Exact solution of the U(5)–O(6) transitional description in the interacting boson model with two-particle and two-hole configuration mixing

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Lianrong Dai, Feng Pan, Ziwei Feng, Yú Zhang, Sai Cui and J. P. Draayer. Exact solution of the U(5)–O(6) transitional description in the interacting boson model with two-particle and two-hole configuration mixing[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/6/064102
Lianrong Dai, Feng Pan, Ziwei Feng, Yú Zhang, Sai Cui and J. P. Draayer. Exact solution of the U(5)–O(6) transitional description in the interacting boson model with two-particle and two-hole configuration mixing[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/6/064102 shu
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Exact solution of the U(5)–O(6) transitional description in the interacting boson model with two-particle and two-hole configuration mixing

    Corresponding author: Feng Pan, daipan@dlut.edu.cn
  • 1. Department of Physics, Liaoning Normal University, Dalian 116029, China
  • 2. Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Abstract: Exact solution of the U(5)-O(6) transitional description in the interacting boson model with two-particle and two-hole configuration mixing is derived based on the Bethe ansatz approach. The Bethe ansatz equations in determining eigenstates and the corresponding eigen-energies of the model are provided. Specific N=2 and N=4 cases are exemplified to demonstrate the feature of the solution. As an example of the application, some low-lying level energies and B(E2) ratios of 108Cd are fitted and compared with the corresponding experimental data.

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    1.   Introduction
    • The interacting boson model (IBM) has been proven to be very successful in the description of both collective valence shell [1] and multi-particle-hole [24] excitations in nuclei. Most noticeably, the IBM Hamiltonian without configuration mixing can be solved analytically in the U(5) (vibrational), O(6) (γ-unstable), SU(3) (rotational) limits [1], and the U(5)–O(6) transitional case [5]. On the other hand, configuration mixing due to multi-particle-hole excitations was considered in understanding shape coexistence phenomena by taking different symmetry limits of the IBM for different configurations [612], which has been proven to be successful in describing intruder states and shape coexistence phenomena in near closed shell nuclei, typically those around proton numbers Z~50 and Z~82 [24]. Very recently, the intruder configuration mixing schemes with $ 2n $-particle and $ 2n $-hole configurations from $ n = 0 $ up to $ n\rightarrow\infty $ in the U(5) (vibrational) and the O(6) (γ-unstable) limits of the IBM-I were proposed [13, 14], of which the simple Hamiltonians suitable to describe the intruder and normal configuration mixing turn to be exactly solvable based on the SU(1,1) coherent states.

      The configuration mixing schemes in the IBM [24] can be considered in both the IBM-II and the IBM-I with no distinction between neutron-type and proton-type bosons as shown in [710, 1316]. In this work, it is shown that the U(5)↔O(6) transitional Hamiltonian of the IBM-I with two-particle and two-hole configuration mixing is also exactly solvable based on the Bethe ansatz approach. The results of $ N = 2 $ and $ N = 4 $ cases are exemplified to demonstrate the feature of the solution. As a use of the theory, the model is applied to fit some low-lying level energies and B(E2) ratios of $ ^{108} {\rm{Cd}}$.

    2.   The model and its exact solution
    • The Hamiltonian of the U(5)–O(6) transitional description in the IBM-I with two-particle and two-hole configuration mixing can be written as [24]

      $ \hat{H} = P_{N}\hat{H}^{(1)}_{0}P_{N}+P_{N+2}\hat{H}^{(2)}_{0}P_{N+2}+P\hat{H}_{\rm{mix}}P, $

      (1)

      where $ P_{N} $ and P are projection operators, in which $ P_{N} $ projects to the N-boson subspace, while P projects to the subspace with N and $ N+2 $ bosons,

      $ \hat{H}^{(i)}_{0} = a^{(i)}_{s} S^{0}_{s}+a_{d}^{(i)}S^{0}_{d}+g^{(i)}S^{+}S^{-} $

      (2)

      for $ i = 1 $ and $ 2 $ are the U(5)–O(6) transitional Hamiltonians [5], and

      $ \hat{H}_{\rm{mix}} = g_{s}\,(S^{+}_{s}+S^{-}_{s})+g_{d}\,(S^{+}_{d}+S^{-}_{d}) $

      (3)

      is the two-configuration mixing term. Here $ S^{+} = S^{+}_{s}-S^{+}_{d} $ with $ S_{s}^{+} = {1\over{2}}s^{\dagger 2} $ and $ S^{+}_{d} = {1\over{2}}d^{+}\cdot d^{+} = {1\over{2}}\sum_{\mu}(-1)^{\mu}d^{\dagger }_{\mu}d^{+}_{-\mu} $, in which $ s^{\dagger} $ (s) and $ d^{\dagger}_{\mu} $ ($ d_{\mu} $) are the creation (annihilation) operators of s- and d-bosons, respectively, $ S^{-}_{\rho} = \left(S^{+}_{\rho}\right)^{\dagger } $ for $ \rho = s $ or d, $ S^{0}_{s} = {1\over{2}}(\hat{n}_{s}+{1\over{2}}) $ and $ S^{0}_{d} = {1\over{2}}(\hat{n}_{d}+{5\over{2}}) $, respectively, with $ \hat{n}_{s} = s^{\dagger }s $ and $ \hat{n}_{d} = \sum_{\mu}d^{\dagger }_{\mu}d_{\mu} $, and $ \alpha^{(i)}_{\rho} $, $ g^{(i)} $, $ g_{s} $ and $ g_{d} $ are real parameters. The U(5)$ \leftrightarrow $O(6) Hamiltonian (2), which is equivalent to the consistent-Q formulism of the IBM but different from that used in [5], is in the U(5) (vibrational) phase when $ \alpha_{\rho}^{(i)}\neq 0 $ and $ g^{(i)} = 0 $ or in the O(6) (γ-unstable) phase when $ \alpha_{\rho}^{(i)} = 0 $ and $ g^{(i)}<0 $. Since $ 2(\alpha_{d}^{(i)}-\alpha_{s}^{(i)}) $ is the energy gap of d- and s-bosons, $ \alpha_{s}^{(1)} $ is taken to be zero. In previous configuration mixing study, $ \Delta = 2\alpha_{d}^{(2)}-2\alpha_{d}^{(1)} = 2\alpha_{s}^{(2)}-2\alpha_{s}^{(1)} $ is taken according to the energy of the lowest intruder state to reduce the number of parameters. The shape phase within the N-boson and $ N+2 $-boson configuration controlled by the two sets of parameters $ \{\alpha_{\rho}^{(i)},\,g^{(i)}\} $ ($ i = 1,2 $) can be different to describe the shape (phase) coexistence.

