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Mailoud S, Borgonovi F, Izrailev FM. Spectrum statistics in the integrable Lieb-Liniger model. Phys Rev E 2021; 104:034212. [PMID: 34654076 DOI: 10.1103/physreve.104.034212] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2021] [Accepted: 08/05/2021] [Indexed: 11/07/2022]
Abstract
We address the old and widely debated question of the spectrum statistics of integrable quantum systems, through the analysis of the paradigmatic Lieb-Liniger model. This quantum many-body model of one-dimensional interacting bosons allows for the rigorous determination of energy spectra via the Bethe ansatz approach and our interest is to reveal the characteristic properties of energy levels in dependence of the model parameters. Using both analytical and numerical studies we show that the properties of spectra strongly depend on whether the analysis is done for a full energy spectrum or for a single subset with fixed total momentum. We show that the Poisson distribution of spacing between nearest-neighbor energies can occur only for a set of energy levels with fixed total momentum, for neither too large nor too weak interaction strength, and for sufficiently high energy. By studying long-range correlations between energy levels, we found strong deviations from the predictions based on the assumption of pseudorandom character of the distribution of energy levels.
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Affiliation(s)
- Samy Mailoud
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
| | - Fausto Borgonovi
- Dipartimento di Matematica e Fisica and Interdisciplinary Laboratories for Advanced Materials Physics, Università Cattolica, via Musei 41, 25121 Brescia, Italy.,Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy
| | - Felix M Izrailev
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico.,Department of Physics and Astronomy, Michigan State University, E. Lansing, Michigan 48824-1321, USA
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Characterization of real-world networks through quantum potentials. PLoS One 2021; 16:e0254384. [PMID: 34255791 PMCID: PMC8277057 DOI: 10.1371/journal.pone.0254384] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2021] [Accepted: 06/24/2021] [Indexed: 11/19/2022] Open
Abstract
Network connectivity has been thoroughly investigated in several domains, including physics, neuroscience, and social sciences. This work tackles the possibility of characterizing the topological properties of real-world networks from a quantum-inspired perspective. Starting from the normalized Laplacian of a network, we use a well-defined procedure, based on the dressing transformations, to derive a 1-dimensional Schrödinger-like equation characterized by the same eigenvalues. We investigate the shape and properties of the potential appearing in this equation in simulated small-world and scale-free network ensembles, using measures of fractality. Besides, we employ the proposed framework to compare real-world networks with the Erdős-Rényi, Watts-Strogatz and Barabási-Albert benchmark models. Reconstructed potentials allow to assess to which extent real-world networks approach these models, providing further insight on their formation mechanisms and connectivity properties.
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Elkamshishy AA, Greene CH. Observation of Wigner-Dyson level statistics in a classically integrable system. Phys Rev E 2021; 103:062211. [PMID: 34271623 DOI: 10.1103/physreve.103.062211] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2020] [Accepted: 05/27/2021] [Indexed: 11/07/2022]
Abstract
Resonances in particle transmission through a 1D finite lattice are studied in the presence of a finite number of impurities. Although this is a one-dimensional system that is classically integrable and has no chaos, studying the statistical properties of the spectrum such as the level spacing distribution and the spectral rigidity shows the same statistics as the one obtained for chaotic systems. Using a dimensionless parameter that reflects the degree of state localization, we demonstrate how the transition from Poisson-level statistics to the Wigner-Dyson is affected by state localization. The resonance positions are calculated using both the Wigner-Smith time delay and a Siegert state method, which are in good agreement. Our results show the dependence of the level statistics on the localization length as it evolves from a Poisson distribution to Wigner-Dyson.
