Mitja Rosina

The variational approach to the two−body density matrix

A variational method for the two−body density matrix is developed for practical calculations of the properties of many−fermion systems with two−body interactions. In this method the energy E = JHijkl ρijkl is minimized using the two−body density matrix elements ρijkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters. The approximation consists in satisfying only a subset of necessary conditions—the nonnegativity of the following matrices: the two−body density matrix, the ’’two−hole matrix’’ Qijkl = 〈Ψ‖ajaia+ka+l‖Ψ〉 and the particle−hole matrix Gijkl = 〈Ψ‖ (a+iaj−ρij)+ (a+kal−ρk) ‖Ψ〉. The idea of the method was introduced earlier; here some further physical interpretation is given and a numerical procedure for calculations within a small single−particle model space is described. The method is illustrated on the ground state of Be atom using 1s, 2s, 2p orbitals.

The variational approach to the density matrix for light nuclei

Abstract A previously introduced variational method for density matrices is adapted for applications in light nuclei. In this method the energy E = ∑ H ijklϱijkl is minimized using the two-body density matrix elements ϱijkl = 〈 Ψ ‖ a j + a i + a k a l ‖ Ψ 〉 as variational parameters. As the approximation, a subset of necessary conditions on the two-body density matrix is satisfied: the non-negativity of the two-body density matrix, the non-negativity of the two-hole density matrix Q ijkl = ϱijkl − δ ikϱjl + δ ilϱjk − δ jlϱik + δ jkϱil + δ ik δ jl − δ il δ jk and the non-negativity of the particle-hole matrix G ijkl = − ϱkjil + δ ikϱjl . The method is applied to the ground states of the nuclei 15 O, 16 O, 17 O and 18 O using the model space spanned by the 1 p 1 2 , 1 d 5 2 , 2 s 1 2 subshells and of the nuclei 20 Ne, 24 Mg and 28 Si using the model space spanned by the 1d, 2s shell. The comparison of the ground state energy with the results of the complete diagonalization in the same model space is encouraging.

The nucleon-nucleon potential in the nonrelativistic quark model

Abstract The effective nucleon-nucleon potential is presented in the model with two three-quark clusters. The potential is nonlocal and nonadiabatic. The shape of the local adiabatic part strongly depends on the definition of the effective potential and on the choice of the subspace used in the calculation. To get some information about the repulsive core and the weak attractive part, one has also to take into account the nonlocal terms. Then for commonly used quark-quark interactions, a repulsive core is obtained, which is not very sensitive to the choice of the parameters of the quark-quark interaction; beyond the core there is a weak attraction. The trial function is constructed as a mixture of NN, ΔΔ and different coloured baryon-coloured baryon configurations. The trial function is an antisymmetric colour singlet with isospin T = 0, spin S = 1 and orbital angular momentum L = 0.

A generator coordinate approach for the description of pairing vibrations in the seniority-zero space

Abstract The application of the generator coordinate method is extended to non-harmonic systems. The many-dimensional Hill-Wheeler integral equation is reduced to a one-dimensional integral equation by expressing all independent parameters in the generator function by a single parameter. It is shown that the subspace spanned by a proper single-parameter family is the same as that spanned by the many-parametric family of generator functions. The subspace of the seniority-zero states relevant to the description of pairing vibrational states is spanned by the N -projected BCS function depending on the gap parameter. It is shown on a schematic three-level system that a good description of energies and two-body transfer amplitudes of the pairing vibrational states can be obtained in a subspace much smaller than the seniority-zero subspace. Having a method working equally well for superconducting nuclei, non-superconducting nuclei and in the intermediate region, we tested how well the seniority-zero subspace is suitable for the description of low-lying 0 + states in Pb and Sn isotopes. It seems to us that only a qualitative description is possible.

A new particle-hole approach to collective states

Abstract The relevance of the particle-hole space is demonstrated by showing that in some commonly used formalisms the first excited state lies entirely within the particle-hole space generated from the correlated ground state. This property is proved for several cases of angularmomentum projection — projected Hartree-Fock method (PHF) — and for the generator coordinate method in the Gaussian overlap approximation, while in other cases it has been verified only numerically. A new method is presented for the approximate calculation of energies and transition amplitudes of particle-hole excited states. Only hermitian one-body operators are used to generate the excited states. The two-body density matrix of a correlated state approximating the ground state is required as input data. The formula is tested on the 2 + and 3 − states of 8 Be and 12 C by using the PHF ground state. Where comparison is possible the method gives better agreement with PHF and experiment than the extended random-phase approximation.

EXCITATIONS AS GROUND-STATE VARIATIONAL PARAMETERS.

Abstract The ground state density matrix can be expressed in terms of transition amplitudes 〈r|a a + a b | g 〉 by the following relation: 〈 g |a a + a c a b + a d |g〉= Σ r 〈 g |a a + a c |r〉〈r|a b + a d | g 〉. A formalism is derived in which these transition amplitudes are used as ground state variational p parameters. The quadratic form with transition amplitudes as variational parameters keeps the above matrix non-negative definite. This represents one necessary N -representability condition. This condition plus symmetry and trace relations are used to bound the two-body density matrix. (It keeps also the one-body density matrix between 0 and 1.) the trial ground state is also restricted in such a way that already half of the particle-hole excited states exhausts the sum over r . The variational equations have the form of a secular equation similar to that in the random phase approximation. The single-particle energies are, however, introduced as Lagrange multipliers for the trace relations of the density matrix. The one-body density matrix is also determined using variation methods. Additional Lagrange multipliers are introduced to guarantee certain symmetry relations, which are disregarded in the RPA. The obtained ground state is then consistent with excited states having different quantum numbers. The calculation for a simple system with a schematic intershell pairing is compared with the exact solution, the RPA and some other methods. The ground state energies and transition amplitudes are good for the entire range of the interaction.

The variational calculation of reduced density matrices

Abstract We discuss the computational problem encountered in making direct variational calculations of the reduced density matrices of many-particle systems. The problem is one of minimizing a linear function within a convex domain defined by a finite set of nonlinear constraints. Two different algorithms are presented for which working programs have been written.

The particle-hole states in some light nuclei calculated with the two-body density matrix of the ground states

Abstract Low-lying excited states are calculated within the particle-hole space using the previously introduced hermitian operator method. Calculations are performed on 16O using the model space spanned by the 1 p 1 2 , 1 d 5 2 , 2 s 1 2 subshells and on the nuclei20Ne, 24Mg, and 28Si using the model space spanned by the 1d, 2s shell. The lowest state for each set of quantum numbers J, P, T is compared with the result of complete diagonalization and agrees in most cases within 1 or 2 MeV. In order to test the variational approach to density matrices in the previous paper, also the approximate two-body density matrix elements are used as input and they give a similar agreement in most cases.

Production and detection of doubly charmed tetraquarks

The feasibility of tetraquark detection is studied. For the ccud tetraquark we show that in present (SELEX, Tevatron, RHIC) and future facilities (LHCb, ALICE) the production rate is promising and we propose some detectable decay channels.

The generator coordinate method as an approximation to the exact diagonalization in a given model space

Abstract The low-lying states of the nuclei 16 O and 20 Ne are calculated with the generator coordinate method. For 16 O two generator coordinates are used, which are related to the quadrupole and to the octupole deformation of the single-particle potential. For 20 Ne the generator coordinates are related to the parameters β and D of the Nilsson potential. In both cases six pairs of coordinate values are chosen. The results are compared with those of the complete diagonalization. The agreement is, for both cases, better than 0.5 MeV for most low-lying states.