Localization–delocalization transition for disordered cubic harmonic lattices

Abstract

We study numerically the disorder-induced localization-delocalization phase transitions that occur for mass and spring constant disorder in a three-dimensional cubic lattice with harmonic couplings. We show that, while the phase diagrams exhibit regions of stable and unstable waves, the universality of the transitions is the same for mass and spring constant disorder throughout all the phase boundaries. The combined value for the critical exponent of the localization lengths of ν = 1.550(-0.017)(+0.020) confirms the agreement with the universality class of the standard electronic Anderson model of localization. We further support our investigation with studies of the density of states, the participation numbers and wave function statistics.

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

“…For instance, the critical points and exponents of the three-dimensional (3D) AM extracted from the quantities α q and α m [43], the level spacing [34], the level number variance [44], and the transfer matrix [45] are close to one another. The critical exponents of the 3D AM estimated by multifractal exponents [42,43] agree well with that of a truncated Lennard-Jones liquid (TLJL) evaluated by the level spacing statistics [36] and that of the disordered lattices with random mass or spring distributions (DLMS) calculated by the transfer matrix [14,15]. Among all approaches, the multifractal finite-size scaling (MFSS) is a recently proposed tool, which combines multifractal analysis with finite-size scaling [42,43].…”

“…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].…”

“…, ν = 1.45 +0.2 −0.2[35], ν = 1.58 +0.03 −0.03[42], ν = 1.59 +0.012 −0.011[43], and ν = 1.57 +0.02 −0.02[45] for the 3D AM, ν = 1.61 +0.07 −0.06 for electrons with topological disorder[62], ν = 1.55 +0.002 −0.002 for disordered phonons[63], ν ≈ 1.5-1.6 for electrons in fcc and bcc lattices[64], ν = 1.550 +0.020 −0.017[14] and ν ≈ 1.57-1.59[15] for phonons in the DLMS of various disorder distributions, ν p = 1.55 +0.09 −0.09 for the real and ν n = 1.60 +0.07 −0.07 for the imaginary INMs of the TLJL[36], and ν = 1.63 +0.05 −0.05[65] and ν = 1.59 +0.01 −0.01…”

Azahelicene Superbases as MAILD Matrices for Acidic Analytes

“…The presence of the GOE statistics near the BP is also a proof for delocalization of the vibrational excitations. As stated in the introduction, the mobility edge for vibrational disordered systems is situated right below the Debye level and not at the BP 18, 42–44. In this regime the DOS is known 59, 60 to decay exponentially.…”

“…The remaining field theory for Λ could in principle be used for deriving the localization properties of the model. It is well known that in stable systems (no negative eigenvalues present) force‐constant disorder leads to localization only for frequencies in the upper band‐tail, i.e ., near the Debye frequency 18, 42–44. Therefore the first step in this scheme, namely, the derivation of the self‐consistent Born approximation (SCBA) from a saddle point is performed only.…”