Predictions Of Random Matrix Theory Research Articles

Structural stability hypothesis of dual unitary quantum chaos

Having spectral correlations that, over small enough energy scales, are described by random matrix theory is regarded as the most general defining feature of quantum chaotic systems as it applies in the many-body setting and away from any semiclassical limit. Although this property is extremely difficult to prove analytically for generic many-body systems, a rigorous proof has been achieved for dual-unitary circuits—a special class of local quantum circuits that remain unitary upon swapping space and time. Here we consider the fate of this property when moving from dual-unitary to generic quantum circuits focusing on the , i.e., the Fourier transform of the two-point correlation. We begin with a numerical survey that, in agreement with previous studies, suggests that there exists a finite region in parameter space where dual-unitary physics is stable and spectral correlations are still described by random matrix theory, although up to a maximal quasienergy scale. To explain these findings, we develop a perturbative expansion: it recovers the random matrix theory predictions, provided the terms occurring in perturbation theory obey a relatively simple set of assumptions. We then provide numerical evidence and a heuristic analytical argument supporting these assumptions. Published by the American Physical Society 2024

Random matrix theory approach to quantum Fisher information in quantum ergodic systems.

We theoretically investigate quantum parameter estimation in quantum chaotic systems. Our analysis is based on an effective description of quantum ergodic systems in terms of a random matrix Hamiltonian. Based on this approach, we derive an analytical expression for the time evolution of the quantum Fisher information (QFI), which we find to have three distinct timescales. Initially, the QFI increase is quadratic in time, characterizing the timescale over which initial information is extractable from local measurements only. This quickly passes into linear increase with slope determined by the decay rate of the measured spin observable. When the information is fully spread among all degrees of freedom, a second quadratic timescale determines the long-time behavior of the QFI. We test our random matrix theory prediction with the exact diagonalization of a nonintegrable spin system, focusing on the estimation of a local magnetic field by measurements of the many-body state. Our numerical calculations agree with the effective random matrix theory approach and show that the information on the local Hamiltonian parameter is distributed throughout the quantum system during the quantum thermalization process.

Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes

Studies of many-body non-Hermitian parity-time (PT)-symmetric quantum systems are attracting a lot of interest due to their relevance in research areas ranging from quantum optics and continuously monitored dynamics to Euclidean wormholes in quantum gravity and dissipative quantum chaos. While a symmetry classification of non-Hermitian systems leads to 38 universality classes, we show that, under certain conditions, PT-symmetric systems are grouped into 24 universality classes. We identify 14 of them in a coupled two-site Sachdev-Ye-Kitaev (SYK) model and confirm the classification by spectral analysis using exact diagonalization techniques. Intriguingly, in 4 of these 14 universality classes, AIIIν, BDIν†, BDI++ν, and CI−−ν, we identify a basis in which the SYK Hamiltonian has a block structure in which some blocks are rectangular, with ν∈N the difference between the number of rows and columns. We show analytically that this feature leads to the existence of ν robust purely eigenvalues, whose level statistics follow the predictions of Hermitian random matrix theory for classes A, AI, BDI, and CI, respectively. We have recently found that this ν is a topological invariant, so these classes are topological. By contrast, nontopological real eigenvalues display a crossover between Hermitian and non-Hermitian level statistics. Similarly to the case of Lindbladian dynamics, the reduction of universality classes leads to unexpected results, such as the absence of Kramers degeneracy in a given sector of the theory. Another novel feature of the classification scheme is that different sectors of the PT-symmetric Hamiltonian may have different symmetries. Published by the American Physical Society 2024

Many-body quantum chaos in mixtures of multiple species.

We study spectral correlations in many-body quantum mixtures of fermions, bosons, and qubits with periodically kicked spreading and mixing of species. We take two types of mixing, namely, Jaynes-Cummings and Rabi, respectively, satisfying and breaking the conservation of a total number of species. We analytically derive the generating Hamiltonians whose spectral properties determine the spectral form factor in the leading order. We further analyze the system-size (L) scaling of Thouless time t^{*}, beyond which the spectral form factor follows the prediction of random matrix theory. The L dependence of t^{*} crosses over from lnL to L^{2} with an increasing Jaynes-Cummings mixing between qubits and fermions or bosons in a finite-size chain, and it finally settles to t^{*}∝O(L^{2}) in the thermodynamic limit for any mixing strength. The Rabi mixing between qubits and fermions leads to t^{*}∝O(lnL), previously predicted for single species of qubits or fermions without total-number conservation.

Closed and open superconducting microwave waveguide networks as a model for quantum graphs.