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Affiliation(s)
- Ahmed A Elkamshishy
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
| | - Chris H Greene
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
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Amoroso N, Bellantuono L, Pascazio S, Lombardi A, Monaco A, Tangaro S, Bellotti R. Potential energy of complex networks: a quantum mechanical perspective. Sci Rep 2020; 10:18387. [PMID: 33110089 PMCID: PMC7592062 DOI: 10.1038/s41598-020-75147-w] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2020] [Accepted: 10/12/2020] [Indexed: 12/26/2022] Open
Abstract
We propose a characterization of complex networks, based on the potential of an associated Schrödinger equation. The potential is designed so that the energy spectrum of the Schrödinger equation coincides with the graph spectrum of the normalized Laplacian. Crucial information is retained in the reconstructed potential, which provides a compact representation of the properties of the network structure. The median potential over several random network realizations, which we call ensemble potential, is fitted via a Landau-like function, and its length scale is found to diverge as the critical connection probability is approached from above. The ruggedness of the ensemble potential profile is quantified by using the Higuchi fractal dimension, which displays a maximum at the critical connection probability. This demonstrates that this technique can be successfully employed in the study of random networks, as an alternative indicator of the percolation phase transition. We apply the proposed approach to the investigation of real-world networks describing infrastructures (US power grid). Curiously, although no notion of phase transition can be given for such networks, the fractality of the ensemble potential displays signatures of criticality. We also show that standard techniques (such as the scaling features of the largest connected component) do not detect any signature or remnant of criticality.
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Affiliation(s)
- Nicola Amoroso
- Dipartimento di Farmacia-Scienze del Farmaco, Università degli Studi di Bari Aldo Moro, 70125, Bari, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70125, Bari, Italy
| | - Loredana Bellantuono
- Dipartimento Interateneo di Fisica "M. Merlin", Università degli Studi di Bari Aldo Moro, 70125, Bari, Italy
| | - Saverio Pascazio
- Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70125, Bari, Italy.
- Dipartimento Interateneo di Fisica "M. Merlin", Università degli Studi di Bari Aldo Moro, 70125, Bari, Italy.
| | - Angela Lombardi
- Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70125, Bari, Italy
| | - Alfonso Monaco
- Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70125, Bari, Italy
| | - Sabina Tangaro
- Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70125, Bari, Italy
- Dipartimento di Scienze del Suolo, della Pianta e degli Alimenti, Università degli Studi di Bari Aldo Moro, 70125, Bari, Italy
| | - Roberto Bellotti
- Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70125, Bari, Italy
- Dipartimento Interateneo di Fisica "M. Merlin", Università degli Studi di Bari Aldo Moro, 70125, Bari, Italy
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Ujfalusi L, Varga I, Schumayer D. Quantum chaos in one dimension? PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:016230. [PMID: 21867296 DOI: 10.1103/physreve.84.016230] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2011] [Revised: 05/25/2011] [Indexed: 05/31/2023]
Abstract
In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest N eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit N→∞ the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.
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Affiliation(s)
- László Ujfalusi
- Elméleti Fizika Tanszék, Fizikai Intézet, Budapesti Műszaki és Gazdaságtudományi Egyetem, Budapest, Hungary
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Schumayer D, van Zyl BP, Hutchinson DAW. Quantum mechanical potentials related to the prime numbers and Riemann zeros. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:056215. [PMID: 19113208 DOI: 10.1103/physreve.78.056215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2008] [Indexed: 05/27/2023]
Abstract
Prime numbers are the building blocks of our arithmetic; however, their distribution still poses fundamental questions. Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the nontrivial zeros of the Riemann zeta(s) function. According to the Hilbert-Pólya conjecture, there exists a Hermitian operator of which the eigenvalues coincide with the real parts of the nontrivial zeros of zeta(s) . This idea has encouraged physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Marchenko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the zeta(s) function. We demonstrate the multifractal nature of these potentials by measuring the Rényi dimension of their graphs. Our results offer hope for further analytical progress.
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Affiliation(s)
- Dániel Schumayer
- Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, 730 Cumberland Street, Dunedin 9016, New Zealand.
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Benet L, Leyvraz F, Seligman TH. Wigner-Dyson statistics for a class of integrable models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:045201. [PMID: 14682991 DOI: 10.1103/physreve.68.045201] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2003] [Indexed: 11/07/2022]
Abstract
We construct an ensemble of second-quantized Hamiltonians with two bosonic degrees of freedom, whose members display with probability one Gaussian orthogonal ensemble (GOE) or Gaussian unitary ensemble (GUE) statistics. Nevertheless, these Hamiltonians have a second integral of motion, namely, the boson number, and thus are integrable. To construct this ensemble we use some "reverse engineering" starting from the fact that n bosons in a two-level system with random interactions have an integrable classical limit by the old Heisenberg association of boson operators to actions and angles. By choosing an n-body random interaction and degenerate levels we end up with GOE or GUE Hamiltonians. Ergodicity of these ensembles completes the example.
