Scaling behavior for a class of quantum phase transitions

Abstract

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 ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ of the controlling parameter $\ensuremath{\lambda}$, depend only on the ratio $\ensuremath{\eta}=({\ensuremath{\lambda}}_{1}\ensuremath{-}{\ensuremath{\lambda}}_{c})/({\ensuremath{\lambda}}_{2}\ensuremath{-}{\ensuremath{\lambda}}_{c})$, where $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ 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.