Ab-initio shell model with a core
A. F. Lisetskiy, B. R. Barrett, M. K. G. Kruse, P. Navratil, I. Stetcu, J. P. Vary
aa r X i v : . [ nu c l - t h ] A ug Ab-initio shell model with a core
A. F. Lisetskiy, ∗ B. R. Barrett, M.K.G. Kruse, P. Navratil, I. Stetcu, and J. P. Vary Department of Physics, University of Arizona, Tucson, AZ 85721 Lawrence Livermore National Laboratory, Livermore, CA 94551 Los Alamos National Laboratory, Los Alamos, NM 87545 Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011 (Dated: October 22, 2018)We construct effective 2- and 3-body Hamiltonians for the p-shell by performing 12¯ h Ω ab initio no-core shell model (NCSM) calculations for A=6 and 7 nuclei and explicitly projecting the many-body Hamiltonians onto the 0¯ h Ω space. We then separate these effective Hamiltonians into 0-,1- and 2-body contributions (also 3-body for A=7) and analyze the systematic behavior of thesedifferent parts as a function of the mass number A and size of the NCSM basis space. The role ofeffective 3- and higher-body interactions for
A >
PACS numbers: 21.10.Hw,23.20.En,23.20.Lv,23.20.-g,27.40.+zKeywords: NCSM, ab-initio, effective interactions
I. INTRODUCTION
Microscopic ab-initio many-body approaches have sig-nificantly progressed in recent years [1, 2, 3, 4, 5, 6, 7, 8].Nowdays, due to increased computing power and noveltechniques, ab-initio calculations are able to reproduce alarge number of observables for atomic nuclei with massup to A=14. The light nuclei have also served as a cru-cial site to recognize the important role of three-bodyforces and three-body correlations. Approaches like theNo-Core Shell Model (NCSM) [5], the Green’s FunctionMonte Carlo (GFMC) [6] and the Coupled-cluster theorywith single and double excitations (CCSD) [8] can be for-mally extended for heavier nuclei. However, the explo-sive growth in computational power, required to achieveconvergent results, severely hinders the detailed ab-initio studies of heavier, A ≥
16, nuclei. In the case of theNCSM, the slow convergence of the calculated energiesis caused by the adoption of a two-body cluster approxi-mation, which does not take many-body correlations intoaccount. Straightforward employment of the three-bodyand higher-body interactions dramatically complicatesthe problem, even for light nuclei.An alternative approach is to construct a small-spaceeffective two-body interaction, which would account forthe many-body correlations for the A-body system in alarge space. Attempts to include many-body correlationsapproximately modifying the one-body part of the ef-fective two-body Hamiltonian and employing a unitarytransformation have been reported recently [9].In this paper we derive a valence space (0¯ h Ω) effec-tive two-body interaction that accounts for all the core-polarization effects available in the ab-initio
NCSM wave-functions.First, in the framework of the NCSM, we constructthe effective Hamiltonians on the two-body cluster level ∗ [email protected] for A=6 systems in the N max ¯ h Ω space. N max representsthe limit on the total oscillator quanta (N) above theminimum configuration. We take N max values from 2 to12. Second, following the original idea of Ref. [10], weemploy an unitary many-body transformation and obtainthe effective two-body Hamiltonian in the 0¯ h Ω space (p-shell space), which exactly reproduces the lowest, 0¯ h Ωspace dominated, eigenstates of the 6-body Hamiltonianin the large N max ¯ h Ω space. Third, we perform NCSMcalculations for A=4 and A=5 systems with the effectiveHamiltonian constructed on the two-body cluster levelfor the A=6 system and determine the core and one-bodyparts of the effective two-body Hamiltonian for A=6 inthe p-shell space. Finally, the procedure is generalizedfor arbitrary mass number A. We analyze the propertiesof the constructed two-body Hamiltonians, investigatetheir efficiency to reproduce the observables of differentA-body systems calculated in large N max ¯ h Ω spaces andstudy the role of the effective p-shell space three-bodyinteraction.
