Determination of scale invariance in random-matrix spectral fluctuations without unfolding

Abstract

We apply the singular value decomposition (SVD) method, based on normal-mode analysis, to decompose the spectra of finite random matrices of standard Gaussian ensembles in trend and fluctuation modes. We use the fact that the fluctuation modes are scale invariant and follow a power law, to characterize the transition between the extreme regular and chaotic cases. Thereby, we quantify the quantum chaos in systems described by random matrix theory in a direct way, without performing any previous unfolding procedure, and therefore, avoiding possible artifacts.

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

“…Plotting the singular values squared λ k = σ 2 k according to their rank is knows as the singular value scree plot [92][93][94] and much information can be gleaned from it. This approach has been applied to the spectrum of disordered systems in several studies 41,[86][87][88][89] , the first few λ k (k ≤ O(1)) correspond to global features…”

“…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 .…”

“…As will be discussed in detail in the appendix, SVD essentially returns a set of modes which can be used to construct the energy spectra of the different realizations in the ensemble. Arranging the modes according to the size of their amplitude squared, λ k (where k = 1 is the largest), the first few λ k (O(1)) correspond to global features of the spectra 41,[86][87][88] . Thus one can globally unfold the spectra by filtering out these modes when reconstructing the spectrum.…”

“…A different way to obtain a comprehensive picture of the behavior of the energy spectrum is to plot λ k vs. k, also known as a scree plot. Usually, a power law behavior λ ∼ k −α , is detected for certain ranges of k. The power law exponent corresponds to the statistics of the energy spectrum with α = 2 for the Poisson behavior, α = 1 for the Wigner regime 41,[86][87][88] . For energies larger than E T h (small values of k) in the metallic regime of a single particle Anderson model α = 1 + d/2 (where d is the dimensionality) 89 .…”

On intermediate statistics across many-body localization transition

“…The distinction between these two parts of ρ(E) is to some extent arbitrary and may lead to dubious outcomes see, for instance, the differences between local unfolding and Gaussian broadening [35]. Over the past years, there is a constant effort to tackle such problems from the mathematical and computational viewpoints [36,37].…”