Trace maps associated with general two-letter substitution rules

M. Kolá and M. K. Ali
Phys. Rev. A 42, 7112 – Published 1 December 1990
PDFExport Citation

Abstract

Spectral properties, as determined by trace maps, of the one-dimensional chains (layered structures) constructed according to general two-letter substitution rules are investigated. In all trace maps thus obtained an important role is played by the quantity I=x2+y2+z2-xyz-4. However, only a very small fraction of all such trace maps are similar to the Fibonacci golden-mean trace map is that I is their invariant. In addition to the known case of the precious-mean lattices (precious means are ratios of the form 1/2[m+(m2+4)1/2], m being any positive integer; m=1 gives the golden mean), we have identified two new large classes of substitution rules that give trace maps with invariant I. One of them is a superset of the precious-mean lattices. All other cases represent a vast assortment of different trace maps (and thus the potential for various hitherto unexplored spectral properties) with a unifying feature that the set I=0 plays the role of an attractor in the trace space. In most (but not all) cases, two chains with identical trace maps (and thus identical spectra) are locally isomorphic. Generally, local isomorphism equivalence classes seem to be subsets of identical spectrum equivalence classes.

  • Received 25 January 1990

DOI:https://doi.org/10.1103/PhysRevA.42.7112

©1990 American Physical Society

Authors & Affiliations

M. Kolá

  • The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama, 351-01 Japan

M. K. Ali

  • Department of Physics, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4

References (Subscription Required)

Click to Expand
Issue

Vol. 42, Iss. 12 — December 1990

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review A

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×