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We study quantum spin systems with quenched Gaussian disorder. We prove thatthe variance of all physical quantities in a certain class vanishes in theinfinite volume limit. We study also replica symmetry breaking phenomena, wherethe variance of an overlap operator in the other class does not vanish in thereplica symmetric Gibbs state. On the other hand, it vanishes in a spontaneousreplica symmetry breaking Gibbs state defined by applying an infinitesimalreplica symmetry breaking field. We prove also that the finite variance of theoverlap operator in the replica symmetric Gibbs state implies the existence ofa spontaneous replica symmetry breaking.

Universal nature of replica symmetry breaking in quantum systems with Gaussian disorder