Wigner Random Banded Matrices with Sparse Structure: Local Spectral Density of States

Fyodorov, Yan V.; Chubykalo, O. A.; Izrailev, F. M.; Casati, Giulio

doi:10.1103/physrevlett.76.1603

Cited by 118 publications

(191 citation statements)

References 23 publications

(38 reference statements)

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“…(7) with Γ = 2πV 2 ρ, provided the eigenstates are localized within the band: Γ < Db. Recently this result has been derived for sparse matrices with a diffuse band [39]. The BW spreading also emerges in the well-known nuclear physics model of a state interacting with a large set of states with ρ = const by means of a constant or weakly fluctuating matrix element [5].…”

Section: Chaotic Eigenstatesmentioning

confidence: 99%

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Flambaum

Gribakina

Gribakin

et al. 1999

Physica D: Nonlinear Phenomena

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Results of an extensive study of a real quantum chaotic many-body system -the Ce atom -are presented. We discuss the origins of the quantum chaotic behaviour of the system, analyse statistical and dynamical properties of the multi-particle chaotic eigenstates and consider matrix elements or transition amplitudes between them. We show that based on the universal properties of the chaotic eigenstates a statistical theory of finite few-particle systems with strong interaction can be developed. We also discuss such important physical effects as enhancement of weak perturbations in many-body quantum chaotic systems, distribution of single-particle occupation numbers and its deviations from the standard Fermi-Dirac shape, and ways of introducing statistical temperature-based description in such systems.

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Section: Chaotic Eigenstatesmentioning

confidence: 99%

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Flambaum

Gribakina

Gribakin

et al. 1999

Physica D: Nonlinear Phenomena

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“…In particular the width Γ gives an energy scale at which the level statistics, for example the number variance Σ 2 (E), changes behavior from the WignerDyson to the Poisson case [9]. It has been also shown that the Breit-Wigner distribution appears in the case of sparse random matrices with preferential basis [10].While the properties of the Breit-Wigner distribution are well understood in random matrix models, the problem of real interacting finite many-body fermionic systems was much less investigated. Indeed, in the latter case the nature of the two-body interaction should be taken into account, since it gives certain restrictions on the structure of matrix elements.…”

mentioning

confidence: 99%

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confidence: 99%

Breit-Wigner Width and Inverse Participation Ratio in Finite Interacting Fermi Systems

Georgeot

Shepelyansky

1997

Phys. Rev. Lett.

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For many-body Fermi systems we determine the dependence of the Breit-Wigner width Γ and inverse participation ratio ξ on interaction strength U ≥ Uc and energy excitation δE ≥ δE ch when a crossover from Poisson to Wigner-Dyson P (s)-statistics takes place. At U ≥ Uc the eigenstates are composed of a large number of noninteracting states and even for U < Uc there is a regime where P (s) is close to the Poisson distribution but ξ ≫ 1.PACS numbers: 05.45.+b, 05.30.Fk, 24.10.Cn In 1955 Wigner [1] introduced the local density of states to study "the properties of the wave functions of quantum mechanical systems which are assumed to be so complicated that statistical considerations can be applied to them". This quantity ρ W (E) characterizes the spreading of eigenstates over the levels of an unperturbed system (for example in the absence of interaction between particles), and allows to estimate how many of these unperturbed states contribute to the real wave function. Generally ρ W (E) has a Breit-Wigner distribution with Lorentzian shape of width Γ which determines the energy spreading over unperturbed states. This concept has been shown since then to be very important in a wide range of physical problems, from nuclear physics and many-electron atoms and molecules to condensed matter.The study of such complex systems has been successfully performed through the theory of random matrices (see for example [2]). Very often the physics of such systems determines some preferential basis in which the Hamiltonian matrix has large diagonal matrix elements, while the non-diagonal elements corresponding to transitions between the basis states are relatively small. The investigation of random matrices of this type has been started only recently [3][4][5][6]. It has been shown that the eigenstates of such superimposed band random matrices (SBRM) are spread over the basis states according to the Breit-Wigner distribution [6]; this has been also confirmed analytically through the supersymmetry approach [7,8]. This spreading determines the number of unperturbed states contributing to a given eigenstate, which can be measured through the inverse participation ratio (IPR) ξ [6-8]. In particular the width Γ gives an energy scale at which the level statistics, for example the number variance Σ 2 (E), changes behavior from the WignerDyson to the Poisson case [9]. It has been also shown that the Breit-Wigner distribution appears in the case of sparse random matrices with preferential basis [10].While the properties of the Breit-Wigner distribution are well understood in random matrix models, the problem of real interacting finite many-body fermionic systems was much less investigated. Indeed, in the latter case the nature of the two-body interaction should be taken into account, since it gives certain restrictions on the structure of matrix elements. A very convenient model to investigate this kind of problem has been introduced some time ago in [11,12]. This model consists of n fermions distributed over m energy orbitals, coupled by ...

