Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nlin/0402011 · Physical Review E

Stringent numerical test of the Poisson distribution for finite quantum integrable Hamiltonians

Armando Relaño, J. Dukelsky, José María Gómez Gómez, J. Retamosa

Published Aug 19, 2004• 27 citations
arXivHTMLPDF

Abstract

Using a class of exactly solvable models based on the pairing interaction, we show that it is possible to construct integrable Hamiltonians with a Wigner distribution of nearest-neighbor level spacings. However, these Hamiltonians involve many-body interactions and the addition of a small integrable perturbation very quickly leads the system to a Poisson distribution. Besides this exceptional case, we show that the accumulated distribution of an ensemble of random integrable two-body pairing Hamiltonians is in perfect agreement with the Poisson limit. These numerical results for quantum integrable Hamiltonians provide a further empirical confirmation of the work of Berry and Tabor in the semiclassical limit.

DOI
10.1103/physreve.70.026208

Related papers

Unexpected non-Wigner behavior in level-spacing distributions of next-nearest-neighbor coupled XXZ spin chains

Kazue Kudo, Tetsuo Deguchi

The level-spacing distributions of $\mathrm{XXZ}$ spin chains with next-nearest-neighbor couplings are studied under periodic boundary conditions. We confirm that integrable $\mathrm{XXZ}$ spin chains mostly have the Poisson distribution as expected. On the contrary, the level-spacing distributions of next-nearest-neighbor coupled $\mathrm{XXZ}$ chains are given by non-Wigner distributions. It is against the expectations, since the models are nonintegrable.

arXiv:cond-mat/0308099•Physical review. B, Condensed matter•Aug 25, 2003•32 citations

Random matrix theory of integrability-to-chaos transition

Ben Craps, Marine De Clerck, Oleg Evnin +1 more

The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.

cond-mat.stat-mechCross-listedarXiv:2604.03669•Apr 7, 2026

Statistical signatures of integrable and non-integrable quantum hamiltonians

Feng He, Arthur Hutsalyuk, Giuseppe Mussardo +1 more

Abstract Integrability has long served as a cornerstone concept in classical mechanics, where it possesses a precise and unambiguous definition. Extending this notion to the quantum domain, however, remains a far more subtle and elusive problem. In particular, deciding whether a given quantum Hamiltonian—viewed simply as a matrix of its elements—does or does not define an integrable system is far from obvious. Yet this question is crucial: it bears directly on non-equilibrium dynamics, spectral correlations, the behaviour of correlation functions, and other fundamental properties of many-body quantum systems. In this work, we develop a statistical framework for addressing quantum integrability from a purely probabilistic standpoint. Our approach begins with the observation that a necessary signature of integrability is the finite probability of encountering vanishing energy gaps in the spectrum. On this basis, we formulate a twofold protocol capable of distinguishing between integrable and non-integrable Hamiltonians. The first step consists of a systematic Monte Carlo decimation of the spectrum, designed to reveal the emergence (or absence) of Poissonian level spacing statistics. The iterative decimation compresses the Hilbert space exponentially, and its termination point determines whether the spectrum is governed by a mixed distribution or approaches the Poisson limit. In the second step, the distinction is instead obtained by analysing the distributions of k -step gaps which can help in discriminating between Poisson and mixed statistics. This procedure applies to Hamiltonians of arbitrary finite size, regardless of whether their structure involves a finite number of blocks or an exponentially fragmented Hilbert space. As a concrete benchmark, we implement the protocol on a class of quantum Hamiltonians constructed from the permutation group S N , thereby demonstrating both its effectiveness and its broad applicability.

arXiv:2510.02440•Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2026•1 citation

Statistical properties of spectral fluctuations of N interacting bosons in a harmonic trap