II. APPROACHA. No Core Shell Model and effective interaction
The starting point of the No Core Shell Model (NCSM)approach is the bare, exact A-body Hamiltonian con-strained by the Harmonic Oscillator (HO) potential [5]: H Ω A = A X j =1 h Ω j + A X j>i =1 V ij (Ω , A ) , (1)where h Ω j is the one-body HO Hamiltonian h Ω j = p j m + 12 m Ω r j (2)and V ij (Ω , A ) is a bare NN interaction V NN ij modified bythe term introducing A- and Ω-dependent corrections tooffset the HO potential present in h Ω j : V ij (Ω , A ) = V NNij − m Ω A ( ~r i − ~r j ) . (3)The eigenvalue problem for the exact A-body Hamilto-nian (1) for A > A = 2 problem is consid-erably simpler. For many realistic NN interactions its so-lution in the relative HO basis with N max = 450 accountswell for the short range correlations and is a precise ap-proximation for the infinite space ( N max → ∞ ) result.This allows one to adopt the two-body cluster approxima-tion to construct the NCSM effective two-body Hamilto-nian H N max , Ω A,a =2 for an A-body system in an N max ¯ h Ω spaceof tractable dimension, where the lower index a standsfor the number of particles in the cluster. This approx-imation consists of solving Eq.(1) for the a = 2 bodysubsystem of A leading to H Ω A,a =2 = h Ω1 + h Ω2 + V (Ω , A ) . (4)The information about the total number of interactingparticles A enters the bare H Ω A,a =2 Hamiltonian throughthe second term in the right hand side of (3). Next,we find the unitary transformation U which reduces thebare H Ω A,a =2 Hamiltonian in the “infinite space” ( N ∞ max =450) to the diagonal form: E Ω A, = U H Ω A, U † , (5)where, for the sake of simplicity, we omit the index A for U and keep only the index a indicating the order ofcluster approximation. The matrix U can be split into4 blocks: U = (cid:18) U ,P U ,P Q U ,QP U ,Q (cid:19) , (6)where the square d P × d P U ,P matrix corresponds to theP-space (or model space) of dimension d P , characterizedby the chosen N max value.Taking into account that the E Ω A, matrix has a diago-nal form E Ω A, = (cid:18) E Ω A, ,P E Ω A, ,Q (cid:19) , (7)one can calculate the effective H N max , Ω A, Hamiltonian us-ing the following formula: H N max , Ω A, = U † ,P q U † ,P U ,P E Ω A, ,P U ,P q U † ,P U ,P . (8)It is easy to show by inserting Eq.(5) into the Eq.(8),and taking into account Eq.(6) that the unitary transfor-mation (8) is equivalent to the commonly used unitary transformation [11, 12] and that Eq.(8) is identical to theEqs.(15,16) from [5]. We note, that, by using Eq.(8) onedoes not need to calculate and store a large number ofmatrix elements of the ω -operator (i.e., U † ,P ω = U † ,P Q ).Furthermore, the decoupling condition QH eff P = 0 is au-tomatically satisfied, which is obvious from the diagonalform of the E Ω A, matrix. We note that our treatment ofcenter-of-mass motion remains the same as in the NCSM(Ref. [5]). We initiate all effective interaction develop-ments at the A-body level, and, through a series of steps,arrive at a smaller space effective interaction appropriatefor the A-body system. For this reason, our derived ef-fective Hamiltonians have their first subscript as “A”. B. Projection of the many-body Hamiltonian
The next step of the traditional NCSM prescription isto construct the full A-body Hamiltonian using the ef-fective two-body Hamiltonian (8) and to diagonalize itin the A-body N max model space. As we increase thenumber of nucleons, the dimension of the corresponding N max model space increases very rapidly. For instance,up-to-date computing resources allow us to go as high as N max = 16 for the lower part of the p-shell (A=5,6) [13],while already for the upper part of the p-shell (A ∼ N max = 8. The computational eigenvalueproblem for many-body systems is complicated becauseof the very large matrix dimensions involved. The largestdimension of the model space that we encountered in thisstudy for Li with N max = 12 exceeds d P = 4 . × .To solve this problem we have used the specialized ver-sion of the shell-model code ANTOINE [14, 15], recentlyadapted for the NCSM [16].In