Two-level interacting boson models beyond the mean field

PHYSICAL REVIEW C 75, 014301 (2007) Two-level interacting boson models beyond the mean field Jos´ eM.Arias, 1Jorge Dukelsky,2Jos´ e Enrique Garc´ ıa-Ramos,3and Julien Vidal4 1Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de F´ ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 2Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain 3Departamento de F´ ısica Aplicada, Universidad de Huelva, E-21071 Huelva, Spain 4Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee, CNRS UMR 7600, Universit´ e Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris Cedex 05, France (Received 31 August 2006; published 2 January 2007) The phase diagram of two-level boson Hamiltonians, including the interacting boson model (IBM), is studied beyond the standard mean field approximation using the Holstein-Primakoff mapping. The limitations of the usual intrinsic state (mean field) formalism concerning finite-size effects are pointed out. The analytic results are compared to numerics obtained from exact diagonalizations. Excitation energies and occupation numbers are studied in different model space regions (Casten triangle for IBM) and especially at the critical points. DOI: 10.1103/PhysRevC.75.014301 PACS number(s): 21.60.Fw, 73.43.Nq I. INTRODUCTION The concepts of phase transition and critical points have been defined, strictly speaking, for macroscopic systems. However, it has been recently suggested that precursors of phase transitions can be observed in finite-size mesoscopic systems [1]. In nuclear physics, the different nuclear shapes and the phase transitions between them are conveniently studied within the interacting boson model (IBM) [2]. This was recognized soon after the introduction of the model [3–7] but has been studied more thoroughly in the last few years [8–18] after the introduction of the concept of critical point symmetries [19–21]. Since the IBM was formulated from the beginning in terms of creation and annihilation boson operators, its geometric interpretation in terms of shape variables is usually done by introducing a boson condensate with two shape parameters, βand γ(order parameters) [3,22]. The parameter βis related to the axial deformation of the system, while γmeasures the deviation from axial symmetry. The equilibrium shape of the system is obtained by minimizing the expectation value of the Hamiltonian in the intrinsic state. Shape phase transitions are studied theoretically using one or more control parameters in the Hamiltonian. These control parameters drive the system in different phases characterized by order parameters and allows one to study in a simple way phase transitions and critical points in nuclear physics. The phase diagram of the IBM has been studied using several approaches [8–12,14,16–18], and it is well known that the dynamical symmetry associated with U(5) corresponds to a spherical shape (β=0), the dynamical symmetry SU(3) is associated with an axially deformed shape (γ=0,π/3,β = 0), and the dynamical symmetry O(6) is related to a γ-unstable deformed shape (β= 0 and γindependent). These symmetry limits are usually represented as the vertices of a triangle (Casten triangle) [23]. Phase transitions between these shapes have been widely studied, and it is known that the phase transition from U(5) to O(6) is second order, while any other transition within the Casten triangle from a spherical to a deformed shape is first order [9,12]. These studies have been performed, as mentioned above, by using the intrinsic state formalism. However, this approximate method is known to be only correct at leading order in a 1/N expansion, where Nis the number of bosons. In this paper, we present a method that goes beyond this order and computes finite-size corrections to several spectroscopic observables. We stress that 1/N corrections obtained with the intrinsic state formalism (or Hartree-Bose method) are in general incorrect, and they give no information on the proper finite-size corrections. The paper is organized as follows. First the model Hamiltonian is introduced in Sec. II. In Sec. III, the HolsteinPrimakoff mapping [24] is performed leading to a boson Hamiltonian in which we retain terms in orders N,N1/2, and N0. Then a Bogoliubov transformation is performed to diagonalize the Hamiltonian and to study both the