Kamalika Roy, Barnali Chakrabarti, V. K. B. Kota

Spectral fluctuations of a system of N weakly interacting bosons in an isotropic harmonic trap are studied, with the focus on the deviations from Poisson spectral statistics, typical of a quantum integrable systems. We have utilized the ideas formulated by Makino et al. [Phys. Rev. E 67, 066205 (2003)1063-651X10.1103/PhysRevE.67.066205] who have extended the approach of Berry and Robnik [J. Phys. A 17, 2413 (1984)JPHAC50305-447010.1088/0305-4470/17/12/013]. Earlier investigations of the Berry conjecture [Proc. R. Soc. London, Ser. A 356, 375 (1977)1364-502110.1098/rspa.1977.0140] of Poisson spectral statistics mainly considered quantum systems whose classical counterparts are integrable. However, the system of N weakly interacting bosons in the external trap has no classical counterpart. Also, it is a realistic and experimentally achievable system with close relation to Bose-Einstein condensation. Thus, a stringent analysis of the applicability of the Berry conjecture to this kind of systems is indeed required. We observe that for small boson number, the system is close to integrability and the nearest-neighbor level spacing distribution and the level number variance exhibit deviations from Poisson statistics similar to those of rational rectangular billiards.

Physical Review E•Nov 18, 2014•1 citation

Integrability of interacting two-level boson systems

Luis Benet, Christof Jung, F. Leyvraz

We prove that a wide class of two-level boson systems interacting through k -body forces is integrable in the semiclassical limit irrespective of the value of the parameters of the model. This class of models includes number-conserving Hamiltonians, and interactions which do not conserve the particle number. Applications to two-component and atom–molecule Bose–Einstein condensates at zero temperature as well as to the bosonic k -body embedded ensembles of Gaussian random matrices in the dense limit are discussed.

Journal of Physics A Mathematical and General•Apr 2, 2003•11 citations

Two measures of "chaos" in quantum many body systems

A. P. Zuker, Rafael A. Molina, M. Dufour

The definition of deterministic chaosextreme sensitivity to boundary or initial conditions-is extended to systems of many-body Hamiltonian matrices of low rank (1+2-body). Two indicators are proposed: X(T) the overlaps ofinitial and final wavefunctions and Z(r) the spreading of an initial wavefunction-measured by the number of principal components in the final basis of exact eigenstates- both under random variations of the Hamiltonian, assimilated to a time evolution T. At small fixed T indicator X coincides with the usual parametrizations of nearest neighbor spacings but it is argued that taken at face-value it suggests that, contrary to the accepted view, systems may be more irregular for separable (Poisson) interactions than for non-separable (Wigner) ones. The idea is supported by the fluctuating ground state energies obtained in mean field approaches, smoothed by the residual non-separable terms of the Hamiltonian. Indicator Z is quite consistent with the idea that the approach of the Wigner limit leads to chaos.

