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Pirogov-Sinai theory for relatively bounded quantum perturbations of classical lattice models

Pirogov–Sinai theory allows us to study in detail low-temperature phase transitions of the first kind in classical spin systems (or contour models) on a lattice starting from the known phase diagram of the ground states [1]. In recent years there have been a number of papers in which Pirogov–Sinai theory has been carried over to quantum models obtained by weak perturbations of the classical ones (see, for example, the survey [2]). In a recent paper [3] the technique of relatively bounded perturbations of classical lattice Hamiltonians was developed; this allows Pirogov–Sinai theory to be applied in situations where no assumptions are made about the structure of excited states of the quantum models except that they are separated from the ground states by a spectral gap. In the simplest case of two phases of Ising type, the proposed extension of Pirogov–Sinai theory looks like this. We consider a quantum spin system on the lattice Z , where the spin space H is assumed to be finite-dimensional (the general theory of quantum spin systems is set out in detail in the book [4], for example). Let H = H(μ) be a translation-invariant (formal) Hamiltonian with interaction of finite range and depending smoothly on an n-dimensional parameter μ defined in a neigbourhood of the origin. We may assume that H = P x hx, where hx is the shift by x of a certain self-adjoint non-negative operator h = h(μ) acting on a finite subset Λ0 of the lattice Z . We shall assume that for μ = 0 the operator h(0) has the following property. In the Hilbert space corresponding to a node of the lattice there are two distinguished orthogonal vectors Ω− and Ω such that the vectors ΩΛ0 ≡ N x∈Λ0 Ω − x and Ω + Λ0 ≡ N x∈Λ0 Ω + x are two zero-energy ground states of h(0), and in their orthogonal complement the spectrum of h(0) is strictly positive. In particular, it follows immediately from this condition that in any finite volume Λ ⊂ Z the vectors N x∈Λ Ω − x and N x∈Λ Ω + x are ground states with non-zero spectral gap of the restriction HΛ(0) of H(0) to this volume (properly defined as a self-adjoint operator acting in H ⊗Λ). We now assume that for small non-zero μ the energies of the ground states differ in the first order of perturbation theory in the sense that