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

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

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

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.

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

Isovector part of nuclear energy density functional from chiral two- and three-nucleon forces

A recent calculation of the nuclear energy density functional from chiral two- and three-nucleon forces is extended to the isovector terms pertaining to different proton and neutron densities. An improved density-matrix expansion is adapted to the situation of small isospin-asymmetries and used to calculate in the Hartree-Fock approximation the density-dependent strength functions associated with the isovector terms. The two-body interaction comprises of long-range multi-pion exchange contributions and a set of contact terms contributing up to fourth power in momenta. In addition, the leading order chiral three-nucleon interaction is employed with its parameters fixed in computations of nuclear few-body systems. With this input one finds for the asymmetry energy of nuclear matter the value $A(\rho_0) \simeq 26.5\,$MeV, compatible with existing semi-empirical determinations. The strength functions of the isovector surface and spin-orbit coupling terms come out much smaller than those of the analogous isoscalar coupling terms and in the relevant density range one finds agreement with phenomenological Skyrme forces. The specific isospin- and density-dependences arising from the chiral two- and three-nucleon interactions can be explored and tested in neutron-rich systems.Comment: 14 pages, 7 figures, to be published in European Physical Journal

Exact calculation of three-body contact interaction to second order

For a system of fermions with a three-body contact interaction the second-order contributions to the energy per particle $\bar E(k_f)$ are calculated exactly. The three-particle scattering amplitude in the medium is derived in closed analytical form from the corresponding two-loop rescattering diagram. We compare the (genuine) second-order three-body contribution to $\bar E(k_f)\sim k_f^{10}$ with the second-order term due to the density-dependent effective two-body interaction, and find that the latter term dominates. The results of the present study are of interest for nuclear many-body calculations where chiral three-nucleon forces are treated beyond leading order via a density-dependent effective two-body interaction.Comment: 9 pages, 6 figures, to be published in European Journal

Towards a Model-Independent Low Momentum Nucleon-Nucleon Interaction

We provide evidence for a high precision model-independent low momentum nucleon-nucleon interaction. Performing a momentum-space renormalization group decimation, we find that the effective interactions constructed from various high precision nucleon-nucleon interaction models, such as the Paris, Bonn, Nijmegen, Argonne, CD Bonn and Idaho potentials, are identical. This model-independent low momentum interaction, called V_{low k}, reproduces the same phase shifts and deuteron pole as the input potential models, without ambiguous assumptions on the high momentum components, which are not constrained by low energy data and lead to model-dependent results in many-body applications. V_{low k} is energy-independent and does not necessitate the calculation of the Brueckner G matrix.Comment: 12 pages, 5 figures, minor changes and additions, to appear in Phys. Lett.

Nuclear energy density functional from chiral two- and three-nucleon interactions

An improved density-matrix expansion is used to calculate the nuclear energy density functional from chiral two- and three-nucleon interactions. The two-body interaction comprises long-range one- and two-pion exchange contributions and a set of contact terms contributing up to fourth power in momenta. In addition we employ the leading order chiral three-nucleon interaction with its parameters $c_E, c_D$ and $c_{1,3,4}$ fixed in calculations of nuclear few-body systems. With this input the nuclear energy density functional is derived to first order in the two- and three-nucleon interaction. We find that the strength functions $F_\nabla(\rho)$ and $F_{so}(\rho)$ of the surface and spin-orbit terms compare in the relevant density range reasonably with results of phenomenological Skyrme forces. However, an improved description requires (at least) the treatment of the two-body interaction to second order. This observation is in line with the deficiencies in the nuclear matter equation of state $\bar E(\rho)$ that remain in the Hartree-Fock approximation with low-momentum two- and three-nucleon interactions.Comment: 16 pages, 12 figures, submitted to Eur. Phys. J.

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