We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite number of the usual exponentials. This procedure leads to a set of stochastic differential equations on the group manifold, which can be further formulated in terms of the supersymmetric effective action. This action has the form of the Witten topological field theory in the continuum limit. As a consequence, we show how it can be used to obtain the exact results for a specific quantum many-body system which can be otherwise solved only by the Bethe ansatz. This represents an example of a many-body system treated exactly using the path-integral formulation. Moreover, our method can deal with time-dependent parameters, which we demonstrate explicitly.

Hybrid photonic nanostructures allow the engineering of novel interesting states of light. One recent example is topological photonic crystals where a nontrivial Berry phase of the photonic band structure gives rise to topologically protected unidirectionally-propagating (chiral) edge states of photons. Here we demonstrate that by coupling an array of emitters to the chiral photonic edge state one can create strongly correlated states of photons in a highly controllable way. These are topologically protected and have a number of remarkable universal properties: the outcome of scattering does not depend on the positions of emitters and is given only by universal numbers, the zeroes of Laguerre polynomials; two-photon correlation functions manifest a well-pronounced evenodd effect with respect to the number of emitters; and the result of scattering is robust with respect to fluctuations in the emitters' transition frequencies.

For a certain class of open quantum systems there exists a dynamical symmetry which connects different time-evolved density matrices. We show how to use this symmetry for dynamics in the Liouville space with time-dependent parameters. This allows us to introduce a concept of generalized coherent states in the Liouville space (i.e. for density matrices). Dynamics of this class of density matrices is characterized by robustness with respect to any time-dependent perturbations of the couplings. We study their dynamical context while focusing on common physical situations corresponding to compact and non-compact symmetries.