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arXiv:0908.3132 · Physical Review E

Fidelity susceptibility in the two-dimensional transverse-field Ising and XXZ models

Wing Chi Yu, Ho-Man Kwok, Junpeng Cao, Shi-Jian Gu

Published Aug 13, 2009• 47 citations
arXivHTMLPDF

Abstract

We study the fidelity susceptibility in the two-dimensional (2D) transverse-field Ising model and the 2D XXZ model numerically. It is found that in both models, the fidelity susceptibility as a function of the driving parameter diverges at the critical points. The validity of the fidelity susceptibility to signal for the quantum phase transition is thus verified in these two models. We also compare the scaling behavior of the extremum of the fidelity susceptibility to that of the second derivative of the ground-state energy. From those results, the theoretical argument that fidelity susceptibility is a more sensitive seeker for a second-order quantum phase transition is also testified in the two models.

DOI
10.1103/physreve.80.021108

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We study the fidelity susceptibility in the two-dimensional (2D) transverse-field Ising model and the 2D XXZ model numerically. It is found that in both models, the fidelity susceptibility as a function of the driving parameter diverges at the critical points. The validity of the fidelity susceptibility to signal for the quantum phase transition is thus verified in these two models. We also compare the scaling behavior of the extremum of the fidelity susceptibility to that of the second derivative of the ground-state energy. From those results, the theoretical argument that fidelity susceptibility is a more sensitive seeker for a second-order quantum phase transition is also testified in the two models.
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We study critical behaviors of the reduced fidelity susceptibility for two neighboring sites in the one-dimensional transverse field Ising model. It is found that the divergent behaviors of the susceptibility take the form of square of logarithm, in contrast with the global ground-state fidelity susceptibility which is power divergence. In order to perform a scaling analysis, we take the square root of the susceptibility and determine the scaling exponent analytically and the result is further confirmed by numerical calculations.
We study critical behaviors of the reduced fidelity susceptibility for two neighboring sites in the one-dimensional transverse field Ising model. It is found that the divergent behaviors of the susceptibility take the form of square of logarithm, in contrast with the global ground-state fidelity susceptibility which is power divergence. In order to perform a scaling analysis, we take the square root of the susceptibility and determine the scaling exponent analytically and the result is further confirmed by numerical calculations.
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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.
The inner product between the ground-state eigenvectors with proximate interaction parameters, namely, the fidelity, plays a significant role in the quantum dynamics. In this paper, the critical behaviors of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum (transverse-field) Ising model are investigated by means of the numerical-diagonalization method; the former susceptibility has been investigated rather extensively. The critical exponents for these fidelity susceptibilities are estimated as α(F)((t))=0.752(24) and α(F)((h))=1.81(13), respectively. These indices are independent, and suffice for obtaining conventional critical indices such as ν=0.624(12) and γ=1.19(13).