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arXiv:1408.2739 · Physical Review B

Fidelity at Berezinskii-Kosterlitz-Thouless quantum phase transitions

Gaoyong Sun, Alexei K. Kolezhuk, T. Vekua

Published Jan 15, 2015• 64 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.91.014418

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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 \ensuremathχc\ensuremath−\ensuremathχ(\ensuremathλ)\ensuremath∼∣\ensuremathλc\ensuremath−\ensuremathλ∣{\ensuremath{\chi}}_{c}\ensuremath{-}\ensuremath{\chi}(\ensuremath{\lambda})\ensuremath{\sim}\sqrt{|{\ensuremath{\lambda}}_{c}\ensuremath{-}\ensuremath{\lambda}|}\ensuremathχc​\ensuremath−\ensuremathχ(\ensuremathλ)\ensuremath∼∣\ensuremathλc​\ensuremath−\ensuremathλ∣​, where \ensuremathλ\ensuremath{\lambda}\ensuremathλ is a parameter driving the transition and \ensuremathχc{\ensuremath{\chi}}_{c}\ensuremathχc​ is the peak value at the transition point \ensuremathλ=\ensuremathλc\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}\ensuremathλ=\ensuremathλc​. 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 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.
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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.
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