Light-induced Floquet spin-triplet Cooper pairs in unconventional magnets

Abstract

The recently predicted unconventional magnets offer a new ground for exploring the formation of nontrivial spin states due to their inherent nonrelativistic momentum-dependent spin splitting. In this work, we consider unconventional magnets with $d$- and $p$-wave parities, and investigate the effect of time-periodic light drives for inducing the formation of spin-triplet phases in the normal and superconducting states. In particular, we consider unconventional magnets without and with conventional superconductivity under linearly and circularly polarized light drives and treat the time-dependent problem within Floquet formalism, which naturally unveils photon processes and Floquet bands determining the emergent phenomena. We demonstrate that the interplay between unconventional magnetism and light gives rise to a non-trivial light-matter coupling which governs the emergence of Floquet spin-triplet states with and without superconductivity that are absent otherwise. We find that photon-assisted processes promote the formation of spin-triplet densities and spin-triplet Cooper pairs between different Floquet sidebands. More precisely, the Floquet sidebands offer an additional quantum number, the Floquet index, which considerably broadens the classification of superconducting correlations that lead to Floquet spin-triplet Cooper pairs as an entirely dynamical phenomenon due to the interplay between light and unconventional magnetism. Furthermore, we discuss how the number of photons is connected to the symmetry of Cooper pairs and also explore how the distinct light drives can be used to manipulate them and probe the angular symmetry of unconventional magnets. Our results therefore unveil the potential of unconventional magnets for realizing nontrivial light-induced superconducting states.

Figures (11)

• Figure 1: (a) Schematics of an unconventional magnet (gray) under a time-periodic light drive $\bm{A}(t)$ (wiggle yellow arrows). (b) An unconventional magnet in proximity to a conventional spin-singlet $s$-wave superconductor (violet) under a time-periodic light drive. (c) Illustration of circularly and linearly polarized light drives, with components $A_{x,y}$. Here, $\phi_A$ denotes the linear polarization angle. (d) Spin-split Fermi surfaces in the $k_x$–$k_y$ plane for $d$-wave and $p$-wave unconventional magnets, with $\theta_J$ representing the orientation angle measured from the $x$-axis. The blue and red colors correspond to up and down spins. (e) Sketch of the isotropic order parameter of a conventional spin-singlet $s$-wave superconductor.
• Figure 2: (a,b) Energy dispersion (a) and spin density along $z$ (b) for a $d_{x^{2}-y^{2}}$-wave altermagnet in the normal state ($\Delta=0$) at $k_{y}=0$, while in (c,d) the same is plotted but for a $p_{x}$-wave magnet. The blue and red colors in (a,c) indicate spin-up and spin-down bands, respectively, while in (b,d) the blue and red colors indicate the positive and negative values of the spin density. In (b,d), the frequency is chosen as $z=0+i10^{-3}$. (e,f) Energy dispersion and real part of the spin-triplet pair amplitude for a $d_{x^{2}-y^{2}}$-wave altermagnet with spin-singlet $s$-wave superconductivity. (g,h) The same quantities as in (e,f) but for a $p_x$-wave magnet with spin-singlet $s$-wave superconductivity. In (e,g), the superconducting energy bands formed by spin-up electrons and spin-down holes are depicted in blue, while the bands formed by spin-down electrons and spin-up holes are shown in red. The blue and red colors in (f,h) mark the positive and negative values of the real part of the spin-triplet pair amplitude; here $z=0.1\Delta+i10^{-3}$. Parameters: $B=1$, $\alpha_d=\alpha_p =0.5$, $\theta _{J}=0$, $\mu =1$; in (e-h) we also consider $\Delta =0.7\mu$ for the $d_{x^{2}-y^{2}}$-wave altermagnet and $\Delta =0.1 \mu$ for $p_{x}$-wave magnet.
• Figure 3: (a) Floquet spin density $S_{\rm F,z}$ as a function of momenta for $d_{x^{2}-y^{2}}$-wave altermagnets under CPL with $\eta=+1$. (b) The same as in (a) but for a $p_{x}$-wave magnet. (c,d) The same as in (a,b) but under LPL with $\phi_A=0$. The black dashed lines in (a) connect the spin-degenerate nodes between the $n^{\text{th}}$ and the $m^{\text{th}}$ Floquet sidebands and $\delta n=n-m$ [Eq. (\ref{['eq_spindenlines']}) and (\ref{['eq_spindenlinesP']})]. Parameters: $B=1$, $\alpha_{d,p} =0.5$, $\theta _{J}=0$, $\mu =1$, $\Omega/\mu = 1$, $k_{A}/k_F=0.5$, and $k_F=\sqrt{\mu/B}=1$; $11$ Floquet sidebands are considered with $n \in [-5,5]$.
• Figure 4: (a,c) Floquet spin density projected to the zero-photon subspace ($S^{(0)}_{z}$) for $d_{x^{2}-y^{2}}$-wave altermagnets (a) and $p_{x}$-wave magnets, both under CPL. (b,d) $S^{(0)}_{z}$ as a function of the driving amplitude $k_{A}$ for the cases shown in (a,b) at $k_{y}=0$ and $k_{x}$ indicated in (a); $k_{x}$ is given by Eq. (\ref{['kxn_d']}) for (b), while by Eq. (\ref{['kxn_p']}) for (d). Parameters as in Fig. \ref{['figure4']}.
• Figure 5: (a) Floquet BdG spectrum of a CPL-driven $d_{x^{2}-y^{2}}$-wave magnet with $s$-wave superconductivity with $\eta=+1$ and $\theta_J=0$. The centers of the superconducting gaps, defined by Eq. (\ref{['eq_ec']}), are marked by blue and red circles corresponding to $\nu = +$ and $\nu = -$, respectively. (b) Floquet BdG spectrum of a CPL-driven $p_{x}$-wave magnet with $s$-wave superconductivity. Thin gray solid (dashed) lines represent electron (hole) branches. In both panels, the driving amplitude is $ak_{A} = 0.5$ and the photon energy is $\hbar \Omega = 1$. All other magnetic and superconducting parameters are identical to those used in Fig. \ref{['figure2']}; $11$ Floquet sidebands are considered with $n \in [-5,5]$.