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arXiv:0710.2586 · Physical Review A

Mode entanglement of an electron in one-dimensional determined and random potentials

Longyan Gong, Peiqing Tong

Published Apr 21, 2005• 16 citations
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Abstract

By using the measure of concurrence, mode entanglement of an electron moving in four kinds of one-dimensional determined and random potentials is studied numerically. The extended and localized states can be distinguished by mode entanglement. There are sharp transitions in concurrence at mobility edges. It provides that the mode entanglement may be a new index for a metal-insulator transition.

DOI
10.1103/physreva.71.042333

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By using the measure of concurrence, mode entanglement of an electron moving in four kinds of one-dimensional determined and random potentials is studied numerically. The extended and localized states can be distinguished by mode entanglement. There are sharp transitions in concurrence at mobility edges. It provides that the mode entanglement may be a new index for a metal-insulator transition.
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By mapping the Fock space of many local fermionic modes isomorphically onto a many-qubit space and using the measure of concurrence, this paper studies numerically the mode entanglement of two spinless electrons with on-site interaction U moving in the one-dimensional Harper model. Generally speaking, for electrons in extended regimes (potential parameter λ < 2), the spectrum-averaged concurrence N〈C〉 first decreases slowly as λ increases until its local minimum, then increases with λ until its peak at λ = 2, while for electrons in localized regimes (λ > 2), N〈C〉 decreases drastically as λ increases. The functions of N 〈C〉 versus λ are different for electrons in extended and localized regimes. The maximum of N〈C〉 occurs at the point λ = 2, which is the critical value in the one-dimensional single-particle Harper model. From these studies it can distinguish extended, localized and critical regimes for the two-particle system. It is also found for the same λ that the interaction U always induce the decreases of concurrence, i.e., the concurrence can reflect the localization effect due to the interaction. All these provide us a new quantity to understand the localization properties of eigenstates of two interacting particles.
We study in this work the ground-state entanglement properties of two models of noninteracting fermions moving in one dimension, that exhibit metal-insulator transitions. We find that entanglement entropy grows either logarithmically or in a power-law fashion in the metallic phase, thus violating the (one-dimensional version of) entanglement area law. No such violation is found in the insulating phase. We further find that characteristics of single fermion states at the Fermi energy (which can not be obtained from the many-fermion Slater determinant) is captured by the lowest energy single fermion mode of the entanglement Hamiltonian; this is particularly true at the metal-insulator transition point. Our results suggest entanglement is a powerful way to detect metal-insulator transitions, without knowledge of the Hamiltonian of the system.
As a hallmark of a pure quantum effect, quantum entanglement has provided unconventional routes to characterize condensed matter systems. Here, from the perspective of quantum entanglement, we disclose exotic quantum physics in non-Hermitian quasicrystals. We study a class of experimentally realizable models for non-Hermitian quasicrystal chains, in which asymmetric hopping and complex potential coexist. We diagnose the global phase diagram by means of entanglement from both the real-space and momentum-space partitions. By measuring the entanglement entropy, we numerically determine the metal-insulator transition point. We combine real-space and momentum-space entanglement spectra to complementarily characterize the delocalization phase and the localization phase. Inspired by the entanglement spectrum, we further analytically prove that a duality exists between the two phase regions. The transition point is self-dual and exact, further validating the numerical result from diagonalizing non-Hermitian matrices. Finally, we identify the mobility edge by means of entanglement.
We consider Anderson localization and the associated metal-insulator transition for non-interacting fermions in D = 1, 2 space dimensions in the presence of spatially correlated on-site random potentials. To assess the nature of the wavefunction, we follow a recent proposal to study momentum-space entanglement. For a D = 1 model with long-range disorder correlations, both the entanglement spectrum and the entanglement entropy allow us to clearly distinguish between extended and localized states based upon a single realization of disorder. However, for other models including the D = 2 case with long-range correlated disorder, we find that the method is not similarly successful. We analyze the reasons for its failure, concluding that the much desired generalization to higher dimensions may be problematic.