Crossover in nonstandard random-matrix spectral fluctuations without unfolding

Abstract

Recently, singular value decomposition (SVD) was applied to standard Gaussian ensembles of random-matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the β-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a crossover between soft and rigid behavior. In this way, possible artifacts introduced applying unfolding techniques are avoided. By using the trend modes, we perform data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes.

Normal mode analysis of spectra of random networks

“…This random network ensemble has already been studied in Refs. [3,36,46] and is constructed as follows. Starting with the standard ER network, we add self-edges and further consider all edges to have random strengths.…”

“…Although this ensemble was already discussed in Ref. [3], there, the α values were chosen in such a way that the crossover in the long-range correlations of the spectral fluctuations, between the GOE and Poisson limits, was clearly visualized in the scree diagrams. The latter, as it is shown there, does not allow to see in detail the transition in the short-range correlations quantified by the NNSD and the P (r) distribution.…”

“…When nonstandard random-matrix ensembles are considered, such as the β-Hermite ensemble and the sparse matrix ensemble, the scale invariance of the fluctuations is lost. In these cases, a crossover, instead of a power law, is observed in the scree diagrams [3].…”

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

“…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.…”