Anderson universality in a model of disordered phonons

Abstract

We consider the localization properties of a lattice of coupled masses and springs with random mass and spring constant values. We establish the full phase diagrams of the system for pure mass and pure spring disorder. The phase diagrams exhibit regions of stable as well as unstable wave modes. The latter are of interest for the instantaneous-normal-mode spectra of liquids and the nascent field of acoustic metamaterials. We show the existence of delocalization-localization transitions throughout the phase diagram and establish, by high-precision numerical studies, that the universality of these transitions is of the Anderson type.

Localization of phonons in mass-disordered alloys: A typical medium dynamical cluster approach

“…However, these are often based on uncontrolled approximations, and their region of validity is always in question. This is where numerically exact methods such as exact diagonalization (ED) 54,55 and transfer matrix method (TMM) 56 prove their mettle and provide very useful benchmarks for approximate theories. Recently, Monthus and Garel 57 use ED for relatively large system sizes to investigate the localization of phonons in mass-disordered systems.…”

Tight-binding chains with off-diagonal disorder: Bands of extended electronic states induced by minimal quasi–one-dimensionality

“…In two dimensions the states retain their exponential decay of amplitude, while in three dimensions a possibility of a metal-insulator transition arises. The results get adequate support from the calculations of the localization length [4,5], density of states [6] and the multi-fractality of the spectra and wave functions of spinless, non-interacting fermionic systems [7][8][9].The path breaking observation by Anderson [1], over the years, has extended its realm well beyond the electronic properties of disordered solid materials, and has been found out to be ubiquitous in a wide variety of systems. For example, one can refer to the field of localization of light, an idea pioneered about three decades ago…”

“…In two dimensions the states retain their exponential decay of amplitude, while in three dimensions a possibility of a metal-insulator transition arises. The results get adequate support from the calculations of the localization length [4,5], density of states [6] and the multi-fractality of the spectra and wave functions of spinless, non-interacting fermionic systems [7][8][9].…”

Localization-delocalization transition of the instantaneous normal modes of liquid water

“…Since interference is a fundamental property of waves, the LDT is expected to exist widely in nature. In addition to matter waves, this transition has been found in a variety of classical systems [3,[7][8][9][10][11][12][13][14][15]. Among others, the vibration of liquids belongs to an essential category [16][17][18][19][20][21][22][23].…”