fact, the NCSM calculation for the A=6 system inthe N max = 12 space yields nearly converged energiesfor the lowest states dominated by the N = 0 compo-nents, while there is incomplete convergence for A ≥ N max = 8 space. Therefore, considering the N max = 12NCSM results as exact solutions for the lowest N = 0dominated 6-body states, we may construct the N max = 0space Hamiltonian for the A=6 system, which exactly re-produces those N max = 12 eigenvalues [10]. Moreover,if it is possible to solve the 6-body problem for A=6,then it is possible to solve the 6-body problem for arbi-trary A, using the corresponding effective Hamiltonian H N max , Ω A, obtained in the two-body cluster approxima-tion. This means that we can determine for any A-bodysystem the effective Hamiltonian in the N max = 0 space,which accounts for 6-body cluster dynamics in the large N max = 12 space.To generalize, we start by defining the procedure fordetermining the effective Hamiltonian matrix elementsfor the a -body cluster in the A-nucleon system. We dothis by constructing the full a -body Hamiltonian usingthe effective 2-body Hamiltonian (8) and diagonalizingit in the N max model space. In the spirit of Eq.(4), thisyields the eigenenergies E N max , Ω A,a of the a -body systemand their corresponding a eigenvectors which make upthe unitary transformation matrix U A,N max a ,P . These a -body results can then be projected into a smaller, sec-ondary P -space, given by N , max ¯ h Ω with N , max = 0,where, similar to Eqs.(6) and (7), E N max , Ω A,a and U A,N max a ,P can be split into parts related to the two spaces, P and Q , where P + Q = P . The new secondary effectiveHamiltonian then takes the following general form: H N , max ,N max A,a = U A, † a ,P q U A, † a ,P U Aa ,P E N max , Ω A,a ,P U Aa ,P q U A, † a ,P U Aa ,P , (9)where the Ω superscript on the left-hand side is omit-ted for the sake of simplicity. As stated earlier, the newindex a determines the order of the cluster approxima-tion in the smaller P space, i.e. , N , max = 0. Becausethe P space has N , max = 0, the projection into thisspace ”freezes” some number of the a nucleons into fixedsingle particle configurations, which can be thought ofas the ”inert core” states in the Standard Shell Model(SSM) approach. Consequently, it is possible to write a as a = A c + a v , where A c is the number of nucleonsmaking up the core configuration, while a v refers to thesize of valence cluster.For instance, in the case of p-shell nuclei, A c = 4,so, if a = 5 ( i.e. the 5-body cluster approximation),then the effective Hamiltonian H N , max =0 ,N max A,a =5 is simplya one-body Hamiltonian ( a v = 1) appropriate for theA-nucleon system. Similarly, for the 6-body cluster ap-proximation, i.e. , a = 6, we obtain the effective Hamil-tonian H N , max =0 ,N max A,a =6 , which is a two-body Hamiltonian( a v = 2) for the A-body system, and, so on for largervalues of a . Whatever the value of a v is, the effec-tive Hamiltonian H N , max =0 ,N max A,a contains the informa-tion about the a -body dynamics in the original large N max ¯ h Ω space, since it reproduces exactly the lowest d P eigenvalues E N max , Ω A,a ,P of the a -body Hamiltonian in the N max ¯ h Ω space, where d P is a dimension of the P space.In the case of a doubly magic closed shell with twoextra nucleons i.e., A = 6 , , etc. , the dimension ofthe effective Hamiltonian H ,N max A,a = A is a 2-body ( a v = 2)Hamiltonian in the p-, sd-, pf-spaces, etc. , respectively.This means that the secondary effective Hamiltonian (9)contains only 1-body and 2-body terms, even after the ex-act A-body cluster transformation. This effective Hamil-tonian (9), which now contains the correlation energy ofall A nucleons, is the correct one-body plus two-bodyHamiltonian to use in a SSM calculation with inert core.The A c = A − A c -body ”core”and the valence 2-body wave functions. This consider-ably simplifies the calculation of the effective Hamilto-nian, because only the 0¯ h Ω part (P -space part) of thecomplete N max ¯ h Ω wave function needs to be specified.
