Spectral statistics of the two-body random ensemble revisited

Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with k-body interactions

Embedded random matrix ensembles with k-body interactions are well established to be appropriate for many quantum systems. For these ensembles the two point correlation function is not yet derived, though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues, say E and E^{'}. Fluctuation measures such as the number variance and Dyson-Mehta Δ_{3} statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with k-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called q-normal distribution. With this, the eigenvalue density can be expanded by starting with the q-normal form and using the associated q-Hermite polynomials He_{ζ}(x|q). Covariances S_{ζ}S_{ζ^{'}}[over ¯] (overline representing ensemble average) of the expansion coefficients S_{ζ} with ζ≥1 here determine the two-point function as they are a linear combination of the bivariate moments Σ_{PQ} of the two-point function. Besides describing all these, in this paper formulas are derived for the bivariate moments Σ_{PQ} with P+Q≤8, of the two-point correlation function, for the embedded Gaussian unitary ensembles with k-body interactions [EGUE(k)] as appropriate for systems with m fermions in N single particle states. Used for obtaining the formulas is the SU(N) Wigner-Racah algebra. These formulas with finite N corrections are used to derive formulas for the covariances S_{ζ}S_{ζ^{'}}[over ¯] in the asymptotic limit. These show that the present work extends to all k values, the results known in the past in the two extreme limits with k/m→0 (same as q→1) and k=m (same as q=0).

Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time

The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.

Exponential Ramp in the Quadratic Sachdev-Ye-Kitaev Model

A long period of linear growth in the spectral form factor provides a universal diagnostic of quantum chaos at intermediate times. By contrast, the behavior of the spectral form factor in disordered integrable many-body models is not well understood. Here we study the two-body Sachdev-Ye-Kitaev model and show that the spectral form factor features an exponential ramp, in sharp contrast to the linear ramp in chaotic models. We find a novel mechanism for this exponential ramp in terms of a high-dimensional manifold of saddle points in the path integral formulation of the spectral form factor. This manifold arises because the theory enjoys a large symmetry group. With finite nonintegrable interaction strength, these delicate symmetries reduce to a relative time translation, causing the exponential ramp to give way to a linear ramp.

Crossover in nonstandard random-matrix spectral fluctuations without unfolding

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.

Eigenstate thermalization in the two-dimensional transverse field Ising model. II. Off-diagonal matrix elements of observables

We study the matrix elements of few-body observables, focusing on the off-diagonal ones, in the eigenstates of the two-dimensional transverse field Ising model. By resolving all symmetries, we relate the onset of quantum chaos to the structure of the matrix elements. In particular, we show that a general result of the theory of random matrices, namely, the value 2 of the ratio of variances (diagonal to off-diagonal) of the matrix elements of Hermitian operators, occurs in the quantum chaotic regime. Furthermore, we explore the behavior of the off-diagonal matrix elements of observables as a function of the eigenstate energy differences and show that it is in accordance with the eigenstate thermalization hypothesis ansatz.

Eigenstate thermalization in the two-dimensional transverse field Ising model

We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. We consider two nonequivalent Hamiltonians: the ferromagnetic 2D-TFIM and the antiferromagnetic 2D-TFIM in the presence of a uniform longitudinal field. We use full exact diagonalization to examine the behavior of quantum chaos indicators and of the diagonal matrix elements of operators of interest in the eigenstates of the Hamiltonian. An analysis of finite size effects reveals that quantum chaos and eigenstate thermalization occur in those systems whenever the fields are nonvanishing and not too large.

Improved unfolding by detrending of statistical fluctuations in quantum spectra

A fundamental relation exists between the statistical properties of the fluctuations of the energy-level spectrum of a Hamiltonian and the chaotic properties of the physical system it describes. This relationship has been addressed previously as a signature of chaos in quantum dynamical systems. In order to properly analyze these fluctuations, however, it is necessary to separate them from the general tendency, namely, its secular part. Unfortunately this process, called unfolding, is not trivial and can lead to erroneous conclusions about the chaoticity of a system. In this paper we propose a technique to improve the unfolding procedure for the purpose of minimizing the dependence on the particular procedure. This technique is based on detrending the fluctuations of the unfolded spectra through the empirical mode decomposition method.

