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arXiv:0803.0814 · Physical Review B

Singularities in ground-state fidelity and quantum phase transitions for the Kitaev model

Jian-Hui Zhao, Huan-Qiang Zhou

Published Jul 1, 2009• 58 citations
arXivHTMLPDF

Abstract

The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.

DOI
10.1103/physrevb.80.014403

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The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.
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.
We study the scaling behavior of the fidelity (F ) in the thermodynamic limit using the examples of a Tomonaga-Luttinger liquid in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow powerlaw scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of F for an anisotropic quantum critical point for both thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The behavior of F inside the gapless phase of the Kitaev model is found to depend strongly on its distance from the anisotropic quantum critical point. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate F through the overlap between the ground states for angle of rotation η and η+ dη, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.
We study the scaling behavior of the fidelity (FFF) in the thermodynamic limit using the examples of a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of FFF for an anisotropic quantum critical point for both the thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The interesting scaling behavior of FFF inside the gapless phase of the Kitaev model is also discussed. Finally, we consider a rotation of each spin in the Kitaev model around the zzz axis and calculate FFF through the overlap between the ground states for the angle of rotation $\ensuremath{\eta}$ and $\ensuremath{\eta}+d\ensuremath{\eta}$, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.
The fidelity per site between two ground states of a quantum lattice system corresponding to different values of the control parameter defines a surface embedded in a Euclidean space. The Gaussian curvature naturally quantifies quantum fluctuations that destroy orders at transition points. It turns out that quantum fluctuations wildly distort the fidelity surface near the transition points, at which the Gaussian curvature is singular in the thermodynamic limit. As a concrete example, the one-dimensional quantum Ising model in a transverse field is analyzed. We also perform a finite size scaling analysis for the transverse Ising model of finite sizes. The scaling behavior for the Gaussian curvature is numerically checked and the correlation length critical exponent is extracted, which is consistent with the conformal invariance at the critical point.
The Kitaev honeycomb model with a topological phase transition at zero temperature is studied using the quantum-information method. Based on the exact solution of the ground state, the mutual information between two nearest sites and between two bonds with the greatest distance are obtained. It is found that the mutual information shows some singularities at the critical point where the system transits from the gapless phase to the gapped phase. Finite-size effects and scaling behavior are also studied. Our results indicate that the mutual information can serve as a good indicator of the topological phase transition. This is because the mutual information is believed to be able to catch some global correlation properties of the system. Meanwhile, this method has the advantages that the phase transition can be determined easily and the order parameters, which are hard to obtain for some topological phase transitions, are not necessarily known.
We compute the quantum Fisher information (QFI) associated with two different local operators in the ground state of the Kitaev honeycomb model, and find divergent behavior in the second derivatives of these quantities with respect to the driving parameter at the quantum phase transition between the gapped and gapless phases for both fully antiferromagnetic and fully ferromagnetic exchange couplings, thus demonstrating that the second derivative a locally defined, experimentally accessible, QFI can detect topological quantum phase transitions. The QFI associated with a local magnetization operator behaves differently from that associated with a local bond operator depending on whether the critical point is approached from the gapped or gapless side. We show how the behavior of the second derivative of the QFI at the critical point can be understood in terms of the diverging length scales associated to the two and four point correlators of the Majorana degrees of freedom. We present critical exponents associated with the divergences of these length scales.