III. EFFECTIVE TWO-BODY P-SHELLINTERACTION
Utilizing the approach outlined above, we have cal-culated effective p-shell Hamiltonians for Li, using the6-body Hamiltonians with N max = 2 , , ..,
12 and Ω = 14MeV, constructed from the INOY (inside nonlocal out-side Yukawa) interaction [18, 19]. This is a new type ofinteraction, which has local behavior appropriate for tra-ditional NN interactions at longer ranges, but exhibits anonlocality at shorter distances. The nonlocality of theNN interaction has been introduced in order to accounteffectively for three-nucleon (NNN) interactions whichcorrectly describe the NNN bound states H and He,whereas still reproducing NN scattering data with highprecision. The corresponding excitation energies of p-shell dominated states and the binding energy of Li areshown in Fig.1 as a function of N max . The dimension of FIG. 1: The excitation energies of the J π states and groundstate energy for Li calculated in the N max ¯ h Ω spaces withthe INOY interaction and ¯ h Ω = 14 MeV. The experimentalspectra and ground state energy are shown for comparison. the configurational space for the N max = 12 case consid-ered is 48 million (M-scheme). A two orders of magnitudeincrease in the size of the model space, as compared tothe previous N max = 6 study [10], allows us to determinea converged value of 31.681 MeV for the Li binding en-ergy. Furthermore, the excitation energy of the highestlying p-space state, J π = 0 +2 , is lowered by an amountof 2.1 MeV in comparison to the N max = 6 case, indicat-ing improved convergence for both the excited states andground state for N max = 12.In the SSM an effective two-body Hamiltonian for anucleus with mass number A is represented in terms ofthree components: H A SSM = H + H + V A , (10)where H is the inert core part associated with the inter-action of the nucleons occupying closed shells, H is theone-body part corresponding to the interaction of valencenucleons with core nucleons, and V A is the two-body partreferring to the interaction between valence particles. Itis usually assumed that the core and one-body parts areconstant for an arbitrary number of valence particles andthat only the two-body part V A may contain mass depen-dence that includes effects of three-body and higher-bodyinteractions.To represent the H ,N max A,a Hamiltonian in the SSM for-mat, we develop a valence cluster expansion (VCE), H ,N max A,a = H A,A c + H A,A c +11 + a v X k =2 V A,A c + kk , (11)where the lower index, k, stands for the k-body interac-tion in the a v -body valence cluster ( a = A c + a v ); thefirst upper index A for the mass dependence; and thesecond upper index, A c + k for the number of particlescontributing to the corresponding k-body part. Thus, weconsider the more general case of allowing the core (k=0),one-body (k=1) and other k-body parts to vary with themass number A. This appears necessary to include theA-dependence of the excitations of the core ( A c ) nucle-ons treated in the original N max basis space. For the A=6case the two-body valence cluster (2BVC) approximationis exact: H ,N max A =6 ,a =6 = H , + H , + V , , (12)where the core part, H , , is defined as the ground state J π = 0 + energy of He calculated in the N max ¯ h Ω spacewith the TBMEs of the primary effective Hamiltonian, H N max , Ω6 , for A=6. Then the one-body part, H , ,is de-termined as H , = H ,N max , − H , . (13)The TBMEs of the one-body part, H , , h ab | H , | cd i JT = ( ǫ a + ǫ b ) δ a,c δ b,d (14)may be represented in terms of single particle energies(SPE) , ǫ a : ǫ pa = E ( Li , j a ) − H , , ǫ na = E ( He , j a ) − H , . (15)where the index a (as well as b,c, and d) denotes theset of single particle HO quantum numbers ( n a , l a , j a ),upper index stands for proton (p) and neutron (n), andthe E( Li,J), E( He,J) are NCSM energies of the low-est J πi = 3 / − and J πi = 1 / − states calculated in the N max ¯ h Ω space for the 5-body system using the TBMEsof the A = 6 effective Hamiltonian, H N max , Ω A =6 , , which in-cludes Coulomb energy. Finally, the two-body part V , is obtained by subtracting of two Hamiltonians: V , = H ,N max , − H ,N max , . (16)It is worth noting that since the Coulomb energy is in-cluded in the original Hamiltonian, the proton-proton (pp), neutron-neutron (nn) and proton-neutron (pn) T =1 TBMEs of the two-body part, V , , have different val-ues. The pn TBMEs of the core, one-body and two-body parts of the expanded Hamiltonian for Li are listedin the Table I. In Table I we also list the values of H N max =12 , Ω6 , with Ω = 14 MeV, so that one can observehow much these values change when the correlations upto 6-bodies are included, so as to obtain the values of H , , .The results presented in Table I indicate that thelargest parts of the effective Hamiltonian are attributedto the interaction among core nucleons (k=0) and theinteraction of valence nucleons with the core nucleons(k=1). However, these two largest contributions par-tially cancel each other. The pure two-body part cor-responding to the interaction of valence nucleons is con-siderably smaller than the individual core and one-bodyparts. Note that one may re-partition the core and sin-gle particle energies by shifting a constant amount from H A, to H A, . A shift of ≈
24 MeV ( ≈
32 MeV) forA=6 (7) produces core and valence energies where thecore matches the He as in the NCSM with A=4.To investigate the balance of the pure two-body, V , ,core, H , , and one-body, H , , parts of the effectiveHamiltonian with the increase of the size of the origi-nal many-body space, we have plotted the sum of coreand one-body parts, H , + H , , as a function of N max in Fig.2. The results in Fig.2 reveal a weak dependence FIG. 2: The diagonal TBMEs of the sum for the core and one-body parts, h ab | H , + H , | ab i , for the effective Hamiltonian, H ,N max , , for Li as a function of N max . The correspondingcurves are labeled by quantum numbers 2 j a j b j a j b . of the sum of the core and one-body parts of the effec-tive Hamiltonian on N max starting at N max = 6. Thismeans that the converged results for core plus one-bodyparts of the effective Hamiltonian are closely approached.The gaps in the curves are governed by the size of thespin-orbit splitting ǫ − ǫ .Plotting the diagonal pn TBMEs of the residual two-body part, V , , of the effective Hamiltonian in Fig.3, weobserve, that they exhibit stronger dependence than the TABLE I: The pn TBMEs of the NCSM H N max =12 , Ω A =6 , Hamiltonian with Ω = 14 MeV, the p-shell effective Hamiltonians H ,N max , and H ,N max , obtained from an N max = 12 NCSM calculation for Li are shown. The core, H A, , one-body, H A, , andresidual two-body, V A, parts for latest two Hamiltonians are presented. The H ,N max , Hamiltonian with A-independent coreand one-body parts is shown in last three columns. H , Ω A, H , A, , (MeV) H , , , (MeV) H , , , (MeV) H , , , (MeV)2j a b c d J T A=6 A=6 A=7 H , H , V , H , H , V , H , H , W , core plus one-body parts with increase of N max . From FIG. 3: The diagonal pn TBMEs of the two-body part, h ab | V , | ab i JT , of the effective Hamiltonian, H ,N max , , as afunction of N max . The corresponding curves are labeled byquantum numbers 2 j a j b j a j b and JT. Fig.3 we observe that the T=0 TBMEs are, on average,attractive, while the T=1 TBMEs are repulsive. Startingat N max =6 the two-body part shows smooth regularity.The results for nondiagonal matrix elements, shown inFig. 4, indicates smooth, regular changes towards smallerabsolute values of these TBMEs. We note that slow con-vergence of TBMEs with increasing N max reminds us of earlier treatment of core polarization [20, 21], where weobserve slow convergence with ”improved” treatments ofcore-polarization within perturbation theory. FIG. 4: The non-diagonal pn TBMEs of the two-body part, h ab | V , | cd i JT , for the effective Hamiltonian, H ,N max , , as afunction of N max . The corresponding curves are labeled byquantum numbers 2 j a j b j c j d and JT. A. Two-body valence cluster approximation for
A > The VCE given by the Eq.(11) would require a three-body part V , of the p-shell effective interaction H ,N max , to reproduce exactly the NCSM results for A=7 nuclei: H ,N max A =7 ,a =7 = H , + H , + V , + V , . (17)Therefore, it is worth knowing how good the 2BVC ap-proximation for A=7 as well as for A > H ,N max A =7 ,a =6 Hamiltonian, using Eq.(9), and expanded it interms of zero-, one- and two-body valence clusters, i.e. omitting the three-body part: H ,N max A =7 ,a =6 = H , + H , + V , . (18)In other words, we have first performed NCSM calcu-lations for the a -body systems ( a = 4 , ,
