Uniqueness of the ground state in weak perturbations of non-interacting gapped quantum lattice systems

We consider a general weak perturbation of it non-interacting quantum lattice system with a non-degenerate gapped ground state. We prove that in a finite volume the dependence of the ground state on the boundary condition exponentially decays with the distance to the boundary, which implies in particular that the infinite-volume ground state is unique. Also, equivalent forms of boundary conditions for ground states of general finite quantum systems are discussed.