Communications in Mathematical Physics

, Volume 134, Issue 1, pp 1–27 | Cite as

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

  • Claudio Albanese


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.


Magnetic Field Neural Network Statistical Physic Complex System Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Claudio Albanese
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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