6) with the H N max , Ω A =7 , Hamiltonian. Thus, H ,a =40 is the He “core”energy and H ,a =51 is the one-body part determined asin Eqs.(13)-(15), but with A=7; and V ,a =62 is obtainedby subtracting H , + H , from H ,N max A =7 ,a =6 .The resulting parts of the H ,N max A =7 , Hamiltonian aregiven in Table I. Comparing the TBMEs for A=6 andA=7 (Table I), we find that they differ considerably.There is a big change separately for the core and one-body parts, but weaker changes for the two-body parts,which tend to become larger in magnitude with increas-ing A. We have then performed SSM calculations for theground state energy of Li, using the zero-, one- and two-body parts in Eq.(18). Namely, the one- and two-bodyparts were employed in a SSM calculation of the groundand excited states energies of the valence nucleons in thep-shell, i.e. , 0¯ h Ω space, to which the He core energy, H , , was added, in order to yield the total energies.These calculations were repeated for N max = 0 , , ... Li with H N max , Ω A =7 , for the same values of N max . The SSM andNCSM results for the ground-state energy are shown inFig.5.It is also of interest to find out what would be theresult if we take the fixed core and one-body parts atvalues which are appropriate for the a = 4 and a = 5systems, respectively, because this is analogous to whatis done in the SSM to determine energies relative to aninert core. To do this we adopt an alternative two-bodyVCE, which assumes that the core and one-body partsare A independent, i.e. , H ,N max A, = H , + H , + W A, , (19)similar to the SSM convention given by Eq.(10). We havethen performed another set of SSM calculations for A=7in the same manner as described previously, but using thedecomposition given in Eq.(19). To distinguish betweenthe two-body part of the VCE given by the Eqs.(11) FIG. 5: The ground state energy, E gs , of Li as a function of N max . The NCSM results with the H N max , Ω A =7 , Hamiltonian areshown by filled circles connected with the solid line. The SSMresults with the effective H ,N max , Hamiltonian decomposedaccording to Eq.(18) are shown by squares connected withthe dashed line. The SSM results with the effective H ,N max , Hamiltonian decomposed according to Eq.(19) are shown byfilled circles connected with a dashed line. and (19), we have introduced the new notation, W A, ,in Eq.(19). The Hamiltonian H , , expanded accordingto the Eq.(19) is shown in last three columns of TableI and the corresponding results are depicted in Fig.5 bythe dots connected with a dashed line. Figure 5 indi-cates that for light systems a realistic balance of core,one-body and two-body parts of the effective interactionmay be achieved only when both the core and one-bodyparts are mass-dependent, contrary to earlier approaches.A-independent core and one-body parts lead to a verystrong two-body part for the valence nucleons and, sub-sequently, to drastic overbinding. It is obvious, that, inorder to compensate for such an effect one would needto introduce a strongly repulsive three-body effective in-teraction with an unrealistic strength of about 10 MeV.Although, the effect on the spectrum is smaller, the VCEwith the A-dependent core and one-body parts also yieldsbetter agreement with the exact NCSM results for theexcited states. The corresponding low-energy spectrumof Li obtained with the NCSM and the A-dependentSSM (using the values in columns 12,13 and 14 of Ta-ble I) are compared in Fig.6. The differences observed inFigs.5 and 6 for the ground state and excited states, re-spectively, may be attributed to the neglected three-bodypart of the effective interaction at the two-body valencecluster level.We have generalized the 2BVC expansion procedure ofEq.(18) for arbitrary mass number A, H ,N max A,a =6 = H A, + H A, + V A, , (20)and applied it to the A=7,8,9, and 10 isobars for N max =6. The difference of the NCSM and SSM ground state en-ergies for different mass number A is plotted as a function FIG. 6: NCSM (solid line) and SSM (using Eq.(18), dashedline) spectra for Li. The states with spin J are marked by2J. of isospin projection T z = ( N − Z ) / FIG. 7: The difference of the NCSM and SSM (Eq.(20)ground state energies for different values of mass number Aas a function of isospin projection T z = ( N − Z ) / shows that the three-body and higher-body correlationsbecome more important with increasing mass number.There is also a very strong isospin dependence of the ob-tained results. For the highest isospin values the SSMsystematically underbinds nuclei in comparison to theNCSM and higher-body correlations appear to be smallfor systems containing only valence neutrons. However,there is an opposite effect in the vicinity of the N = Z line where SSM yields considerably more binding energythan the NCSM.Thus, the residual a -body correlations with a ≥ A ≥ TABLE II: The 3-body T=3/2 parts of the p-shell effectiveHamiltonian, H ,N max , , obtained from an N max = 6 NCSMcalculation for He is shown in column 9. The 3-body nnnparts of the p-shell effective Hamiltonians, H ,N max A, , for A=8,9 and 10 are shown in columns 10,11 and 12, respectively. V A, , (MeV)2j a b c d e f
