Level Statistics of SU(3)-SU(3)* Transitional Region

The level statistics of SU(3)-SU(3)* transitional region of IBM is described by the nearest neighbor spacing distribution statistics. The energy levels are determined by using the SO(6)representation of eigenstates. By employing the MLE technique, the parameter of Abul-Magd distribution is estimated where suggest less regular dynamics for transitional region in compare to dynamical symmetry limits. Also, the O(6)dynamical symmetry which is known as the critical point of this transitional region, describes a deviation to more regular dynamics.


Introduction
The investigation of significant changes in energy levels and electromagnetic transition rates resulting in the shape phase transitions [1][2][3][4] from one kind of collective behavior to another has received a lot of attention in recent years. The new symmetries called (5) X and (5) E are obtained within the framework of the collective model have employed to describe atomic nuclei at the critical points [5][6][7]. In the interacting boson model (IBM) framework [8][9][10][11], a very simple twoparameter description has been used leading to a symmetry triangle describing many atomic nuclei. This model describes the nuclear structure of the even-even nuclei within the (6) U symmetry, possessing the (5) U , (3) SU and (6) O dynamical symmetry limits. No phase transition is found between the (3) SU and (6) O vertices of the triangle. However, as discussed in Refs. [12][13] in the context of catastrophe theory, an analysis of the separatrix of the IBM-1 Hamiltonian in the coherent state formalism shows that there is a phase transition in between oblate ( (3) SU ) and prolate ( (3) SU ) deformed nuclei. This phase transition and its critical point symmetry, which in fact, coincides by the O(6) limit has described from the standpoint of physical observables in Refs. [14][15][16] by Jolie et al.
On the other hand, the energy level statistics [17][18][19] has employed as a new observable to analyze the evolution of energy spectra along the two dynamical symmetry limits. It's well- 3 known [20][21], the energy spectra of nuclei correspond to any definite phase, governed due to the dynamical symmetry limits, exhibit more regularity. On the other hand, in the transitional region which contains the critical point of quantum phase transition, system goes from a symmetry limit to another dynamical symmetry limit. Consequently, a combination of different symmetries visualizes by nuclei and a deviation of regular dynamics occurs in spectra.
In the present paper, we investigate the fluctuation properties of energy spectra for (3) (3) SU SU  transitional region in the Nearest Neighbor Spacing Distribution (NNSD) statistics framework. To determine the energy spectra of considered region, the (6) SO representation of eigenstates [22][23]  This paper is organized as follows: section 2 briefly summarizes the theoretical aspects of considered Hamiltonian and (6) SO representations of eigenstates. In section 3, details about statistical investigation are presented which includes unfolding procedure and MLE technique which applied to Abul-Magd distribution. Numerical results are presented in Section 4. Section 5 is devoted to summarize and some conclusion based on the results given in section 4.

Theoretical framework
The phase transition have been studied widely in Refs. [14][15][16], are those of the ground state deformation. In the Interacting Boson Model (IBM), one would achieve a very simple two parameters description leading to a symmetry triangle which is known as extended Casten triangle [11]. There are four dynamical symmetries of the IBM called (5) U , The Algebraic structure of IBM has been described in detail in Refs. [22][23]. Here, we briefly outline the basic ansatz and summarize the results have obtained in this paper for our considered representation. The classification of states in the IBM (6) SO limit is [26][27] The multiplicity label  in the On the other hand, coupled two body operators are of the form represent coupling to angular momentum () l and the C coefficients are known isoscalar factors [28]. These processes lead to the normalized two-boson (6) SO representation displayed in Tables (1-2) for systems with total boson number N  3 and 4, respectively.
Table1. The (6) SO representation of eigenstates for systems with total boson number () representation of eigenstates for systems with total boson number () Now, with using these eigenstates, one can determine energy spectra for considered systems as Where  denote the matrix elements of quadrupole term in Hamiltonian as presented in Tables (3-4)  Now, we are proceeding to determine the energy spectra by parameter free method, i.e. up to over all scale factors while we have considered all levels in our sequences.

