Scaling behavior for a class of quantum phase transitions.

We show that for quantum phase transitions with a single bosonic zero mode at the critical point, like the Dicke model and the Lipkin-Meshkov-Glick model, metric quantities such as fidelity, that is, the overlap between two ground states corresponding to two values λ(1) and λ(2) of the controlling parameter λ, depend only on the ratio η = (λ(1) -λ(c))/(λ(2) -λ(c)), where λ = λ(c) at the critical point. This scaling property is valid also for time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.