symmetric (spherical) and the broken (deformed) phases. All this is done in general for two-level boson models in which the lowest level is a scalar sboson while the upper level is an arbitrary Lboson. The IBM corresponds to the particular case L=2 (dµbosons). We also present results for the case L=0asan illustration of the general method. In Sec. IV, we compare the analytical results with exact numerical diagonalizations for different paths along the Casten triangle. Finally, Sec. V presents the summary and conclusions. II. THE MODEL As already noted in Ref. [14], the experimental exploration of the shape transition and critical points in nuclei is difficult because of the lack of a continuous control parameter. However, in theoretical studies, this limitation is overcome by using a Hamiltonian written in terms of one or more control parameters that can vary continuously. In this work, we consider a two-level boson model in which the lowest level is characterized by a zero angular momentum (sboson), while the upper level has an arbitrary angular momentum L. The Hamiltonian proposed is a generalization of the IBM 0556-2813/2007/75(1)/014301(10) 014301-1 ©2007 The American Physical Society ARIAS, DUKELSKY, GARC´ IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007) consistent-Qformalism (CQF) [25], which depends on two control parameters xand χ, H=xn L−1−x NQχ·Qχ,(1) where nL=µL† µLµis the operator for the number of bosons in the upper level, Nis the total number of bosons, the symbol ·stands for the scalar product defined as a·b= +L µ=−L(−1)µaµb−µ, and Qχis a multipole operator written as Qχ µ=(s†˜ L+L†s)(L) µ+χ[L†×˜ L](L) µ,(2) where ˜ Lµ=(−1)µL−µ.ForL=2(dbosons), the Hamiltonian (1) is the well-known CQF Hamiltonian for IBM. Though it is not the most general IBM Hamiltonian, it captures the most important low-energy properties of a wide range of nuclei [26–28]. In particular, it is general enough to describe different nuclear phases and quantum phase transitions, and it has been used for that purpose at the mean field level [9,10,12]. The Hamiltonian (1) comprises different models depending on the value of L. For instance, for L=1 the Hamiltonian is appropriate for studying the phase diagram of the vibron model [29] of interest in molecular physics. III. MEAN FIELD AND BEYOND The usual way of getting the phase diagram of the model (1) is to introduce shape variables. This can be done by considering the intrinsic state formalism, also called the Hartree-Bose approximation [3,5,22,30]. In this approach, the ground state is a variational state built out of a condensate of “dressed” bosons, which are independent bosons moving in the average nuclear field. For L=2, these bosons are defined as † c=1 1+β2s†+βcos γd † 0+1 √2βsin γ(d† 2+d† −2) , (3) and the Nboson condensate is |c= 1 √N!(† c)N|0.(4) The variational variables βand γare the order parameters of the system, and their equilibrium values are fixed by minimizing the expectation value of the energy. The expression of this energy can be found in many references [3,22,30,31] and can be written schematically as E(N,β,γ,x,χ)=NF(1)(N,β,γ,x,χ) +(N−1)F(2)(N,β,γ,x,χ),(5) where F(1)(N,β,γ,x,χ) is the matrix element of the onebody operators divided by N, and F(2)(N,β,γ,x,χ)isthe matrix element of the two-body operators divided by N−1. Note that there is no N2dependence in the two-body operator because of the definition of the Hamiltonian. Actually, the only relevant contribution is the leading one (order N), since the next one (N0, for instance) are incomplete as explained below. For the standard IBM Hamiltonian (L=2), with an attractive quadrupole interaction, the nucleus always becomes axially deformed, either prolate (γ=0) for χ<0 or oblate (γ=π/3) for χ>0. As a consequence, the parameter γ can be incorporated in the value of β.