The potential of fidelity in accurately detecting unconventional quantum phase transitions (QPTs) remains an open question. Specific attention is directed towards high-order phase transitions, including the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the Gaussian transition. In this study, we extend the concept of fidelity susceptibility to fixed-point fidelity susceptibility in the thermodynamic limit and assess this measure using tensor-network algorithms. Our results demonstrate that the fixed-point fidelity susceptibility effectively identifies the BKT and Gaussian phase transitions, a capability lacking in the traditional finite-fidelity approach. Furthermore, we establish a correlation between the fixed-point fidelity method and the entanglement method, demonstrating their equivalence. This research not only confirms the efficacy of fidelity in detecting QPTs but also enhances the understanding of fidelity-related methodologies.
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 $S\ensuremath{\ge}2$. We show that the height of the peak in the fidelity susceptibility (${\ensuremath{\chi}}_{F}$) 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 ${\ensuremath{\chi}}_{F}$ 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 (${S}_{vN}^{\ensuremath{'}}$) has a peak height that diverges as
ln2(L) for
S=1 and
ln3(L) for $S\ensuremath{\ge}2$ and can be used to locate both kinds of transitions accurately. We include higher-order corrections for finite-size scalings and obtain the value of the central charge consistent with
c=1 predicted by conformal field theory. The crossing point of ${\ensuremath{\chi}}_{F}$ between different system sizes is at the IOG point for
S=1 but is inside the gapped phase for $S\ensuremath{\ge}2$, while those of ${S}_{vN}^{\ensuremath{'}}$ are at the phase-transition points for all
S truncations. Our work elaborates on how to use the finite-size scaling of ${\ensuremath{\chi}}_{F}$ or ${S}_{vN}^{\ensuremath{'}}$ to detect infinite-order quantum phase transitions and discusses the efficiency and accuracy of the two methods.
We investigate the one-dimensional Heisenberg
XXZ model involving the Dzyaloshinskii-Moriya (DM) interaction from the viewpoint of quantum information. By using the density-matrix--renormalization-group technique, we study the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the ground-state fidelity, the fidelity susceptibility, and the entanglement entropy. We argue that the BKT phase transition can be well characterized in our model, and the critical line induced by the DM interaction can also be detected.
Fidelity has been widely used to detect various types of quantum phase transitions (QPTs). However, challenges remain in locating the transition points with precision in several models with unconventional phases. We propose the fidelity map approach to detect QPTs with higher accuracy and sensitivity than the conventional fidelity measures. Our scheme extends the fidelity concept from a single-dimensional to a multidimensional quantity and uses a metaheuristic algorithm to search for the critical points that globally maximize the fidelity within each phase. We verify the scheme in several interacting models that possess unconventional phases and QPTs, namely, the spin-1 Kitaev-Heisenberg model, the spin-
21 XXZ model, and the interacting Su-Schrieffer-Heeger model. The fidelity map can capture a wide range of phase transitions accurately even in small systems, thus providing a convenient tool to study QPTs in unseen models without prior knowledge of the symmetry of the system.
We study slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-
1/2 J1-
J2 antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\ensuremath{\chi}}_{\ensuremath{\rho}}$, ${\ensuremath{\chi}}_{D}$, and ${\ensuremath{\chi}}_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the
J1-
J2 model. We show that they all can accurately identify a quantum critical point in this model occurring at ${J}_{2}^{c}\ensuremath{\sim}0.241{J}_{1}$ between a gapless Heisenberg phase for
J2<J2c and a dimerized phase for
J2>J2c. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.
We clarify the long-standing controversy concerning the behavior of the ground-state fidelity in the vicinity of a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type in one-dimensional systems. Contrary to the prediction based on the Gaussian approximation of the Luttinger-liquid approach, it is shown that the fidelity susceptibility does not diverge at the transition but has a cusplike peak ${\ensuremath{\chi}}_{c}\ensuremath{-}\ensuremath{\chi}(\ensuremath{\lambda})\ensuremath{\sim}\sqrt{|{\ensuremath{\lambda}}_{c}\ensuremath{-}\ensuremath{\lambda}|}$, where $\ensuremath{\lambda}$ is a parameter driving the transition and ${\ensuremath{\chi}}_{c}$ is the peak value at the transition point $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$. Numerical claims of the logarithmic divergence of fidelity susceptibility with the system size (or temperature) are explained by logarithmic corrections due to marginal operators, which is supported by numerical calculations for large systems.
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 $|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36$, where
B2 is a transition width controlling the asymptotic form of the correlation length $\ensuremath{\xi}\ensuremath{\sim}exp(\ensuremath{-}B/\sqrt{|J\ensuremath{-}{J}_{c}|})$ 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 nonuniversal background.