Floquet Engineering Magnetism and Superconductivity in the Square-Lattice Hubbard Model

TL;DR

This work investigates how circularly polarized light can Floquet-engineer magnetism and superconductivity in the square-lattice Hubbard model under sub-bandwidth driving. Using a Floquet generalization of the random-phase approximation within the Floquet-Keldysh formalism, the authors quantify how drive-induced changes to fermiology and spin fluctuations modify the magnetic instability and generate unconventional superconducting orders. They find drive-tunable switching between antiferromagnetic and ferromagnetic orders, topological chiral d-wave states near AFM instabilities, and, near the Floquet Lifshitz transition, spin-triplet p+ip pairing, revealing a versatile pathway to engineer topological superconductivity out of a correlated metal. The results highlight the interplay between multi-photon processes and band deformation as a robust mechanism for controlling magnetic and superconducting orders in driven correlated materials, with potential relevance to cuprates, graphene moirés, and moiré superconductors.

Abstract

We study the interplay of magnetic order and superconductivity in the square-lattice Hubbard model under periodic driving with circularly polarized light. Formulating diagrammatic techniques based on the random-phase approximation in terms of Floquet Green's functions, allows us to analyze fluctuation-driven unconventional pairing for weak-to-moderate interactions. The interplay of repulsive interactions and photo-assisted hopping of electrons gives rise to a rich magnetic phase diagram featuring an antiferromagnetic-to-ferromagnetic crossover prior to a Floquet Lifshitz transition. Close to the antiferromagnetic transition, topological $d+id$-wave superconductivity prevails in the phase diagram for a wide range of drive parameters. At intermediate-to-high frequency driving near the Floquet Lifshitz transition, superconducting orders are tuned from spin-singlet $d$-wave to spin-triplet $p$-wave character, providing an effective protocol for Floquet engineering topological superconductivity.

Figures (5)

• Figure 1: Time averaged driven non-interacting susceptibility at $\omega=0$ and spectral function at $\omega=\mu$. Panels (a)-(b): Normalized particle-hole susceptibility in momentum space plotted over drive amplitude. (a) shows low drive frequency, (b) high drive frequency. Gray dots mark the maximum of the susceptibility for a given amplitude. The blue dashed lines mark the Floquet Lifshitz transition (FLT) and the black dashed lines mark the dynamical localization point (DLP). Panels (c), (d) show the Fermi surfaces for $\Omega=2.0$ at the transition point of the susceptibility maximum. Panels (e), (f) show the same for the high-frequency case ($\Omega=8.0$, at-bandwidth driving). Panels (g), (h) show the time-averaged spectral function at $\omega=\mu$ in momentum space over drive amplitude for both low and high frequency. Again, the dashed lines mark the Floquet-Lifshitz transition and the dynamical localization point. Panels (i), (j) show the first harmonic ($n=1$ Floquet component) of the spectral function.
• Figure 2: Interplay of magnetic order and superconductivity in the weak-to-moderate interacting regime. (a) The critical interaction strength for the onset of magnetic order under circularly polarized light as a function of $E_0/\Omega$. Empty dots show results obtained by an expansion of $\chi^{\mathrm{ph}}_{00}(\mathbf{q}=\mathbf{q}_C, \omega=0)$ up to second order $\mathcal{O}((E_0/\Omega)^2)$ (see SM for details SM). Dashed line shows the first order expansion ($\mathcal{O}(E_0/\Omega)$) for $\Omega=8.0$. (b)-(c) Tentative nonequilibrium phase diagram of the time averaged Floquet steady state at low and intermediate-frequency driving. AFM and FM order are identified by the position of the peak of the susceptibility in momentum space. AFM1 is conventional Néel-type AFM order on the square lattice, while AFM2 also entails spin anti-alignment but on the enlarged and tilted lattice given by the next-nearest neighbors (the susceptibility diverges at the $X$-point), leading to stripe ordering. The superconducting regimes $d\mathrm{SC}1$ and $d\mathrm{SC}2$ correspond to the respective AFM transitions and show different mixing of $d_{x^2-y^2}$ and $d_{xy}$-symmetry components. Triplet $p+ip$-wave superconductivity ($p\mathrm{SC}$) is found close to the FM transition at intermediate drive frequency (panel (c)). Insets: Decomposition of the respective symmetry components of the superconducting order.
• Figure 3: The linearized gap equation as a general eigenvalue problem. The hat signifies $\hat{\Delta}$ being a matrix in Keldysh indices, while $\Gamma^{(4)}\equiv\Gamma^{1'2'|12}$ is the vertex projected into the pairing channel with all possible Keldysh indices. We only consider zero-momentum Cooper pairs, thus setting $\mathbf{q} = 0$.
• Figure 4: The time-averaged effective pairing vertex in momentum space (spin-singlet) for representative drive parameters. Panel a) shows a sub-bandwidth drive protocol where the vertex is still peaked close to the $M$-point, while panels b) and c) show the vertex after the response transition and after the Floquet Lifshitz transition respectively. Panel d) shows the same for high frequency, where the peaks close to the $M$-point are enhanced. Panel e) shows the vertex close to the Floquet Lifshitz transition. Panel f) shows the vertex close to the dynamical localization point ($t_{\mathrm{eff}}/t'_{\mathrm{eff}}\ll1$), where the system is effectively an enlarged square lattice tilted by $\pi/4$, thus, the peaks sit at the $X$-point.
• Figure 5: The time-averaged superconducting singlet/triplet gaps for representative drive parameters. The columns show from left to right: 1) the absolute value of the time-averaged superconducting gap as obtained from the linearized gap equation, together with the real part of the corresponding (largest) eigenvalue, 2) the phase of the superconducting gap, 3) the Berry curvature computed with Fukui's method Fukui_2005 after plugging the gap into the Floquet BdG-Hamiltonian, 4) the Floquet bands around $\mu$ of the gapped BdG model. The rows show from top to bottom: i) the equilibrium case with topologically trivial $d_{x^2-y^2}$-symmetry ii) low frequency, low amplitude driving with chiral $d+id$-symmetry, iii) intermediate frequency, low amplitude driving with chiral $d+id$-symmetry, iv) high frequency (at bandwidth $\Omega=W$), high amplitude driving at the dynamical localization point, showing the second kind of chiral $d+id$-symmetry with $d_{xy}$ being dominant, v) high frequency, high amplitude driving (just before the Floqet Lifshitz transition) showing chiral spin triplet $p+ip$-wave superconductivity.