β>0 corresponds to γ=0, while negative βimplies γ=π/3. In the case χ=0, the nucleus becomes γunstable; i.e., the energy is independent of γ. In this framework, one-phonon excitations above the ground state are constructed by directly replacing in the ground state (4) a condensate boson by an excited boson, this procedure is known as the Tamm-Dancoff approximation (TDA) method, or by including ground state fluctuations, which is the random phase approximation (RPA) method [5,31,32]. For L=2, there are five excited phonons that are characterized by their angular momentum projection Kand can be labeled as βexcitation with K=0,γ excitations with K=±2, and finally two K=±1 excitations. Note that not all the excited phonons are always physical, some of them become spurious Goldstone bosons associated with broken symmetries. This is the case for axially deformed nuclei; the K=±1 excitations are spurious Goldstone bosons because the state constructed with this excitation corresponds to an O(3) rotation of the whole system. In the case of γ-unstable nuclei, the K= ±2 excitations also become Goldstone bosons and are related to O(5) rotations of the ground state. In the case of L=0, only a K=0 excitation exists, and it is directly related, as we will see, with the βband of the IBM [33]. The mean field description of the ground state energy just mentioned is only valid at order N. The first quantum corrections can be obtained within the RPA formalism. Alternatively, the Holstein-Primakoff expansion [24] offers a simple and natural expansion in powers of 1/N. The advantages of this transformation are that it is Hermitian, preserves the boson commutation relation, and provides a correct expansion in powers of N. Furthermore, its leading order coincides with the mean field contribution. The Holstein-Primakoff expansion eliminates the sboson transforming the bilinear boson operators in the following way: L† µLν=b† µbν,(6) L† µs=N1/2b† µ(1 −nb/N)1/2=(s†Lµ)†,(7) s†s=N−nb,(8) where the bbosons satisfy [bµ,b † ν]=δµ,ν. The mapping fulfills the commutation relations at each order in Nin the Taylor expansion of the square root. We next introduce the cbosons through a shift transformation b† µ=√Nλ∗ µ+c† µ,(9) where the λµare complex numbers that form a (2L+1)- dimensional vector. This shift allows for a macroscopic occupation number nb. Thus, it allows one to consider at the same time the spherical phase, setting λµ=0 for all µ, and the deformed phase, λµ= 0. In this latter situation, we shall only consider the case λ0= 0 without loss of generality. 014301-2 TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007) The Hamiltonian then reads H=N1λ2 05x−4−4(x−1)λ2 0+(x−1)χα(L) 0,0λ0 ×41−λ2 01/2+χα(L) 0,0λ0+N1/2λ0(c† 0+c0) ×5x−4−8λ2 0(x−1) +2(x−1)χα(L) 0,0λ0 ×−4λ2 0+3 1−λ2 01/2+χα(L) 0,0λ0 +N03x−2−6λ2 0(x−1)nc+(x−1)(2L+1) −(2L+3)λ2 0+1−λ2 0(P† c+Pc) −4λ2 0c†2 0+2c† 0c0+c2 0+2χ(x−1) ×λ0(1 −λ2 0)1/2+L  µ=−L α(L) 0,µ +2c† µcµ(−1)µα(L) µ,−µ +α(L) 0,µ+(−1)µα(L) µ,0(c† µc† −µ+cµc−µ −λ3 0α(L) 0,0 21−λ2 01/22+3c†2 0+2c† 0c0+c2 0+2nc −λ5 0α(L) 0,0 41−λ2 03/21+c†2 0+2c† 0c0+c2 0 +χ2(x−1)λ2 01++L  µ=−L 2c† µcµ(−1)µα(L) 0,0α(L) µ,−µ +α(L) 0,µ 2+(−1)µα(L) 0,µ 2(c† µc† −µ+cµc−µ) +O(1/√N),(10) =H1+H1/2+H0+O(1/√N),(11) where α(L) µ,ν =L, µ;Lν|L, µ +νand P† c=c†·c†=(Pc)†. The term of order N(H1) is exactly the mean field energy. Setting λ=β/1+β2,one gets E(N,β,x,y)=Nβ2 (1 +β2)2[5x−4+xβ2 +βy(x−1)(4 +βy)],(12) where y=χα(L) 0,0. In the case of L=2, Eq. (12) reduces to the IBM ground state energy. Note that H1only depends on L through the Clebsch-Gordan coefficient L, µ;Lν|L, µ +ν, although this dependence can be absorbed in the parameter y. H1provides the mean field energy and therefore the equilibrium values of the order parameter. Figure 1depicts the phase diagram corresponding to H1. For given parameters x and χ(y), the first step consists in minimizing H1with respect to β(λ), getting the equilibrium value β0(λ0). The study of these minima has been shown in several publications [4,7], but for completeness we summarize here the main features: (i) β=0 is always a stationary point. For x<4/5,β = 0 is a maximum, while for x>4/5,it becomes a minimum. In the case of x=4/5,β =0 is an inflection S D O(6) SU(3)U(5) φ ρ antispinodalcriticalspinodal FIG. 1. Qualitative phase diagram for the Hamiltonian (1)and L=2. Insets show typical energy surfaces vs deformation parameter βin each of the phases and at the phase borders. Control variables are defined as ρ=1−xand φ=2π 3√7(√7 2+χ). point. x=4/5 is the point at which a minimum at β=0 starts to develop and defines the antispinodal line. (ii) For χ= 0(y= 0), there exists a region, where two minima, one spherical and one deformed, coexist. This region is defined by the point at which the β=0 minimum appears (antispinodal point) and the point at which the β= 0 minimum appears (spinodal point). The spinodal line is defined by the implicit equation 3x 3x−4=A B1−1+B A3 2,(13) where A=[4 −3x+2(x−1) y2]2and B= 36 y2(x−1)2. The SU(3) case, χ=− √7/2, provides x≃0.820361. (iii) In the coexistence region, the critical point is defined as the situation in which both minima (spherical and deformed) are degenerate. At the critical point, the two degenerated minima are at β0=0 and β0= α(L) 0,0χ/2(β0=y/2) and their energy is equal to zero. The critical point line can be calculated to be xc=4+y2 5+y2=4+χ2L, 0; L0|L, 02 5+χ2L, 0; L0|L, 02.