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We study effects of perturbation Hamiltonian to quantum spin systems whichcan include quenched disorder. Model-independent inequalities are derived,using an additional artificial disordered perturbation. These inequalitiesenable us to prove that the variance of the perturbation Hamiltonian densityvanishes in the infinite volume limit even if the artificial perturbation isswitched off. This theorem is applied to spontaneous symmetry breakingphenomena in a disordered classical spin model, a quantum spin model withoutdisorder and a disordered quantum spin model.

Zero-variance of perturbation Hamiltonian density in perturbed spin systems