We explore a distinguishability
D and a visibility
V, satisfying the relation
D2+V2=1, to measure a single electron's particlelike and wavelike behaviors, respectively. For several typical one-dimensional nonuniform lattice systems, we find that
D2 is less than, equal to, and greater than
V2 for delocalized, critical, and localized states, respectively. In this sense, the Anderson transition can be viewed as a transition from relatively more particlelike behavior to relatively more wavelike behavior.
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.
The clock operator U and shift operator V are higher-dimensional Pauli operators. Just recently, tighter uncertainty relations with respect to U and V were derived, and we apply them to study the electron localization properties in several typical one-dimensional nonuniform lattice systems. We find that uncertainties Δ U 2 are less than, equal to, and greater than uncertainties Δ V 2 for extended, critical, and localized states, respectively. The lower bound LB of the uncertainty relation is relatively large for extended states and small for localized states. Therefore, in combination with traditional quantities, for instance inverse participation ratio, these quantities can be as novel indexes to reflect Anderson localization.