Trace maps, invariants, and some of their applications

TL;DR

This paper explores the algebraic structure of trace maps in two-letter substitution rules, revealing new invariants and applying these findings to various physical systems like electronic spectra and spin models.

Contribution

It introduces new invariants beyond the Fricke character for trace maps and discusses their applications in physical models.

Findings

Identification of new invariants for trace maps

Application to electronic spectra and gap-labeling

Analysis of kicked two-level systems and 1D Ising model

Abstract

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x,y,z) = x^2 + y^2 + z^2 - 2 x y z - 1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.