Fidelity and entanglement entropy for infinite-order phase transitions

Abstract

We study the fidelity and the entanglement entropy for the ground states of quantum systems that have infinite-order quantum phase transitions. In particular, we consider the quantum O(2) model with a spin-$S$ truncation, where there is an infinite-order Gaussian (IOG) transition for $S=1$ and there are Berezinskii-Kosterlitz-Thouless (BKT) transitions for $S\ge 2$. We show that the height of the peak in the fidelity susceptibility (${\chi }_F$) converges to a finite thermodynamic value as a power law of $1/L$ for the IOG transition and as $1/\ln (L)$ for BKT transitions. The peak position of ${\chi }_F$ resides inside the gapped phase for both the IOG transition and BKT transitions. On the other hand, the derivative of the block entanglement entropy with respect to the coupling constant ($S_{vN}^{\prime }$) has a peak height that diverges as ${\ln }^2(L)$ [${\ln }^3(L)$] for $S=1$ ($S\ge 2$) and can be used to locate both kinds of transitions accurately. We include higher-order corrections for finite-size scalings and crosscheck the results with the value of the central charge $c=1$. The crossing point of ${\chi }_F$ between different system sizes is at the IOG point for $S=1$ but is inside the gapped phase for $S\ge 2$, while those of $S_{vN}^{\prime }$ are at the phase-transition points for all $S$ truncations. Our work elaborates how to use the finite-size scaling of ${\chi }_F$ or $S_{vN}^{\prime }$ to detect infinite-order quantum phase transitions and discusses the efficiency and accuracy of the two methods.

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