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Register a new userCombining symmetry breaking and restoration with configuration interaction: extension to z-signature symmetry in the case of the Lipkin Model
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Background: Ab initio many-body methods whose numerical cost scales polynomially with the number of particles have been developed over the past fifteen years to tackle closed-shell mid-mass nuclei. Open-shell nuclei have been further addressed by implementing variants based on the concept of spontaneous symmetry breaking (and restoration). Purpose: In order to access the spectroscopy of open-shell nuclei more systematically while controlling the numerical cost, we design a novel many-body method that combines the merit of breaking and restoring symmetries with those brought about by low-rank individual excitations. Methods: The recently proposed truncated configuration-interaction method based on optimized symmetry-broken and -restored states is extended to the z-signature symmetry associated with a discrete subgroup of SU(2). The highly-truncated N-body Hilbert subspace within which the Hamiltonian is diagonalized is spanned by a z-signature broken and restored Slater determinant vacuum and associated low-rank excitations. Results: The proposed method provides an excellent reproduction of the ground-state energy and of low-lying excitation energies of various z-signatures and total angular momenta. In doing so, the successive benefits of (i) breaking the symmetry, (ii) restoring the symmetry, (iii) including low-rank particle-hole excitations and (iv) optimizing the amount by which the underlying vacuum breaks the symmetry are illustrated. Conclusions: The numerical cost of the newly designed variational method is polynomial with respect to the system size. The present study confirms the results obtained previously for the attractive pairing Hamiltonian in connection with the breaking and restoration of U(1) global gauge symmetry. These two studies constitute a strong motivation to apply this method to realistic nuclear Hamiltonians.
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Background: Ab initio many-body methods have been developed over the past ten years to address mid-mass nuclei... As progress in the design of inter-nucleon interactions is made, further efforts must be made to tailor many-body methods. Methods: We formulate a truncated configuration interaction method that consists of diagonalizing the Hamiltonian in a highly truncated subspace of the total N-body Hilbert space. The reduced Hilbert space is generated via the particle-number projected BCS state along with projected seniority-zero two and four quasi-particle excitations. Furthermore, the extent by which the underlying BCS state breaks U(1) symmetry is optimized in presence of the projected two and four quasi-particle excitations... The quality of the newly designed method is tested against exact solutions of the so-called attractive pairing Hamiltonian problem. Results: By construction, the method reproduce exact results for N=2 and N=4. For N=(8,16,20) the error on the ground-state correlation energy is less than (0.006, 0.1, 0.15) % across the entire range of inter-nucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage to automatically access the low-lying spectroscopy, which it does with high accuracy. Conclusions: The numerical cost of the newly designed variational method is polynomial (N$^6$) in system size. It achieves an unprecedented accuracy on the ground-state correlation energy, effective pairing gap and one-body entropy as well as on the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a strong enough motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate and versatile ab initio description of mid-mass open-shell nuclei in the future.
We explore the breaking of rotational symmetry on the lattice for bound state energies and practical methods for suppressing this breaking. We demonstrate the general problems associated with lattice discretization errors and finite-volume errors using an $alpha$ cluster model for $^8$Be and $^{12}$C. We consider the two and three $alpha$-particle systems and focus on the lowest states with non-zero angular momentum which split into multiplets corresponding to different irreducible representations of the cubic group. We examine the dependence of such splittings on the lattice spacing and box size. We find that lattice spacing errors are closely related to the commensurability of the lattice with the intrinsic length scales of the system. We also show that rotational symmetry breaking effects can be significantly reduced by using improved lattice actions, and that the physical energy levels are accurately reproduced by the weighted average of a given spin multiplets.
We study the breaking of rotational symmetry on the lattice for irreducible tensor operators and practical methods for suppressing this breaking. We illustrate the features of the general problem using an $alpha$ cluster model for $^{8}$Be. We focus on the lowest states with non-zero angular momentum and examine the matrix elements of multipole moment operators. We show that the physical reduced matrix element is well reproduced by averaging over all possible orientations of the quantum state, and this is expressed as a sum of matrix elements weighted by the corresponding Clebsch-Gordan coefficients. For our $alpha$ cluster model we find that the effects of rotational symmetry breaking can be largely eliminated for lattice spacings of $aleq 1.7$ fm, and we expect similar improvement for actual lattice Monte Carlo calculations.
Effects of the isospin-symmetry breaking (ISB) beyond mean-field Coulomb terms are systematically studied in nuclear masses near the $N=Z$ line. The Coulomb exchange contributions are calculated exactly. We use extended Skyrme energy density functionals (EDFs) with proton-neutron-mixed densities, to which we add new terms breaking the isospin symmetry. Two parameters associated with the new terms are determined by fitting mirror and triplet displacement energies (MDEs and TDEs) of isospin multiplets. The new EDFs reproduce MDEs for the $T=frac12$ doublets and $T=1$ triplets, and TDEs for the $T=1$ triplets. Relative strengths of the obtained isospin-symmetry-breaking terms {em are not} consistent with the differences in the $NN$ scattering lengths, $a_{nn}$, $a_{pp}$, and $a_{np}$. Based on low-energy experimental data, it seems thus impossible to delineate the strong-force ISB effects from beyond-mean-field Coulomb-energy corrections.
The conventional Skyrme interaction is generalized by adding zero-range charge-symmetry-breaking and charge-independence-breaking terms, and the corresponding energy density functional is derived. It is shown that the extended model accounts for experimental values of mirror and triplet displacement energies (MDEs and TDEs) in sd-shell isospin triplets with, on average, about 100~keV precision using only two additional adjustable coupling constants. Moreover, the model is able to reproduce, for the first time, the A=4n versus A=4n+2 staggering of the TDEs.
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