A Shielding Property for Thermal Equilibrium States on the Quantum Ising Model

Abstract

We show that Gibbs states of non-homogeneous transverse Ising chains satisfy a \emph{shielding} property. Namely, whatever the fields on each spin and exchange couplings between neighboring spins are, if the field in one particular site is null, the reduced states of the subchains to the right and to the left of this site are \emph{exactly} the Gibbs states of each subchain alone. Therefore, even if there is a strong exchange coupling between the extremal sites of each subchain, the Gibbs states of the each subchain behave as if there is no interaction between them. In general, if a lattice can be divided into two disconnected regions separated by an interface of sites with zero applied field, we can guarantee a similar result only if the surface contains a single site. Already for an interface with two sites we show an example where the property does not hold. When it holds, however, we show that if a perturbation of the Hamiltonian parameters is done in one side of the lattice, the other side is completely unchanged, with regard to both its equilibrium state and dynamics.