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U,T Each metastable free energy fj , j ∈ Q, defines a tangent functional αj : for all β,u,T +ηK ∂ fj |η=0 . Notice that item (c) ensures boundedness K ∈ Br , we set αj (K) = ∂η 2 of the tangent functional. We show now that these tangent functionals are linearly independent, and that any other tangent functional is a linear combination of these ones. We examine the manifold where q phases coexist; without loss of generality, we can choose u˜ ∈ MQ with Q = {1, . . , q}. 12) with k1 , . . , kq−1 being q − 1 different indices.

The approach was however different and involved studying the Gibbs states, which is more intricate and does not easily extend to the quantum case. It is simpler to look at tangent functionals, and then to use existing results on their equivalence with DLR or KMS states. Notice that the Pirogov–Sinai theory also provides various extra information, such as the fact that the limit of U (q) , as T → 0 and β → ∞, is equal to U (q) . Also, the extremal equilibrium states can be shown to be exponentially clustering.

1. Spiral order in Z2 embedding of FA into FB : An operator K ∈ FA corresponds to the operator K ⊗ 1HB\A in FB . In the following we denote by K both operators. 2) A Zν (the limit being taken through a sequence of increasing subsets of Zν , where increasing refers to the (spiral) ordering defined above). The algebras FA contain the observable algebras OA which have the same embedding properties as the field algebras and, moreover, satisfy the following commutativity condition: If A ∩ B = ∅, then for any K ∈ FA , L ∈ OB we have [K, L] = 0.