Weinberg Eigenvalues and Pairing with Low-Momentum Potentials

The nonperturbative nature of nucleon-nucleon interactions evolved to low momentum has recently been investigated in free space and at finite density using Weinberg eigenvalues as a diagnostic. This analysis is extended here to the in-medium eigenvalues near the Fermi surface to study pairing. For a fixed value of density and cutoff Lambda, the eigenvalues increase arbitrarily in magnitude close to the Fermi surface, signaling the pairing instability. When using normal-phase propagators, the Weinberg analysis with complex energies becomes a form of stability analysis and the pairing gap can be estimated from the largest attractive eigenvalue. With Nambu-Gorkov Green's functions, the largest attractive eigenvalue goes to unity close to the Fermi surface, indicating the presence of bound states (Cooper pairs), and the corresponding eigenvector leads to the self-consistent gap function.Comment: 16 pages, 9 figure

Algebraic characterization of differential operators of Calabi-Yau type

We give an algebraic characterization of Picard-Fuchs operators attached to families of Calabi-Yau manifolds with a point of maximally unipotent monodromy and discuss possibilities for their differential Galois groups.Comment: 20 page

Variational Calculations using Low-Momentum Potentials with Smooth Cutoffs

Recent variational calculations of the deuteron and the triton illustrate that simple wave function ansatze become more effective after evolving the nucleon-nucleon potential to lower momentum (``V_lowk''). However, wave function artifacts from the use of sharp cutoffs in relative momentum decrease effectiveness for small cutoffs (< 2 fm^-1) and slow down convergence in harmonic oscillator bases. These sharp cutoff artifacts are eliminated when V_lowk is generated using a sufficiently smooth cutoff regulator.Comment: 11 pages, 4 figure

Microscopically-constrained Fock energy density functionals from chiral effective field theory. I. Two-nucleon interactions

The density matrix expansion (DME) of Negele and Vautherin is a convenient tool to map finite-range physics associated with vacuum two- and three-nucleon interactions into the form of a Skyme-like energy density functional (EDF) with density-dependent couplings. In this work, we apply the improved formulation of the DME proposed recently in arXiv:0910.4979 by Gebremariam {\it et al.} to the non-local Fock energy obtained from chiral effective field theory (EFT) two-nucleon (NN) interactions at next-to-next-to-leading-order (N$^2$LO). The structure of the chiral interactions is such that each coupling in the DME Fock functional can be decomposed into a cutoff-dependent coupling {\it constant} arising from zero-range contact interactions and a cutoff-independent coupling {\it function} of the density arising from the universal long-range pion exchanges. This motivates a new microscopically-guided Skyrme phenomenology where the density-dependent couplings associated with the underlying pion-exchange interactions are added to standard empirical Skyrme functionals, and the density-independent Skyrme parameters subsequently refit to data. A Mathematica notebook containing the novel density-dependent couplings is provided.Comment: 28 pages, 12 figures. Mathematica notebook provided with submission

Three-Body Forces Produced by a Similarity Renormalization Group Transformation in a Simple Model

A simple class of unitary renormalization group transformations that force hamiltonians towards a band-diagonal form produce few-body interactions in which low- and high-energy states are decoupled, which can greatly simplify many-body calculations. One such transformation has been applied to phenomenological and effective field theory nucleon-nucleon interactions with success, but further progress requires consistent treatment of at least the three-nucleon interaction. In this paper we demonstrate in an extremely simple model how these renormalization group transformations consistently evolve two- and three-body interactions towards band-diagonal form, and introduce a diagrammatic approach that generalizes to the realistic nuclear problem.Comment: 25 pages, 18 figures, minor typos corrected and references update

Low-momentum interactions with smooth cutoffs

Nucleon-nucleon potentials evolved to low momentum, which show great promise in few- and many-body calculations, have generally been formulated with a sharp cutoff on relative momenta. However, a sharp cutoff has technical disadvantages and can cause convergence problems at the 10-100 keV level in the deuteron and triton. This motivates using smooth momentum-space regulators as an alternative. We generate low-momentum interactions with smooth cutoffs both through energy-independent renormalization group methods and using a multi-step process based on the Bloch-Horowitz approach. We find greatly improved convergence for calculations of the deuteron and triton binding energies in a harmonic oscillator basis compared to results with a sharp cutoff. Even a slight evolution of chiral effective field theory interactions to lower momenta is beneficial. The renormalization group preserves the long-range part of the interaction, and consequently the renormalization of long-range operators, such as the quadrupole moment, the radius and 1/r, is small. This demonstrates that low-energy observables in the deuteron are reproduced without short-range correlations in the wave function.Comment: 29 pages, 19 figure

Convergence of the Born Series with Low-Momentum Interactions

The nonperturbative nature of nucleon-nucleon interactions as a function of a momentum cutoff is studied using Weinberg eigenvalues as a diagnostic. This investigation extends an earlier study of the perturbative convergence of the Born series to partial waves beyond the 3S1-3D1 channel and to positive energies. As the cutoff is lowered using renormalization-group or model-space techniques, the evolution of nonperturbative features at large cutoffs from strong short-range repulsion and the iterated tensor interaction are monitored via the complex Weinberg eigenvalues. When all eigenvalues lie within the unit circle, the expansion of the scattering amplitude in terms of the interaction is perturbative, with the magnitude of the largest eigenvalue setting the rate of convergence. Major decreases in the magnitudes of repulsive eigenvalues are observed as the Argonne v18, CD-Bonn or Nijmegen potentials are evolved to low momentum, even though two-body observables are unchanged. For chiral EFT potentials, running the cutoff lower tames the impact of the tensor force and of new nonperturbative features entering at N3LO. The efficacy of separable approximations to nuclear interactions derived from the Weinberg analysis is studied as a function of cutoff, and the connection to inverse scattering is demonstrated.Comment: 21 pages, 15 figures, minor additions, to appear in Nucl. Phys.