A novel quantum Shannon entropy as a sensitivity of Anderson transitions: It simultaneously relates to periodic and antiperiodic boundary conditions
Abstract
In general, extended states are more sensitive to boundary conditions (BCs) than localized ones. We expand an eigenstate |βp > of a Hamiltonian with periodic BC as the superposition of all eigenstates |β_a> of the same Hamiltonian but with antiperiodic BC, i.e., |βp >=Σβ_\textit{a=1}N |βa> <βa |βp>=Σβa=1N Cβa|βa> , where Cβa=<βa |βp> and N is the Hilbert space dimension. Thus, we propose a novel quantum Shannon entropy S_b=-Σβa=1N |Cβa|^2log |Cβa|2 , which simultaneously relates to periodic and antiperiodic BCs. It measures the information of |βp > contained in all the N eigenstates |β_a> . Based on it, we define a reduced quantum Shannon entropy S_b^r=1-S_b/log{N} , where 0≤ S_b^r≤1 . We apply it to several typical models. Results show that < S_b^r> are relatively smaller for delocalized states and are less than and close to one for localized states as N increases. There are rapid changes in < S_b^r> around Anderson transition points. Therefore, S_br can be used as a sensitivity measure to characterize Anderson transitions.
- Publication:
-
EPL (Europhysics Letters)
- Pub Date:
- May 2018
- DOI:
- 10.1209/0295-5075/122/37002
- Bibcode:
- 2018EL....12237002G