Quasi-state Decompositions for Quantum Spin Sytems

I discuss the concept of quasi-state decompositions for ground states and equilibrium states of quantum spin systems. Some recent results on the ground states of a class of one-dimensional quantum spin models are summarized and new work in progress is presented. I also outline some challenging open problems and conjectures. INTRODUCTION The aim of this contribution is threefold. First I would like to review the notion of quasi-state decomposition as it was developed and used in (Aizenman and Nachtergaele, 1993a). I then show how interesting quasi-state decompositions for the ground states of a class quantum spin models can be obtained using a Poisson integral representation of the Gibbs kernel e . Secondly I will try to give a short review of the results we have recently obtained using quasi-state decompostions. These results mainly concern a class of one-dimensional quantum spin Hamiltonians introduced by Affleck (Affleck, 1985). The third and last part is devoted to a discussion of work in progress and some open problems in the subject as well as some stimulating conjectures. 2. QUASI-STATE DECOMPOSITIONS AND POISSON INTEGRAL REPRESENTATIONS A quantum system is determined by a C*-algebra of observables A and a one-parameter group of automorphisms of A representing the dynamics. Here we are mainly interested in quantum spin systems defined on a finite lattice Γ for which A ≡ AΓ = ⊗ x∈ΓA{x} with A{x} ∼= M(d,C) (the complex d× d matrices), for all x ∈ Γ. Often Γ is a subset of an infinite lattice (e.g. 6 6 ) and one is then interested in properties of the system as Γ tends to the full lattice (the thermodynamic limit). The dynamics is usually defined in terms of a Hamiltionian H = H ∈ A and we will give several examples of interesting Hamiltonians below. A state of the system is a normalized positive linear functional of A. A class of physically interesting states are the equilibrium states: for any β ≥ 0 one defines a state ωβ by ωβ(A) = TrAe Tr e−βH (2.1) for all A ∈ A. In this paper by ground state we always mean a state which is a limiting point of (2.1) as β → ∞, possibly under specified boundary conditions. In most examples discussed here (2.1) will converge to a unique limit for finite systems and the thermodynamic limit will be taken afterwards. It has been remarked many times that analyzing the structure of equilibrium states and ground states of a quantum system poses extra difficulties as compared to the situation in classical equilibrium statistical mechanics where the states are probability measures. Sure enough these probability measures can be highly non-trivial as well, but the fact that there is an underlying configuration space which can be visualized in a concrete way helps a lot in understanding their behaviour. The aim of this section is to explain how at least for some quantum spin models one can obtain a pictorial representation of the ground states and equilibrium states which resembles somewhat the Copyright c © 1993 by the author. Faithful reproduction of this article by any means is permitted for non-commercial purposes. * Partially supported by NSF Grants No. PHY92–14654 and PHY90–19433 A02. 1 situation for classical systems. More precisely we will derive a decomposition of ωβ of the form ωβ(A) = ∫