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Trace and antitrace maps for aperiodic sequences: Extensions and applications

Wang, Xiaoguang; Grimm, Uwe and Schreiber, Michael (2000). Trace and antitrace maps for aperiodic sequences: Extensions and applications. Physical Review B, 62(21) pp. 14020–14031.

DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1103/PhysRevB.62.14020
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Abstract

We study aperiodic systems based on substitution rules by means of a transfer-matrix approach. In addition to the well-known trace map, we investigate the so-called “antitrace” map, which is the corresponding map for the difference of the off-diagonal elements of the 2x2 transfer matrix. The antitrace maps are obtained for various binary, ternary, and quaternary aperiodic sequences, such as the Fibonacci, Thue-Morse, period-doubling, Rudin-Shapiro sequences, and certain generalizations. For arbitrary substitution rules, we show that not only trace maps, but also antitrace maps exist. The dimension of our antitrace map is r(r+1)/2, where r denotes the number of basic letters in the aperiodic sequence. Analogous maps for specific matrix elements of the transfer matrix can also be constructed, but the maps for the off-diagonal elements and for the difference of the diagonal elements coincide with the antitrace map. Thus, from the trace and antitrace map, we can determine any physical quantity related to the global transfer matrix of the system. As examples, we employ these dynamical maps to compute the transmission coefficients for optical multilayers, harmonic chains, and electronic systems.

Item Type: Journal Article
ISSN: 1098-0121
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 12317
Depositing User: Uwe Grimm
Date Deposited: 14 Nov 2008 10:59
Last Modified: 02 Dec 2010 20:15
URI: http://oro.open.ac.uk/id/eprint/12317
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