Inhomogeneous quantum quenches in the sine-Gordon theory

We study inhomogeneous quantum quenches in the attractive regime of the sine-Gordon model. In our protocol, the system is prepared in an inhomogeneous initial state in finite volume by coupling the topological charge density operator to a Gaussian external field. After switching off the external field, the subsequent time evolution is governed by the homogeneous sine-Gordon Hamiltonian. Varying either the interaction strength of the sine-Gordon model or the amplitude of the external source field, an interesting transition is observed in the expectation value of the soliton density. This affects both the initial profile of the density and its time evolution, and can be summarised as a steep transition between behaviors reminiscent of the Klein-Gordon, and the free massive Dirac fermion theory with initial external fields of high enough magnitude. The transition in the initial state is also displayed by the classical sine-Gordon theory and hence can be understood by semi-classical considerations in terms of the presence of small amplitude field configurations and the appearance of soliton excitations, which are naturally associated with bosonic and fermionic excitations on the quantum level, respectively. Features of the quantum dynamics are also consistent with this correspondence and comparing them to the classical evolution of the density profile reveals that quantum effects become markedly pronounced during the time evolution. These results suggest a crossover between the dominance of bosonic and fermionic degrees of freedom whose precise identification is rather non-trivial. Nevertheless, their interplay is expected to influence the sine--Gordon dynamics in arbitrary inhomogeneous settings.