Fidelity and entanglement entropy for infinite-order phase transitions

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 $χ_F$ 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 χ_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 $S^prime_vN$ has a peak height that diverges as ln²(L) \[ln³(L)\] 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 χ_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^prime_vN are at the phase-transition points for all S truncations. Our work elaborates how to use the finite-size scaling of χ_F or S^prime_vN to detect infinite-order quantum phase transitions and discusses the efficiency and accuracy of the two methods.