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Affiliation(s)
- L Benet
- Centro de Ciencias Físicas, UNAM, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, Mexico
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van Zyl BP, Hutchinson DAW. Riemann zeros, prime numbers, and fractal potentials. ACTA ACUST UNITED AC 2003; 67:066211. [PMID: 16241330 DOI: 10.1103/physreve.67.066211] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2003] [Indexed: 11/07/2022]
Abstract
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels.
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Affiliation(s)
- Brandon P van Zyl
- Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada
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Emary C, Brandes T. Chaos and the quantum phase transition in the Dicke model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:066203. [PMID: 16241322 DOI: 10.1103/physreve.67.066203] [Citation(s) in RCA: 160] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2003] [Indexed: 05/04/2023]
Abstract
We investigate the quantum-chaotic properties of the Dicke Hamiltonian; a quantum-optical model that describes a single-mode bosonic field interacting with an ensemble of N two-level atoms. This model exhibits a zero-temperature quantum phase transition in the N --> infinity limit, which we describe exactly in an effective Hamiltonian approach. We then numerically investigate the system at finite N, and by analyzing the level statistics, we demonstrate that the system undergoes a transition from quasi-integrability to quantum chaotic, and that this transition is caused by the precursors of the quantum phase transition. Our considerations of the wave function indicate that this is connected with a delocalization of the system and the emergence of macroscopic coherence. We also derive a semiclassical Dicke model that exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos.
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Affiliation(s)
- Clive Emary
- Department of Physics, UMIST, Manchester, UK.
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Wu H, Sprung DW. Reply to "Fractal potentials from energy levels". PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:6327. [PMID: 9963388 DOI: 10.1103/physreve.51.6327] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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11
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Ramani A, Grammaticos B, Caurier E. Fractal potentials from energy levels. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:6323-6326. [PMID: 9963387 DOI: 10.1103/physreve.51.6323] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Brack M, Bhaduri RK, Law J, Maier C, Murthy MVN. Effect of a magnetic flux line on the quantum beats in the Henon-Heiles level density. CHAOS (WOODBURY, N.Y.) 1995; 5:317-329. [PMID: 12780185 DOI: 10.1063/1.166080] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The quantum density of states of the Henon-Heiles potential displays a pronounced beating pattern. This has been explained by the interference of three isolated classical periodic orbits with nearby actions and periods. A singular magnetic flux line, passing through the origin, drastically alters the beats even though the classical Lagrangian equations of motion remain unchanged. Some of the changes can be easily understood in terms of the Aharonov-Bohm effect. However, we find that the standard periodic orbit theory does not reproduce the diffraction-like quantum effects on those classical orbits which intersect the singular flux line, and argue that corrections of relative order variant Planck's over 2pi are necessary to describe these effects. We also discuss the changes in the distribution of nearest-neighbor spacings in the eigenvalue spectrum, brought about by the flux line. (c) 1995 American Institute of Physics.
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Affiliation(s)
- M. Brack
- Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, GermanyDepartment of Physics and Astronomy, McMaster University, Hamilton L8S 4M1, CanadaDepartment of Physics, University of Guelph, Guelph, Ontario N1G 2W1, CanadaInstitute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, GermanyInstitute of Mathematical Sciences, Madras 600 113, India
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Date G, Jain SR, Murthy MV. Rectangular billiard in the presence of a flux line. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:198-203. [PMID: 9962632 DOI: 10.1103/physreve.51.198] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Khalfin I, Berkovits R, Gitterman M. Chaos in double-barrier heterostructures. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:4708-4711. [PMID: 9961769 DOI: 10.1103/physreve.49.4708] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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15
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Wu H, Sprung DW. Riemann zeros and a fractal potential. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:2595-2598. [PMID: 9960893 DOI: 10.1103/physreve.48.2595] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Wu H, Sprung DW, Feng DH, Vallières M. Modeling chaotic quantum systems by tridiagonal random matrices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:4063-4066. [PMID: 9960480 DOI: 10.1103/physreve.47.4063] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Wu H, Sprung DW, Vallières M, Feng DH. Suppression of energy fluctuations in the classical counterpart of quantum models. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:2444-2447. [PMID: 9960276 DOI: 10.1103/physreve.47.2444] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lopac V, Brant S, Paar V. Dependence of chaotic behavior on the residual interaction in the odd-odd nucleus 106Ag. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:3503-3506. [PMID: 9907397 DOI: 10.1103/physreva.45.3503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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