      The Hamiltonian (1) can be diagonalized in the $ N\oplus (N+2) $-boson subspace, of which the complete basis vectors in each configuration can be taken as those of U(6)$\supset $U(5)$\supset $O(5)$\supset $O(3) with $ \vert N\,n_{d}\, \nu_{d}\,\eta\, L\, M\rangle $, where N is the total number of bosons, $ n_{d} $ is the number of d-bosons, $ \nu_{d} $ is the d-boson seniority number labeling the irrep of $ O(5) $, L is the angular momentum quantum number, M is the quantum number of the third component of the angular momentum, and η is an additional quantum number needed to distinguish different states with the same L. Moreover, the two sets of operators $ \{{S}^{\pm}_{\rho},\; {S}^{0}_{\rho}\} $ ($ \rho = s,d $), which are two copies of SU(1,1) algebra, satisfy the commutation relations

      $ [S^{0}_{\rho'},\; {S}^{\pm}_{\rho}] = \pm\,\delta_{\rho'\rho}{S}^{\pm}_{\rho},\; [{S}^{-}_{\rho},\; {S}^{+}_{\rho'}] = 2\delta_{\rho'\rho}S^{0}_{\rho}. $

      (4)

      Equivalently, for given N, $ n_{d} $, $ \nu_{d} $, η, L, and M, the orthonormalized basis vectors $ \vert N\,n_{d}\, \nu_{d}\,\eta\, L\, M\rangle $ can also be expressed as those of $ {{SU}}_{d} $(1,1)$ \otimes {{SU}} _{s} $(1,1) with

      $\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle = (-1)^{\xi} {\cal N} (S^{+}_{s})^{{N-\nu_{d}-\nu_{s}\over{2}}-\xi}(S^{+}_{d})^{\xi} \vert \nu_{s};\,\nu_{d}\,\eta\, L\,M\rangle, $

      (5)

      in which $ n_{d} = 2\xi+\nu_{d} $ and $ \xi = 0,1,2,\cdots, {1\over{2}}(N-\nu_{d}-\nu_{s}) $ with $ \nu_{s} = 0 $ or 1, where the normalization constant

      $ {\cal N} = \left( {2^{N-\nu_{d}-\nu_{s}-\xi}(2\tau+3)!!\over{\xi!(N-\nu_{d}-2\xi)!(2\nu_{d}+2\xi+3)!!}}\right)^{1\over{2}}. $

      (6)

      The conventional phase factor $ (-1)^{\xi} $ shown in (5) for $ {{SU}}_{d} $(1,1) is adopted, which is consistent with the generalized pairing operator $ S^{\pm} = S^{+}_{s}- S^{\pm}_{d} $ used in (2). The matrix representations of $ {{SU}}_{d}(1,1)\otimes $ $ {{SU}}_{s} $(1,1) under the basis vectors (5) are given by

      $ \begin{split} S^{+}_{d}\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\, \eta\,L\, M\rangle =& -{1\over{2}}\sqrt{(2\xi+2)(2\nu_{d}+2\xi+5)}\,\vert N +2,\,\xi+1\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle,\\ S^{-}_{d}\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle =& -{1\over{2}}\sqrt{2\xi(2\nu_{d}+2\xi+3)}\,\vert N-2,\xi-1\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle,\\ S^{0}_{d}\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle =& {1\over{2}}(\nu_{d}+2\xi+{5\over{2}})\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle, \end{split} $

      (7)

      and

      $ \begin{split} S^{+}_{s}\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle =& {1\over{2}}\sqrt{(N-\nu_{d}-2\xi+2)(N-\nu_{d}-2\xi+1)}\,\vert N+2,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle,\\ S^{-}_{s}\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\, \eta\,L\, M\rangle = &{1\over{2}}\sqrt{(N-\nu_{d}-2\xi)(N-\nu_{d}-2\xi-1)}\,\vert N-2,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle,\\ S^{0}_{s}\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle =& {1\over{2}}(N-\nu_{d}-2\xi+{1\over{2}})\,\vert N,\,\xi\,\nu_{s}\,\nu_{d}\,\eta\, L\, M\rangle. \end{split} $

      (8)

      It can be proven that the eigenstate of (1) can be written as

      $\begin{split} \vert \zeta\, \nu_{s};\nu_{d}\,\eta\, L\,M\rangle =& \left(\alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}{\mathop \prod \limits_{\rho = 1}^k }S^{+}(x^{(\zeta)}_{\rho})\right.\\&\left.+\beta^{(\zeta)}_{\nu_{s}, \nu_{d},\eta,L}{\mathop \prod \limits_{\rho = 1}^{k + 1} }S^{+}(y^{(\zeta)}_{\rho}) \right)\vert \nu_{s};\nu_{d}\,\eta\, L\,M\rangle, \end{split} $

      (9)

      where $ \alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $ and $ \beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $, in general, are complex numbers to be determined, ζ labels the ζ-th set of solution $ \{x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}; y^{(\zeta)}_{1},\cdots,y^{(\zeta)}_{k+1}\} $, $ \alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $, $ \beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $, $ \vert \nu_{s};\nu_{d}\,\eta\, L\,M\rangle $ is the boson pairing vacuum state satisfying $ S^{-}_{\rho}\vert \nu_{s}; \nu_{d}\,\eta\, L\,M\rangle $ for $ \rho = s $ and d, in which $ \nu_{s}( = 0 $ or 1), and

      $ S^{+}(x) = x\, S_{s}^{+}+ S_{d}^{+}, $

      (10)

      which is equivalent to the form used in [5] with a linear transformation for x, where x is the spectral parameter to be determined. Using the commutation relations (4), one can directly check that

      $ [g_{s} S^{-}_{s}+g_{d} S^{-}_{d},\,S^{+}(x)] = 2g_{s}\,x\,S^{0}_{s}+2g_{d}\,S^{0}_{d}, $

      (11)

      $\begin{split} [[g_{s}\, S^{-}_{s}+g_{d}\, S^{-}_{d},\,S^{+}(x)],\, S^{+}(y)] =& {2y\,(g_{s}\,x-g_{d})\over{x-y}}S^{+}(x)\\&+{2x\,(g_{s}\,y-g_{d})\over{y-x}}S^{+}(y), \end{split}$

      (12)

      $ [\alpha_{s} S^{0}_{s}+\alpha_{d} S^{0}_{d},\,S^{+}(x)] = {(\alpha_{s}-\alpha_{d})\, x\over{1+x}}S^{+}+{\alpha_{s}\,x+\alpha_{d}\over{1+x}} S^{+}(x), $

      (13)

      $ [S^{-},\,S^{+}(x)] = 2x\, S^{0}_{s}-2S^{0}_{d}, $

      (14)

      $ [[S^{-},\,S^{+}(x)], \,S^{+}(y)] = {2y\,(1+x)\over{x-y}}S^{+}(x)+{2x\,(1+y)\over{y-x}}S^{+}(y). $

      (15)

      There are also useful identities:

      $ g_{s}\,S^{+}_{s}+g_{d}\,S^{+}_{d} = {g_{s}-x\, g_{d}\over{1+x}}S^{+}+{g_{s}+g_{d}\over{1+x}}S^{+}(x), $

      (16)

