Approaching Thouless Energy and Griffiths Regime in Random Spin Systems By Singular Value Decomposition

Abstract

We employ singular value decomposition to study the eigenvalue spectrums of random spin systems. It's shown the singular values λ k of the sample matrix in the ergodic phase contain two branches that both follow power-law behavior but with different power exponents α. By analyzing the singular vectors, it is verified the higher part of λ k with k > kTh are universal that follows random matrix theory, where kTh estimates the Thouless energy. We further demonstrate that α corresponds only to the exponential part of the level spacing distribution while is insensitive to the level repulsion, or equivalently the system's symmetry. Consequently, the power exponent α gives an underestimation for the many-body localization transition point, which, however, reveals the Griffiths regime in a transparent way.

Large scale behavior of the energy spectra of the quantum random antiferromagnetic Ising chain with mixed transverse and longitudinal fields

“…A similar difference between the ratio statistics corresponding to level spacing scale and scree plot behavior was very recently noted in Ref. 90 for the Heisenberg chain.…”

“…The small energy scales show all the universal features expected in the metallic regime, while higher energy scales clearly are nonuniversal. A non Poissonian behavior at large energies has also been very recently seen for an other canonical microscopic model for MBL, the Heisenberg chain 90 . Nevertheless, E Ex is not observed there since only a small range of eigenvalues were considered there, similar to the P = 2048 case depicted in Fig.…”

“…In order to circumvent these problems we will use a different method to study the properties of the spectra, known as singular value decomposition (SVD). This method has been successfully applied to analyze the transition from Wigner to Poisson statistics in the Anderson transition [86][87][88] , to characterizing the NEE in the GRP model 41 , to study the large energy scale spectrum behavior beyond the Thouless energy in metallic systems 89 , and very recently to the MBL transition in the Heisenberg chain 90 .…”