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arXiv:0803.1292 · Physical Review A

Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model

Shuo Yang, Shi-Jian Gu, Chang-Pu Sun, Hai-Qing Lin

Published Jul 2, 2008• 147 citations
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Abstract

We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped $A$ phase with Abelian anyon excitations to a gapless $B$ phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/\ensuremath{\xi}=2\text{ }{\text{sinh}}^{\ensuremath{-}1}[\sqrt{2{J}_{z}\ensuremath{-}1}/(1\ensuremath{-}{J}_{z})]$, which diverges around the critical point ${J}_{z}={(1/2)}^{+}$.

DOI
10.1103/physreva.78.012304

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We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped AAA phase with Abelian anyon excitations to a gapless BBB phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be 1/\ensuremathξ=2 sinh\ensuremath−1[2Jz\ensuremath−1/(1\ensuremath−Jz)]1/\ensuremath{\xi}=2\text{ }{\text{sinh}}^{\ensuremath{-}1}[\sqrt{2{J}_{z}\ensuremath{-}1}/(1\ensuremath{-}{J}_{z})]1/\ensuremathξ=2 sinh\ensuremath−1[2Jz​\ensuremath−1​/(1\ensuremath−Jz​)], which diverges around the critical point Jz=(1/2)+{J}_{z}={(1/2)}^{+}Jz​=(1/2)+.
We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be 1/ξ=2sinh⁡−1[2Jz−1/(1−Jz)]1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]1/ξ=2sinh−1[2Jz​−1​/(1−Jz​)], which diverges around the critical point Jz=(1/2)+J_z=(1/2)^+Jz​=(1/2)+.
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.
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 analyze ground-state behaviors of fidelity susceptibility (FS) and show that the FS has its own distinct dimension instead of real system's dimension in general quantum phases. The scaling relation of the FS in quantum phase transitions (QPTs) is then established on more general grounds. Depending on whether the FS's dimensions of two neighboring quantum phases are the same or not, we are able to classify QPTs into two distinct types. For the latter type, the change in the FS's dimension is a characteristic that separates two phases. As a non-trivial application to the Kitaev honeycomb model, we find that the FS is proportional to L 2 ln L in the gapless phase, while L 2 in the gapped phase. Therefore, the extra dimension of ln L can be used as a characteristic of the gapless phase.
We study reduced fidelity and reduced fidelity susceptibility in the Kitaev honeycomb model. It is shown that the nearest-two-site reduced fidelity susceptibility manifests itself as a peak at the quantum phase transition point, although the one-site reduced fidelity susceptibility vanishes. Our results directly reveal that the reduced fidelity susceptibility can be used to characterize the quantum phase transition in the Kitaev honeycomb model, which suggests that, despite its local nature, the reduced fidelity susceptibility is an accurate marker of the topological phase transition when it is properly chosen.
Temperature evolution of the spin correlation and excitation spectrum is investigated for a Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of quantum spins into Majorana fermions, in addition to a phase transition associated with time reversal symmetry breaking. We show that the spin correlation develops around the crossover temperatures, whereas it shows a slight change at the critical temperature, as in other Kitaev-type models. We also calculate the excitation spectrum in terms of Majorana fermions, and find that the excitation gap in the non-Abelian phase is fragile against thermal fluctuations of the Z2Z_2Z2​ fluxes, while that in the Abelian phase remains open.