      $ \prod \limits_{\mu = 1}^{k}S^{\dagger }(y_{\mu}) = \sum\limits_{j = 1}^{k+1} {{\mathop \prod \nolimits_{\mu = 1}^k }(x_{j}-y_{\mu})\over{ {\mathop \prod \nolimits_{t ( \ne j)}^{k + 1} }(x_{j}-x_{t}) }} \prod \limits_{\rho\,(\neq j)}^{k+1}S^{\dagger }(x_{\rho}) $

      (17)

      or

      $ \prod \limits_{\mu = 1}^{k}S^{\dagger }(x_{\mu}) = \sum\limits_{j = 1}^{k+1} {{\mathop \prod \nolimits_{\mu = 1}^k }(y_{j}-x_{\mu})\over{ {\mathop \prod \nolimits_{t ( \ne j)}^{k + 1} }(y_{j}-y_{t}) }} \prod \limits_{\rho\,(\neq j)}^{k+1}S^{\dagger }(y_{\rho}), $

      (18)

      which can be proven by using mathematical induction on k due to the fact that $ S^{+}(x) $ and $ S^{+}(y) $ are binomials of $ S_{s}^{+} $ and $ S_{d}^{+} $. Using (17) with $ x_{k+1} = -1 $, we also have

      $\begin{split} \prod \limits_{\rho\,(\neq j)}^{k+1}S^{\dagger }(y_{\rho}) =& {{\mathop \prod \nolimits_{\mu {\kern 1pt} ( \ne j)}^{k + 1} }(1+y_{\mu})\over{ \prod^{k}_{t = 1}(1+x_{t}) }} \prod \limits_{\rho = 1}^{k}S^{\dagger }(x_{\rho}) \\&-\sum\limits_{i = 1}^{k} {\prod^{k+1}_{\mu \,(\neq j)}(x_{i}-y_{\mu})\over{ {\mathop \prod \nolimits_{t{\kern 1pt} ( \ne i)}^k }(x_{i}-x_{t})(x_{i}+1) }} S^{+}\prod \limits_{\rho\,(\neq i)}^{k}S^{\dagger }(x_{\rho}). \end{split}$

      (19)

      Similar to the U(5)–O(6) case shown in [5], using the above commutation relations and identities, one can verify that

      $ \begin{split}& P_{N}\hat{H}^{(1)}_{0}P_{N}{\mathop \prod \nolimits_{\rho = 1}^k }S^{+}(x_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle = \left( \sum_{j = 1}^{k}{\alpha^{(1)}_{s} x_{j}+\alpha_{d}^{(1)}\over{1+x_{j}}}+ {\alpha^{(1)}_{s}\overline{S^{0}}_{s}+\alpha^{(1)}_{d}\overline{S^{0}}_{d}}\right){\mathop \prod \nolimits_{\rho = 1}^k }S^{+}(x_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle\\&\quad+ \sum_{j = 1}^{k}\left({(\alpha^{(1)}_{s}-\alpha_{d}^{(1)})x_{j}\over{1+x_{j}}}+g^{(1)}(2x_{j}\,\overline{S^{0}}_{s}-2\overline{S^{0}}_{d})+ g^{(1)}\sum_{j'(\neq j)}^{k}{2x_{j}(x_{j'}+1)\over{x_{j'}-x_{j}}} \right)S^{+}\prod^{k}_{\rho(\neq j)}S^{+}(x_{\rho})\vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle, \end{split} $

      (20)

      $ \begin{split}& P_{N+2}\hat{H}^{(1)}_{0}P_{N+2}{\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle = \left( \sum_{j = 1}^{k+1}{\alpha^{(2)}_{s} y_{j}+\alpha_{d}^{(2)}\over{1+y_{j}}}+ {\alpha^{(2)}_{s}\overline{S^{0}}_{s}+\alpha^{(2)}_{d}\overline{S^{0}}_{d}}\right) {\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle\\ &\quad+\sum_{j = 1}^{k+1}\left({(\alpha^{(2)}_{s}-\alpha_{d}^{(2)})y_{j}\over{1+y_{j}}}+g^{(2)}(2y_{j}\,\overline{S^{0}}_{s}-2\overline{S^{0}}_{d})+ g^{(2)}\sum_{j'(\neq j)}^{k+1}{2y_{j}(y_{j'}+1)\over{y_{j'}-y_{j}}} \right)S^{+}\prod^{k+1}_{\rho(\neq j)}S^{+}(y_{\rho})\vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle, \end{split}$

      (21)

      $ \begin{split}& P\hat{H}_{\rm{mix}}P {\mathop \prod \nolimits_{\rho = 1}^k }S^{+}(x_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle = \sum_{j = 1}^{k+1}{\prod \nolimits_{\mu = 1}^{k}(y_{j}-x_{\mu})(g_{s}-y_{j}\,g_{d})\over{\prod \nolimits_{t(\neq j)}^{k+1}(y_{j}-y_{t})(1+y_{j})}}S^{+}\prod^{k+1}_{\rho(\neq j)}S^{+}(y_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle\\ &\quad+\sum_{j = 1}^{k+1}{\prod \nolimits_{\mu = 1}^{k}(y_{j}-x_{\mu})(g_{s}+g_{d})\over{\prod \nolimits_{t(\neq j)}^{k+1}(y_{j}-y_{t})(1+y_{j})}}{\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y_{\rho}) \vert \nu_{s};\nu_{d} \, \eta\, L\, M\rangle, \end{split}$

      (22)

      where the identities (18) and (12) for $ y_{j} $ within the summation over j are used,

      $ \begin{split}& P\hat{H}_{\rm{mix}}P {\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y_{\rho}) \vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle = (g_{s}\,S^{-}_{s}+g_{d}\,S^{-}_{d}){\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y_{\rho}) \vert \nu_{s};\nu_{d} \, \eta\, L\, M\rangle\\ &\quad= \sum_{j = 1}^{k+1}\left( 2g_{s}\,y_{j}\, \overline{S^{0}}_{s}+2g_{d}\,\overline{S^{0}}_{d}+\sum_{j'(\neq j)}^{k+1}{2y_{j}(g_{s} y_{j'}-g_{d}) \over{y_{j'}-y_{j}}}\right)\prod^{k+1}_{\rho(\neq j)}S^{+}(y_{\rho})\vert \nu_{s};\nu_{d}\, \eta\, L\, M\rangle\\ &\quad = \sum_{j = 1}^{k+1}\left( 2g_{s}\,y_{j}\, \overline{S^{0}}_{s}+2g_{d}\,\overline{S^{0}}_{d}+\sum_{j'(\neq j)}^{k+1}{2y_{j}(g_{s}\,y_{j'}-g_{d})\over{y_{j'}-y_{j}}}\right)\\&\quad \times \left( {{\mathop \prod \nolimits_{\mu {\kern 1pt} ( \ne j)}^{k + 1} }(1+y_{\mu})\over{ \prod^{k}_{t = 1}(1+x_{t}) }} \prod \nolimits_{\rho = 1}^{k}S^{\dagger }(x_{\rho}) -\sum_{i = 1}^{k} {\prod^{k+1}_{\mu \,(\neq j)}(x_{i}-y_{\mu})\over{ {\mathop \prod \nolimits_{t{\kern 1pt} ( \ne i)}^k }(x_{i}-x_{t})(x_{i}+1) }} S^{+}\prod \nolimits_{\rho\,(\neq i)}^{k}S^{\dagger }(x_{\rho}) \right)\vert \nu_{s};\nu_{d} \, \eta\, L\, M\rangle, \end{split} $

      (23)

      where the identity (19) is used.

