Finite-momentum bound pairs of two electrons in an altermagnetic metal

TL;DR

This work solves the two-electron problem on a square lattice with $d$-wave altermagnetism and attractive $U$ and $V$ interactions, showing that a finite center-of-mass momentum $\mathbf{Q}$ minimizes the bound-state energy due to the altermagnetic spin-splitting. Using a separable channel decomposition of the nearest-neighbor interaction and an exact two-body ansatz, the authors derive a $5\times5$ secular equation whose solutions yield bound states below the two-particle continuum and reveal that the bound pairs track the lower edge $E_{2p}^{(0)}(\mathbf{Q})$. When $V<0$ is strong, multiple bound states appear with varying symmetry; the altermagnetic coupling $\lambda$ splits degeneracies and induces a pronounced mixing between spin-singlet and spin-triplet channels, evidenced in the momentum-space wavefunctions. The results provide a concrete two-body mechanism for altermagnetism-induced FFLO superconductivity and suggest that, in a many-electron setting, the order parameter could exhibit coherent mixtures of singlet and triplet components, with potential implications for unconventional superconductivity in altermagnetic materials.

Abstract

We solve the two-electron problem on a square lattice with $d$-wave altermagnetism, considering both on-site and nearest-neighbor attractive interactions. The altermagnetic spin-splitting in the single-particle dispersion naturally gives rise to a ground state of two-electron bound pairs with nonzero center-of-mass momentum. This finite-momentum pairing can be interpreted as a two-body mechanism underlying the recently proposed altermagnetism-induced Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) superconducting state. Additionally, when the nearest-neighbor attraction is strong, the resulting finite-momentum bound pairs exhibit a mixture of both spin-singlet and spin-triplet characteristics, suggesting the possibility of unconventional superconductors, where spin-singlet and spin-triplet pairings coexist.

Figures (7)

• Figure 1: The energy of the bound pairs as a function of the center-of-mass momentum $\mathbf{Q}=(Q_{x},Q_{y})$, at the on-site attractive interaction $U=-5t$ with the $d_{xy}$-wave altermagnetic coupling $\lambda=3t$: (a) a 2D contour plot in the $Q_{x}$-$Q_{y}$ plane, and (b) a 1D plot along the high-symmetry lines in the first Brillouin zone, as indicated in the inset. The grey line shows the lower bound of the two-fermion continuum $E_{2p}^{(0)}(\mathbf{Q})$. Here, we set $V=0$.
• Figure 2: The same as in Fig. \ref{['fig1']}(b), but with different $d_{xy}$-wave altermagnetic coupling strengths: $\lambda=2t$ (a) and $\lambda=4t$ (b).
• Figure 3: The energy of the bound pairs as a function of the center-of-mass momentum $\mathbf{Q}=(Q_{x},Q_{y})$ along the line cut $\Gamma$-$X$-$M$-$\Gamma$, at the nearest-neighbor attractive interaction $V=-8t$, without (upper panel) and with the $d_{xy}$-wave altermagnetic coupling $\lambda=3t$ (lower panel). Here, we take the on-site interaction $U=2t$. The strong nearest-neighbor attraction leads to four bound states, which can be well categorized as the extended $s$-wave, $p$-wave (two-fold degenerate), and $d$-wave at the $\Gamma$ point. The arrow in the lower panel highlights the minimum of the second lowest bound state along the $\Gamma$-$M$ line.
• Figure 4: (a) The energy of the bound pairs at $\mathbf{Q}=(Q_{x},Q_{y})=(0.6\pi,0.6\pi)$ as a function of the $d_{xy}$-wave altermagnetic coupling. The two $p$-wave states split with increasing altermagnetic coupling. The weight of the partial-wave components ($\tilde{M}_{\eta}$, as explicitly indicated) of the four bound pairs: the ground-state pair (a), the first-excited pair (b), the second-excited pair (c) and the third-excited pair (d). Here, we take $U=2t$ and $V=-8t$, as in Fig. \ref{['fig3']}.
• Figure 5: The wave-functions $\Phi_{\mathbf{k}}$ of the four bound pairs at $\mathbf{Q}=(Q_{x},Q_{y})=(0.6\pi,0.6\pi)$, without (upper panel) and with the $d_{xy}$-wave altermagnetic coupling $\lambda=3t$ (lower panel), ordered by increasing energy. Here, we take $U=2t$ and $V=-8t$, as in Fig. \ref{['fig3']}.