Unitary dressing transformations and exponential decay below threshold for quantum spin systems. Parts I and II

Abstract

We consider a class of quantum spin systems defined on connected graphs of which the following HeisenbergXY-model with a variable magnetic field gives an example:

$$H_\lambda = \sum\limits_{x \in \mathbb{Z}^d } {h_x \sigma _x^{(3)} + \lambda } \sum\limits_{< x,y > \subset \mathbb{Z}^d } {(\sigma _x^{(1)} \sigma _y^{(1)} + \sigma _x^{(2)} \sigma _y^{(2)} )} .$$

We treat first the case in whichh _x =±1 for all sitesx and we introduce a unitary dressing transformation to control the spectrum for λ small. Then, we consider a situation in which |h _x | can be less than one forx in a finite setL and prove exponential decay away fromL of dressed eigenfunctions with energy below the one-quasiparticle threshold. If the ground state is separated by a finite gap from the rest of the spectrum, this result can be strengthened and one can compute a second unitary transformation that makes the ground state of compact support. Finally, a case in which the singular setL is of finite density, is considered. The main technical tools we use are decay estimates on dressed Green's functions and variational inequalities.