      Therefore, the eigen-equation

      $ \hat{H}\vert \zeta, \nu_{s};\nu_{d}\,\eta\, L\,M\rangle = E^{(\zeta)}_{\nu_{s},\nu_{d},L}\vert \zeta, \nu_{s};\nu_{d}\,\eta\, L\,M\rangle $

      (24)

      is fulfilled if and only if

      $ \begin{split} &\alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\left({(\alpha^{(1)}_{s}-\alpha_{d}^{(1)})x^{(\zeta)}_{j}\over{1+x^{(\zeta)}_{j}}} +g^{(1)}(2x^{(\zeta)}_{j}\,\overline{S^{0}}_{s}-2\overline{S^{0}}_{d})+ g^{(1)}\sum_{j'(\neq j)}^{k}{2x^{(\zeta)}_{j}(x^{(\zeta)}_{j'}+1)\over{x^{(\zeta)}_{j'}-x^{(\zeta)}_{j}}}\right) \\ &\quad -\beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\sum_{i = 1}^{k+1}\left( 2g_{s}\,y^{(\zeta)}_{i}\, \overline{S^{0}}_{s}+2g_{d}\,\overline{S^{0}}_{d}+\sum_{i'(\neq i)}^{k+1}{2y^{(\zeta)}_{i}(g_{s}\,y^{(\zeta)}_{i'}-g_{d})\over{y^{(\zeta)}_{i'}-y^{(\zeta)}_{i}}}\right) {\prod^{k+1}_{\mu \,(\neq i)}(x^{(\zeta)}_{j}-y^{(\zeta)}_{\mu})\over{ \prod^{k}_{t\,(\neq j)}(x^{(\zeta)}_{j}-x^{(\zeta)}_{t})(x^{(\zeta)}_{j}+1) }} = 0\; \; {\rm{for}}\; \; j = 1,2,\cdots,k, \end{split}$

      (25)

      $ \begin{split}&{l}\beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\left({(\alpha^{(2)}_{s}-\alpha_{d}^{(2)})y^{(\zeta)}_{j}\over{1+y^{(\zeta)}_{j}}} +g^{(2)}(2y^{(\zeta)}_{j}\,\overline{S^{0}}_{s}-2\overline{S^{0}}_{d})+ g^{(2)}\sum_{j'(\neq j)}^{k+1}{2y^{(\zeta)}_{j}(y^{(\zeta)}_{j'}+1)\over{y^{(\zeta)}_{j'}-y^{(\zeta)}_{j}}} \right)+\alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}{\prod \nolimits_{\mu = 1}^{k}(y^{(\zeta)}_{j}-x^{(\zeta)}_{\mu})(g_{s}-y^{(\zeta)}_{j}\,g_{d})\over{\prod \nolimits_{t(\neq j)}^{k+1}(y^{(\zeta)}_{j}-y^{(\zeta)}_{t})(1+y^{(\zeta)}_{j})}} = 0\\&\quad {\rm{for}}\; \; j = 1,2,\cdots,k+1, \end{split}$

      (26)

      and

      $ \begin{split} &\alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\left(E^{(\zeta)}_{k,\nu_{s},\nu_{d},L}-\sum_{j = 1}^{k}{ \alpha^{(1)}_{s}x^{(\zeta)}_{j} +\alpha_{d}^{(1)}\over{1+x^{(\zeta)}_{j}}}- {\alpha^{(1)}_{s}\overline{S^{0}}_{s}-\alpha^{(1)}_{d}\overline{S^{0}}_{d}}\right) \\ &\quad=\beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\sum_{j = 1}^{k+1}\left( 2g_{s}\,y^{(\zeta)}_{j}\, \overline{S^{0}}_{s}+2g_{d}\,\overline{S^{0}}_{d}+\sum_{j'(\neq j)}^{k+1}{2y^{(\zeta)}_{j}(g_{s}\,y^{(\zeta)}_{j'}-g_{d})\over{y^{(\zeta)}_{j'}-y^{(\zeta)}_{j}}}\right){\prod^{k+1}_{\mu(\neq j)}(1+y^{(\zeta)}_{\mu})\over{\prod \nolimits_{t = 1}^{k}(1+x^{(\zeta)}_{t})}}, \end{split} $

      (27)

      $\beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} \left(E^{(\zeta)}_{k,\nu_{s},\nu_{d},L}- \sum_{j = 1}^{k+1}{ \alpha^{(2)}_{s} y^{(\zeta)}_{j} +\alpha_{d}^{(2)}\over{1+y^{(\zeta)}_{j}}}- {\alpha^{(2)}_{s}\overline{S^{0}}_{s}-\alpha^{(2)}_{d}\overline{S^{0}}_{d}}\right) = \alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} \sum_{j = 1}^{k+1}{\prod \nolimits_{\mu = 1}^{k}(y^{(\zeta)}_{j}-x^{(\zeta)}_{\mu}) (g_{s}+g_{d})\over{\prod \nolimits_{t(\neq j)}^{k+1}(y^{(\zeta)}_{j}-y^{(\zeta)}_{t})(1+y^{(\zeta)}_{j})}}. $

      (28)

      Eqs. (27) and (28) are nothing but the eigen-equation

      $ \left( {\begin{array}{*{20}{c}}A &B\cr C &D\cr\end{array}} \right) \left( {\begin{array}{*{20}{c}}\alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\cr \beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\cr\end{array}} \right) = E^{(\zeta)}_{k,\nu_{s},\nu_{d},L}\left( {\begin{array}{*{20}{c}}\alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\cr \beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L}\cr\end{array}} \right) $

