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A Hamiltonian in the IBM framework is constructed from two types of boson operators: s-boson with
$ J^\pi = 0^+ $ and d-boson with$ J^\pi = 2^+ $ [1]. To discuss different scenarios in the IBM, it is convenient to adopt the consistent-Q Hamiltonian [33], which can be expressed as$ \begin{equation} \hat{H}(\eta,\; \chi) = \varepsilon \left[ (1-\eta)\hat{n}_{d} - \frac{\eta}{4N}\hat{Q}^{\chi} \cdot \hat{Q}^{\chi} \right] \, . \end{equation} $
(1) In the Hamiltonian,
$ \hat{n}_{d} = d^\dagger\cdot\tilde{d} $ is the d-boson number operator,$ \hat{Q}^{\chi} = (d^{\dagger} s + s^{\dagger} \tilde{d})^{(2)} + \chi (d^{\dagger} \tilde{d})^{(2)} $ is the quadrupole operator, η and χ are the control parameters with$ \eta\in[0,1] $ and$ \chi\in[-\sqrt{7}/2,0] $ , and ε is a scale factor, set as 1 for convenience. Subsequently, the different dynamical scenarios in the IBM are characterized by the different values of the control parameters η and χ. Specifically, the Hamiltonian is in the U(5) DS when$ \eta = 0 $ ; it is in the O(6) DS when$ \eta = 1 $ and$ \chi = 0 $ ; it is in the SU(3) DS when$ \eta = 1 $ and$\chi = - {\sqrt{7}}/{2}$ . Three DSs in the IBM describe three typical nuclear shapes: the spherical (U(5)), axially-deformed (SU(3)), and γ-unstable (O(6)).The mean-field structure of the IBM can be established by using the coherent state defined as [1]
$ \begin{aligned}[b] |\beta, \gamma, N\rangle = &\frac{1}{\sqrt{N! (1 + \beta^2)^N}} [s^\dagger + \beta \mathrm{cos} \gamma\; d_0^\dagger\, \\&+ \frac{1}{\sqrt{2}} \beta \mathrm{sin} \gamma (d_2^\dagger + d_{ - 2}^\dagger)]^N |0\rangle\, . \end{aligned} $
(2) The scaled potential surface corresponding to the Hamiltonian (1) in the large-N limit is then given as [34]
$ \begin{aligned}[b] V(\beta, \gamma)= &\frac{1}{N} \langle\beta, \gamma, N | \hat{H}(\eta,\; \chi) | \beta, \gamma, N\rangle|_{N\rightarrow\infty} \\ = & (1-\eta) \frac{\beta^2}{1+\beta^2} - \frac{\eta}{4(1 + \beta^2)^2}\\ &\times[4\beta^2 - 4\sqrt{\frac{2}{7}}\chi\beta^3 \mathrm{cos3} \gamma + \frac{2}{7}\chi^2\beta^4]\, . \end{aligned} $
(3) By minimizing this potential with respect to β and γ, we can prove that the first-order ground state quantum phase transitions (GSQPTs) may occur at the parameter points with [34]
$ \begin{equation} \eta_\mathrm{c} = \frac{14}{28+\chi^2},\; \; \; \chi\in[-\frac{\sqrt{7}}{2},\; 0)\, \end{equation} $
(4) and the second-order GSQPT occurs only at the point
$ \eta_\mathrm{c} = 1/2 $ with$ \chi = 0 $ on the U(5)-O(6) transitional line.The two-dimensional phase diagram of the IBM can be mapped into a triangle. As shown in Fig. 1, each vertex of the triangle corresponds to a given DS, and the entire area is cut into two regions by the first-order transitional line: the spherical and deformed. Fig. 1 further shows that the AW arc denoted by the dotted curve extends its trajectory from the SU(3) vertex to the interior of the triangle with the parameter trajectory being approximately described by
Figure 1. (color online) Triangle phase diagram of the IBM, with the dashed black line denoting the critical points of the first-order GSQPTs described by (4) and six parameter points A, B, C, D, E, and F with
$(\eta,\; \chi) = (0.38,\; -\sqrt{7}/2),\; (0.47,\; -\sqrt{7}/2),\; $ $ (0.75,\; -\sqrt{7}/2),\; (0.88,\; -1.032), \; (0.88,\; -0.7)$ , and$(1.0,\; -0.9)$ selected to analyze the spectral fluctuations in different scenarios. In addition, the chaotic region of the IBM identified previously [10] are schematically illustrated by two shaded areas, with a red curve signifying the trajectory of the AW arc of regularity passing through the chaotic region.$ \begin{equation} \chi = \frac{4+(\sqrt{7}-4)\eta}{6\eta-8}\, , \end{equation} $
(5) which can be determined from either a minimal fraction of the chaotic phase-space volume [10] or the minimal values of the entropy-ratio product [16]. The fraction of chaotic-space volume [10], which is defined as an phase-space integral under given conditions, is a measure of classical chaos and is therefore applied to test the chaocity generated by the classical limit of the Hamiltonian. It is often calculated using Monte Carlo methods. The smaller the fraction, the more regular the system is. In contrast, the entropy-ratio product, which is defined based on the Shannon information entropy of wave function [16], may be considered as a quantum measure of chaos. The wave functions and all the reference bases must be known to calculate the entropy-ratio product. Similarly, the smaller the entropy-ration product, the more regular the system. In practice, the two methods agree with each other very well in characterizing the trajectory of the regular arc in the IBM (see, for example, Fig. 8 in [16]), which embodies the consistency between classical and quantum chaos appearing in an IBM system. More details of the two methods are available in [6, 10, 15, 16]. In contrast, another approximate parametrization of the arc can be obtained as
Figure 8. (color online) Same as in Fig. 7 but taking those bound above different energy cutoffs.
