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arXiv:2008.12366 · Physical review. B./Physical review. B

Revealing divergent length scales using quantum Fisher information in the Kitaev honeycomb model

James Lambert, Erik Schwartz Sørensen

Published Dec 1, 2020• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.102.224401

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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.
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)+.
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.
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 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 $1/\ensuremath{\xi}=2\text{ }{\text{sinh}}^{\ensuremath{-}1}[\sqrt{2{J}_{z}\ensuremath{-}1}/(1\ensuremath{-}{J}_{z})]$, which diverges around the critical point Jz=(1/2)+{J}_{z}={(1/2)}^{+}Jz​=(1/2)+.
In this work we evaluate quantum Fisher information (QFI) of a two-qubit reduced state in the Ising-XXZ diamond structure at finite temperatures, which is applied to demonstrate quantum criticality and quantum phase transition in this model. Here we adopt the average QFI with respect to the local orthonormal observable bases [Phys. Rev. A. 88, 014,301 (2013)]. The results show that QFI around quantum critical points decreases or increases steeper when the system temperature is relatively lower. Meanwhile the singularity of QFI’ first-derivatives is becoming more and more obvious as the temperature is approaching zero. By the finite-temperature QFI and its scaling relation with temperature, quantum phase transitions can be well detected from the entangled state in ferrimagnetic phase to an unentangled state in ferrimagnetic phase or to an unentangled state in ferromagnetic phase. Moreover it is further found that the two kinds of phase transitions can be distinguished by the different scaling behaviors. It has been proved that QFI has enough capability to characterize quantum phase transition at finite temperature.
We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories based on different Majorana fermion representations of the S=1/2 spin operators. First, we use a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite snake string operator. To ensure that the fermionized Hamiltonian remains local, we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Second, we use the conventional Kitaev representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both representations, we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin liquids, the JWT fails to correctly describe the magnetic instability and finite-temperature behavior. Our results show that the magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale, we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures, in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures. Published by the American Physical Society 2024