      (29)

      with

      $ \begin{split}A =& \sum_{j = 1}^{k}{ \alpha^{(1)}_{s}x^{(\zeta)}_{j} +\alpha_{d}^{(1)}\over{1+x^{(\zeta)}_{j}}}+ {\alpha^{(1)}_{s}\overline{S^{0}}_{s}+\alpha^{(1)}_{d}\overline{S^{0}}_{d}},\\ B =& \sum_{j = 1}^{k+1}\left( 2g_{s}\,y^{(\zeta)}_{j}\, \overline{S^{0}}_{s}+2g_{d}\,\overline{S^{0}}_{d}+\sum_{j'(\neq j)}^{k+1}{2y^{(\zeta)}_{j}(g_{s}\,y^{(\zeta)}_{j'}-g_{d})\over{y^{(\zeta)}_{j'}-y^{(\zeta)}_{j}}}\right){\prod^{k+1}_{\mu(\neq j)}(1+y^{(\zeta)}_{\mu})\over{\prod \nolimits_{t = 1}^{k}(1+x^{(\zeta)}_{t})}},\\ C =& \sum_{j = 1}^{k+1}{\prod \nolimits_{\mu = 1}^{k}(y^{(\zeta)}_{j}-x^{(\zeta)}_{\mu}) (g_{s}+g_{d})\over{\prod \nolimits_{t(\neq j)}^{k+1}(y^{(\zeta)}_{j}-y^{(\zeta)}_{t})(1+y^{(\zeta)}_{j})}},\\ D =& \sum_{j = 1}^{k+1}{ \alpha^{(2)}_{s} y^{(\zeta)}_{j} +\alpha_{d}^{(2)}\over{1+y^{(\zeta)}_{j}}}+ {\alpha^{(2)}_{s}\overline{S^{0}}_{s}+\alpha^{(2)}_{d}\overline{S^{0}}_{d}}. \end{split} $

      (30)

      Thus, $ \alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $, $ \beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $, and $ E^{(\zeta)}_{k,\nu_{s},\nu_{d},L} $ can be expressed in terms of the variables $ \{x^{(\zeta)}_{j}\} $ ($ j = 1,2,\cdots,k $) and $ \{y^{(\zeta)}_{j}\} $ ($ j = 1,2,\cdots,k+1 $). Once $ \alpha^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $, $ \beta^{(\zeta)}_{\nu_{s},\nu_{d},\eta,L} $ and $ E^{(\zeta)}_{k,\nu_{s},\nu_{d},L} $ are expressed in terms of A, B, C, and D shown in (30), the variables $ \{x^{(\zeta)}_{j}\} $ ($ j = 1,2,\cdots,k $) and $ \{y^{(\zeta)}_{j}\} $ ($ j = 1,2,\cdots,k+1 $) are determined by the Eqs. (25) and (26). It is obvious that (25)-(28) become

      ${(\alpha^{(1)}_{s}-\alpha_{d}^{(1)})\,x^{(\zeta)}_{j}\over{1+x^{(\zeta)}_{j}}} +g^{(1)}(2x^{(\zeta)}_{j}\,\overline{S^{0}}_{s}-2\overline{S^{0}}_{d})+ g^{(1)}\sum_{j'(\neq j)}^{k}{2x^{(\zeta)}_{j}(x^{(\zeta)}_{j'}+1)\over{x^{(\zeta)}_{j'}-x^{(\zeta)}_{j}}} = 0\; \; {\rm{for}}\; \; j = 1,2,\cdots,k, $

      (31)

      $ E^{(\zeta)}_{k,\nu_{s},\nu_{d},L} = \sum_{j = 1}^{k}{ \alpha^{(1)}_{s} x^{(\zeta)}_{j}+\alpha_{d}^{(1)}\over{1+x^{(\zeta)}_{j}}}+ {\alpha^{(1)}_{s}\overline{S^{0}}_{s}+\alpha^{(1)}_{d}\overline{S^{0}}_{d}},\; \alpha^{(\zeta)}_{\nu_{s},\nu_{d},L}\neq0,\; \; \beta^{(\zeta)}_{\nu_{s},\nu_{d},L} = 0, $

      (32)

      or

      $ {(\alpha^{(2)}_{s}-\alpha_{d}^{(2)})y^{(\zeta)}_{j}\over{1+y^{(\zeta)}_{j}}} +g^{(2)}(2y^{(\zeta)}_{j}\,\overline{S^{0}}_{s}-2\overline{S^{0}}_{d})+ g^{(2)}\sum_{j'(\neq j)}^{k+1}{2y^{(\zeta)}_{j}(y^{(\zeta)}_{j'}+1)\over{y^{(\zeta)}_{j'}-y^{(\zeta)}_{j}}} = 0\; \; {\rm{for}}\; \; j = 1,2,\cdots,k+1, $

      (33)

      $ E^{(\zeta)}_{k,\nu_{s},\nu_{d},L} = \sum_{j = 1}^{k+1}{ \alpha^{(2)}_{s} y^{(\zeta)}_{j} +\alpha_{d}^{(2)}\over{1+y^{(\zeta)}_{j}}}+ {\alpha^{(2)}_{s}\overline{S^{0}}_{s}+\alpha^{(2)}_{d}\overline{S^{0}}_{d}},\; \alpha^{(\zeta)}_{\nu_{s},\nu_{d},L} = 0,\; \; \beta^{(\zeta)}_{\nu_{s},\nu_{d},L}\neq0, $

      (34)

      when $ g_{s} = g_{d} = 0 $ without configuration mixing, which are the Bethe ansatz equations and the corresponding eigen-energy of the U(5)–O(6) transitional case for the N-boson normal states and the $ N+2 $-boson intruder states, respectively.

      Similar to the results shown in [18], there are extended Heine-Sieltjes polynomials $ y^{(i)}(x) $ related to (31) and (33) satisfying

      $ F(x){{\rm d}^2 y^{(i)}(x)\over{{\rm d}x^2}}+G^{(i)}(x){{\rm d} y^{(i)}(x)\over{{\rm d}x}}+V^{(i)}(x)\,y^{(i)}(x) = 0 $

      (35)

      for $ i = 1 $ or $ i = 2 $, where $ F(x) = x\,(1+x)^2 $,

      $\begin{split} G^{(i)}(x)/F(x) =& {1\over{1+x}}\left({2\overline{S^{0}}_{d}\over{x}}-2\overline{S^{0}}_{s}\right.\\&\left.+{\alpha^{(i)}_{d}-\alpha^{(i)}_{s}\over{g^{(i)}(1+x)}}-2k+2\right), \end{split} $

      (36)

      and $ V^{(i)}(x) $ is a linear function of x determined by (35). The roots of (31) or (33) are zeros of $ y^{(1)}(x) $ or $ y^{(2)}(x) $. Hence, the polynomial approach shown in [18] applies to this case as well, which can be used to get solution of (31) and (33) when there is no configuration mixing with $ g_{s} = g_{d} = 0 $.

