Models of intermediate spectral statistics

Bogomolny, E.; Gerland, Ulrich; Schmit, C.

doi:10.1103/physreve.59.r1315

Cited by 266 publications

(400 citation statements)

References 23 publications

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“…The critical statistics, therefore, governs the spectral fluctuations that are weaker than for the Poisson statistics (Σ 2 (N ) =N ) but much stronger than for the Wigner-Dyson statistics, (Σ 2 (N ) = lnN ). Later on remarkable similarities were found between the spectral statistics of a number of dynamical systems e.g pseudointegrable billiards and the critical statistics near the mobility edge [36]. However such a critical region being inaccessible either perturbatively or semiclassically, a different tool was required to probe into it.…”

Section: Critical Ensemble and Multifractality Of Eigenvectorsmentioning

confidence: 99%

Alternative technique for complex spectra analysis

Shukla

2000

Phys. Rev. E

117

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The choice of a suitable random matrix model of a complex system is very sensitive to the nature of its complexity. The statistical spectral analysis of various complex systems requires, therefore, a thorough probing of a wide range of random matrix ensembles which is not an easy task. It is highly desirable, if possible, to identify a common mathematical structure among all the ensembles and analyze it to gain information about the ensemble-properties. Our successful search in this direction leads to Calogero Hamiltonian, a one-dimensional quantum Hamiltonian with inverse-square interaction, as the common base. This is because both, the eigenvalues of the ensembles, and, a general state of Calogero Hamiltonian, evolve in an analogous way for arbitrary initial conditions. The varying nature of the complexity is reflected in different form of the evolution parameter in each case. A complete investigation of Calogero Hamiltonian can then help us in the spectral analysis of complex systems. Recent statistical studies in various branches of theoretical physics, ranging from Calogero model of 1-d fermionic system [1], random matrix (RM) model of disordered systems, matrix models of random surfaces to non-linear sigma model of quantum chaotic systems have revealed the presence of a common mathematical structure [2][3][4]. The connecting-web of these various models with each other is well-described in [3]. However, so far, the connection of RM model with other models was established only for standard Gaussian ensembles (SGE), that is, Gaussian ensembles invariant under unitary transformation. This was achieved by showing that distribution of the eigenvalues of the ensemble is governed by a Fokker-Planck (F-P) equation [6,7] similar to that of Dyson's "Brownian" motion model [5]. Through the reduction of F-P equation to the Schrodinger equation, the latter model is already known to be connected to Calogero Hamiltonian and thereby to various other models [6][7][8]. In this paper, we explore RM models with non-invariant distributions, and, following the same route as in the case of SGE, connect them to Calogero Hamiltonian. This gives us a new technique to analyze the spectral behavior of the quantum operators of complex systems.The connection between Complex systems and Calogero Hamiltonian seems to be wide-ranging. The eigenvalue dynamics of Hermitian operators, for example, Hamiltonians of complex quantum systems e.g. chaotic systems, disordered systems seems to have an intimate connection with the particle-dynamics of Calogero-Moser (CM) Hamiltonian. The latter describes the dynamics of particles interacting via pairwise inverse square interaction and confined to move along a real line [1]here µ i is the position of the i th particle and V (µ i ) is the confining potential. Similarly the level-dynamics of the unitary operators e.g. time-evolution operator is connected to Calogero-Sutherland (CS) Hamiltonian [9]:where particles are confined to move in a circle thus mimicking the similar confinement of eigenvalues due t...

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Section: Critical Ensemble and Multifractality Of Eigenvectorsmentioning

confidence: 99%

Alternative technique for complex spectra analysis

Shukla

2000

Phys. Rev. E

117

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“…In Equation (56), a given electron interacts with any other electron in the system. Restricting the interaction to neighboring pairs only, one obtains (Bogomolny et al 1999)…”

