Fulde-Ferrell-Larkin-Ovchinnikov States and Topological Bogoliubov Fermi Surfaces in Altermagnets: an Analytical Study

Abstract

We present an analytical study of the ground-state phase diagram for dilute two-dimensional spin-1/2 Fermi gases exhibiting $d$-wave altermagnetic spin splitting under $s$-wave pairing. Within the Bogoliubov-de Gennes mean-field framework, four distinct phases are identified: a Bardeen-Schrieffer-Cooper-type superfluid, a normal metallic phase, a nodal superfluid with topological Bogoliubov Fermi surfaces (TBFSs), and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states with finite center-of-mass momentum. Among these, the FFLO states and TBFSs exemplify two unconventional forms of superconductivity. Considering the simplicity of this model, with only one band, zero net magnetization, and $s$-wave paring, the emergence of both unconventional phases underscores the pivotal role of altermagnetic spin splitting in enabling exotic pairing phenomena. This analytical study not only offers a valuable benchmark for future numerical simulations, but also provides a concrete experimental roadmap for realizing FFLO states and TBFSs in altermagnets.

Figures (7)

• Figure 1: (a) Zero temperature phase diagram at fixed chemical potential $\mu$ (normalized to the two-body binding energy $\varepsilon_{B}$) versus non-relativistic spin splitting $t_{\textrm{AM}}$. The solid black/dashed black/solid red curve giving the phase boundary is determined by Eq. (\ref{['eq: t_am_c-mu']})/Eq. (\ref{['eq: upper-t_AM-0-mu']})/Eq. (\ref{['eq: t_AM-FFLO-mu']}), respectively. (b) Zero temperature phase diagram at fixed total particle number $N$ as a function of the binding energy $\varepsilon_{B}$ normalized to the Fermi energy $\varepsilon_{F}$. The solid black/solid red curve is given by Eq. (\ref{['eq: t_am_c-N']})/Eq. (\ref{['eq: t_AM-FFLO-N']}), respectively. The black curves in both figures show the Pauli-limit, upper altermagnetic coupling strength, where a SF with zero center-of-mass momentum $\mathbf{q}=0$ becomes unstable towards a normal state, when the possibility of an inhomogeneous FFLO is not considered. In the weak-coupling regime (i.e., for large $\mu/\varepsilon_{B}$ or small $\varepsilon_{B}/\varepsilon_{F}$), the solid black curve in (a) and (b) provides an excellent estimate for the phase boundary separating a TBFS phase and a FFLO state at fixed $\mu$, and a BCS SF phase and a FFLO state at fixed $N$, respectively.
• Figure 2: (a) Schematic $d_{xy}$-wave spin splitting obtained by plotting $\xi_{\bf{k}}+s(\sigma)J_{\bf{k}}=0$. Here $\mu=1$. $t_{\textrm{AM}}=0$ gives the dashed circle, and $t_{\textrm{AM}}=0.5$ gives the red and blue ellipses, corresponding to spin-up and spin-down channels, respectively. The semi-major and semi-minor of each ellipse is denoted as $a$ and $b$. (b) Contour plot of $E_{\bf{k}\pm}=0$ (see Eq. (\ref{['eq: BdG-spectrum']}) for definition) at normal phase, here $\mu=1$, $t_{\textrm{AM}}=0.5$, and $\Delta=0$. The dash line corresponds to $\sum_{\eta}E_{\bf{k} \eta}=0$. Cyan and blue colored regions correspond to negative $E_{\bf{k} +}$ and $E_{\bf{k} -}$. (c) Plot of $E_{\bf{k}}$ (see Eq. (\ref{['eq: BdG-spectrum-min']}) for definition) at different $t_{\textrm{AM}}$ values, here $\mu=1,\Delta=0.2$. When $t_{AM}$ is small, $E_{\bf{k}\pm}$ is always positive, indicating a BCS SF phase. When $t_{AM}$ is large, $E_{\bf{k}\pm}$ will vanishing, meaning the emergence of Bogoliubov Fermi surfaces. (d) Plot of $\bar{t}_{\textrm{AM}}$ (see Eq. (\ref{['eq: upper-t_AM-0']}) for definition) as a function of $\Delta/\mu$. (e)-(f) Anisotropic Bogoliubov quasiparticle energy $E_{\bf{k} +}$ and $E_{\bf{k} -}$ at BCS SF phase. The white solid lines represent the Bogoliubov Fermi surfaces $E_{\bf{k} \pm} = 0$, which separate the whole $\bf{k}$ domain into two regions labelled by $\bf{k}_{\pm}$. Here $\mu=1$, $\Delta=0.2$, and $t_{\textrm{AM}}=0.5$, which corresponds to the red dot in (d).
• Figure 3: Plot of $S(t_{\textrm{AM}},Q,0)/\mathcal{A}$ (see Eq. (\ref{['eq: SQ']}) for definition) at different $t_{\textrm{AM}}$ values, here $\varepsilon_{B}=0.05$ and $\mu=1$.
• Figure 4: (a) Illustration of Fermi surface nesting for $\mathbf{q}$ along (1, 1) direction; (b) Bogoliubov Fermi surfaces for $\mathbf{q}=(q_{c}/\sqrt{2},q_{c}/\sqrt{2})$, here $q_c = a - b$. (c) Fermi surface nesting for $\mathbf{q}$ along (1, 0) direction. (d) Bogoliubov Fermi surfaces for $\mathbf{q}=(q_{c},0)$, here $q_c$ is given by Eq. (\ref{['eq: q_c-10']}). $\Delta=0$, $\mu=1$, and $t_{AM}=0.5$ are chosen for plotting Bogoliubov Fermi surfaces in (b) and (d).
• Figure 5: (a)-(b) Sign distribution of $w_{+}^{2}-4$ and $w_{-}^{2}-4$ (see Eq. (\ref{['eq: wpm-discriminant']}) for definition) on the ($k, Q$) domain. The two curves $Q_{c1}(k)$ and $Q_{c1}(k)$ are given by Eq. (\ref{['eq: critical-Q']}). (c) When $w_{+}^{2}-4>0$ and $w_{-}^{2}-4>0$, $z_{1}$ and $z_{3}$ are within the unit circle while $z_{2}$ and $z_{4}$ are outside the unit circle. At $w_{+}^{2}-4=0$, $z_{1}$ and $z_{2}$ coincide with each other at $z=1$. After that, $w_{+}^{2}-4<0$ and $w_{-}^{2}-4>0$, we have (d) where $z_{1}$ and $z_{2}$ are on the unit circle, while $z_{3}$ ($z_{4}$) is inside (outside) the unit circle. When $w_{-}^{2}-4=0$, $z_{3}$ and $z_{4}$ coincide with each other at $z=-1$. After that, $w_{+}^{2}-4<0$ and $w_{-}^{2}-4<0$, we have (e) where all four poles are on the unit circle. In (d)-(e), small arcs (blue and red solid line) are added to the unit circle to form a closed contour (solid line) to remove the poles on the unit circle.