      It is clear that the pairing operators $ {\mathop \prod \nolimits_{\rho = 1}^k }S^{+}(x^{(\zeta)}_{\rho}) $ and $ {\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y^{(\zeta)}_{\rho}) $ used in (9) are symmetric with respect to any permutation among $ \{x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}\} $ and $ \{y^{(\zeta)}_{1},\cdots,y^{(\zeta)}_{k+1}\} $. Therefore, there are $ k!(k+1)! $ identical roots of (25) and (26), of which only one is needed to construct the eigenstate (9). Once the ζ-th root $ \{x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}\} $ and $ \{y^{(\zeta)}_{1},\cdots,y^{(\zeta)}_{k+1}\} $ of (25) and (26) are obtained, the pairing operators $ {\mathop \prod \nolimits_{\rho = 1}^k }S^{+}(x^{(\zeta)}_{\rho}) $ and $ {\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(y^{(\zeta)}_{\rho}) $ in (9) can be expressed as

      $ \begin{split}& {\mathop \prod \nolimits_{\rho = 1}^k }S^{+}(x^{(\zeta)}_{\rho}) = \sum_{\mu = 0}^{k}S^{(\mu)}(x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k})\,S^{+\,\mu}_{s}S^{+\, k-\mu}_{d},\\ & {\mathop \prod \nolimits_{\rho = 1}^{k + 1} }S^{+}(x^{(\zeta)}_{\rho}) = \sum_{\mu = 0}^{k+1}S^{(\mu)}(y^{(\zeta)}_{1},\cdots,y^{(\zeta)}_{k+1})\,S^{+\,\mu}_{s}S^{+\, k+1-\mu}_{d}, \end{split}$

      (37)

      where $ S^{(\mu)}(x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}) = \sum_{1\leq i_{1}<\cdots< i_{\mu}\leq k}\prod \nolimits_{q = 1}^{\mu}x^{(\zeta)}_{i_{q}} $ starting with $ S^{(0)}(x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}) = 1 $, and similarly for $ S^{(\mu)}(y^{(\zeta)}_{1},\cdots,y^{(\zeta)}_{k+1}) $, is the $ \mu $-th elementary symmetric polynomial of the k root-components $ \{x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}\} $ of (25), which is helpful for calculating matrix elements of physical quantities of the system.

    3.   The solution exemplified:
    • Since the solution of (1) can be derived by using the extended Heine-Sieltjes polynomial approach shown in (35) when there is no configuration mixing with $ g_{s} = g_{d} = 0 $, from which one obtains the roots $ \{x^{(\zeta)}_{01},\cdots,x^{(\zeta)}_{0k}\} $ and $ \{y^{(\zeta)}_{01},\cdots,y^{(\zeta)}_{0k+1}\} $ for this case, where the total number of the roots is $ 2k+3 $, for which $ N = 2k+v_{d}+v_{s} $ for allowed angular momentum quantum number L determined by the reduction rule $ (\nu_{d}\,0)\downarrow L $ of O(5)$\supset $O(3). Then, for small values of $ g_{s} $ and $ g_{d} $, the root $ \{x^{(\zeta)}_{1},\cdots,x^{(\zeta)}_{k}\} $ and $ \{y^{(\zeta)}_{1},\cdots,y^{(\zeta)}_{k+1}\} $ of (25) and (26) for given ζ can be obtained by using $ \{x^{(\zeta)}_{01},\cdots,x^{(\zeta)}_{0k}\} $ and $ \{y^{(\zeta)}_{01},\cdots,y^{(\zeta)}_{0k+1}\} $ as the initial root to find the solution. Repeatedly dosing so progressively, one can find $ 2k+3 $ sets of the roots for any real values $ g_{s} $ and $ g_{d} $. It should be noted that all roots are real when $ g_{s} = g_{d} = 0 $, which is a common feature of the general SU(1,1) Gaudin models without configuration mixing studied previously [5, 17, 18]. For $ g_{s}\neq 0 $ and/or $ g_{d}\neq0 $, however, complex roots occur in the middle part of the spectrum when the mixing of the N-boson and $ N+2 $-boson configurations is relatively strong, especially when k is large, which is common when the configuration mixing strengths $ g_{s} $ and $ g_{d} $ become sufficiently large. Since $ -1 $ and $ 0 $ are singular points of (25) and (26), the real part of the root components lies in the union $ (-\infty,-1)\bigcup(-1,0)\bigcup(0,\infty) $, where no any pair of the root components are the same. Actually, similar to the solution of the pairing model [19], all root components are always symmetric with respect to the real axis on the complex plane, namely, if a root component is complex, there must be the conjugate root component involved.

      To demonstrate the feature of the solution, we consider an example with $ \alpha_{s}^{(1)} = 0 $, $ \alpha_{d}^{(1)} = 0.3 $ MeV, $ g^{(1)} = -0.5 $ MeV, $ \alpha_{s}^{(2)} = 1.5 $ MeV, $ \alpha_{d}^{(2)} = 1.8 $ MeV, $ g^{(2)} = -0.2 $ MeV, $ g_{s} = g_{d} = 0.2 $ MeV. Only $ \nu_{s} = \nu_{d} = 0 $ case is exemplified in the following. As shown in Table 1, all roots in this case are real for $ N = 2 $. It is obvious that the first 2 roots mainly lie in the $ N = 2 $ configuration as indicated by the small $ \beta^{(\zeta)}/\alpha^{(\zeta)} $ values, while the last 3 roots mainly lie in the $ N+2 = 4 $ configuration as indicated by the relatively larger $ \beta^{(\zeta)}/\alpha^{(\zeta)} $ values. For $ N = 4 $, the pattern of the roots is similar. The first 3 roots mainly lie in the $ N = 4 $ configuration, while the last 4 roots mainly lie in the $ N+2 = 6 $ configuration. As shown in Table 2, the third root components with $ {y^{(3)}_{1} = y^{(3)*}_{2}} $ and the fifth root components with $ x^{(5)}_{1} = x^{(5)*}_{2} $ are complex in this case. In fact, with further increasing of the configuration mixing strengths $ g_{s} $ and/or $ g_{d} $, complex root also occurs for the $ N = 2 $ case. Since the 2 roots with relatively small $ \beta^{(\zeta)}/\alpha^{(\zeta)} $ values mainly lie in the $ N = 2 $ configuration, while the 3 roots with relatively larger $ \beta^{(\zeta)}/\alpha^{(\zeta)} $ values mainly lie in the $ N+2 = 4 $ configuration, the root of the ground state in this case becomes complex when $ g_{s} $ and/or $ g_{d} $ become sufficiently large. For example, if we keep other parameters the same as those shown in Table 1, the root of the ground state becomes complex when $ g_{s} = g_{d}\geqslant 6.06492 $ MeV with $ x_{1}^{(1)} = 0.56635 $, $ y^{(1)}_{1} = 0.72673+0.00027\,I $, $ y^{(1)}_{2} = 0.72673- 0.00027\,I $, $ \beta^{(1)}/\alpha^{(1)} = -0.3067 $, $ E^{(1)} = -14.8053 $ MeV for $ g_{s} = g_{d} = 6.06492 $ MeV. Therefore, complex solution is likely to occur when the configuration mixing is sufficiently strong.