$ \begin{equation} \chi = \frac{2\sqrt{2}-(2\sqrt{2}+\sqrt{7})\eta}{2\eta}\, , \end{equation} $
(6) by connecting the approximate SU(3) symmetry with the AW arc of regularity [22]. We can observe that Eq. (6) describes the parameter trajectory being very close to the one described by (5) for
$ \eta\in[0.5,\; 1.0] $ .As schematically illustrated in Fig. 1, the trajectory of the AW arc may pass through two chaotic regions in the IBM phase diagram described by the parameters
$ (\eta,\; \chi) $ [10], whereas the two chaotic regions may cover most of the deformed area of the phase diagram. To analyze the spectral fluctuations in different scenarios, we select six parameter points to perform statistical calculations. These parameter points correspond to$(\eta,\; \chi) = (0.38,\; -{\sqrt{7}}/{2})$ ,$(0.47,\; -{\sqrt{7}}/{2})$ ,$(0.75,\; -{\sqrt{7}}/{2})$ ,$ (0.88,\; -1.032) $ ,$(0.88, -0.7)$ , and$ (1.0,\; -0.9) $ , which have been denoted by A, B, C, D, E, and F, respectively, as shown in Fig. 1. The points A, B, and C are used to illustrate three typical scenarios in the U(5)-SU(3) GSQPT, namely the spherical phase, critical point, and deformed phase; points D and E are selected to indicate the scenarios inside the triangle but lying on and off the AW arc; point F represents a typical case on the SU(3)-O(6) line, which in the large-N limit corresponds to a crossover [1].We can prove that the extreme values of the potential function (3) appear at either
$ \gamma = 0^\circ $ or$ \gamma = 60^\circ $ , whereas the latter can be equivalently realized by obtaining a negative β value owing to$ V(\beta,\gamma = 60^\circ) = V(-\beta,\gamma = 0^\circ) $ . The stationary points discussed here are defined as the extreme value points of the potential function$V(\beta)\equiv V(\beta,\gamma = 0^\circ)$ , namely those with$ \dfrac{\partial V(\beta)}{\partial\beta} = 0 $ . As an example, the potential curve at a typical parameter point inside the triangle is shown in Fig. 2 to illustrate the stationary point structure. Generally, the stationary points of a potential curve include the global minimum, global maximum, local minimum, local maximum, and$ |\beta|\rightarrow\infty $ limit point, which are denoted as$ V_{\mathrm{gmin}} $ ,$ V_{\mathrm{gmax}} $ ,$ V_{\mathrm{lmin}} $ ,$ V_{\mathrm{lmax}} $ , and$ V_{\mathrm{lim}} $ , respectively, as shown in Fig. 2. Among them,$ V_{\mathrm{lmin}} $ may correspond to a saddle point of$ V(\beta,\gamma) $ if we observe it from the degree of freedom of β and γ. At the mean-field level,$ V_{\mathrm{gmin}} $ and$ V_{\mathrm{gmax}} $ correspond only to the ground-state and highest excited energies, respectively. Therefore, more emphasis should be placed on$ V_{\mathrm{lmin}} $ ,$ V_{\mathrm{lmin}} $ , and$ V_{\mathrm{lmin}} $ . In terms of excitation energy, the related “critical” energies are further expressed asFigure 2. (color online) Potential curve
$V(\beta)$ (in any unit) solved using (3) with$(\eta,\; \chi) = (0.9,-1.3)$ . The stationary points on the curve are signified by the dashed lines.$ E_{\mathrm{lmin}} = V_{\mathrm{lmin}}-V_{\mathrm{gmin}}, $
(7) $E_{\mathrm{lmax}} = V_{\mathrm{lmax}}-V_{\mathrm{gmin}}, $
(8) $E_{\mathrm{lim}} = V_{\mathrm{lim}}-V_{\mathrm{gmin}}\, . $
(9) We easily know that the energy scale in the “critical” energies is as same as the excitation energy per boson
$ E/N $ (in any unit), which means that the “critical” energies defined above can be directly used to compare the excitation energy$ E/N $ solved from the same Hamiltonian. -
To measure the spectral fluctuations in the IBM, we adopt two statistical measures here: the nearest neighbor level spacing distribution
