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arXiv:1111.4341

Bell Function Values Approach to Topological Quantum Phase Transitions

Dong-Ling Deng, Chunfeng Wu, Jing‐Ling Chen, Shi-Jian Gu, Sixia Yu, C. H. Oh

Published Nov 18, 2011• 2 citations
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Abstract

We investigate the relation between Bell function values (BFV) of the reduced density matrix and the topological quantum phase transitions in the Kitaev-Castelnovo-Chamon model. % [Phys. Rev. B \textbf{77}, %054433 (2008)]. We find that the first order derivative of BFV exhibits singular behavior at the critical point and we propose that it can serve as a good and convenient marker for the transition point. More interestingly, the value of the critical point can be analytically obtained in this approach. Since the BFV serves as a measure of nonlocality when it is greater than the classical bound of the correlation functions, our work has established a link between quantum nonlocality and phase transitions.

DOI
10.48550/arxiv.1111.4341

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We investigate the relation between Bell function values (BFV) of the reduced density matrix and the topological quantum phase transitions in the Kitaev-Castelnovo-Chamon model. % [Phys. Rev. B \textbf{77}, %054433 (2008)]. We find that the first order derivative of BFV exhibits singular behavior at the critical point and we propose that it can serve as a good and convenient marker for the transition point. More interestingly, the value of the critical point can be analytically obtained in this approach. Since the BFV serves as a measure of nonlocality when it is greater than the classical bound of the correlation functions, our work has established a link between quantum nonlocality and phase transitions.
We investigate the critical behaviors of the Bell nonlocality, which are measured by the Bell-function values, in conventional and topological quantum-phase transitions. First, by using the quantum-renormalization-group method, we show that the Bell function value of a three-qubit Bell inequality in the anisotropic spin-1/21/21/2 XYXYXY model exhibits singular behaviors at the critical point. Moreover, for topological quantum-phase transitions, we find that the first-order derivative of the Bell-function value of the well-known Clauser-Horne-Shimony-Holt inequality in the Kitaev-Castelnovo-Chamon model behaves singularly at the topological-phase-transition point. Our results have established a link between quantum nonlocality and phase transitions.
A useful approach to characterize and identify quantum phase transitions lies in the concept of multipartite entanglement. In this paper, we consider well-known measures of multipartite (global) entanglement, i.e., average linear entropy of one-qubit and two-qubit reduced density matrices, in order to study topological quantum phase transition (TQPT) in the Kitaev Toric code Hamiltonian with a nonlinear perturbation. We provide an exact mapping from aforementioned measures in the above model to internal energy and energy-energy correlations in the classical Ising model. Accordingly, we find that the global entanglement shows a continuous and sharp transition from a maximum value in the topological phase to zero in the magnetized phase in a sense that its first-order derivative diverges at the transition point. In this regard, we conclude that not only can the global entanglement serve as a reasonable tool to probe quantum criticality at TQPTs, but it also can reveal the highly entangled nature of topological phases. Furthermore, we also introduce a conditional version of global entanglement which becomes maximum at the critical point. Therefore, regarding a general expectation that multipartite entanglement reaches maximum value at the critical point of quantum many-body systems, our result proposes that the conditional global entanglement can be a good measure of multipartite entanglement in TQPTs.
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\n\nWe apply the Wigner-Yanase skew information approach to analyze two typical models that exhibit a topological quantum phase transition. Based on the exact solutions of the ground states, the Wigner-Yanase skew information between two nearest sites for each of the two models is obtained. For the one-dimensional Kitaev chain model, the first-order derivative of the Wigner-Yanase skew information is non-analytical around the critical point. The scaling behavior and the universality are verified numerically. In particular, the skew information can also detect the factorization transition in such a model. For the two-dimensional Kitaev honeycomb model, the first-order derivative of the Wigner-Yanase skew information shows some singularities at the critical points where the system transits from the gapless phase to the gapped one. Our results suggest that the Wigner-Yanase skew information can serve as a good indicator of the topological phase transitions in these models and shed considerable light on the relationships between topological quantum phase transition and information theory.\n
We study the quantum phase transitions (QPTs) in the Kitaev spin model on a triangle-honeycomb lattice. In addition to the ordinary topological QPTs between Abelian and non-Abelian phases, we find new QPTs which can occur between two phases belonging to the same topological class, namely, either two non-Abelian phases with the same Chern number or two Abelian phases with the same Chern number. Such QPTs result from the singular behaviors of the nonlocal spin-spin correlation functions at the critical points.