2J 2T A = 7 A = 8 A = 9 A = 10nnn nnn nnn nnn3 3 1 3 3 1 1 3 -0.055 0.181 0.354 0.4713 3 3 3 3 3 3 3 -0.366 -0.181 -0.080 -0.0263 3 1 3 3 1 3 3 -0.504 -0.280 -0.126 -0.0303 1 1 3 1 1 3 3 -0.306 -0.197 -0.081 0.0103 3 3 3 3 1 3 3 0.290 0.281 0.270 0.2613 3 3 3 1 1 3 3 -0.246 -0.202 -0.165 -0.1353 3 1 3 1 1 3 3 0.388 0.356 0.317 0.2833 3 1 3 3 1 5 3 -0.209 -0.038 0.066 0.124 B. Beyond the two-body valence cluster expansion
The analysis of the A=7 systems may allow us to de-rive an effective three-body Hamiltonian for the p-shelland to give an idea about the strength of the three-bodyinteraction. To derive the three-body effective Hamilto-nian, we employ the three-body valence cluster expansion(3BVC) approximation, H ,N max A,a =7 = H A, + H A, + V A, + V A, , (21)which is the exact one for A = 7 systems. ComparingEqs.(18) and (21), we find the the following result forthe three-body part V A, of the effective Hamiltonian: V A, = H ,N max A, − H ,N max A, . (22)Using Eq.(9), we derive the A=7 Hamiltonian, H ,N max , ,employing a = 7 NCSM eigenvectors and eigenvalues,obtained with the H N max , Ω7 , interaction. The same pro-cedure is then repeated to calculate the A=7 Hamilto-nian, H ,N max , , employing a = 6 NCSM eigenvectorsand eigenvalues, obtained with the H N max , Ω6 , interaction.Then, the residual three-body part V , is calculated ac-cording to Eq.(22). The same scheme can be applied for A > N max = 6 are given in Table II.On average, the nnn T = 3 / T = 1 TBMEs for A=7 (see Table I) andhave an opposite sign. Performing the same procedure,we have obtained the 3BMEs for the A=8, 9 and 10 sys-tems, which are also listed in Table II. Comparing nnn3BMEs for different A, we note that diagonal 3BMEs be-come more repulsive, while there are only small changes TABLE III: Results for He, He and He from SSM calcu-lations with the effective 2BVC and 3BVC Hamiltonians andfrom exact NCSM calculation for N max = 6 with the INOYinteraction. J πi E( He), (MeV) J πi E( He), (MeV)2BVC 3BVC NCSM 2BVC 3BVC NCSM0 +1 -26.323 -26.542 -26.604 1 / − -22.328 -22.342 -22.8352 +1 -21.608 -21.609 -21.752 3 / − -17.429 -17.452 -17.9611 +1 -18.555 -19.224 -19.386 E( He), (MeV)0 +2 -16.108 -16.644 -16.843 0 + -21.219 -19.720 -21.0862 +2 -14.736 -15.681 -15.682 for non-diagonal 3BMEs; however their magnitudes be-come smaller for larger mass. This is in contrast to whatwe observed in the previous section for the two-body ef-fective interaction.The T=3/2 3BMEs can be represented in terms ofT=1 TBMEs using the coefficients of fractional parent-age (CFP) for the 3-body to 2-body reduction problem.Following this idea, we have calculated 3-body correc-tions for the corresponding TBMEs using T=3/2 3BMEsshown in Table II. It is worth noting, that this is notan exact way to treat the 3-body degrees of freedombut an approximation which estimates average 3-bodyeffect. Using the 3-body corrected neutron TBMEs, wehave performed SSM calculations for He, He and He,which have no valence protons and 4, 5 and 6 valence neu-trons, respectively, in the p-shell. Since there are only va-lence neutrons in the case of He isotopes, only the T=3/2three-body coupling is possible, and, thus, the T=1/23BMEs are not required for calculations. As an example,the results of the SSM calculations for He, He and Hewith effective interactions obtained in 2BVC and 3BVCapproximations from INOY interaction are compared toexact NCSM results in Table III and Fig. 8.Obtained results indicate that accounting for the effec-tive 3-body interactions considerably improves the agree-ment with the exact NCSM for the He, does not bringmuch change for He and yields worse results for He(see Fig.8). Performing a similar calculation with the ef-fective interaction obtained in the 3BVC approximationstarting from the CD-Bonn interaction [22], we obtainedresults which are shown in Fig.9. Note, that the effectiveCD-Bonn interaction constructed in the 2BVC approxi-mation considerably underbinds the He isotopes in com-parison to the exact NCSM results. The subsequent em-ployment of the 3BVC approximation compensates theselarge differences and yields much better results for He.However, to draw more quantitative conclusion about the3-body and higher-body effective interactions, one needsto perform exact diagonalization using the 3BMEs. Wewill evaluate this effect in future studies.