Revista Mexicana de Física•Dec 1, 2004•1 citation
Using a class of exactly solvable models based on the pairing interaction, we show that it is possible to construct integrable Hamiltonians with a Wigner distribution of nearest-neighbor level spacings. However, these Hamiltonians involve many-body interactions and the addition of a small integrable perturbation very quickly leads the system to a Poisson distribution. Besides this exceptional case, we show that the accumulated distribution of an ensemble of random integrable two-body pairing Hamiltonians is in perfect agreement with the Poisson limit. These numerical results for quantum integrable Hamiltonians provide a further empirical confirmation of the work of Berry and Tabor in the semiclassical limit.
The level-spacing distributions of XXZ\mathrm{XXZ}XXZ spin chains with next-nearest-neighbor couplings are studied under periodic boundary conditions. We confirm that integrable XXZ\mathrm{XXZ}XXZ spin chains mostly have the Poisson distribution as expected. On the contrary, the level-spacing distributions of next-nearest-neighbor coupled XXZ\mathrm{XXZ}XXZ chains are given by non-Wigner distributions. It is against the expectations, since the models are nonintegrable.
The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.
Abstract Integrability has long served as a cornerstone concept in classical mechanics, where it possesses a precise and unambiguous definition. Extending this notion to the quantum domain, however, remains a far more subtle and elusive problem. In particular, deciding whether a given quantum Hamiltonian—viewed simply as a matrix of its elements—does or does not define an integrable system is far from obvious. Yet this question is crucial: it bears directly on non-equilibrium dynamics, spectral correlations, the behaviour of correlation functions, and other fundamental properties of many-body quantum systems. In this work, we develop a statistical framework for addressing quantum integrability from a purely probabilistic standpoint. Our approach begins with the observation that a necessary signature of integrability is the finite probability of encountering vanishing energy gaps in the spectrum. On this basis, we formulate a twofold protocol capable of distinguishing between integrable and non-integrable Hamiltonians. The first step consists of a systematic Monte Carlo decimation of the spectrum, designed to reveal the emergence (or absence) of Poissonian level spacing statistics. The iterative decimation compresses the Hilbert space exponentially, and its termination point determines whether the spectrum is governed by a mixed distribution or approaches the Poisson limit. In the second step, the distinction is instead obtained by analysing the distributions of k -step gaps which can help in discriminating between Poisson and mixed statistics. This procedure applies to Hamiltonians of arbitrary finite size, regardless of whether their structure involves a finite number of blocks or an exponentially fragmented Hilbert space. As a concrete benchmark, we implement the protocol on a class of quantum Hamiltonians constructed from the permutation group S N , thereby demonstrating both its effectiveness and its broad applicability.
Spectral fluctuations of a system of N weakly interacting bosons in an isotropic harmonic trap are studied, with the focus on the deviations from Poisson spectral statistics, typical of a quantum integrable systems. We have utilized the ideas formulated by Makino et al. [Phys. Rev. E 67, 066205 (2003)1063-651X10.1103/PhysRevE.67.066205] who have extended the approach of Berry and Robnik [J. Phys. A 17, 2413 (1984)JPHAC50305-447010.1088/0305-4470/17/12/013]. Earlier investigations of the Berry conjecture [Proc. R. Soc. London, Ser. A 356, 375 (1977)1364-502110.1098/rspa.1977.0140] of Poisson spectral statistics mainly considered quantum systems whose classical counterparts are integrable. However, the system of N weakly interacting bosons in the external trap has no classical counterpart. Also, it is a realistic and experimentally achievable system with close relation to Bose-Einstein condensation. Thus, a stringent analysis of the applicability of the Berry conjecture to this kind of systems is indeed required. We observe that for small boson number, the system is close to integrability and the nearest-neighbor level spacing distribution and the level number variance exhibit deviations from Poisson statistics similar to those of rational rectangular billiards.
We prove that a wide class of two-level boson systems interacting through k -body forces is integrable in the semiclassical limit irrespective of the value of the parameters of the model. This class of models includes number-conserving Hamiltonians, and interactions which do not conserve the particle number. Applications to two-component and atom–molecule Bose–Einstein condensates at zero temperature as well as to the bosonic k -body embedded ensembles of Gaussian random matrices in the dense limit are discussed.
The definition of deterministic chaosextreme sensitivity to boundary or initial conditions-is extended to systems of many-body Hamiltonian matrices of low rank (1+2-body). Two indicators are proposed: X(T) the overlaps ofinitial and final wavefunctions and Z(r) the spreading of an initial wavefunction-measured by the number of principal components in the final basis of exact eigenstates- both under random variations of the Hamiltonian, assimilated to a time evolution T. At small fixed T indicator X coincides with the usual parametrizations of nearest neighbor spacings but it is argued that taken at face-value it suggests that, contrary to the accepted view, systems may be more irregular for separable (Poisson) interactions than for non-separable (Wigner) ones. The idea is supported by the fluctuating ground state energies obtained in mean field approaches, smoothed by the residual non-separable terms of the Hamiltonian. Indicator Z is quite consistent with the idea that the approach of the Wigner limit leads to chaos.