We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of equal width but incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equationis effectively one-dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90^{∘} and 120^{∘}, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions, strongly depends on frequency and is nonsymmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the other hand, at a Y junction, similarity can be achieved in a certain frequency range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.

Decomposing cryptocurrency high-frequency price dynamics into recurring and noisy components

This paper investigates the temporal patterns of activity in the cryptocurrency market with a focus on Bitcoin, Ethereum, Dogecoin, and WINkLink from January 2020 to December 2022. Market activity measures—logarithmic returns, volume, and transaction number, sampled every 10 s, were divided into intraday and intraweek periods and then further decomposed into recurring and noise components via correlation matrix formalism. The key findings include the distinctive market behavior from traditional stock markets due to the nonexistence of trade opening and closing. This was manifested in three enhanced-activity phases aligning with Asian, European, and U.S. trading sessions. An intriguing pattern of activity surge in 15-min intervals, particularly at full hours, was also noticed, implying the potential role of algorithmic trading. Most notably, recurring bursts of activity in bitcoin and ether were identified to coincide with the release times of significant U.S. macroeconomic reports, such as Nonfarm payrolls, Consumer Price Index data, and Federal Reserve statements. The most correlated daily patterns of activity occurred in 2022, possibly reflecting the documented correlations with U.S. stock indices in the same period. Factors that are external to the inner market dynamics are found to be responsible for the repeatable components of the market dynamics, while the internal factors appear to be substantially random, which manifests itself in a good agreement between the empirical eigenvalue distributions in their bulk and the random-matrix theory predictions expressed by the Marchenko–Pastur distribution. The findings reported support the growing integration of cryptocurrencies into the global financial markets.

Average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians.

To which degree the average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians agrees with that of random pure states is a question that has attracted considerable attention in the recent years. While there is substantial evidence that the leading (volume-law) terms are identical, which and how subleading terms differ between them is less clear. Here we carry out state-of-the-art full exact diagonalization calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking terms to address this question in the absence and presence of U(1) symmetry, respectively. We first introduce the notion of maximally chaotic regime, for the chain sizes amenable to full exact diagonalization calculations, as the regime in Hamiltonian parameters in which the level spacing ratio, the distribution of eigenstate coefficients, and the entanglement entropy are closest to the random matrix theory predictions. In this regime, we carry out a finite-size scaling analysis of the subleading terms of the average entanglement entropy of midspectrum eigenstates when different fractions ν of the spectrum are included in the average. We find indications that, for ν→0, the magnitude of the negative O(1) correction is only slightly greater than the one predicted for random pure states. For finite ν, following a phenomenological approach, we derive a simple expression that describes the numerically observed ν dependence of the O(1) deviation from the prediction for random pure states.

Unfolding a composed ensemble of energy spectra using singular value decomposition

In comparing the behavior of an energy spectrum to the predictions of random matrix theory one must transform the spectrum such that the averaged level spacing is constant, a procedure known as unfolding. Once energy spectra belong to an ensemble where there are large realization-to-realization fluctuations the canonical methods for unfolding fail. Here we show that singular value decomposition can be used even for the challenging situations where the ensemble is composed out of realizations originating from a different range of parameters resulting in a non-monotonous local density of states. This can be useful in experimental situations for which the physical parameters cannot be tightly controlled, or for situations for which the local density of states is strongly fluctuating.

Experimental study of the elastic enhancement factor in a three-dimensional wave-chaotic microwave resonator exhibiting strongly overlapping resonances.

We study the elastic enhancement factor and the two-point correlation function of the scattering matrix obtained from measurements of reflection and transmission spectra of a three-dimensional (3D) wave-chaotic microwave cavity in regions of moderate and large absorption. They are used to identify the degree of chaoticity of the system in the presence of strongly overlapping resonances, where other measures such as short- and long-range level correlations cannot be applied. The average value of the experimentally determined elastic enhancement factor for two scattering channels agrees well with random-matrix theory predictions for quantum chaotic systems, thus corroborating that the 3D microwave cavity exhibits the features of a fully chaotic system with preserved time-reversal invariance. To confirm this finding we analyzed spectral properties in the frequency range of lowest achievable absorption using missing-level statistics.