(14) In the case of L=2, xc=4+2 7χ2 5+2 7χ2,(15) being in the SU(3) limit (χ=− √7/2),x c=9/11. (iv) According to the previous analysis, a first-order phase transition appears for χ= 0(y= 0); while for χ= 0(y=0), an isolated point of second-order phase transition occurs at x=4/5. In this last case, antispinodal, spinodal, and critical points collapse into a single point. 014301-3 ARIAS, DUKELSKY, GARC´ IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007) The substitution of β0(λ0) in the Hamiltonian (11) implies that the term of order N1/2vanishes because it is proportional to the derivative of H1with respect to λ. More precisely, one has that ∂H1/∂λ =2H1/2. The first quantum correction comes from the N0term which is a simple quadratic form in the c-boson operators. It can thus be diagonalized through a Bogoliubov transformation. This transformation depends on the phase, spherical or deformed, and in the next subsections both will be treated separately. A. Bogoliubov transformation in the spherical phase In the spherical phase, β=0(λµ=0 for all µ) and x> 4/5. In this case, the Hamiltonian (11) reads as H=(3x−2)nc+(x−1)[(2L+1) +(P† c+Pc)] +O(1/N), (16) which is straightforwardly diagonalized via a Bogoliubov transformation c† µ=uµξ† µ+vµ˜ ξµ,(17) ˜ cµ=uµ˜ ξµ+vµξ† µ, where the coefficients verify u2 µ−v2 µ=1, with uµ=u−µand vµ=v−µ. The phases of the coefficients are chosen so as to minimize the mean field energy, leading to H=2L+1 2[−x+(x)1/2]+nξ(x)1/2+O(1/N),(18) where we have introduced (x)=x(5x−4),and nξis the number operator for ξbosons. Note that in the spherical phase, the mean field energy is equal to zero. In this phase, which is only defined for 4/5⩽x⩽1, the spectrum is, at this order, independent of χ(y) and has a trivial dependence on L.As shown in Ref. [33]forL=0, one has to diagonalize Hat next order (1/N) to see the role played by this parameter. In this phase there exists a (2L+1) times degenerated phonon (5 in the IBM case), ξ. The Hamiltonian is completely harmonic, and therefore the two phonon excitation energy is exactly twice the one phonon excitation energy. Another observable of interest that can be calculated easily is the number of Lbosons in each state. For the calculation of such an observable, the Hellmann-Feynman theorem can be used. It establishes that the derivative of the eigenvalue of a given operator, e.g., the Hamiltonian, is equal to the expectation value of the derivative of this operator with the corresponding eigenfunction. This leads to nL= ∂ ∂θ [(1 +θ)H],(19) where θ=x 1−x. In this case, the contribution from the mean field is zero, and the first nonvanishing contribution comes from the term proportional to N0in the energy. Therefore, nLgs =2L+1 23x−2 (x)−1+O(1/N),(20) nLpξ =nLgs +p3x−2 (x)+O(1/N),(21) where nLgs stands for the expectation value of the number of Lbosons in the ground state, pis the number of excited ξbosons and nLpξ stands for the expectation value of the number of Lbosons in the state with pexcited ξbosons. The N0correction is singular at x=4/5 as already noted in similar models [34,35]. Note that here we have chosen β0as an order parameter, but we could have taken nLgs equivalently. Indeed, in the thermodynamic limit, this quantity is only nonvanishing in the deformed phase, as we shall now see. B. Bogoliubov transformation in the deformed phase In the deformed phase, where β0= 0(λ0= 0), the situation is more complicated and strongly depends on L.Inthe following, we will discuss separately the two cases L= 0,2,but we emphasize that the form (11) of the expanded Hamiltonian allows the study of arbitrary L. 1. The case L =0 The case L=0 has recently attracted much attention because at the mean field level, it reproduces exactly the IBM phase diagram (although, of course, it does not include K=2 excitations). In Ref. [33], we computed the finite-size