Universality of modulation length (and time) exponents

TL;DR

This paper introduces a universal exponent, nuL, describing the crossover from fixed to varying modulation lengths in systems with a parameter lambda, applicable across various physical models including Fermi systems and magnetic models.

Contribution

It defines and analyzes a new universal exponent nuL for modulation length crossovers, extending the concept to diverse physical systems and response functions.

Findings

nuL generally equals 1/2 across systems

nuL can take rational values in special cases

The framework applies to Fermi systems, magnetic models, and topological insulators

Abstract

We study systems with a crossover parameter lambda, such as the temperature T, which has a threshold value lambda* across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We report on a new exponent, nuL, characterizing the universal nature of this crossover. These exponents, similar to standard correlation length exponents, are obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex k-plane (or omega-plane) as the parameter lambda is varied. Near the crossover, the characteristic modulation wave-vector KR on the variable modulation length "phase" is related to that on the fixed modulation length side, q via |KR-q|\propto|T-T*|^{nuL}. We find, in general, that nuL=1/2. In some special instances, nuL may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value, lambda*. We discuss extensions of this result to multiple other arenas. These include the ANNNI model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times) but also to the standard correlation lengths (or times). We introduce the notion of a Josephson timescale. We comment on the presence of "chaotic" modulations in "soft-spin" and other systems. These relate to glass type features. We discuss applications to Fermi systems - with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in strongly correlated electronic systems.