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Register a new userSingular Value Decomposition and Similarity Renormalization Group Evolution of Nuclear Interactions
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One of the main challenges for ab initio nuclear many-body theory is the growth of computational and storage costs as calculations are extended to heavy, exotic, and structurally complex nuclei. Here, we investigate the factorization of nuclear interactions as a means to address this issue. We perform Singular Value Decompositions of nucleon-nucleon interactions in partial wave representation and study the dependence of the singular value spectrum on interaction characteristics like regularization scheme and resolution scales. We develop and implement the Similarity Renormalization Group (SRG) evolution of the factorized interaction, and demonstrate that this SVD-SRG approach accurately preserves two-nucleon observables. We find that low-resolution interactions allow the truncation of the SVD at low rank, and that a small number of relevant components is sufficient to capture the nuclear interaction and perform an accurate SRG evolution, while the Coulomb interaction requires special consideration. The rank is uniform across all partial waves, and almost independent of the basis choice in the tested cases. This suggests an interpretation of the relevant singular components as mere representations of a small set of abstract operators that can describe the interaction and its SRG flow. Following the traditional workflow for nuclear interactions, we discuss how the transformation between the center-of-mass and laboratory frames creates redundant copies of the partial wave components when implemented in matrix representation, and we discuss strategies for mitigation. Finally, we test the low-rank approximation to the SRG-evolved interactions in many-body calculations using the In-Medium SRG. By including nuclear radii in our analysis, we verify that the implementation of the SRG using the singular vectors of the interaction does not spoil the evolution of other observables.
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An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.
The similarity renormalization group (SRG) has been successfully applied to soften interactions for ab initio nuclear calculations. In almost all practical applications in nuclear physics, an SRG generator with the kinetic energy operator is used. With this choice, a fast convergence of many-body calculations can be achieved, but at the same time substantial three-body interactions are induced even if one starts from a purely two-nucleon (NN) Hamiltonian. Three-nucleon (3N) interactions can be handled by modern many-body methods. However, it has been observed that when including initial chiral 3N forces in the Hamiltonian, the SRG transformations induce a non-negligible four-nucleon interaction that cannot be currently included in the calculations for technical reasons. Consequently, it is essential to investigate alternative SRG generators that might suppress the induction of many-body forces while at the same time might preserve the good convergence. In this work we test two alternative generators with operators of block structure in the harmonic oscillator basis. In the no-core shell model calculations for 3H, 4He and 6Li with chiral NN force, we demonstrate that their performances appear quite promising.
Efforts to describe nuclear structure and dynamics from first principles have advanced significantly in recent years. Exact methods for light nuclei are now able to include continuum degrees of freedom and treat structure and reactions on the same footing, and multiple approximate, computationally efficient many-body methods have been developed that can be routinely applied for medium-mass nuclei. This has made it possible to confront modern nuclear interactions from Chiral Effective Field Theory, that are rooted in Quantum Chromodynamics with a wealth of experimental data. Here, we discuss one of these efficient new many-body methods, the In-Medium Similarity Renormalization Group (IMSRG), and its applications in modern nuclear structure theory. The IMSRG evolves the nuclear many-body Hamiltonian in second-quantized form through continuous unitary transformations that can be implemented with polynomial computational effort. Through suitably chosen generators, we drive the matrix representation of the Hamiltonian in configuration space to specific shapes, e.g., to implement a decoupling of low- and high-energy scales, or to extract energy eigenvalues for a given nucleus. We present selected results from Multireference IMSRG (MR-IMSRG) calculations of open-shell nuclei, as well as proof-of-principle applications for intrinsically deformed medium-mass nuclei. We discuss the successes and prospects of merging the (MR-)IMSRG with many-body methods ranging from Configuration Interaction to the Density Matrix Renormalization Group, with the goal of achieving an efficient simultaneous description of dynamic and static correlations in atomic nuclei.
We have developed a fully consistent framework for calculations in the Quasiparticle Random Phase Approximation (QRPA) with $NN$ interactions from the Similarity Renormalization Group (SRG) and other unitary transformations of realistic interactions. The consistency of our calculations, which use the same Hamiltonian to determine the Hartree-Fock-Bogoliubov (HFB) ground states and the residual interaction for QRPA, guarantees an excellent decoupling of spurious strength, without the need for empirical corrections. While work is under way to include SRG-evolved 3N interactions, we presently account for some 3N effects by means of a linearly density-dependent interaction, whose strength is adjusted to reproduce the charge radii of closed-shell nuclei across the whole nuclear chart. As a first application, we perform a survey of the monopole, dipole, and quadrupole response of the calcium isotopic chain and of the underlying single-particle spectra, focusing on how their properties depend on the SRG parameter $lambda$. Unrealistic spin-orbit splittings suggest that spin-orbit terms from the 3N interaction are called for. Nevertheless, our general findings are comparable to results from phenomenological QRPA calculations using Skyrme or Gogny energy density functionals. Potentially interesting phenomena related to low-lying strength warrant more systematic investigations in the future.
We present a pedagogical discussion of Similarity Renormalization Group (SRG) methods, in particular the In-Medium SRG (IMSRG) approach for solving the nuclear many-body problem. These methods use continuous unitary transformations to evolve the nuclear Hamiltonian to a desired shape. The IMSRG, in particular, is used to decouple the ground state from all excitations and solve the many-body Schrodinger equation. We discuss the IMSRG formalism as well as its numerical implementation, and use the method to study the pairing model and infinite neutron matter. We compare our results with those of Coupled cluster theory, Configuration-Interaction Monte Carlo, and the Self-Consistent Greens Function approach. The chapter concludes with an expanded overview of current research directions, and a look ahead at upcoming developments.
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