Assessing periodicity of periodic leg movements during sleep

BACKGROUND

Periodic leg movements (PLM) during sleep consist of involuntary periodic movements of the lower extremities. The debated functional relevance of PLM during sleep is based on correlation of clinical parameters with the PLM index (PLMI). However, periodicity in movements may not be reflected best by the PLMI. Here, an approach novel to the field of sleep research is used to reveal intrinsic periodicity in inter movement intervals (IMI) in patients with PLM.

METHODS

Three patient groups of 10 patients showing PLM with OSA (group 1), PLM without OSA or RLS (group 2) and PLM with RLS (group 3) are considered. Applying the "unfolding" procedure, a method developed in statistical physics, enhances or even reveals intrinsic periodicity of PLM. The degree of periodicity of PLM is assessed by fitting one-parameter distributions to the unfolded IMI distributions. Finally, it is investigated whether the shape of the IMI distributions allows to separate patients into different groups.

RESULTS

Despite applying the unfolding procedure, periodicity is neither homogeneous within nor considerably different between the three clinically defined groups. Data-driven clustering reveals more homogeneous and better separated clusters. However, they consist of patients with heterogeneous demographic data and comorbidities, including RLS and OSA.

CONCLUSIONS

The unfolding procedure may be necessary to enhance or reveal periodicity. Thus this method is proposed as a pre-processing step before analyzing PLM statistically. Data-driven clustering yields much more reasonable results when applied to the unfolded IMI distributions than to the original data. Despite this effort no correlation between the degree of periodicity and demographic data or comorbidities is found. However, there are indications that the nature of the periodicity might be determined by long-range interactions between LM of patients with PLM and OSA.

Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems

We study how the proximity to an integrable point or to localization as one approaches the atomic limit, as well as the mixing of symmetries in the chaotic domain, may affect the onset of thermalization in finite one-dimensional systems. We consider systems of hard-core bosons at half-filling with nearest-neighbor hopping and interaction, and next-nearest-neighbor interaction. The latter breaks integrability and induces a ground-state superfluid to insulator transition. By full exact diagonalization, we study chaos indicators and few-body observables. We show that when different symmetry sectors are mixed, chaos indicators associated with the eigenvectors, contrary to those related to the eigenvalues, capture the onset of chaos. The results for the complexity of the eigenvectors and for the expectation values of few-body observables confirm the validity of the eigenstate thermalization hypothesis in the chaotic regime, and therefore the occurrence of thermalization. We also study the properties of the off-diagonal matrix elements of few-body observables in relation to the transition from integrability to chaos and from chaos to localization.

Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization

By means of full exact diagonalization, we study level statistics and the structure of the eigenvectors of one-dimensional gapless bosonic and fermionic systems across the transition from integrability to quantum chaos. These systems are integrable in the presence of only nearest-neighbor terms, whereas the addition of next-nearest-neighbor hopping and interaction may lead to the onset of chaos. We show that the strength of the next-nearest-neighbor terms required to observe clear signatures of nonintegrability is inversely proportional to the system size. Interestingly, the transition to chaos is also seen to depend on particle statistics, with bosons responding first to the integrability breaking terms. In addition, we discuss the use of delocalization measures as main indicators for the crossover from integrability to chaos and the consequent viability of quantum thermalization in isolated systems.

The influence of static correlations on multivariate correlation analysis of the EEG

The choice of the EEG reference strongly influences the results derived from different correlation measures. Such a dependence may easily mislead the interpretation of the correlation structure of the brain activity. We provide a systematic study of the influence of the choice of reference on linear multivariate EEG correlation patterns as determined by sensitive correlation measures derived from the equal-time correlation matrix. In addition, an effective algorithm to extract the effect of static correlations is developed. The eigenvalues of the correlation matrix and their spacing statistics are studied for artificial time series with known correlation structure and for an epileptic EEG in various montages. The correction method proposed in this paper works with varying quality for different choices of the EEG reference. Furthermore, the optimal choice of the reference depends also on the correlation structure of the underlying system.

Localized short-range correlations in the spectrum of the equal-time correlation matrix

We suggest a procedure to identify those parts of the spectrum of the equal-time correlation matrix C where relevant information about correlations of a multivariate time series is induced. Using an ensemble average over each of the distances between eigenvalues, all nearest-neighbor distributions can be calculated individually. We present numerical examples, where (a) information about cross correlations is found in the so-called "bulk" of eigenvalues (which generally is thought to contain only random correlations) and where (b) the information extracted from the lower edge of the spectrum of C is statistically more significant than that extracted from the upper edge. We apply the analysis to electroencephalographic recordings with epileptic events.

