My main interests are the different phenomena that distinguish quantum many-body systems from their few-body counterparts, specifically in various out-of-equilibrium settings. Unfortunately, accurate descriptions of many-body systems are generally out of reach due to the exponential scaling of the Hilbert space, such that various approximate methods need to be developed and used in the description of these systems. Some of the topics I have looked into are:
Integrability. Integrable systems present a special class of quantum many-body systems exhibiting a large amount of (mathematical) symmetries. On the physical level, the resulting extensive amount of conservation laws leads to non-ergodic behaviour, whereas on the mathematical level exact wave functions can be efficiently obtained as Bethe ansatz states. These can then also be used as a starting point in the perturbative description of non-integrable systems.
Floquet dynamics. When systems are being driven far from equilibrium, it is possible to engineer new physics not present in static systems, all encoded in the so-called Floquet Hamiltonian -- e.g. by tuning the driving frequency of the system, it is possible to prepare the system in targeted resonances of quantum many-body states.
Counterdiabatic driving. As an additional application of Floquet driving, the introduction of high-frequency oscillations can be used to engineer and speed up slow (adiabatic) quantum processes through counterdiabatic driving, where diabatic losses are counteracted using an effective counterdiabatic field. This additional field then has some interesting connections with quantum control and quantum geometry.