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arXiv:1906.05307 · Physical review. B./Physical review. B

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

Łukasz Cincio, Marek M. Rams, Jacek Dziarmaga, Wojciech Hubert Zurek

Published Aug 19, 2019• 13 citations
arXivHTMLPDF

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 ${J}_{c}$ to ${J}^{*}$ in the gapped phase by a value $|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36$, where ${B}^{2}$ 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 ${J}_{c}=0.3048(3)$. In the spin-$3/2$ XXZ model the shift from ${J}_{c}=1$ to ${J}^{*}=1.0021$ is small but the narrow universal peak is much more pronounced over the nonuniversal background.

DOI
10.1103/physrevb.100.081108

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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{J}_{c}Jc​ to J∗{J}^{*}J∗ in the gapped phase by a value ∣J∗\ensuremath−Jc∣=B2/36|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36∣J∗\ensuremath−Jc​∣=B2/36, where B2{B}^{2}B2 is a transition width controlling the asymptotic form of the correlation length \ensuremathξ\ensuremath∼exp(\ensuremath−B/∣J\ensuremath−Jc∣)\ensuremath{\xi}\ensuremath{\sim}exp(\ensuremath{-}B/\sqrt{|J\ensuremath{-}{J}_{c}|})\ensuremathξ\ensuremath∼exp(\ensuremath−B/∣J\ensuremath−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{J}^{*}=0.212J∗=0.212 that is significantly different from Jc=0.3048(3){J}_{c}=0.3048(3)Jc​=0.3048(3). In the spin-3/23/23/2 XXZ model the shift from Jc=1{J}_{c}=1Jc​=1 to J∗=1.0021{J}^{*}=1.0021J∗=1.0021 is small but the narrow universal peak is much more pronounced over the nonuniversal background.
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 discuss how to locate critical points in the Berezinskii-Kosterlitz-Thouless (BKT) universality class by means of gap-scaling analyses. While accurately determining such points using gap extrapolation procedures is usually challenging and inaccurate due to the exponentially small value of the gap in the vicinity of the critical point, we show that a generic gap-scaling analysis, including the effects of logarithmic corrections, provides very accurate estimates of BKT transition points in a variety of spin and fermionic models. As a first example, we show how the scaling procedure, combined with density-matrix-renormalization-group simulations, performs extremely well in a nonintegrable spin-3/23/23/2 XXZ model, which is known to exhibit strong finite-size effects. We then analyze the extended Hubbard model, whose BKT transition has been debated, finding results that are consistent with previous studies based on the scaling of the Luttinger-liquid parameter. Finally, we investigate an anisotropic extended Hubbard model, for which we present the first estimates of the BKT transition line based on large-scale density-matrix-renormalization-group simulations. Our work demonstrates how gap-scaling analyses can help to locate accurately and efficiently BKT critical points, without relying on model-dependent scaling assumptions.
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-SSS truncation, where there is an infinite-order Gaussian (IOG) transition for S=1S=1S=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/L1/L1/L for the IOG transition and as 1/ln(L)1/ln(L)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){ln}^{2}(L)ln2(L) for S=1S=1S=1 and ln3(L){ln}^{3}(L)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=1c=1c=1 predicted by conformal field theory. The crossing point of ${\ensuremath{\chi}}_{F}$ between different system sizes is at the IOG point for S=1S=1S=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 SSS 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 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-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}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{J}_{1}J1​-J2{J}_{2}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{J}_{2}<{J}_{2}^{c}J2​<J2c​ and a dimerized phase for J2>J2c{J}_{2}>{J}_{2}^{c}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.
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.