$ P(S) $ [12] and the$ \Delta_3 $ statistics of Dyson and Mehta [13]. The spectral statistics should be performed to the so-called unfolded spectrum to be consistent with the requirements of the GOE [8]. First, we construct the staircase function of the spectrum,$ N(E) $ , defined as the number of levels below E with the level energies E solved from the Hamiltonian (1).$ N(E) $ is further separated into average and fluctuating parts$ \begin{equation} N(E) = N_{\mathrm{av}}(E)+N_{\mathrm{fluct}}(E)\, .\ \end{equation} $
(10) The average part can be expanded as a polynomial of sixth order in E [8, 14]:
$ \begin{equation} N_{\mathrm{av}}(E) = a_0+a_1E+a_2E^2+a_3E^3+a_4E^4+a_5E^5+a_6E^6\, \end{equation} $
(11) with the expanding parameters
$ a_i $ determined from the best fit to$ N(E) $ . Subsequently, the unfolded spectrum is obtained via the mapping$ \tilde{E_i} = N_{\mathrm{av}}(E_i) $ . With the unfolded spectrum, the nearest neighbor level spacings are obtained from$ \begin{equation} S_i = \tilde{E}_{i+1}- \tilde{E}_i\, , \end{equation} $
(12) and the distribution
$ P(S) $ is then given as the probability of two neighboring levels to be separated by a distance S. Specifically,$ P(S) $ will be shown as the histogram of the normalized spacing, and the results are further fitted to the Brody distribution [12]$ \begin{equation} P_\omega(S) = \alpha(1 + \omega)S^\omega \mathrm{exp}(-\alpha S^{1+\omega})\, \end{equation} $
(13) with
$ \alpha = \Gamma[(2 + \omega)/(1 + \omega)]^{1+\omega} $ . The Brody distribution interpolates between Poisson statistics$ (\omega = 0) $ characterizing a fully regular system and the Wigner distribution$ (\omega = 1) $ indicating a completely chaotic system [8]. As a result, the intermediate value with$ \omega\in[0,\; 1] $ provides a quantitative estimation of the quantum chaos in the spectrum.Spectral rigidity,
$ \Delta_3(L) $ , is a measure of the deviation of the staircase function from a straight line [13]. It is defined by$ \begin{equation} \Delta_3 (a,L) = \frac{1}{L} \mathrm{min}_{A,B}\int_a^{a+L} [N(\tilde{E})-A\tilde{E}- B]^2 {\rm d}\tilde{E}\, , \end{equation} $
(14) where A and B give the best local fit to
$ N(\tilde{E}) $ in the interval$ a\leq\tilde{E}\leq a+L $ , where L is the energy length of the interval. A rigid spectrum should provide a smaller$ \Delta_3 $ , whereas a soft spectrum provides a larger$ \Delta_3 $ . A smoother$ \Delta_3(L) $ can be obtained by averaging$ \Delta_3(a,L) $ over$ n_a $ intervals$ (a,\; a + L) $ :$ \begin{equation} \Delta_3(L) = \frac{1}{n_a}\sum_a\Delta_3(a,L)\, . \end{equation} $
(15) The successive intervals are set to overlap by
$ L/2 $ . In the concrete calculations, a useful formula [8]$ \begin{aligned}[b] \Delta_3(a,L) = &\frac{n^2}{16}-\frac{1}{L^2}\Bigg(\sum_{i = 1}^n\tilde{\epsilon}_i\Bigg)^2+\frac{3n}{2L^2}\Bigg(\sum_{i = 1}^n\tilde{\epsilon}_i^2\Bigg)\\ &-\frac{3}{L^4}\Bigg(\sum_{i = 1}^n\tilde{\epsilon}_i^2\Bigg)+\frac{1}{L}\Bigg(\sum_{i = 1}^n(n-2i+1)\tilde{\epsilon}_i\Bigg)\, \end{aligned} $
(16) with
$ \tilde{\epsilon}_i = \tilde{E}_i-(a+L/2) $ is often adopted. For the Poisson statistics, it is given by$ \begin{equation} \Delta_3^\mathrm{P}(L) = \frac{L}{15}\, , \end{equation} $
(17) whereas for the GOE (chaotic) case, it is approximately given by
$ \begin{equation} \Delta_3^{\mathrm{GOE}}(L) = \frac{1}{\pi^2}(\mathrm{log}L-0.0687)\, \end{equation} $
(18) for
$ L\gg1 $ . The exact form of$ \Delta_3 $ statistics for the GOE and Poisson limit can be solved using the integral [8]$ \begin{equation} \Delta_3(L) = \frac{2}{L^4}\int_0^L(L^3-2L^2r+r^3)\Sigma^2(r){\rm d}r\, \end{equation} $
