Floquet engineering spin triplet states in unconventional magnets

Abstract

We consider unconventional magnets with and without spin-singlet $s$-wave superconductivity and demonstrate the emergence of spin triplet states due to light drives. In particular, we find that a high-frequency linearly polarized light drive induces a spin-triplet density in $d$-wave altermagnets which does not exist in the static regime and can directly reveal the strength of the altermagnetic field. In this high-frequency regime, we also show that linearly polarized light enables the formation of odd-frequency spin-triplet superconducting correlations possessing $d$-wave and $s$-wave parities, which can be controlled by the light drive and accessed by measuring the spin density. Moreover, for low-frequency linearly and circularly polarized light drives, we obtain that the types of superconducting correlations are broadened due to the presence of Floquet bands, enabling spin-triplet pairs in $d$- and $p$-wave unconventional magnets, which are absent in the undriven phase.

Figures (4)

• Figure 1: (a) Sketch of an UM (blue) under a time-periodic light drive $\bm{A}(t)$ (orange winkle arrow) with circular or linear polarization. (b) Spin densities along $z$ in the static regime of $d$-wave AMs and $p$-wave UMs as a function of $k_{x}$ and $k_{y}$. (c) Sketch of an UM coupled to a conventional spin-singlet $s$-wave superconductor (green) under a time-periodic light drive $\bm{A}(t)$. (d) Real part of the mixed spin-triplet pair amplitude in the static regime of $d$-wave AMs and $p$-wave UMs with conventional superconductivity. Parameters: for (b) $M=0.5$, $\mu=1$, $k_{\rm F}=1$, $\hbar^{2}/2m=1$, $\theta_{d,p}=0$ at frequency $z=0+i 10^{-3}$; for the $d$-wave AM and $p$-wave UM in (d), $\Delta=0.7\mu$ and $\Delta=0.25\mu$, respectively, at $z=0.1\Delta+i10^{-3}$.
• Figure 2: (a,b) Integrated spin density $\bar{\rho}_z^\text{eff}$ (green curve) as a function of $A_{0}$ for $M/\mu=0.5$ (a) and $M/\mu=1.2$ (b), both at $\omega/\mu=-0.9$ for a $d_{x^{2}-y^{2}}$-wave AM under LPL. The dotted blue and dashed red curves show $\bar{\rho}_{z,M}^\text{eff}$ and $\bar{\rho}_{z,\Omega}^\text{eff}$, respectively. (c) $\bar{\rho}_{z,M}^\text{eff}(\omega)$ and $\bar{\rho}_{z,\Omega}^\text{eff}(\omega)$ as a function of $\omega$ and $A_{0}$ at $M/\mu=0.5$, where black dashed lines indicate $\omega/\mu=-0.9$. (d) Spin density $\bar{\rho}_z^\text{eff}$ as a function of $M$ and $\omega$ at $eA_{0}/(\hbar k_{\rm F})=0.5$, while (e) shows line cuts at distinct values of $M$ marked in (d) and depicted by blue, gray, and orange curves for $M/\mu<1$, $M/\mu=-1$, and $M/\mu>1$, respectively. (f) Energy versus $k_{x}$ at $k_{y}=0$ for the distinct values corresponding to the distinct values of $M$ in (e). The spin-up (spin-down) bands are denoted in blue (red). Parameters: $\mu=1$, $\hbar^2/(2m)=1$ and $k_{\rm F}= 1$, $\theta_d=0$ and $\phi=0$, $eA_0/(\hbar k_{\rm F})=0.5$.
• Figure 3: (a) Real part of the integrated spin-triplet pair amplitude ${\rm Re}\bar{F}^{\rm eff}_{t}$ as a function of $\omega$ and $A_{0}$ at $M/\mu=0.2$ for a $d_{x^{2}-y^{2}}$-wave AM with superconductivity under LPL. $\omega_{\pm}^{\pm}$ and $\omega_{\Omega}^{\pm}$ depicted by dashed curves are the negative band edges of the spectrum of Eq. (\ref{['HeffSCLPL']}). (b) Line cuts of (a) as a function of $\omega$ at $eA_{0}/(\hbar k_{\rm F})=0.5$, where dashed (dotted) curve shows $\bar{F}^{\rm eff}_{t,\rm M(\Omega)}$. (c) Real part of $\bar{F}^{\rm eff}_{t}$, $\bar{F}^{\rm eff}_{t,\rm M}$ and $\bar{F}^{\rm eff}_{t, \Omega}$ of (b) as a function of $A_{0}$ at $\omega$ indicated by markers in (a). (d) ${\rm Re}\bar{F}^{\rm eff}_{t}$ as a function of $\omega$ and $M$ at $eA_{0}/(\hbar k_{\rm F})=0.5$. (e) Line cuts of (d) at $M/\mu=1.5$. (f) $|\bar{F}^{\rm eff}_{t}|^{2}$ and its singlet and triplet contributions as a function of $\omega$ at $M/\mu=0.2$ and $eA_{0}/(\hbar k_{\rm F})=0.5$. The gray curve shows $|\bar{F}^{\rm eff}_{t}|^{2}$ at $A_{0}=0$. Parameters: $\Delta=0.5$, the rest same as in Fig. \ref{['figure2']}.
• Figure 4: (a) Spin-triplet odd-Floquet odd-frequency odd-parity pair amplitude pair amplitude $\bar{F}_{\rm low- \Omega}$ integrated in $\bm{k}$ as a function of $A_{0}$ for distinct $M$ in a $d_{x^{2}-y^{2}}$-wave AM with conventional superconductivity under low-$\Omega$ LPL. (b) Same as in (a) but for an emergent spin-triplet odd-Floquet even-frequency even-parity pairing in a $p_{x}$-wave UM. The insets show the momentum dependence of $F_{\rm low- \Omega}$. Parameters: $\phi=0$, $\Delta=0.5$, and $\omega = \Omega/2$, while the rest same as in Fig. \ref{['Figure3']}.