Detection and characterization of changes of the correlation structure in multivariate time series

We propose a method based on the equal-time correlation matrix as a sensitive detector for phase-shape correlations in multivariate data sets. The key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix. Consequently, detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors. The performance of the technique is demonstrated by application to N(f)-tori, autoregressive models, and coupled chaotic systems. The high sensitivity, the comparatively small computational effort, and the excellent time resolution of the method recommend it for application to the analysis of complex, spatially extended, nonstationary systems.

1/f noise in the two-body random ensemble

We show that the spectral fluctuations of the two-body random ensemble exhibit 1/f noise. This result supports a recent conjecture stating that chaotic quantum systems are characterized by 1/f noise in their energy level fluctuations. After suitable individual averaging, we also study the distribution of the exponent alpha in the 1/ f(alpha) noise for the individual members of the ensemble. Almost all the exponents lie inside a narrow interval around alpha=1, suggesting that also individual members exhibit 1/f noise, provided they are individually unfolded.

Duality between the weak and strong interaction limits for randomly interacting fermions

We establish the existence of a duality transformation for generic models of interacting fermions with two-body interactions. The eigenstates at weak and strong interaction U possess similar statistical properties when expressed in the U=0 and U= infinity eigenstates bases, respectively. This implies the existence of a duality point U(d) where the eigenstates have the same spreading in both bases. U(d) is surrounded by an interval of finite width which is characterized by a non-Lorentzian spreading of the strength function in both bases. Scaling arguments predict the survival of this intermediate regime as the number of particles is increased.

Group symmetries in two-body random matrix ensembles generating order out of complexity

The two-body random matrix ensembles with spin TBRE-s and in a single j shell TBRE-j introduced recently in the context of ground state structures in complex interacting particle systems, possess U(N) superset U(N/2)multiply sign in circle SU(2) and U(N)superset O(3) group symmetries, respectively, with N the number of single particle states. It is shown that both these group symmetries give rise to simplicities in the ground state structures but in different ways.

Entropy production and wave packet dynamics in the Fock space of closed chaotic many-body systems

Highly excited many-particle states in quantum systems such as nuclei, atoms, quantum dots, spin systems, quantum computers, etc., can be considered as "chaotic" superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is due to a very high level density of many-body states that are easily mixed by a residual interaction between particles (quasiparticles). For such systems, we have derived simple analytical expressions for the time dependence of the energy width of wave packets, as well as for the entropy, number of principal basis components, and inverse participation ratio, and tested them in numerical experiments. It is shown that the energy width Delta(t) increases linearly and very quickly saturates. The entropy of a system increases quadratically, S(t) approximately t(2), at small times, and afterward can grow linearly, S(t) approximately t, before saturation. Correspondingly, the number of principal components determined by the entropy N(pc) approximately exp[S(t)] or by the inverse participation ratio increases exponentially fast before saturation. These results are explained in terms of a cascade model which describes the flow of excitation in the Fock space of basis components. Finally, the striking phenomenon of damped oscillations in the Fock space at the transition to equilibrium is discussed.

Dynamics of quantum systems

A relation between the eigenvalues of an effective Hamilton operator and the poles of the S matrix is derived that holds for isolated as well as for overlapping resonance states. The system may be a many-particle quantum system with two-body forces between the constituents or it may be a quantum billiard without any two-body forces. Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points in the complex plane. Under certain conditions, these branch points appear as double poles of the S matrix. They influence the dynamics of open as well as of closed quantum systems. The dynamics of the two-level system is studied in detail analytically as well as numerically.

Random versus realistic interactions for low-lying nuclear spectra

We compare the shell-model results for realistic interactions with those obtained for various ensembles of random matrix elements. We show that, although the quantum numbers of the ground states in the even-even nuclei have a high probability ( approximately 60%) to be J(pi)T = 0(+)0, the overlap of those states with the realistic wave functions is very small in average. The transition probabilities B(E2) predicted with random interactions are also too small. The presence of the regular pairing is shown to be a significant element of realistic physics not reproduced by random interactions.