=0.5t$ causes $\gamma_{1}(\mathbf{Q})$ to peak at a nonzero $\mathbf{Q}$. (b) The eigenvectors associated with $\gamma_{1}(\mathbf{Q})$ are shown for $\mathbf{Q}$ along the $x$-axis.
• Figure 4: The results are identical to that presented in Fig. \ref{['fig3']}, with the exception that they are calculated at a larger lattice filling factor of $\nu=0.7$.
• Figure 5: The peak value of the dominant eigenvalue of the inverse vertex function matrix, $\gamma_{1}(\mathbf{Q}_{\textrm{max}})$, is shown as a function of the $d_{xy}$-wave altermagnetic coupling strength $\lambda$ for two lattice filling factors: $\nu=0.2$ (black solid line) and $\nu=0.7$ (red dashed line). The arrows highlight the critical altermagnetic coupling at which $\gamma_{1}(\mathbf{Q}_{\textrm{max}})$ vanishes. The inset shows the magnitude of the $\mathbf{Q}$ vector at which this peak occurs.