Localization Properties of a Quasiperiodic Ladder under Physical Gain and Loss: Tuning of Critical Points, Mixed-Phase Zone and Mobility Edge

David Laroze,Santanu K. Maiti,Souvik Roy,Judith Helena Ojeda Silva,Laura M. Pérez

doi:10.3390/ma15020597

Abstract

We explore the localization properties of a double-stranded ladder within a tight-binding framework where the site energies of different lattice sites are distributed in the cosine form following the Aubry–André–Harper (AAH) model. An imaginary site energy, which can be positive or negative, referred to as physical gain or loss, is included in each of these lattice sites which makes the system a non-Hermitian (NH) one. Depending on the distribution of imaginary site energies, we obtain balanced and imbalanced NH ladders of different types, and for all these cases, we critically investigate localization phenomena. Each ladder can be decoupled into two effective one-dimensional (1D) chains which exhibit two distinct critical points of transition from metallic to insulating (MI) phase. Because of the existence of two distinct critical points, a mixed-phase (MP) zone emerges which yields the possibility of getting a mobility edge (ME). The conducting behaviors of different energy eigenstates are investigated in terms of inverse participation ratio (IPR). The critical points and thus the MP window can be selectively controlled by tuning the strength of the imaginary site energies which brings a new insight into the localization aspect. A brief discussion on phase transition considering a multi-stranded ladder was also given as a general case, to make the present communication a self-contained one. Our theoretical analysis can be utilized to investigate the localization phenomena in different kinds of simple and complex quasicrystals in the presence of physical gain and/or loss.

Highlights

The phenomenon of Anderson localization [1,2] was explored almost seven decades ago and it remains a highly active area of research in the discipline of condensed matter physics [3,4] since the localization behavior helps us understand how the mobility of carriers is affected by disorder
We focus on the central feature of our analysis, i.e., the tuning of critical points and the localization phenomena by means of the NH parameter δ
The critical points of phase transition for both the two decoupled chains, i.e., Wc1 and Wc2 become reduced with the increase in the interaction parameter δ, as it enhances more scattering at the lattice sites

Summary

Introduction

The phenomenon of Anderson localization [1,2] was explored almost seven decades ago and it remains a highly active area of research in the discipline of condensed matter physics [3,4] since the localization behavior helps us understand how the mobility of carriers is affected by disorder. We add imaginary site energies at all distinct lattice sites, along with the cosine modulation following the AAH form, and studied the interplay between the environmental interaction (viz, the NH quantity) [46–51] and the correlated disorder on the localization phenomena. Both balanced and imbalanced NH distributions are taken into account [52,53] for a comprehensive analysis. For all these four types of NH AAH ladders, we critically investigate the localization properties and specific role of the NH factor.

Decoupling of Two-Stranded AAH Ladder into Two Effective 1D Chains

Theoretical Prescription

Numerical Results and Discussion

Two-Stranded AAH Ladders in Presence of Physical Gain and/or Loss

Eigenvalue Spectrum

Tuning of Critical Points, Mixed-Phase Zone and Mobility Edge

M-Stranded Ladder: A General Case

Closing Remarks