A Quantum Spin System with Random Interactions I

[摘要] We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups ${mathcal{T}_t(𝜔)}_{𝜔in𝛺}$ of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space 𝛺, is used to prove the almost sure independence of the Arveson spectrum $mathrm{Sp}(mathcal{T}(𝜔))$ of $mathcal{T}_t(𝜔)$. As a consequence, for any family of $(mathcal{T}(𝜔), 𝛽)$-KMS states {ρ(𝜔)}, the spectrum of the generator of the group of unitaries which implement $mathcal{T}(𝜔)$ in the GNS representation is also almost surely independent of 𝜔.

[学科分类] 数学（综合）

[关键词] Spin;system;quasi-local;random;dynamics;evolution;independent;Arveson;KMS.