corrections up to 1/N order in the spherical phase. Here, we will now treat the deformed phase at order (1/N)0.Atthis order, for L=0, the Hamiltonian is easily diagonalized via a Bogoliubov transformation over the cscalar boson, and one gets H=E(x,y,β0)+1 21+β2 0−x+(7x−8)β2 0 +2(x−1)yβ0−2−yβ0+2β2 0+(x,y,β0)1/2 2 +nξ(x,y,β0)1/2+O(1/N),(22) where E(x,y,β0) is given by Eq. (12), (x,y,β0)=x−(3x−4)β2 0+2(x−1)yβ02+yβ0−β2 05x−4−(19x−20)β2 0+2(x−1)yβ06+3yβ0−7β2 0−β4 0 1+β2 02 (23) 014301-4 TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007) and y=χ. In this case, one has a single phonon excitation with K=0. For β0=0, one recovers expression (18) setting L=0. Regarding the expectation value for the number of Lbosons, it can be calculated as before through the HellmannFeynman theorem [see Eq. (19)]. Note that in the deformed phase, a contribution proportional to Ncomes from the mean field energy. More precisely, one has nLgs =Nβ2 0 1+β2 0+(1 −x)2∂ ∂x 1 21+β2 0(1 −x) ×−x+(7x−8)β2 0+2(x−1)yβ0 ×−2−yβ0+2β2 0+(x,y,β0)1/2 2(1 −x),(24) nLpξ =nLgs +p(1 −x)2∂ ∂x (x,y,β0)1/2 1−x,(25) where we used the same notation as in Eqs. (20) and (21), and pis the number of excited ξbosons. 2. The case L =2 In this section, we will focus on the IBM case, i.e., L=2. For arbitrary L= 0, the Hamiltonian (11)mustbe diagonalized separately for each value of µ. Indeed, one has H0=C++2  µ=−2 Hµ,(26) where Cis a constant and Hµ=H−µ. As can be seen in Eq. (11), Hµdepends not only on µbut also on the angular momentum Lvia the Clebsch-Gordan coefficients α(L) µ,ν. We diagonalize separately the modes µ=0,µ=±1, and µ=±2 which correspond to the βphonon (K=0), a Goldstone phonon (K=1 two-fold degenerate), and the γphonon (K=2 two-fold degenerate), respectively. After the diagonalization via a Bogoliubov transformation, the full diagonal Hamiltonian in the deformed phase reads H=E(x,y,β0)+1 2(1 +β2 0)−5x+(19x−24)β2 0 +12(x−1)yβ3 0++2  µ=−2 µ(x,y,β0)1/2 2 +nξµ1/2 µ(x,y,β0)+O(1/N),(27) with 0(x,y,β0) =x−(3x−4)β2 0+2(x−1)yβ02+yβ0−β2 05x−4−(19x−20)β2 0+2(x−1)yβ06+3yβ0−7β2 0−β4 0 1+β2 02,(28) ±1(x,y,β0)=x−(3x−4)β2 0+(x−1)yβ02+yβ0−2β2 05x−4−(3x−4)β2 0+2(x−1)yβ03+yβ0−β2 0 1+β2 02, (29) ±2(x,y,β0)=x−(3x−4)β2 0−2(x−1)yβ02+yβ0+β2 05x−4−(3x−4)β2 0+2(x−1)yβ0−6+yβ0−β2 0 1+β2 02, (30) where y=− √2/7χ.Forβ0=0, the symmetry between modes is restored [µ(x,y,0) =(x)] and one recovers the expression (18) with L=2. For β0= 0, the phonon excitations depend on µ. The excitation for µ=0 bosons, which corresponds to βbosons, is the same as in the L=0 case, namely, 014301-5 ARIAS, DUKELSKY, GARC´ IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007) 0(x,y,β0)=(x,y,β0). In addition, the excitation energy for µ=±1 modes vanishes, since for β0= 0, one has ±1(x,y,β0)∝∂E(x,y,β) ∂β |β0=0. This is in agreement with the fact that the µ=±1 excitation corresponds to a rotation of the ground state, i.e., to a Goldstone phonon. Finally, the µ=±2 excitation corresponds to a γexcitation, which is two-fold degenerate. For the calculation of the expectation value for the number of dbosons, we proceed as in the L=0 case and obtain nLgs =Nβ2 0 1+β2 0+(1 −x)2∂ ∂x   1 21+β2 0(1 −x) ×−5x+(19x−24)β2 0+12(x−1)yβ3 0 + µ=+2  µ=−2 µ(x,y,β0)1/2 2(1 −x) (31) nLpξµ=nLgs +p(1 −x)2∂ ∂x µ(x,y,β0)1/2 1−x,(32) where, again, the same notation as in Eqs. (20) and (21)is used, and pcorresponds to the number of ξµexcited bosons (βor γbosons in the L=2 case). IV. NUMERICAL RESULTS In this section, we compare the analytical results obtained in previous sections with numerical calculations. Note that for clarity only the first 0+and 2+states are plotted as members of the different bands. A. The case L=0 In this case, we perform the numerical calculations using the technique presented in Refs. [33,36]. It allows us to easily deal with a large number of bosons, up to a few thousands. One can reach such a number of bosons because of the underlying O(5) symmetry which allows the use of a seniority scheme that reduces considerably the dimension of the matrices to be diagonalized. In Fig. 2we compare the analytical with the numerical excitation energies for a large number of bosons, N=5000 (all the following calculations for L=0 are performed for N=5000), and χ=− √7/2. Note that single and two phonon excitations are equally well described. The left part of the figure corresponds to the deformed phase; the right part