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“…These methods include banded random matrix models [3,17,18,19], models of a single-level with constant couplings to a "picketfence" spectrum [4,20], and perturbative techniques with partial summations over diagrams to infinite order [21]. Under inequivalent assumptions these approaches affirm a generic Breit-Wigner shape for the LDOS profile,…”

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confidence: 99%

“…1 applies when a given quantized classically chaotic model is subjected to a perturbation of interest. From the BGS conjecture [24] and studies of (banded) random matrix models [3,17,18,19], it is generally expected that for fully chaotic models with generic perturbations Eq. 1 applies with,…”

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confidence: 99%

Estimation of the local density of states on a quantum computer

et al. 2004

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We report an efficient quantum algorithm for estimating the local density of states (LDOS) on a quantum computer. The LDOS describes the redistribution of energy levels of a quantum system under the influence of a perturbation. Sometimes known as the "strength function" from nuclear spectroscopy experiments, the shape of the LDOS is directly related to the survivial probability of unperturbed eigenstates, and has recently been related to the fidelity decay (or "Loschmidt echo") under imperfect motion-reversal. For quantum systems that can be simulated efficiently on a quantum computer, the LDOS estimation algorithm enables an exponential speed-up over direct classical computation.PACS numbers: 05.45.Mt, 03.67.Lx A major motivation for the physical realization of quantum information processing is the idea, intimated by Feynman, that the dynamics of a wide class of complex quantum systems may be simulated efficiently by these techniques [1]. For a quantum system with Hilbert space size N , an efficient simulation is one that requires only Polylog(N ) gates. This situation should be contrasted with direct simulation on a classical processor, which requires resources growing at least as N 2 . However, complete measurement of the final state on a quantum processor requires O(N 2 ) repetitions of the quantum simulation. Similarly, estimation of the eigenvalue spectrum of a quantum system admitting a Polylog(N ) circuit decomposition requires a phase-estimation circuit that grows as O(N ) [2]. As a result there still remains the important problem of devising methods for the efficient readout of those characteristic properties that are of practical interest in the study of complex quantum systems. In this Letter we introduce an efficient quantum algorithm for estimating, to 1/Polylog(N ) accuracy, the local density of states (LDOS), a quantity of central interest in the description of both many-body and complex fewbody systems. We also determine the class of physical problems for which the LDOS estimation algorithm provides an exponential speed-up over known classical algorithms given this finite accuracy.The LDOS describes the profile of an eigenstate of an unperturbed quantum system over the eigenbasis of perturbed version of the same quantum system. In the context of manybody systems the LDOS was introduced to describe the effect of strong two-particle interactions on the single particle (or single hole) eigenstates [3,4,5,6,7]. More recently, the LDOS has been studied to characterize the effect of imperfections (due to residual interactions between the qubits) in the operation of quantum computers [8,9]. This profile plays a fundamental role also in the analysis of system stability for few-body systems subject to a sudden perturbation [10], such as the onset of an external field, and has been studied extensively in the context of quantum chaos and dynamical localization [11,12]. Quite generally the LDOS is related to the survival probability of the unperturbed eigenstate [4,10,13], and there has been considerable re...

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Wigner Random Banded Matrices with Sparse Structure: Local Spectral Density of States

Cited by 118 publications

References 23 publications

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Breit-Wigner Width and Inverse Participation Ratio in Finite Interacting Fermi Systems

Estimation of the local density of states on a quantum computer

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