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 $λ$, only depend on the ratio $η=(λ_1-λ_c)/(λ_2-λ_c)$, where $λ=λ_c$ at the critical point. Such 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.

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