We consider M(v):, models of Euclidean quantum field theory, P(v) = X',t, akyk and define for them translation-invariant ground states. It appears that the number of them is not more than two provided Xim,2 is large.
1. PHASE DIAGRAMS AND GROUND STATES FOR CLASSICAL LATTICE SPIN SYSTEMS
Let us consider a d-dimensional lattice spin system. The spin variables q(s), s E Zd will take values in a finite set @. A configuration of such a system is a mapping y: Zd + @. The space of all configurations will be denoted by &, while the restriction of a configuration q~ E J?' to an arbitrary set V will be denoted by 9(V). A configuration v E JI is called a periodic configuration if it is invariant under some subgroup 2 C Zd of finite index. The set of all periodic configurations with arbitrary subgroups 2 will be denoted by &!(per). By a Hamiltonian with finite range of interaction of the spin system we shall mean the formal expression where W,(S) is a cube with center in the point s E Zd containing (2R + I)d points of the lattice. The Hamiltonian is called periodic (resp., translation invariant) if US = US+, for all t belonging to some subgroup Z C Zd of finite index (resp., for all t E Zd). In what follows we shall deal only with periodic Hamiltonians. Using the Hamiltonian (1.1) one can define in the usual way conditional probabilities p(q~(V)/y(Z~ -V)) for any finite set V and introduce Gibbs states corresponding to the Hamiltonian H (see [I, 21). The main problem in the theory of phase transitions consists of giving a description of the whole set of limit Gibbs distributions corresponding to H. This general problem is very difficult and one can hardly hope that it will be solved in the near future. Often one includes a Hamiltonian in a family of Hamiltonians depending on several parameters and tries to investigate the problem for some domains in the space of parameters using a version of the theory of perturbations.
The problem was considered in this spirit in [3], for Hamiltonians /3H, 6 being sufficiently large. As has been mentioned many times in the physical literature, the structure of the set of Gibbs states in this case depends on the structure of the set of 393