to the spherical one. In this case, a first-order phase transition appears, and at the critical point xc=9/11, the ground state and the first excited state are degenerated, one corresponding to the spherical and the other to the deformed ground state. In Fig. 3we repeat the same comparison for the case χ=0. In this case, a second-order phase transition appears. The energy of the first excited state becomes zero in the deformed phase and in the spherical phase at the critical point, xc=4/5. At this point, regarding the analytical calculations, the deformed βexcitation transforms into the spherical one phonon excitation. However, concerning the numerical results, the 0+ 2 0.78 0.8 0.82 0.84 x 0 1 2 Excitation energy 0+ 2 0+ 3 1 ph 2 ph FIG. 2. (Color online) Behavior of one and two phonon excitation energies, in arbitrary units, for L=0 as a function of xnear the critical point for χ=− √7/2. Lines are analytical results; dots, numerical calculations. state, identified with the βband in the deformed sector, transforms into the two phonon excitation in the spherical sector. Note that although in the spherical phase the N0correction is independent of χ(y),there is a noticeable difference between Figs. 2and 3because for each xvalue, only the phase that corresponds to the lowest mean field energy is plotted. The spherical phase only becomes the most stable from x>9/11 on for χ=− √7/2; while in the case χ=0,it is from x>4/5 on. Note also that in the deformed phase for χ=0, there appear degenerate doublets of levels because of the extra parity symmetry in the Hamiltonian in this case. Thus, the βband is connected to two and three phonon excitation in the spherical phase, while the ββ band is related to the four and five (not shown in Fig. 3) phonon excitation in the spherical phase. For the number of bosons, we compare the analytical formulas with the numerical results for the case of χ=− √7/2 0.79 0.795 0.8 0.805 0.81 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Excitation energy 0+ 1 ph 2 ph FIG. 3. (Color online) Same as Fig. 2,butforχ=0. Excited phonon in the deformed region (x<4/5) is equivalent to the β excitation in IBM. The lowest 0+corresponds to 0+ 2. 014301-6 TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007) -0.001 0 0.001 <n0>/ N - <n0 >mf/ N Num. Analyt. 0 0.2 0.4 0.6 0.8 1 x -0.001 0 Num. Analyt. FIG. 4. Variation of (nL=0gs −nL=0mf gs )/N as a function of x near the critical point for L=0. Full line is for analytical and circles for numerical results. Upper figure corresponds to χ=0 and lower to χ=− √7/2. and χ=0inFig.4. In particular, we are interested in the study of the N0corrections for the ground state; therefore, we subtract the mean field contribution, n0mf gs /N, from both analytical and numerical results. As expected, we observe how the N0correction improves the description of n0gs, especially near the critical point. B. The case L=2 For L=2, the numerical calculations have been carried out with an IBM code [37] which has been modified to allow calculations up to N=100 bosons. All numerical calculations for IBM presented below are performed for N=100. For L=2,the case χ=0 reduces to the L=0 situation already discussed. In particular, the analytical ground state energy is the same in both cases, although the N0correction differs; there exists only one kind of excitation: the β, while the µ=±1 and γexcitations become spurious Goldstone bosons. The βexcitation energy is equivalent to Eq. (23). On the exact diagonalization side, the Hamiltonian (1) can be rewritten in 0 0.2 0.4 0.6 0.8 1 x 0 2 4 6 8 10 12 Excitation energy β γ ββ γγ βγ 1 ph 2 ph FIG. 5. (Color online) Excitation energies (analytical), in arbitrary units, for one and two phonon states as a function of xfor L=2andχ=− √7/2. 