Ground state overlap and quantum phase transitions

We present a characterization of quantum phase transitions in terms of the the overlap function between two ground states obtained for two different values of external parameters. On the examples of the Dicke and $XY$ models, we show that the regions of criticality of a system are marked by the extremal points of the overlap and functions closely related to it. Further, we discuss the connections between this approach and the Anderson orthogonality catastrophe as well as with the dynamical study of the Loschmidt echo for critical systems.

Figure 1

(Color online) The overlap function $⟨g\mid \tilde{g}⟩$, Eq. (3), as a function of $\lambda <{\lambda }_c$, taken for the resonant case ${\omega }_0=\omega =1$ and $\delta \lambda ={10}^{-6}$. Note the dramatic decreasing of the function as we approach the point of criticality.

Figure 2

(Color online) (a) The overlap function $⟨g\left(q\right)\mid g\left(\tilde{q}\right)⟩$, as a function of $\lambda$ and $\gamma$, for $N={10}^6$ and $\delta \lambda =\delta \gamma ={10}^{-6}$. Note the clear dips of the plot in the regions of criticality. (b) $S_N^{\lambda }(\lambda ,\gamma )$. (c) $S_N^{\gamma }(\lambda ,\gamma )$.