(19) with
$ \Sigma^2(L) = L $ given for the Poisson statistics and$ \begin{aligned}[b] \Sigma^2(L) = &\frac{2}{\pi^2}\Big[\mathrm{ln}(2\pi L)+\bar{\gamma}+1+\frac{1}{2}\Big(\mathrm{Si}(\pi L)\Big)^2\\ &-\frac{\pi}{2}\mathrm{Si}(\pi L)-\mathrm{cos}(2\pi L)-\mathrm{Ci}(2\pi L)\\ &+\pi^2L\Big(1-\frac{2}{\pi}\mathrm{Si}(2\pi L)\Big)\Big]\, \end{aligned} $
(20) given for the GOE [8, 35]. In Eq. (20),
$ \bar{\gamma} $ is the Euler constant and Si (Ci) is the sine (cosine) integral. Similar to the Brody distribution, we can fit the calculated$ \Delta_3(L) $ with the parameterization [8, 12, 14, 36]$ \begin{equation} \Delta_3^{q}(L) = \Delta_3^{\mathrm{Poisson}}[(1-q)L]+\Delta_3^{\mathrm{GOE}}(qL),\; \; \; \; q\in[0,\; 1]\, \end{equation} $
(21) to obtain a quantitative estimation of the possible deviation of
$ \Delta_3(L) $ from the regular ($ q = 0 $ ) or chaotic limit ($ q = 1 $ ).To exemplify the
$ P(S) $ and$ \Delta_3 $ statistics, the calculated results for the spectra with$ L^\pi = 0^+,\; 2^+,\; 3^+ $ in the SU(3) limit are shown in Fig. 3. Since the SU(3) limit in the IBM corresponds to a completely integrable scenario, its dynamics are expected to be close to the regular limit (the Poisson statistics). As shown in Fig. 3, the results agree with such an expectation, resulting in$ \omega\sim0.12 $ ($ q\sim0.40 $ ) for the$ 0^+ $ spectrum and$ \omega\sim0.16 $ ($ q\sim0.43 $ ) for the$ 3^+ $ spectrum, which thus provides an example of regular system for reference. For the$ 2^+ $ spectrum in the SU(3) limit, we observe that the$ P(S) $ statistics indicate a negative value of ω, and the$ \Delta_3 $ statistics present values even larger than the Poisson statistics (the regular limit). This is a result of degeneracies and is related to missing labels according to the analysis given in [8]. In the SU(3) limit, the missing label is the quantum number K, which often provides two possible values$ K = 0,\; 2 $ for the$ 2^+ $ states in a given SU(3) irrep$ (\lambda,\; \mu) $ , but this may not affect the statistics for the$ 0^+ $ and$ 3^+ $ spectra as only one value is possible ($ K = 0 $ or$ K = 2 $ , respectively) for$ L^\pi = 0^+ $ and$ L^\pi = 3^+ $ [8]. Similar scenarios can also occur in the other symmetry limits or even in the cases that are close to a dynamical symmetry [6, 8], where the degeneracies accordingly become the approximate ones. Since the q values fitted from Eq. (21) cannot be negative, they are set as$ q = 0 $ in the following discussions if the$ \Delta_3 $ statistics is larger than the Poisson limit, similar to those shown in Fig. 3(f). Comparatively, ω appears to be always a reliable indicator of the spectral fluctuations in different cases.Figure 3. (color online) (a)
$P(S)$ statistics for the states with$L^\pi = 0^+$ in the SU(3) limit with ω fitted from Eq. (13) are shown to compare the regular limit (RL) and chaotic limit (CL), which are denoted by the dotted and dashed lines, respectively; (b) Same as in (a) but for the$\Delta_3(L)$ statistics with q fitted from Eq. (21). Panels (c)-(f) show the same statistics as in (a) or in (b) but for$L^\pi = 3+$ and$L^\pi = 2^+$ . The calculation is performed for$N = 200$ .
Spectral fluctuations in the interacting boson model
- Received Date: 2022-04-18
- Available Online: 2022-10-15
Abstract: The energy dependence of the spectral fluctuations in the interacting boson model (IBM) and its connections to the mean-field structures are analyzed by adopting two statistical measures: the nearest neighbor level spacing distribution