Convergence Time in 1D Def ρ β has correlation length ξ if for every f, g f g Cor For a 1D Hamiltonian, ρ β has correlation length ξ, then for t = O( 2 O(l) 2 O(ξ) n), l = O(log(n/ε)) thm (Araki '69) For every 1D Hamiltonian, ρ β has ξ = O(β) Thus: Can prepare 1D Gibbs states in time poly(2 β, n) No phase trans.
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