Uniform bound of the entanglement for the ground state of the one-dimensional quantum Ising model with non-homogeneous transverse field

Abstract

We consider the ground state of the one-dimensional quantum Ising model with transverse field hx in one dimension depending on the site x∈Z in a finite volume Λm ≔ {−m, −m + 1, …, m + L}. We make suitable assumptions on the regions where the field is small and prove that if the field is sufficiently large on the complementary set, then the entanglement of the interval Λ0≔0,..,L relative to its complement Λm\Λ0 is bounded uniformly in m and L. The result applies in particular to periodic transverse fields. The bound is established by means of a suitable cluster expansion.