A novel quantum Shannon entropy as a sensitivity of Anderson transitions: It simultaneously relates to periodic and antiperiodic boundary conditions

Longyan Gong; Jingjing Zhang; Kaixin Ma; Xingfei Zhou; Xuechao Zhai; Weiwen Cheng; Shengmei Zhao

doi:10.1209/0295-5075/122/37002

All issues

Volume 122 / No 3 (May 2018)

EPL, 122 3 (2018) 37002

Abstract

Issue		EPL Volume 122, Number 3, May 2018


Article Number		37002
Number of page(s)		6
Section		Condensed Matter: Electronic Structure, Electrical, Magnetic and Optical Properties
DOI		https://doi.org/10.1209/0295-5075/122/37002
Published online		27 June 2018

EPL, 122 (2018) 37002

A novel quantum Shannon entropy as a sensitivity of Anderson transitions: It simultaneously relates to periodic and antiperiodic boundary conditions

Longyan Gong¹^,2^,3^(a), Jingjing Zhang¹^,2, Kaixin Ma¹^,2, Xingfei Zhou¹^,3, Xuechao Zhai¹^,3, Weiwen Cheng² and Shengmei Zhao²

¹ Department of Applied Physics, Nanjing University of Posts and Telecommunications - Nanjing, 210003, China
² Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunications Nanjing, 210003, China
³ New Energy Technology Engineering Laboratory of Jiangsu Province, Nanjing University of Posts and Telecommunications - Nanjing, 210003, China

^(a) lygong@njupt.edu.cn (corresponding author)

Received: 23 March 2018
Accepted: 31 May 2018

Abstract

In general, extended states are more sensitive to boundary conditions (BCs) than localized ones. We expand an eigenstate $|\beta_p \rangle$ of a Hamiltonian with periodic BC as the superposition of all eigenstates $|\beta_a\rangle$ of the same Hamiltonian but with antiperiodic BC, i.e., $|\beta_p \rangle=\Sigma_{\beta_\textit{a}=1}^N |\beta_{a}\rangle \langle\beta_{a} |\beta_{p}\rangle=\Sigma_{\beta_{a}=1}^N C_{\beta_{a}}|\beta_{a}\rangle$ , where $C_{\beta_{a}}=\langle\beta_{a} |\beta_{p}\rangle$ and N is the Hilbert space dimension. Thus, we propose a novel quantum Shannon entropy $S_b=-\Sigma_{\beta_{a}=1}^N |C_{\beta_{a}}|^2\log |C_{\beta_{a}}|^2$ , which simultaneously relates to periodic and antiperiodic BCs. It measures the information of $|\beta_p \rangle$ contained in all the N eigenstates $|\beta_a\rangle$ . Based on it, we define a reduced quantum Shannon entropy $S_b^r=1-S_b/\log{N}$ , where $0\leq S_b^r\leq1$ . We apply it to several typical models. Results show that $\langle S_b^r\rangle$ 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 $\langle S_b^r\rangle$ around Anderson transition points. Therefore, $S_b^r$ can be used as a sensitivity measure to characterize Anderson transitions.

PACS: 72.15.Rn – Localization effects (Anderson or weak localization) / 71.23.An – Theories and models; localized states / 71.30.+h – Metal-insulator transitions and other electronic transitions

© EPLA, 2018

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