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Physical Review C

Chaotic behavior in the cranking and particles-rotor models

András T. Kruppa, Károly F. Pál, N. Rowley

Published Oct 1, 1995• 12 citations
Publisher

Abstract

We investigate the consequences on the order/chaos transition of replacing the scalar rotational energy of the particles-rotor model by a one-body cranking term. The nearest-neighbor distribution of energy levels and the spectral rigidity are studied as a function of the spin or cranking frequency, respectively. Exact energies for our statistical analyses are obtained from a full diagonalization of a many-body Hamiltonian with two-body forces for six particles in an ${\mathit{i}}_{13/2}$ shell coupled to a deformed core. In this model, the competing physical effects (two-body interaction strength, quadrupole deformation, and moment of inertia) are easily parametrized and studied. The dependence of the chaoticity on model parameters reveals the stabilizing influence of a large quadrupole deformation. For low spins, the considerable effects of the (effectively two-body) recoil term give significant differences between the two descriptions.

DOI
10.1103/physrevc.52.1818

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We investigate the consequences on the order/chaos transition of replacing the scalar rotational energy of the particles-rotor model by a one-body cranking term. The nearest-neighbor distribution of energy levels and the spectral rigidity are studied as a function of the spin or cranking frequency, respectively. Exact energies for our statistical analyses are obtained from a full diagonalization of a many-body Hamiltonian with two-body forces for six particles in an i13/2{\mathit{i}}_{13/2}i13/2​ shell coupled to a deformed core. In this model, the competing physical effects (two-body interaction strength, quadrupole deformation, and moment of inertia) are easily parametrized and studied. The dependence of the chaoticity on model parameters reveals the stabilizing influence of a large quadrupole deformation. For low spins, the considerable effects of the (effectively two-body) recoil term give significant differences between the two descriptions.
Shell-model results for the electric quadrupole (E2),(E2),(E2), magnetic dipole (M1),(M1),(M1), and Gamow-Teller transition strength sums and occupation numbers are calculated using different valence spaces and compared to the embedded Gaussian orthogonal ensemble (EGOE) predictions. It is shown that transition strength sums can be considered as a new statistic able to distinguish between regular and chaotic motion. Moreover it is established that the EGOE (but not the GOE) provides the good description of the shell-model strength sums in the chaotic domain. To get this result we have studied the behavior of strength sums in order to chaos transitions generated by means of a family of Hamiltonians $H(\ensuremath{\lambda})=h(1)+\ensuremath{\lambda}V(2),$ built from realistic one- and two-body interactions.
We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.
The energies and wave functions of stationary many-body states are analyzed to look for the signatures of quantum chaos. Shell model calculations with the Wildenthal interaction are performed in the J−TJ-TJ−T scheme for 12 particles in the sdsdsd-shell. The local level statistics are in perfect agreement with the GOE predictions. The analysis of the amplitudes of the eigenvectors in the shell model basis with the aid of the informational entropy and moments of the distribution function shows evidence for local chaos with a localization length reaching 90\% of the total dimension in the middle of the spectrum. The degree of chaoticity is sensitive to the the strength of the residual interaction as compared to the single particle energy spacing.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in the spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges. For single-particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc. R. Soc. A 400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of the spectral pair correlation function) from semiclassics has been completed by Müller et al. [Phys. Rev. Lett. 93, 014103 (2004)]. In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming to the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed the "many-body localized phase" and "ergodic phase." In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide a clear theoretical explanation for these observations. We compute K(t) in the leading two orders in t and show its agreement with RMT for nonintegrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin-1/2 models in a periodically kicking transverse field. In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t; hence, the leading-order RMT behavior K(t)≃2t is a consequence of translation and reflection symmetry of the Ising partition function.Received 5 February 2018Revised 12 April 2018DOI:https://doi.org/10.1103/PhysRevX.8.021062Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosQuantum statistical mechanicsStatistical PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics
It is shown that the moment of inertia of a noninteracting many-body fermion system moving under periodic boundary conditions has the classical or rigid value when calculated on the "cranking" model of Inglis. By investigating the analogous "pushing" case for the inertial mass we show that the rigid moment can be associated with rigid rotation in spite of apparent surface currents. The effect of particle-particle forces is investigated in the lowest order of perturbation theory. The terms corresponding to a level shift or effective mass are just compensated by other terms and there is no change in the moment of inertia. The possible general validity of these results and their consequence is discussed.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.