      $ x^{(\zeta)}_{1} $$ y^{(\zeta)}_{1} $$ y^{(\zeta)}_{2} $$ \beta^{(\zeta)}/\alpha^{(\zeta)} $$ E^{(\zeta)} $
      $ \zeta=1 $$ -1.21628 $$ -1.23643 $$ 0.92428 $$ -0.03496 $$ -0.92692 $
      $ \zeta=2 $$ 4.13257 $$ 1.27972 $$ 3.02724 $$ -0.04105 $$ 0.38300 $
      $ \zeta=3 $$ 0.38403 $$ -2.88886 $$ -1.23803 $$ 4.99472 $$ 4.45235 $
      $ \zeta=4 $$ -0.83687 $$ -1.38586 $$ 1.96679 $$ 3.92177 $$ 4.93409 $
      $ \zeta=5 $$ 3.65856 $$ 0.66637 $$ 9.05257 $$ 2.33688 $$ 5.88248 $

      Table 1.  The roots of (25) and (26), the ratio $ \beta^{(\zeta)}/\alpha^{(\zeta)} $ used in the engen-state (9), and the corresponding eigen-energy $ E^{(\zeta)} $ (in MeV) of the model for $ N = 2 $, where the parameters of (1) are taken as $ \alpha_{s}^{(1)} = 0 $, $ \alpha_{d}^{(1)} = 0.3 $ MeV, $ g^{(1)} = -0.5 $ MeV, $ \alpha_{s}^{(2)} = 1.5 $ MeV, $ \alpha_{d}^{(2)} = 1.8 $ MeV, $ g^{(2)} = -0.2 $ MeV, $ g_{s} = g_{d} = 0.2 $ MeV.

      $ x^{(\zeta)}_{1} $$ x^{(\zeta)}_{2} $$ y^{(\zeta)}_{1} $$ y^{(\zeta)}_{2} $$ y^{(\zeta)}_{3} $$ \beta^{(\zeta)}/\alpha^{(\zeta)} $$ E^{(\zeta)} $
      $ \zeta=1 $$ -1.18956 $$ -1.25809 $$ -1.29710 $$ -1.18038 $$ 0.95864 $$ -0.02389 $$ -3.19779 $
      $ \zeta=2 $$ -1.12636 $$ 3.76299 $$ -1.14255 $$ 1.17349 $$ 3.10565 $$ -0.02637 $$ -1.94532 $
      $ \zeta=3 $$ 0.78722 $$ \; 11.78400 $$ 0.79059-0.5954{I} $$ 0.79059+0.5954{I} $$ \; 10.41310 $$ -0.03314 $$ 0.48860 $
      $ \zeta=4 $$ -2.51113 $$ 1.06583 $$ -3.9343 $$ -1.55612 $$ -1.05363 $$ 3.72052 $$ 4.72435 $
      $ \zeta=5 $$ -0.38146-1.1214I $$ -0.38146+1.1214I $$ -1.73495 $$ -1.11698 $$ 1.49812 $$ 3.10755 $$ 5.25399 $
      $ \zeta=6 $$ -1.13035 $$ 3.19011 $$ -1.24602 $$ 0.613486 $$ 8.04287 $$ 2.37150 $$ 6.15037 $
      $ \zeta=7 $$ 0.73826 $$ \; 10.59300 $$ 0.331345 $$ 1.70441 $$ 18.8986 $$ 1.51102 $$ 7.55080 $

      Table 2.  The same as Table 1, but for $ N = 4 $, in which $ I = \sqrt{-1}. $

      As a use of the theory, the Hamiltonian (1) is applied to describe low-lying spectrum of $ ^{108} {\rm{Cd}}$ with $ N = 6 $ bosons, for which the term $ \hat{H}_{L} = f L\cdot L $ is added to (1) in order to lift degeneracy of the levels with the same seniority but different angular momentum quantum numbers. The E2 operator is simply chosen as

      $ T_{\mu}({\rm{E2}}) = q_{2}{P}_{N}\left( d^{\dagger }_{\mu}s+s^{\dagger }\tilde{d}_{\mu}\right){P}_{N}+ q'_{2}{P}_{N+2}\left( d^{\dagger }_{\mu}s+s^{\dagger }\tilde{d}_{\mu}\right){P}_{N+2} $

      (38)

      with which the B(E2) values are given by

      $ {\rm{B}}({\rm{E2}};L_{i}\rightarrow L_{f} ) = {2L_{f}+1\over{2L_{i}+1}} \left\vert\langle \zeta_{f};\nu^{\prime}_{d}\,\eta^{\prime}\,L_{f}\| T({\rm{E2}})\| \zeta_{i},;\nu_{d}\,\eta\,L_{i}\rangle\right\vert^2, $

      (39)

      where $ q_{2} $ and $ q^{\prime}_{2} $ are effective charge parameters of the normal and intruder configuration, respectively, and the reduced matrix element is defined in terms of the CG coefficient, so that $ \langle\zeta_{f},\nu^{\prime}_{d}\,\eta^{\prime}\,\eta\,L_{f}\vert\vert \hat{I}\vert\vert \zeta_{i},\nu_{d}\,L_{i}\rangle = \delta_{ \zeta_{f},\zeta_{i}}\delta_{ \nu^{\prime}_{d},\nu_{d}} \delta_{L_{f},L_{i}} $ with unit identity operator $ \hat{I} $.

      The same as those shown in [13], the level energies up to the three-phonon states in the normal bands and the intruder states $ 0^{+}_{1}({ i}) $, $ 2^{+}_{1}({ i}) $, and $ 4^{+}_{1}({ i}) $ of $ ^{108} {\rm{Cd}}$ deduced in [20] are considered. The model parameters are produced by a best global fit to the experimental level energies only, from which we get $ \alpha_{s}^{(1)} = 0 $, $ \alpha_{d}^{(1)} = 1.261 $ MeV, $ g^{(1)} = -1 $ keV, $ \alpha_{s}^{(2)} = 300 $ keV, $ \alpha_{d}^{(2)} = 1.416 $ MeV, $ g^{(2)} = -51 $ keV, $ g_{s} = 220 $ keV, $ g_{d} = 200 $ keV, $ f = 5 $ keV, and $ q'_{2}/q_{2} = -0.38 $. Then, the experimentally measured B(E2) ratios, $ R( L_{i}\rightarrow L_{f}) = B({\rm E}2; L_{i}\rightarrow L_{f})/B({\rm E}2; 2^{+}_{1}\rightarrow 0^{+}_{g} ) $, provided in [20] are fitted by only adjusting the ratio $ q_{2}^{\prime}/q_{2} $. The fitted low-lying level energies and B(E2) ratios are shown in Table 3, in which the corresponding results of the $ 2n $-particle and $ 2n $-hole configuration mixing from $ n = 0 $ to $ n\rightarrow\infty $ in the U(5) limit of the IBM (CM5) [13] are also provided. The ratio $ q_{2}^{\prime}/q_{2} = 2.9 $ is mainly determined according to the lower limit of the experimental ratio $ R( 4^{+}_{1}({ i})\rightarrow 2^{+}_{1}({ i})) $. As far as the level energies are concerned, it seems that the U(5)–O(6) transitional description is a little better than the CM5, while the B(E2) ratios produced in the two models are quite the same for the transitions among the normal states. However, though the E2 decays out of the intruder band are still weaker [20], the B(E2) ratio $ R( 2^{+}_{1}({ i})\rightarrow 0^{+}_{1}({ i})) $ and those for transitions form the intruder states to the normal states predicted in this model are far larger than those of the CM5 as shown in III. It can be expected that these values can be reduced when configuration mixing with $ N+2n $ bosons for $ n\geqslant 2 $ are taken into account. Since the E2 operator is simply chosen as the generator of O(6) as shown in (38) and in the CM5 case [13], the E2 selection rules are quite the same as those given in the U(5) or the O(6) limit without configuration mixing, which are given by $ \Delta\nu_{d} = \pm 1 $. Therefore, $ \Delta\nu_{d} = \pm 2 $ transitions, such as $ B({\rm E}2; 2^{+}_{2}\rightarrow 0^{+}_{ g}) $, and $ \Delta\nu_{d} = 0 $ transitions, such as $ B({\rm E}2; 2^{+}_{1}({ i})\rightarrow 2^{+}_1) $ and $ B({\rm E}2; 4^{+}_{1}({ i})\rightarrow 4^{+}_{1}) $, are always zero. In order to improve the theory, O(6) symmetry breaking terms, such as $ (d^{\dagger }\tilde{d})^{2}_{\mu} $ term, which allows $ \Delta\nu_{d} = 0 $ transitions, need to be added in the E2 operator. Alternatively, high order interactions, such as those adopted in [14], may be considered.

