Lie-Schwinger block-diagonalization and gapped quantum chains: Analyticity of the ground-state energy

We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have proven that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant; see [5]. The main result of this paper is that, under the same hypotheses, the ground-state energy is analytic for values of the coupling constant belonging to a sufficiently small disk in the complex plane centred at the origin, uniformly in the length of the chain. Furthermore, assuming that the interaction potentials are invariant under translations, we prove that, in the thermodynamic limit, the energy per site is analytic for values of the coupling constant in the same disk. In our proof we use a new method introduced in [7], which is based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. Indeed, to do this we prove a rather strong result concerning complex Hamiltonians corresponding to complex values of the coupling constant.