Properties of eigenmodes and quantum-chaotic scattering in a superconducting microwave Dirac billiard with threefold rotational symmetry

We report on experimental studies that were performed with a microwave Dirac billiard (DB), that is, a flat resonator containing metallic cylinders arranged on a triangular grid, whose shape has a threefold rotational (${C}_{3}$) symmetry. Its band structure exhibits two Dirac points (DPs) that are separated by a nearly flat band. We present a procedure that we employed to identify eigenfrequencies and to separate the eigenstates according to their transformation properties under rotation by $\frac{2\ensuremath{\pi}}{3}$ into the three ${C}_{3}$ subspaces. This allows us to verify previous numerical results of Zhang and Dietz [Phys. Rev. B 104, 064310 (2021)], thus confirming that the properties of the eigenmodes coincide with those of artificial graphene around the lower DP, and they are well described by a tight-binding model for a honeycomb-kagome lattice of corresponding shape. Above all, we investigate the properties of the wave-function components in terms of the fluctuation properties of the measured scattering matrix, which are numerically not accessible. They are compared to random-matrix theory predictions for quantum-chaotic scattering systems exhibiting extended or localized states in the interaction region, that is, the DB. Even in regions where the wave functions are localized, the spectral properties coincide with those of typical quantum systems with chaotic classical counterparts.

Distributions of the Wigner reaction matrix for microwave networks with symplectic symmetry in the presence of absorption.

We report on experimental studies of the distribution of the reflection coefficients, and the imaginary and real parts of Wigner's reaction (K) matrix employing open microwave networks with symplectic symmetry and varying size of absorption. The results are compared to analytical predictions derived for the single-channel scattering case within the framework of random-matrix theory (RMT). Furthermore, we performed Monte Carlo simulations based on the Heidelberg approach for the scattering (S) and K matrix of open quantum-chaotic systems and the two-point correlation function of the S-matrix elements. The analytical results and the Monte Carlo simulations depend on the size of absorption. To verify them, we performed experiments with microwave networks for various absorption strengths. We show that deviations from RMT predictions observed in the spectral properties of the corresponding closed quantum graph and attributed to the presence of nonuniversal short periodic orbits does not have any visible effects on the distributions of the reflection coefficients and the K and S matrices associated with the corresponding open quantum graph.

Level dynamics and avoided level crossings in driven disordered quantum dots

The statistical properties of the dynamics of energy levels are investigated in the case of two two-dimensional disordered quantum dot models with nearest neighbor hopping subjected to external time-dependent perturbations. While in the first model the external drivings are realized by a continuous variation of the on-site energies, in the second one it is generated by deformations of a parabolic potential. We concentrate on the effects of the potential on the localization properties and investigate the statistics of the energy level velocities and curvatures regarding their typical magnitudes and domain of agreement with the predictions of Random Matrix Theory (RMT) for the Gaussian Orthogonal, Unitary and Symplectic ensembles. Moreover, the statistical properties of the avoided level crossings are investigated in terms of the corresponding Landau-Zener parameters. We find that the strength of the Landau-Zener transitions exhibits universal behavior which also implies universal single-particle dynamics for slow perturbations independent of the disorder and potential strength, the system size and the symmetry class. These results can be verified experimentally by measurements of single-particle energy spectra in quantum dots.

Random matrix analysis of deep neural network weight matrices.

Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the weight matrices of trained deep neural networks using methods from random matrix theory (RMT) and show that the statistics of most of the singular values follow universal RMT predictions. This suggests that they are random and do not contain system specific information, which we investigate further by comparing the statistics of eigenvector entries to the universal Porter-Thomas distribution. We find that for most eigenvectors the hypothesis of randomness cannot be rejected, and that only eigenvectors belonging to the largest singular values deviate from the RMT prediction, indicating that they may encode learned information. In addition, a comparison with RMT predictions also allows to distinguish networks trained in different learning regimes-from lazy to rich learning. We analyze the spectral distribution of the large singular values using the Hill estimator and find that the distribution cannot in general be characterized by a tail index, i.e., is not of power-law type.

Zero energy states clustering in an elemental nanowire coupled to a superconductor

Nanoelectronic hybrid devices combining superconductors and a one-dimensional nanowire are promising platforms to realize topological superconductivity and its resulting exotic excitations. The bulk of experimental studies in this context are transport measurements where conductance peaks allow to perform a spectroscopy of the low lying electronic states and potentially to identify signatures of the aforementioned excitations. The complexity of the experimental landscape calls for a benchmark in an elemental situation. The present work tackles such a task using an ultra-clean carbon nanotube circuit. Specifically, we show that the combination of magnetic field, weak disorder and superconductivity can lead to states clustering at low energy, as predicted by the random matrix theory predictions. Such a phenomenology is very general and should apply to most platforms trying to realize topological superconductivity in 1D systems, thus calling for alternative probes to reveal it.

Tight-binding billiards.

