Correlation Analysis of Quantum Fluctuations and Repulsion Effects of Classical Dynamics in SU(3) model

499 Progress of Theoretical Physics, Vol. 110, No. 3, September 2003 Correlation Analysis of Quantum Fluctuations and Repulsion Effects of Classical Dynamics in SU (3) model Shigeyasu Fujiwara ∗) and Fumihiko Sakata ∗∗) Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan (Received December 4, 2002) §1. Introduction Random matrix theory has played an important role in the study of the quantum characteristics of chaos. Many investigations of level statistics have attracted much attention in conjunction with such a consideration that quantum level fluctuations may reflect nonintegrability in the corresponding classical system. As is well known, the quantum level statistics of classically integrable systems exhibit Poisson distributions except for special cases like a harmonic oscillator with a rational frequency ratio, while the quantum spectra of classically nonintegrable systems are characterized by Gaussian orthogonal ensembles (GOE),1) like the two-dimensional coupled Morse-oscillator system,3) the homogeneous quartic potential system,4) and the hydrogen atom in a uniform magnetic field.5) As is also known, a family of Brody distributions2) interpolates between the Poisson and Wigner distributions. The Brody distribution is characterized by a Brody’s repulsion parameter, which is evaluated in terms of the level fluctuation properties. In the spectrum statistics, a chaotic system possessing time reversal symmetry generally shows such a fluctuation that follows the GOE statistics. The GOE is based on the random matrix theory originally developed by Wigner for the description of highly excited atomic nuclei. Here, it should be noted that the GOE is derived under such a statistical assumption that the matrix elements of the Hamiltonian might be regarded as random numbers and representation-independent.1), 6), 7) Since this assumption is an additional ansatz put by hand not contained in the original quantum system, we try to propose a new method of using a correlation coefficient in evaluating the fluctuations, without introducing any artificial assumptions, and to show its validity through numerical simulations. Here it should be mentioned that our present analysis is quite different from the conventional analysis, because we are ∗) ∗∗) E-mail: E-mail: In many quantum systems, random matrix theory has been used to characterize quantum level fluctuations, which is known to be a quantum correspondent to a regular-to-chaos transition in classical systems. We present a new qualitative analysis of quantum and classical fluctuation properties by exploiting correlation coefficients and variances. It is shown that the correlation coefficient of the quantum level density is roughly inversely proportional relation to the variance of consecutive phase-space point spacings on the Poincaré section plane. 500 S. Fujiwara and F. Sakata §2. Symplectic structure of TDHF phase space at SU (3) model An adopted model is a modified SU (3) Hamiltonian introduced by Li, Klein, and Dreizler.8) It consists of three single-particle levels with energies
0 ,
1 and
2 satisfying
0 <
1 <
2 , and each level has N -fold degeneracy. Hereafter, we treat an even number N -particle system. Using a state |φ0
for which the lowest level with i = 0 is completely occupied, one may introduce a particle and hole operators as â1m |φ0
= â2m |φ0
= b̂m |φ0
= 0 . (2.1) The model Hamiltonian is given by ĤF = 2
i=0 κ̂i0 = N
m=1 2 1

i κ̂ii + Vi (κ̂i0 κ̂i0 + h.c.) , 2 i=1 â†im b̂†m , κ̂ij = N
m=1 â†im âjm , κ̂00 = N
m=1 b̂m b̂†m . (2.2) The generators κ̂ij satisfy the Lie algebra of the SU (3): [κ̂ij , κ̂kl ] = δjk κ̂il − δil κ̂kj . (2.3) The most general single Slater determinant is defined through |φ(f )
= exp(F̂ (t))|φ0
, F̂ (t) = 2
i=1 (fi †i − fi∗ Âi ) , †i = N
m=0 â†im b̂†m = κ̂i0 . The TDHF equation is given by the following variational equation:   ∂ exp(F̂ )|φ
= 0 . δφ| exp(−F̂ ) Ĥ − i ∂t (2.4) (2.5) going to pay our attention on how the energy levels are distributed rather than how the nearest-neighbor level spacings are distributed. In order to show the classical-quantum correspondence, we evaluate the fluctuations for a spacing of consecutive phase space points on the Poincaré section plane, and compare it with the quantum correlation coefficient. From this analysis, one may draw such a conclusion that the magnitude of level fluctuations observed in the quantum system is proportional to that of the repulsion between consecutive phase space points on the Poincaré section plane. It is worth mentioning that our analysis of the level fluctuations differs from the random matrix approach in such a way that no probability assumptions are included in our consideration. This paper is organized as follows. We consider the structure of TDHF phase space for the quantum SU (3) model in §2. The SU (3) Hamiltonian and its correlation analysis are discussed in §3. In §4, we explore the properties of consecutive phase space point spacings, by putting special emphasis on the relationship of the repulsion properties between the quantum spectra and the structure of TDHF phase space. Summary and discussion are presented in §5. Correlation Analysis of Quantum Fluctuations and Repulsion Effects 501 Here, we use the convention
= 1. The canonical variables (Cj∗ , Cj ; j = 1, 2) are defined in terms of the particle-hole amplitudes (fi∗ , fi ; i = 1, 2) in Eq. (2·4) through the following relation  ∗ √ sin f fi qj + ipj ≡ N fj  i ∗ i , Cj = √ 2 i fi fi  √ sin f ∗ fi qj − ipj (2.6) Cj∗ = √ ≡ N fj∗  i ∗ i . 2 i fi fi A set of parameters (Cj∗ , Cj ) satisfies the canonical variables condition.9) TDHF phase space possesses the symplectic structure10) described by ∂H ∂H = {Cj , H}P.B. , iĊj∗ = − = {Cj∗ , H}P.B. , ∗ ∂Cj ∂Cj where H = H(C, C ∗ ) ≡ φ(C, C ∗ )|Ĥ|φ(C, C ∗ )
and 
 ∂A ∂B ∂B ∂A . − {A , B}P.B. = ∂Cj ∂Cj∗ ∂Cj ∂Cj∗ (2.7) (2.8) j We can use the 6th-order symplectic integrators11) to integrate the classical canonical equations of motion. The SU (3) Hamiltonian in the TDHF symplectic manifold is given by H(q1 , p1 ; q2 , p2 ) = φ0 | exp(−F̂ )Ĥ exp(F̂ )|φ0
(q 2 + p21 ) (q 2 + p22 ) 2(N − 1) 2 2 = 2
0 r2 +
1 1 +
2 2 + V1 r (q1 − p21 ) 2 2 N 2(N − 1) 2 2 r (q2 − p22 ) , + V2 N  1 r≡ 2N − q12 − p21 − q22 − p22 . (2.9) 2 In our numerical calculation, we used N = 30 and the single particle energies were
0 = 0,
1 = 1 and
2 = 2, with which our energy scale is non-dimensional. When the interaction V is sufficiently small, crossing points on the Poincaré section plane12) turn out to be a closed curve for almost every set of initial conditions shown in Figs. 1(a)−(c), where√the variables (q1√ , p1 ; q2 , p2 )√in Eq. (2·9) are trans√ formed as (q1 , p1 ; q2 , p2 )−→(q1 / 2N , p1 / 2N ; q2 / 2N , p2 / 2N ). In these Poincaré section planes, we have used samples with initial value from q1 = 0.096 to 0.88 in steps of 0.001 al (...truncated)