Section: Modeling Headwaysmentioning

confidence: 99%

Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling

Nagel¹,

Wagner

Woesler

2003

Operations Research

288

165

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Certain aspects of traffic flow measurements imply the existence of a phase transition. Models known from chaos and fractals, such as nonlinear analysis of coupled differential equations, cellular automata, or coupled maps, can generate behavior which indeed resembles a phase transition in the flow behavior. Other measurements point out that the same behavior could be generated by geometrical constraints of the scenario. This paper looks at some of the empirical evidence, but mostly focuses on different modeling approaches. The theory of traffic jam dynamics is reviewed in some detail, starting from the well-established theory of kinematic waves and then veering into the area of phase transitions. One aspect of the theory of phase transitions is that, by changing one single parameter, a system can be moved from displaying a phase transition to not displaying a phase transition. This implies that models for traffic can be tuned so that they display a phase transition or not.This paper focuses on microscopic modeling, i.e., coupled differential equations, cellular automata, and coupled maps. The phase transition behavior of these models, as far as it is known, is discussed. Similarly, fluid-dynamical models for the same questions are considered. A large portion of this paper is given to the discussion of extensions and open questions, which makes clear that the question of traffic jam dynamics is, albeit important, only a small part of an interesting and vibrant field. As our outlook shows, the whole field is moving away from a rather static view of traffic toward a dynamic view, which uses simulation as an important tool.

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“…3(a) we show the distribution of the spacings between adjacent singular values for reversible cases (heating period performed with I 2 gates) and irreversible ones (heating period performed with I 3 gates). The difference is striking: while the data points for the irreversible case match quite closely the distribution of spacings of the Gaussian orthogonal ensemble (GOE) of random matrices [19], the data points for the reversible case show a weaker repulsion and follow the so-called semi-Poisson statistics, which has been proposed for the energy spectra of systems at metal-insulator transitions [18]. The difference in behavior is also manifest in the spectral rigidity function ∆ 3 (L), which measures, for a given interval L, the least-square deviation of the spectral staircase from the best-fitting straight line [20].…”

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confidence: 99%

Emergent Irreversibility and Entanglement Spectrum Statistics

2014

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We study the problem of irreversibility when the dynamical evolution of a many-body system is described by a stochastic quantum circuit. Such evolution is more general than a Hamiltonian one, and since energy levels are not well defined, the well-established connection between the statistical fluctuations of the energy spectrum and irreversibility cannot be made. We show that the entanglement spectrum provides a more general connection. Irreversibility is marked by a failure of a disentangling algorithm and is preceded by the appearance of Wigner-Dyson statistical fluctuations in the entanglement spectrum. This analysis can be done at the wave-function level and offers an alternative route to study quantum chaos and quantum integrability.In closed quantum systems, evolution is unitary and both irreversibility and nonintegrability are elusive notions. Because of unitarity, evolution is always stable under errors in initial conditions. Thus, in quantum mechanics irreversibility is defined by the vanishing of the probability (known as fidelity) of returning to an initial state under arbitrarily small imperfections in the Hamiltonian during the reversed time evolution [2]. Nonintegrability is associated to a Wigner-Dyson distribution of the energy-level spacings that shows level repulsion [3] and nonintegrable Hamiltonians in this context are irreversible. Integrable Hamiltonians, instead, tend to show clustering of energy levels but can be either reversible or irreversible [4,5]. When the time evolution is not governed by a Hamiltonian, or when the Hamiltonian is time dependent, energy levels are not well defined and these associations cease to be meaningful. How can one relate nonintegrability and irreversibility in these more general cases of quantum evolution?In this Letter we show that one can answer this question by looking at the wave function alone. This route allows one to study generic quantum evolutions even when energy is not well defined. We show that by studying the level statistics of the entanglement spectrum one can determine whether the evolution is irreversible or not through a protocol that we call entanglement cooling. It turns out that the onset of irreversibility is marked by the presence of Wigner-Dyson statistics in the entanglement spectrum.The quantum system we consider contains n qubits and evolves unitarily from an initial factorized state of the form |Ψ 0 = |ψ 1 ⊗ |ψ 2 ⊗ · · · ⊗ |ψ n where each single-qubit state is defined as |ψ j = cos(θ j /2)|0 + sin(θ j /2)e iφj |1 , with θ j and φ j arbitrary. Formally, the evolution is obtained by applying a unitary matrix U to the state vector, |Ψ t = U |Ψ 0 = x Ψ t (x) |x , where the states |x ≡ |x 1 x 2 . . . x n form the computational basis, with x j = 0, 1 for j = 1, . . . , n. Using the language of quantum computing, we assume that this uni- tary matrix is represented by gates. We recall that the two-qubit CNOT gate and arbitrary one-qubit rotations are sufficient for universal quantum computing [6]. In what follows, we shall restrict the ga...

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Models of intermediate spectral statistics

Cited by 266 publications

References 23 publications

Alternative technique for complex spectra analysis

Alternative technique for complex spectra analysis

Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling

Emergent Irreversibility and Entanglement Spectrum Statistics

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