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Laser Physics Letters

Scaling of geometric phases close to the topological quantum phase transition in Kitaev's quantum wire model

Chuan-Jia Shan, Jin-Xin Li, W. W. Cheng, Ji-Bing Liu, Tang-Kun Liu

Published Jan 29, 2014• 3 citations
Publisher

Abstract

We present a unified study of the topological quantum phase transition in Kitaev's 1D p-wave spinless quantum wire model in terms of its ground state geometric phase (GP). We also analyze zero-temperature and finite-temperature scaling parameters, extracted from the GP near the critical point. The derivative of the ground-state GP is nonanalytic at the phase boundary. A finite-size scaling and finite-temperature scaling analysis are carried out and, furthermore, the scaling behavior and universality are verified numerically.

DOI
10.1088/1612-2011/11/3/035202

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We present a unified study of the topological quantum phase transition in Kitaev's 1D p-wave spinless quantum wire model in terms of its ground state geometric phase (GP). We also analyze zero-temperature and finite-temperature scaling parameters, extracted from the GP near the critical point. The derivative of the ground-state GP is nonanalytic at the phase boundary. A finite-size scaling and finite-temperature scaling analysis are carried out and, furthermore, the scaling behavior and universality are verified numerically.
Quantum phase transition is one of the most interesting aspects in quantum many-body systems. Recently, geometric quantum discord has been introduced to signature the critical behavior of various quantum systems. However, it is well-known that topological quantum phase transition can not be described by the conventional Landau's symmetry breaking theory, and thus it is unknown that whether previous study can be applicable in this case. Here, we study the topological quantum phase transition in Kitaev's 1D p-wave spinless quantum wire model in terms of its ground state geometric quantum discord. The derivative of geometric quantum discord is nonanalytic at the critical point, in both zero temperature and finite temperature cases. The scaling behavior and the universality are verified numerically. Therefore, our results clearly show that all the key ingredients of the topological phase transition can be captured by the nearest neighbor and long-range geometric quantum discord.
Quantum phase transition is one of the most interesting aspects in quantum many-body systems. Recently, geometric quantum discord has been introduced to signature the critical behavior of various quantum systems. However, it is well-known that topological quantum phase transition can not be described by the conventional Landau’s symmetry breaking theory, and thus it is unknown that whether previous study can be applicable in this case. Here, we study the topological quantum phase transition in Kitaev’s 1D p-wave spinless quantum wire model in terms of its ground state geometric quantum discord. The derivative of geometric quantum discord is nonanalytic at the critical point, in both zero temperature and finite temperature cases. The scaling behavior and the universality are verified numerically. Therefore, our results clearly show that all the key ingredients of the topological phase transition can be captured by the nearest neighbor and long-range geometric quantum discord.
We have applied the real space quantum renormalization group approach to study the topological quantum phase transition in the one-dimensional chain of a spinless p-wave superconductor. We investigate the behavior of local compressibility and ground-state fidelity of the Kitaev chain. We show that the topological phase transition is signaled by the maximum of local compressibility at the quantum critical point tuned by the chemical potential. Moreover, a sudden drop of the ground-state fidelity and the divergence of fidelity susceptibility at the topological quantum critical point have been used as a proper indicators for the topological quantum phase transition, which signals the appearance of Majorana fermions. We also present the scaling analysis of ground-state fidelity near the critical point that manifests the universal information about the topological phase transition.
We investigate the effect of next-nearest-neighbor hopping on topological quantum phase transitions (QPTs) that are characterized by the number of Majorana zero modes in one-dimensional (1D) p-wave superconducting systems. We also numerically analyze the scaling behavior and the universality of the Berry phase (BP) of the ground state close to the critical point. For critical line (I), the derivative of the ground-state BP is nonanalytic at the phase boundaries. For the phase boundary (II), a noncontractible BP of the ground state itself is also a witness of quantum phase transition.
The geometry of quantum states can be an indicator of criticality, yet it remains less explored under non-Hermitian topological conditions. In this work, we unveil diverse scalings of the quantum geometry over the ground-state manifold close to different topological phase transitions in a non-Hermitian long-range extension of the Kitaev chain. The derivative of the geometric phase, as well as its scaling behavior, shows that systems with different long-range couplings can belong to a distinct critical class. Near certain criticalities, we further find that the Wannier state correlation function associated with the extended Berry connection of the ground state exhibits spatially anomalous behaviors. Finally, we analyze the scaling of the quantum geometric tensor near phase transitions across exceptional points, shedding light on the emergence of a distinct critical class.
Multipartite nonlocality and Bell-type inequalities are used to characterize topological quantum phase transitions (QPTs) in a spinless fermion quantum wire, where both uniform potentials and incommensurate potentials are considered. First, the nonlocality measures show clear signals at the critical points in both the uniform model and the incommensurate model. It indicates that these QPTs are accompanied by dramatic changes of multipartite quantum correlations in the ground states. Second, finite-size scaling analysis is carried out. In particular, in the incommensurate model where translation invariance is broken, with some rescaling techniques, we successfully establish the scaling formula in the large-LLL limit. Finally, the full phase diagram of the model with mixed potentials is figured out. We find a region which is featured with strong randomness in the large-LLL limit. The structure of this region is revealed by analyzing the energy spectrum, and an efficient approach to characterize this region is proposed.