      level energy/MeVthis workExp. [20, 21]CM5 [13]
      $ E(2^{+}_{1}) $0.789$ 0.632 $$ 0.718 $
      $ E(4^{+}_{1}) $1.574$ 1.508 $$ 1.484 $
      $ E(2^{+}_{2}) $1.504$ 1.601 $$ 1.470 $
      $ E(0^{+}_{2}) $1.350$ 1.913 $$ 1.264 $
      $ E(3^{+}_{1}) $2.234$ 2.146 $$ 2.268 $
      $ E(4^{+}_{2}) $2.274$ 2.239 $$ 2.276 $
      $ E(0^{+}_{3}) $2.174$ 2.375 $$ 1.896 $
      $ E(2^{+}_{3}) $2.059$ 2.486 $$ 1.982 $
      $ E(6^{+}_{1}) $2.384$ 2.541 $$ 2.298 $
      $ E(0^{+}_{1}({ i})) $1.755$ 1.720 $$ 1.720 $
      $ E(2^{+}_{1}({ i})) $2.367$ 2.163 $$ 2.438 $
      $ E(4^{+}_{1}({ i})) $2.797$ 2.739 $$ 3.084 $
      $ E(2^{+}_{2}({ i})) $2.727$ 2.366 $$ 3.070 $
      $ E(0^{+}_{2}({ i})) $2.639$ 2.740{*} $$ 2.984 $
      $ E(0^{+}_{3}({ i})) $2.856$ 2.936{*} $$ 2.984 $
      $ R( L_{i}\rightarrow L_{f}) $this workExp. [20, 21]CM5 [13]
      $ R(4^{+}_{1}\rightarrow 2^{+}_1) $$ 1.475 $$ 1.5639 $$ 1.6688 $
      $ R(2^{+}_{2}\rightarrow 2^{+}_1) $$ 1.475 $$ 0.6579 $$ 1.6688 $
      $ R(2^{+}_{2}\rightarrow 0^{+}_{ g}) $$ 0 $$ 0.0676 $$ 0 $
      $ R(2^{+}_{1}({ i})\rightarrow 0^{+}_{1}({ i})) $1.378$ \geq0.338 $$ 0.3428 $
      $ R(4^{+}_{1}({ i})\rightarrow 2^{+}_{1}({ i})) $0.213$ \geq0.226 $$ 0.5901 $
      $ R(2^{+}_{1}({ i})\rightarrow 0^{+}_{ g}) $0.338$ \geq0.002 $$ 0.0029 $
      $ R(4^{+}_{1}({ i})\rightarrow 2^{+}_1) $0.234$ \geq0.005 $$ 0.0043 $
      $ R(2^{+}_{1}({ i})\rightarrow 2^{+}_1) $$ 0 $$ \geq0.015 $$ 0 $
      $ R(4^{+}_{1}({ i})\rightarrow 4^{+}_{1}) $$ 0 $$ \geq0.015 $$ 0 $

      Table 3.  Some low-lying level energies and B(E2) ratios $ R( L_{i}\rightarrow L_{f}) = B({\rm E}2; L_{i}\rightarrow L_{f})/B({\rm E}2; 2^{+}_{1}\rightarrow 0^{+}_{g} ) $ of $ ^{108} {\rm{Cd}}$, where $ {*} $ indicates that the corresponding spin assignment is not fully confirmed. The spin of both $ 0^{+}_{2}({ i}) $ and $ 0^{+}_{3}({ i}) $ states was assigned with $ (0^{+},1^{+}, 2^{+}) $ as shown in [21]. The model parameters are taken as $ \alpha_{s}^{(1)} = 0 $, $ \alpha_{d}^{(1)} = 1.261 $ MeV, $ g^{(1)} = -1 $ keV, $ \alpha_{s}^{(2)} = 300 $ keV, $ \alpha_{d}^{(2)} = 1.416 $ MeV, $ g^{(2)} = -51 $ keV, $ g_{s} = 220 $ keV, $ g_{d} = 200 $ keV, $ f = 5 $ keV, and $ q'_{2}/q_{2} = 2.9 $.

    4.   Summary
    • In this work, it is shown that the U(5)-O(6) transitional Hamiltonian of the interacting boson model with two-particle and two-hole configuration mixing is exactly solvable. An exact solution is derived based on the Bethe ansatz approach, of which the Bethe ansatz equations in determining eigenstates and the corresponding eigen-energies are provided. The features of the solution are numerically exemplified by the $ N = 2 $ and $ N = 4 $ cases. As an example of application, some low-lying level energies and B(E2) ratios of $ ^{108} {\rm{Cd}}$ are fitted and compared with the corresponding experimental data.

      Since the solution of the Hamiltonian without configuration mixing can be easily derived by using the extended Heine-Stieltjes polynomials, the roots of the Bethe ansatz equations for the cases with small mixing parameters can be approximately found by using roots of the equations without configuration mixing as the initial values. Therefore, one can establish a progressive approach to get the solution of the model with arbitrary mixing parameters. Though the solution is only demonstrated for the $ N\oplus (N+2) $ configuration mixing, it can be expected that the model with $ 2n $-particle and $ 2n $-hole configuration mixing for $ n = 0 $ up to a finite n is also exactly solvable by using the identities and the procedures shown in Sec. 2, which is due to the fact that the eigenstates of the model can always be expressed in terms of binomials of s- and d-boson pair operators, though the equations involved will become more complicated. Similar extension to the IBM-II case is also straightforward. Moreover, a chain of isotopes or isotones in the vibrational to γ-soft transitional region may be analyzed by using the model to reveal their shape phase coexistence and evolution, for example, as the analysis for Cd isotopes shown in [2224], which will be considered in our future work.

      One of the authors (F. Pan) is grateful to Professor P. Van Isacker for stimulating discussions on the subject and his suggestion on doing this work.

Reference (24)

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