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Chinese Physics Letters

Von Neumann Entropy of an Electron in One-Dimensional Determined Potentials

Gong Long-Yan, Tong Pei-Qing

Published Oct 27, 2005• 10 citations
Publisher

Abstract

By using the measure of von Neumann entropy, we numerically investigate quantum entanglement of an electron moving in the one-dimensional Harper model and in the one-dimensional slowly varying potential model. The delocalized and localized eigenstates can be distinguished by von Neumann entropy of the individual eigenstates. There are drastic decreases in von Neumann entropy of the individual eigenstates at mobility edges. In the curve of the spectrum averaged von Neumann entropy as a function of potential parameter λ, a sharp transition exists at the metal–insulator transition point λ c = 2. It is found that the von Neumann entropy is a good quantity to reflect localization and metal–insulator transition.

DOI
10.1088/0256-307x/22/11/009

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By using the measure of von Neumann entropy, we numerically investigate quantum entanglement of an electron moving in the one-dimensional Harper model and in the one-dimensional slowly varying potential model. The delocalized and localized eigenstates can be distinguished by von Neumann entropy of the individual eigenstates. There are drastic decreases in von Neumann entropy of the individual eigenstates at mobility edges. In the curve of the spectrum averaged von Neumann entropy as a function of potential parameter λ, a sharp transition exists at the metal–insulator transition point λ c = 2. It is found that the von Neumann entropy is a good quantity to reflect localization and metal–insulator transition.
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With the help of von Neumann entropy, we study numerically the localization properties of two interacting particles (TIP) with on-site interactions in one-dimensional disordered, quasiperiodic, and slowly varying potential systems, respectively. We find that for TIP in disordered and slowly varying potential systems, the spectrum-averaged von Neumann entropy ⟨Ev⟩⟨{E}_{v}⟩⟨Ev​⟩ first increases with interaction UUU until its peak, then decreases as UUU gets larger. For TIP in the Harper model [S. N. Evangelou and D. E. Katsanos, Phys. Rev. B 56, 12797 (1997)], the functions of ⟨Ev⟩⟨{E}_{v}⟩⟨Ev​⟩ versus UUU are different for particles in extended and localized regimes. Our numerical results indicate that for these two-particle systems, the von Neumann entropy is a suitable quantity to characterize the localization properties of particle states. Moreover, our studies propose a consistent interpretation of the discrepancies between previous numerical results.
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The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.