OAI.Heisenberg.spontaneous_magnetization

The theorem states that, for integers d ≥ 3 and ℓ ≥ 1, two things hold for the quantum Heisenberg model on the lattice ℤ^d, in which each site carries a spin with ℓ+1 levels (magnetic quantum numbers k − ℓ/2 for k = 0,…,ℓ, so spin ℓ/2) and the quasi-local algebra is the norm closure of the star algebra generated by the single-site matrix units acting on ℓ² of finitely supported configurations. The Hamiltonian on a finite set Λ is the ferromagnetic nearest-neighbour sum −Σ (S^x_x S^x_y + S^y_x S^y_y + S^z_x S^z_y) over unordered pairs of lattice sites at ℓ¹-distance one inside Λ, and the finite-volume dynamics conjugates an observable A by exp(itH) over boxes [−n,n]^d. First, the dynamics converges: for every real t and every quasi-local A, the box dynamics tends in norm, as n → ∞, to the infinite-volume dynamics at time t. Second, there is a threshold β₀ > 0 such that for every inverse temperature β ≥ β₀ there exists a state ω, meaning a normalized positive continuous linear functional on the quasi-local algebra, that is invariant under all lattice translations and satisfies the KMS condition at β. The KMS condition requires that for all observables A and B there is a bounded function F, continuous on the strip 0 ≤ Im z ≤ β and holomorphic inside it, with F(t) = ω(A·τ_t(B)) and F(t + iβ) = ω(τ_t(B)·A) for all real t. Moreover, the origin spin component S^z has real expectation under ω that is at least ℓ/8.