Universal shift of the fidelity susceptibility peak away from the critical point of the Berezinskii-Kosterlitz-Thouless quantum phase transition

Abstract

We show that the peak which can be observed in fidelity susceptibility around the Berezinskii-Kosterlitz-Thouless transition is shifted from the quantum critical point (QCP) at Jc to J * in the gapped phase by a value |J * −Jc| = B 2 /36, where B 2 is a transition width controlling the asymptotic form of the correlation length ξ ∼ exp(−B/ |J − Jc|) in that phase. This is in contrast to the conventional continuous QCP where the maximum is an indicator of the position of the critical point. The shape of the peak is universal, emphasizing the close connection between fidelity susceptibility and the correlation length. We support those arguments with numerical matrix product state simulations of the one-dimensional Bose-Hubbard model in the thermodynamic limit, where the broad peak is located at J * = 0.212 that is significantly different from Jc = 0.3048(3). In the spin-3/2 XXZ model the shift from Jc = 1 to J * = 1.0021 is small but the narrow universal peak is much more pronounced over the non-universal background.

Fidelity and entanglement entropy for infinite-order phase transitions

“…( 9) to extrapolate accurate values of BKT points for clock models. On the other hand, χ F being finite indicates that one can formulate a scaling hypothesis for the log fidelity as a function of correlation lengths ξ(D), ξ(D+δ), and show that the peak of χ F for BKT transitions is shifted into the gapped phase [51], which has been checked numerically. The only possibility to resolve the contradiction is that the FSS in Eq.…”

Topological charge scaling at a quantum phase transition

“…The value of B is related to the KT transition width. The transition in the considered system is much broader than that undergoing in the 1D Bose-Hubbard model (B = 0.261), in XXZ spin- 3 2 model (B = 1.61) [4] or in 2D XY model (B = 1.5) [13]. This is also the reason why the maximum of χ T is shifted quite far from λ c .…”

“…However, some difficulties were encountered with finite-size scaling (FSS) of the fidelity susceptibility χ F for topological QPTs: It was unclear whether the maxima of χ F obey FSS, and some attempts were made to interpret the emerging discrepancies [3] as logarithmic corrections to scaling. It turned out recently [4] that the maximum of χ F , is shifted relative to the Kosterlitz-Thouless (KT) quantum critical point λ c by a universal constant B 2 36 towards the gapped phase in which the correlation length ξ falls exponentially, ξ(λ) ∼ exp(B/ |λ − λ c |). For this reason, the maximum in question does not scale with the system size L as expected, see, e.g., Eq.…”

“…From the position of this maximum, we can independently determine the numerical value of parameter B for the second time using the main result of Ref. [4]: The maximum of fidelity susceptibility χ F should be shifted with respect to λ c by the universal constant B 2 36 . To prove this, the authors of Ref.…”

“…To prove this, the authors of Ref. [4] assumed that the singular part of the fidelity Ψ 0 (λ 1 )|Ψ 0 (λ 2 ) near the critical point is a homogeneous function with respect to ξ(λ 1 ) and ξ(λ 2 ) and scales as [4]…”