Improved unfolding by detrending of statistical fluctuations in quantum spectra

Exploring universality of the β-Gaussian ensemble in complex networks via intermediate eigenvalue statistics

The eigenvalue statistics are an important tool to capture localization to delocalization transition in physical systems. Recently, a β-Gaussian ensemble is being proposed as a single parameter to describe the intermediate eigenvalue statistics of many physical systems. It is critical to explore the universality of a β-Gaussian ensemble in complex networks. In this work, we study the eigenvalue statistics of various network models, such as small-world, Erdős-Rényi random, and scale-free networks, as well as in comparing the intermediate level statistics of the model networks with that of a β-Gaussian ensemble. It is found that the nearest-neighbor eigenvalue statistics of all the model networks are in excellent agreement with the β-Gaussian ensemble. However, the β-Gaussian ensemble fails to describe the intermediate level statistics of higher order eigenvalue statistics, though there is qualitative agreement till n<4. Additionally, we show that the nearest-neighbor eigenvalue statistics of the β-Gaussian ensemble is in excellent agreement with the intermediate higher order eigenvalue statistics of model networks.

Eigenvalue ratio statistics of complex networks: Disorder versus randomness

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (w_{c}) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate w_{c} with the time taken by maximum entropy random walker to reach the steady state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behavior.

Diagnosing first- and second-order phase transitions with probes of quantum chaos

We explore quantum phase transitions using two probes of quantum chaos: out-of-time-order correlators (OTOCs) and the r-parameter obtained from the level spacing statistics. In particular, we address p-spin models associated with quantum annealing or reverse annealing. Quantum annealing triggers first-order or second-order phase transitions, which is crucial for the performance of quantum devices. We find that the time-averaging OTOCs for the ground state and the average r-parameter change behavior around the corresponding transition points, diagnosing the phase transition. Furthermore, they can identify the order (first or second) of the phase transition by their behavior at the quantum transition point, which changes abruptly (smoothly) in the case of first-order (second-order) phase transitions.

Effects of scarring on quantum chaos in disordered quantum wells

The suppression of chaos in quantum reality is evident in quantum scars, i.e. in enhanced probability densities along classical periodic orbits. They provide opportunities in controlling quantum transport in nanoscale quantum systems. Here, we study energy level statistics of perturbed two-dimensional quantum systems exhibiting recently discovered, strong perturbation-induced quantum scarring. In particular, we focus on the effect of local perturbations and an external magnetic field on both the eigenvalue statistics and scarring. Energy spectra are analyzed to investigate the chaoticity of the quantum system in the context of the Bohigas-Giannoni-Schmidt conjecture. We find that in systems where strong perturbation-induced scars are present, the eigenvalue statistics are mostly mixed, i.e. between Wigner-Dyson and Poisson pictures in random matrix theory. However, we report interesting sensitivity of both the eigenvalue statistics to the perturbation strength, and analyze the physical mechanisms behind this effect.

Criticality in Brownian ensembles

A Brownian ensemble appears as a nonequilibrium state of transition from one universality class of random matrix ensembles to another one. The parameter governing the transition is, in general, size-dependent, resulting in a rapid approach of the statistics, in infinite size limit, to one of the two universality classes. Our detailed analysis, however, reveals the appearance of a new scale-invariant spectral statistics, nonstationary along the spectrum, associated with multifractal eigenstates, and different from the two end-points if the transition parameter becomes size-independent. The number of such critical points during transition is governed by a competition between the average perturbation strength and the local spectral density. The results obtained here have applications to wide-ranging complex systems, e.g., those modeled by multiparametric Gaussian ensembles or column constrained ensembles.

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

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.

Random-matrix spectra as a time series

Spectra of ordered eigenvalues of finite random matrices are interpreted as a time series. Data-adaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding possible artifacts introduced by standard unfolding techniques. The fluctuation modes are scale invariant and follow different power laws for Poisson and Gaussian ensembles, which already during the unfolding allows one to distinguish the two cases.

Criticality and long-range correlations in time series in classical and quantum systems

We present arguments which indicate that a transitional state in between two different regimes implies the occurrence of 1/f time series and that this property is generic in both classical and quantum systems. Our study focuses on two particular examples: the one-dimensional module-1 logistic map and nuclear excitation spectra obtained with a schematic shell-model Hamiltonian. We suggest that a transitional point is characterized by the long-range correlations implied by 1/f time series. We apply a Fourier spectral analysis and the detrended fluctuation analysis method to study the fluctuations to each system.