0.4 0.45 0.5 0.55 0.6 x 2 3 4 5 6 7 8 Excitation energy β γ ββ γγ βγ 0+ 2+ FIG. 6. (Color online) Excitation energies (analytical and numerical), in arbitrary units, of one and two phonon states as a function of x for L=2andχ=− √7/2 in the deformed phase. Lines correspond to analytical, and dots to numerical results. The lowest 0+state corresponds to 0+ 2,and the 2+states are, respectively, 2+ 3(lowest) and 2+ 5(highest). terms of the generators of an SU(1,1) algebra [33,36]inthe same way as that in the L=0 case. Consequently, in this section we will only consider L=2 with χ= 0. Any χvalue can be analyzed, but, as an illustration, here we will present results for the case χ=− √7/2 that gives the U(5)-SU(3) leg in the Casten triangle. First, we plot the analytical results corresponding to one and two phonon excitations (Fig. 5). In the deformed phase, the bosons are βand γexcitations, while in the spherical phase they are spherical harmonic phonons. At the critical point, the βand the ββ bands transform into one and two phonon bands, 0.75 0.8 0.85 x 0 1 2 3 Excitation energy β γ ββ γγ βγ 1 ph 2 ph 0+ 2+ FIG. 7. (Color online) Excitation energies (analytical and numerical), in arbitrary units, of one and two phonon states as a function of xfor L=2andχ=− √7/2 in the region around the critical point. Dots correspond to numerical results. The lowest 0+state corresponds to 0+ 2and the lowest 2+state corresponds to 2+ 1. 014301-7 ARIAS, DUKELSKY, GARC´ IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007) 0.85 0.9 0.95 1 x 0 1 2 Excitation energy 1 ph 2 ph 0+ 2 2+ 1 FIG. 8. (Color online) Excitation energies of one and two phonon states, in arbitrary units, as a function of xfor L=2andχ=− √7/2. Lines correspond to analytical and dots to numerical results. respectively. However, the γ,βγ, and γγ bands apparently disappear when entering the spherical phase. Indeed, that disappearance happens because βand γexcitations become degenerate for β=0. The spherical phonon excitation is a five degenerate excitation where the deformed βand γexcitations collapse together with the Goldstone boson with projection ±1 (which is at zero energy in the deformed phase). In order to compare analytical and numerical results, we will split the analysis into three different regions: deformed phase (Fig. 6), critical region (Fig. 7), and spherical phase (Fig. 8). The harmonic character of the results is observed in all these plots. 1. Deformed phase In the deformed phase (Fig. 6), one and two phonon excitations are clearly separated in energy. Note that the excitation energy for the γband is higher than the corresponding one for the βband, although for x=0 [SU(3) limit] they are degenerated. Also note that the γγ excitation carries the angular momentum projections K=0,±4, which in this approach are degenerated. The correspondence between numerical and analytical states is as follows: the βband is identified with 0+ 2,γ with 0.4 0.45 0.5 0.55 0.6 x 0.52 0.54 0.56 0.58 0.6 0.62 <nd>/N 0+ 2 2+ 3 β γ FIG. 9. (Color online) nd/N as a function of xfor L=2and χ=− √7/2, in the deformed phase for one phonon states. Lines correspond to analytical and dots to numerical results. 0.7956 0.7958 0.796 0.7962 x 1.18 1.19 1.2 Excitation energy 2+ 3 2+ 4 FIG. 10. (Color online) Excitation energies (numerical) of 2+ 3and 2+ 4states at the region of closest approach (see text) for L=2and χ=− √7/2. 2+ 3,ββ with 0+ 3,βγ with 2+ 5,and γγ with 0+ 4. Note that the state 2+ 2belongs to the βband, while 2+ 4is with the ββ band. The overall agreement between analytical and numerical results is satisfactory and improves the description given in Ref. [31] for single and double phonon excitations, although in the present approach, no mixing appears among the different kinds of excitations. The average number of dbosons in the deformed phase, normalized to the total number of bosons, is depicted in Fig. 9for one phonon states (the results for two phonon states are not presented for clarity). It can be observed that a smooth decrease of ndoccurs when xincreases, as is expected when approaching the spherical phase. 2. Critical area The comparison around the critical area (Fig. 7) becomes complicated because one and two phonon states have comparable energies and there appears an interchange of character between states. For example, at the critical point, the ββ is at lower energy than the γexcitation. Starting at x=0.75, the correspondence between analytical and numerical states is similar to the one given in the preceding section; but already at x=0.8, different states interchange their character. The correspondence between states is presented in Table I. Clearly, an interchange of charTABLE I. Correspondence between analytical and numerical states for three values of x:x=0.75 deformed phase, x=9/11 critical point, and x=0.85 spherical phase. Only 0+and 2+states are indicated explicitly. x=0.75 x=9/11 x=0.85 β0+ 2,2+ 20+ 2,2+ 2 γ2+ 32+ 4 ββ 0+ 3,2+ 40+ 