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arXiv:1805.05560 · Physical review. E

Fidelity susceptibility of the anisotropic XY model: The exact solution

Qiang Luo, Jize Zhao, Xiaoqun Wang

Published Aug 7, 2018• 29 citations
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Abstract

We derive several closed-form expressions for the fidelity susceptibility (FS) of the anisotropic XY model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents ν associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.

DOI
10.1103/physreve.98.022106

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We derive several closed-form expressions for the fidelity susceptibility (FS) of the anisotropic XY model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents ν associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
By means of the density matrix renormalization group technique, the scaling relation of the fidelity susceptibility proposed recently is verified for the spin-one XXZXXZXXZ spin chain with an on-site anisotropic term. Moreover, from the results of both the fidelity susceptibility and the entanglement entropy, the critical points and some of the corresponding critical exponents are determined through a proper finite-size scaling analysis, and these values agree with the findings in the literature. Thus our work provides a numerical support of the use of the fidelity in detecting quantum phase transitions.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.
The square-lattice J 1 - J 2 transverse-field (TF) Ising model was investigated with the exact diagonalization (ED) method. In order to analyze the TF-driven phase transition, we applied the longitudinal-field fidelity susceptibility χ F ( h ) , which is readily evaluated via the ED scheme. Here, the longitudinal field couples with the absolute value of the magnetic moment | M | rather than the raw M so that the remedied fidelity susceptibility exhibits a peak around the critical point; note that the spontaneous magnetization does not appear for the finite-size systems. As a preliminary survey, the modified fidelity susceptibility χ F ( h ) is applied to the analysis of criticality for J 2 = 0 , where a number of preceding results are available. Thereby, properly scaling the distance from the multi-criticality, η = 0 . 5 − J 2 , the χ F ( h ) data were cast into the crossover-scaling formula, and the multi-critical exponent for χ F ( h ) is estimated. The result is cross-checked by the numerically evaluated β -function behavior.
The critical properties of the three-dimensional XY model are investigated through measurements of the two-point correlation functions via a worm-algorithm Monte Carlo calculation, which automatically crunches the correlation functions while tracing worms, in large systems. This method has the advantage of directly measuring the order parameter, as well as the correlation length, the susceptibility, and their associated critical exponents. From the finite-size scaling behavior, we find 2 β/ν = 1.033(5) and γ/ν = 1.967(5), confirming the hyperscaling relations.