Recent works have established universal entanglement properties and demonstrated validity of single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians. However, a common property of all quantum-chaotic quadratic Hamiltonians studied in this context so far is the presence of random terms that act as a source of disorder. Here we introduce tight-binding billiards in two dimensions, which are described by noninteracting spinless fermions on a disorder-free square lattice subject to curved open (hard-wall) boundaries. We show that many properties of tight-binding billiards match those of quantum-chaotic quadratic Hamiltonians: The average entanglement entropy of many-body eigenstates approaches the random matrix theory predictions and one-body observables in single-particle eigenstates obey the single-particle eigenstate thermalization hypothesis. On the other hand, a degenerate subset of single-particle eigenstates at zero energy (i.e., the zero modes) can be described as chiral particles whose wave functions are confined to one of the sublattices.

Fluctuation properties of the eigenfrequencies and scattering matrix of closed and open unidirectional graphs with chaotic wave dynamics.

We present experimental and numerical results for the fluctuation properties in the eigenfrequency spectra and of the scattering matrix of closed and open unidirectional quantum graphs, respectively. Unidirectional quantum graphs, that are composed of bonds connected by reflectionless vertices, were introduced by Akila and Gutkin [Akila and Gutkin, J. Phys. A: Math. Theor. 48, 345101 (2015)1751-811310.1088/1751-8113/48/34/345101]. The nearest-neighbor spacing distribution of their eigenvalues was shown to comply with random-matrix theory predictions for typical chaotic systems with completely violated time-reversal invariance. The occurrence of short periodic orbits confined to a fraction of the system, that lead in conventional quantum graphs to deviations of the long-range spectral correlations from the behavior expected for typical chaotic systems, is suppressed in unidirectional ones. Therefore, we pose the question whether such graphs may serve as a more appropriate model for closed and open chaotic systems with violated time-reversal invariance than conventional ones. We compare the fluctuation properties of their eigenvalues and scattering matrix elements and observe especially in the long-range correlations larger deviations from random-matrix theory predictions for the unidirectional graphs. These are attributed to a loss of complexity of the underlying dynamic, induced by the unidirectionality.

Estimating the spectral density of unstable scars

In quantum chaos, the spectral statistics generally follows the predictions of random matrix theory (RMT). A notable exception is given by scar states, that enhance probability density around unstable periodic orbits of the classical system, therefore causing significant deviations of the spectral density from RMT expectations. In this work, the problem is considered of both RMT-ruled and scarred chaotic systems coupled to an opening. In particular, predictions are derived for the spectral density of a chaotic Hamiltonian scattering into a single- or multiple channels. The results are tested on paradigmatic quantum chaotic maps on a torus. The present report develops the intuitions previously sketched in Lippolis (2019 EuroPhys. Lett. 126 10003).

Statistical and dynamical properties of the quantum triangle map

We study the statistical and dynamical properties of the quantum triangle map, whose classical counterpart can exhibit ergodic and mixing dynamics, but is never chaotic. Numerical results show that ergodicity is a sufficient condition for spectrum and eigenfunctions to follow the prediction of random matrix theory, even though the underlying classical dynamics is not chaotic. On the other hand, dynamical quantities such as the out-of-time-ordered correlator (OTOC) and the number of harmonics, exhibit a growth rate vanishing in the semiclassical limit, in agreement with the fact that classical dynamics has zero Lyapunov exponent. Our finding show that, while spectral statistics can be used to detect ergodicity, OTOC and number of harmonics are diagnostics of chaos.

Correlation structure analysis of the global agricultural futures market

This paper adopts the random matrix theory (RMT) to analyze the correlation structure of the global agricultural futures market from 2000 to 2020. It is found that the distribution of correlation coefficients is asymmetric and right skewed, and many eigenvalues of the correlation matrix deviate from the RMT prediction. The largest eigenvalue reflects a collective market effect common to all agricultural futures, the other largest deviating eigenvalues can be implemented to identify futures groups, and there are modular structures based on regional properties or agricultural commodities among the significant participants of their corresponding eigenvectors. Except for the smallest eigenvalue, other smallest deviating eigenvalues represent the agricultural futures pairs with highest correlations. This paper can be of reference and significance for using agricultural futures to manage risk and optimize asset allocation.

The sixth power moment of Dirichlet L-functions over rational function fields

We prove an asymptotic formula, with power savings, for the sixth moment of Dirichlet L-functions over rational function fields Fq(t), which is the exact function field analogue of the result by J.B. Conrey, H. Iwaniec, and K. Soundararajan for the corresponding family and averaging over Q. Our formula confirms also the existing random matrix theory predictions.