3,2+ 3 βγ 2+ 52+ 6 γγ 0+ 50+ 7 1 phonon 2+ 1 2 phonons 0+ 2,2+ 2 014301-8 TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007) 0.7956 0.7958 0.796 0.7962 x 0.18 0.19 0.2 0.21 0.22 0.23 <nd>/ N 2+ 3 2+ 4 FIG. 11. (Color online) nd/N (numerical) of 2+ 3and 2+ 4states at the region of closest approach (see text) for L=2andχ=− √7/2. acter exists among the states corresponding to the γ,ββ,βγ, and γγ bands. An interesting question that arises is whether the interchange of character is due either to level crossing or to level repulsion. We have to take into account that the transition between SU(3) and U(5) is not an integrable path [12]; i.e., a complete set of mutually commuting Hermitian operators does not exist. This implies that crossings are forbidden and only repulsion is allowed. In particular, in the thermodynamic limit, the repulsion becomes anticrossing, i.e., infinite repulsion. In Fig. 10, we show a closeup of one apparent crossing in Fig. 7between 2+states in the region around x=0.796; it is clearly seen that the levels indeed repel each other as expected. To illustrate this result and show how the two involved levels interchange their character, we present in Fig. 11 the expectation value of the dboson number in both states. It is clearly observed that the states interchange their properties at the point of closest approach. The average number of dbosons, normalized to the total number of bosons, in the region around the critical point is depicted in Fig. 12 for the ground state (left panel) and for the βband (right panel). One important feature is the discontinuity appearing at xc=9/11 due to the existence of a first-order phase transition. In the evolution of the β band, a kink appears in the numerical results at the critical point. This behavior at the critical point has been already observed for other observables such as isomer shifts [1], 0.775 0.8 0.825 0.85 x 02 + β 2 ph 0.75 0.775 0.8 0.825 x 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 <nd> / N 0+ 1 gs analyt. FIG. 12. (Color online) nd/N as a function of x,forL=2 and χ=− √7/2, at the critical area for one phonon states. Lines correspond to analytical and dots to numerical results. 0.85 0.9 0.95 1 x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 <nd>/N 02 + 21 + 1 ph 2 ph FIG. 13. (Color online) nd/N as a function of xfor L=2and χ=− √7/2, in the spherical phase for one and two phonon states. Lines correspond to analytical and dots to numerical results. derivatives of the ratios of 4+ 1/2+ 1excitation energies [38], or B(E2; 4+ 1→2+ 1)/B(E2; 2+ 1→0+ 1)[15]. Also note that the 0+ 2state transforms into a two phonon band when passing to the spherical phase. 3. Spherical phase The last region of interest is the spherical phase (Fig. 8). Here, there exists a five degenerated phonon excitation. The correspondence between the analytical and the numerical results is clear: one phonon excitation corresponds to the state 2+ 1; two phonon excitation to the state 0+ 2(also to 2+ 2and 4+ 1 states). The average number of dbosons, normalized to the total number of bosons, in the spherical region is depicted in Fig. 13 for the one and two phonon states. The main discrepancies between numerical and analytical results, as expected, appear close to the critical point. Note that the structure of the states is very simple and that already for x=0.9,the number of dbosons is fixed to 1 and 2 for one and two phonon states, respectively. V. SUMMARY AND CONCLUSIONS In this paper, we have studied two-level boson models characterized by a lowest scalar sboson and an excited Lboson through a Holstein-Primakoff transformation that allows us to treat explicitly order by order an Nexpansion. This treatment shows that only the leading Nterm of the ground state energy is correct in a mean field (or Hartree-Bose) approach. We stress that the equilibrium nuclear shape corresponding to an IBM Hamiltonian should be obtained only when considering the leading Nterm of the ground state energy. Depending on the value of L, models of interest in different fields can be obtained. Thus, L=0 is related to the Lipkin model first introduced in nuclear physics and then used in many fields, L=1 is the vibron model of interest in molecular physics, L=2 is the interacting boson model of nuclear structure, etc. We have presented a method for going accurately 014301-9