FIG. 8: Comparison of spectra for He, He and He fromSSM calculations using the effective 2BVC and 3BVC Hamil-tonians and from exact NCSM calculation for N max = 6 andΩ=14 MeV using the INOY interaction.FIG. 9: Comparison of spectra for He, He and He fromSSM calculations using the effective 2BVC and 3BVC Hamil-tonians and from exact NCSM calculation for N max = 6 andΩ=20 MeV using CD-Bonn interaction. IV. CONCLUSION
Within the NCSM approach we can calculate, by ex-act projection, full A-nucleon dependent TBMEs (and3BMEs). These A-dependent TBMEs (and 3BMEs) canbe separated into core, one-body and two-body (andthree-body) parts, all of which are also A-dependent, con-trary to the SSM approach. When these A-dependent ef-fective one- and two-body (and three-body) interactionsare employed in SSM calculations, they exactly reproducefull NCSM calculations for A=6 (A=7) isobars and yieldresults in good agreement with full NCSM calculationsfor
A >
A >
7, which include the 3-body effective interaction, in-dicate that 3- and higher-body effective interactions mayplay an important role in determining their binding ener-gies and spectra. Future investigations will be extendedto include effective 3-body interactions exactly and toexplore other physical operators, such as transition op-erators and EM moments.
V. ACKNOWLEDGMENTS
We thank the Institute for Nuclear Theory at the Uni-versity of Washington for its hospitality and the De-partment of Energy for partial support during the de- velopment of this work. B.R.B. and A.F.L. acknowl-edge partial support of this work from NSF grantsPHY0244389 and PHY0555396; P.N. acknowledges sup-port in part by the U.S. DOE/SC/NP (Work ProposalN. SCW0498) and U.S Department of Energy GrantDE-FG02-87ER40371; J.P.V. acknowledges support fromU.S. Department of Energy Grants DE-FG02-87ER40371and DE-FC02-07ER41457; and the work of I.S. wasperformed under the auspices of the U.S. DOE. Pre-pared by LLNL under Contract DE-AC52-07NA27344.B.R.B. thanks the Gesellschaft f¨ur Schwerionenforschungmbh Darmstadt, Germany, for its hospitality during thepreparation of this manuscript and the Alexander vonHumboldt Stiftung for its support. [1] P. Navratil, V. G. Gueorgiev, J.P. Vary, W. E. Ormand,and A. Nogga, Phys. Rev. Lett. , 042501 (2007).[2] A. Nogga, P.Navratil, B.R.Barrett, J.P.Vary, Phys. Rev.C. , 064002 (2006).[3] I. Stetcu, B.R.Barrett, P.Navratil, J.P.Vary, Phys. Rev.C. , 044325 (2005).[4] P. Navratil and W.E.Ormand, Phys. Rev. Lett. ,152502 (2002); Phys. Rev. C. , 034305 (2003).[5] P. Navratil, J.P.Vary, B.R.Barrett, Phys. Rev. Lett. ,5728 (2000); Phys. Rev. C. , 054311 (2000).[6] S. C. Pieper, K. Varga, and R. B. Wiringa, Phys. Rev.C 66, 044310 (2002).[7] S. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci. , 53 (2001).[8] K.Kowalski, D.J.Dean, M.Hjorth-Jensen, T.Papenbrock,P.Piecuch, Phys. Rev. Lett. 92, 132501 (2004).[9] S. Fujii, T. Mizusaki, T. Otsuka, T. Sebe and A. Arima,Phys. Lett. B650 , 9 (2007).[10] P. Navratil, M. Thoresen, and B. R. Barrett, Phys. Rev.C. , R573 (1997).[11] Susumu Okubo, Prog. Theor. Phys. Vol. 12, No.5, 603(1954).[12] Kenji Suzuki, Prog. Theor. Phys. Vol. 68, No.1, 246 (1982).[13] P. Navratil and E.Caurier, Phys. Rev. C. , 014311(2004).[14] E. Caurier and F. Nowacki, Acta. Phys. Pol. B30, (1999)705.[15] E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves,J. Retamosa, and A. P. Zuker, Phys. Rev. C , 2033(1999).[16] E. Caurier, P. Navratil, W. E. Ormand, and J.P. Vary,Phys. Rev. C , 051301(R) (2001).[17] D. R. Entem and R. Machleidt, Phys. Rev. C. ,041001(R) (2003).[18] P. Doleschall, Phys. Rev. C. , 054001 (2004).[19] P. Doleschall, I. Borbely, Z. Papp, and W. Plessas, Phys.Rev. C. , 0064005 (2003).[20] B. R. Barrett and M.W. Kirson, Nucl. Phys. A , 145(1970).[21] J. P. Vary, P. U. Sauer and C. W. Wong, Phys. Rev. C. , 1776 (1973).[22] R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev.C.53