The method of Statistical analysis
The spectral fluctuations of low-lying nuclear levels have been considered by different statistics such as Nearest Neighbor Spacing Distribution (NNSD) [17], linear coefficients between adjacent spacing [18] and Dyson-Mehta 3 () L  statistics [29][30] which based on the comparison of statistical properties of nuclear spectra with the predictions of Random Matrix Theory (RMT). The NNSD, or () Ps functions, is the observable most commonly used to analyze the short-range fluctuation properties in the nuclear spectra. The NNSD statistics would perform by complete (few or no missing levels) and pure (few or no unknown spin-parities) level scheme [18] where these conditions are available for a limited number of nuclei. Therefore, to obtain the statistically relevant samples, we in need to combine different level schemes to construct sequences. To compare the different sequences to each other, each set of energy levels must be converted to a set of normalized spacing, namely, each sequence must be unfolded. To unfold our spectrum, we had to use some levels with same symmetry. For a given spectrum{} i E , it is necessary to separate it into the fluctuation part and the smoothed average part, whose behavior is nonuniversal and cannot be described by RMT [17]. To do so, we count the number of the levels below E and write it as  [31][32]. One of popular distribution is Abul-Magd distribution [24] which is derived by assuming that, the energy level spectrum is a product of the superposition of independent subspectra, which are contributed respectively from localized eigenfunctions onto invariant (disjoint) phase space. This distribution is based on the Rosenzweig and Porter random matrix model. The exact form of this model is complicated and its simpler form is proposed by Abul-Magd et al in Ref. [24] as:

5) 24
ss P s q q q q q s q q           Where interpolates between Poisson ( 0) f  and Wigner ( 1) f  distributions. In common considerations, one can handle a least square fit (LSF) of Abul-Magd distribution to sequence while the value of distribution's parameter describes the chaotic or regular dynamics [18]. The LSF-based estimated values have some unusual uncertainties and also exhibit more deviation to chaotic dynamics. Consequently, it is almost impossible to carry out any reliable statistical analysis in some sequences. Recently [25], we have employed the Maximum Likelihood Estimation (MLE) technique to estimate every distribution's parameter which provides more precisions with low uncertainties, i.e. estimated values yield accuracies which are closer to Cramer-Rao Lowe Bound (CRLB). Also, this technique yields results which are almost exact in all sequences, even in cases with small sample sizes, where other estimation methods wouldn't achieve the appropriate results. Consequently, we analyzed the spectral statistics of considered systems with about 10 or more samples in each sequence with more precision. The MLE estimation procedure has been described in detail in Ref [25]. Here, we outline the basic ansatz and summarize the results.

 The ML-based results for Abul-Magd distribution
The MLE method provides an opportunity for estimating exact result with minimum variation. In order to estimate the parameter of distribution, Likelihood function is considered as product of all () Ps functions, 2 (

Numerical results
In the present study, we consider the statistical properties of (3)  Since the investigation of the majority of short sequences yields an overestimation about the degree of chaoticity measured by the "q" ( Abul-Magd distribution's parameter), therefore, we wouldn't concentrate only on the implicit value of " q " and examine a comparison between the amounts of " q " for different  values. It means, the smallest values of "q" explain more regular dynamics and vice versa.
The ML-based estimated "q" values are listed in Table5 where the variations of this quantity for considered systems are presented in Figure2. The matrix elements of quadrupole operator in transitional Hamiltonian (2.1) and consequently, the energy spectra of considered systems, i.e. Eq.(2.7), are symmetric with respect to the control parameter"  "(where only even powers of  appeared in them). The symmetric variations of chaocity degrees for considered systems with respect to  are in agreement with this property of spectra. The "q" values and also ). Our results may be interpreted that the some special values of control parameter (  ) which describe the level crossing for considered systems, explore a deviation to regular dynamics due to the partial dynamical symmetries in these transitional regions [35]. Also, 0   which describe (6) O dynamical symmetry limit and is known as (5) Z critical point of this transitional region [36][37], exhibit more regularity than the predictions for (3) SU (or (3) SU ) dynamical symmetry limit, e.g. we have suggested same results in the Ref. [25] by employing sequences prepared by nuclei which provide empirical evidences for these dynamical symmetry limits.

Summary and conclusion
In this paper, the spectral statistics of (3) (3) SU SU  transitional region was described in NNSD statistics framework. In the parameter free approach, energy spectra have determined by using the (6) SO representation of eigenstates for systems with total boson number N  3 and 4. By employing the MLE technique to estimate with more accuracy, the parameter of Abul-Magd's distribution was estimated where proposed a deviation to less regularity for transitional region between dynamical symmetry limits, namely (6) ( ). These results may be interpreted a less regular dynamics for transitional region due to symmetry broken or a combination of different symmetries in this regions. Also, some deviations to regularity may caused by partial dynamical symmetries in these regions while 0   which describe (6) O dynamical symmetry limits, i.e. (5) Z critical point, explore more regular dynamics.
Works in these directions are in progress.