Driving fundamentally challenges the established equilibrium paradigms of interacting quantum many-body systems. Quantum spin systems exposed to periodic, quasiperiodic and direct-current driving can enter non-equilibrium phases of matter with no equilibrium analog. A striking example is a Floquet time crystal, a phase of matter stabilized by periodic driving, that breaks time-translation symmetry and is thus at odds with thermal equilibrium. The time crystal owes its robustness against perturbations to the interactions between the particles. It is thus a representative case that highlights the important role of interactions in the presence of driving.
This thesis employs the quantum Ising model as a platform to explore driven many-body dynamics. First, we study the dynamics of correlation functions of the periodically driven quantum Ising model. In the open chain, periodic driving stabilizes Floquet Majorana zero modes (MZMs) and Majorana π modes (MPMs) at its boundaries, implying characteristic level pairings throughout the many-body spectrum. We show that the level pairing statistics differ markedly between MZMs and MPMs in the presence of random symmetry-breaking fields, with implications for the boundary spin correlations. In the coexisting regime of MZMs and MPMs, we construct a composite boundary mode as their operator product. We analyze the resilience of the composite mode against integrability-breaking perturbations, surpassing the stability of the individual boundary modes. Next, we apply a quasiperiodic Fibonacci drive to the quantum Ising chain. The boundaries can host Majorana golden-ratio modes (MGMs) unique to the quasiperiodic setting. We map out the dynamic phase diagram which contains self-similar structures under time evolution. Returning to periodic drives, we revisit the relation of the Floquet time crystal to the spectral π pairings of the many-body Floquet operator. Our work shows that the level pairing statistics provides analytical expressions for the temporal spin correlations in Floquet time crystals.
Finally, we turn to magnetic impurities in superconductors. We treat the spins of a Yu-Shiba- Rusinov dimer fully quantum in a zero-bandwidth approximation, accounting for the complex interplay of screening, superconducting correlations, magnetic anisotropies and substrate-mediated interactions. The dimer shows rich phases and excitation spectra which underscore the role of interactions between spins under direct-current driving in a solid-state platform.
