arXiv subjects
Publications and source records attributed to R. J. Furnstahl.
The similarity renormalization group (SRG) is based on unitary transformations that suppress off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables, and provides a method to consistently evolve many-body potentials and other operators.
These lectures give an overview of the ongoing application of effective field theory (EFT) and renormalization group (RG) concepts and methods to density functional theory (DFT), with special emphasis on the nuclear many-body problem.
Conventional nucleon-nucleon potentials with strong short-range repulsion require contributions from high-momentum wave function components even for low-energy observables such as the deuteron binding energy. This can lead to the misconception that reproducing high-energy phase shifts is important for such observables. Interactions derived via the similarity renormalization group decouple high-energy and low-energy physics while preserving the phase shifts from the starting potential. They are used to show that high-momentum components (and high-energy phase shifts) can be set to zero when using low-momentum interactions, without losing information relevant for low-energy observables.
Effective field theory (EFT) methods for a uniform system of fermions with short-range, natural interactions are extended to include pairing correlations, as part of a program to develop a systematic Kohn-Sham density functional theory (DFT) for medium and heavy nuclei. An effective action formalism for local composite operators leads to a free-energy functional that includes pairing by applying an inversion method order by order in the EFT expansion. A consistent renormalization scheme is demonstrated for the uniform system through next-to-leading order, which includes induced-interaction corrections to pairing.
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.
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.
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.
The nonperturbative nature of inter-nucleon interactions is explored by varying the momentum cutoff of a two-nucleon potential. Conventional force models, which have large cutoffs, are nonperturbative because of strong short-range repulsion, the iterated tensor interaction, and the presence of bound or nearly-bound states. But for low-momentum interactions with cutoffs around 2 fm^{-1}, the softened potential combined with Pauli blocking leads to corrections in nuclear matter in the particle-particle channel that are well converged at second order in the potential, suggesting that perturbation theory can be used in place of Brueckner resummations. Calculations of nuclear matter using the low-momentum two-nucleon force V_{low k} with a corresponding leading-order three-nucleon (3N) force from chiral effective field theory (EFT) exhibit nuclear binding in the Hartree-Fock approximation, and become less cutoff dependent with the inclusion of the dominant second-order contributions. The role of the 3N force is essential to obtain saturation, and the contribution to the total potential energy is compatible with EFT power-counting estimates.
The study of the three-body problem with short-range attractive two-body forces has a rich history going back to the 1930's. Recent applications of effective field theory methods to atomic and nuclear physics have produced a much improved understanding of this problem, and we elucidate some of the issues using renormalization group ideas applied to precise nonperturbative calculations. These calculations provide 11-12 digits of precision for the binding energies in the infinite cutoff limit. The method starts with this limit as an approximation to an effective theory and allows cutoff dependence to be systematically computed as an expansion in powers of inverse cutoffs and logarithms of the cutoff. Renormalization of three-body bound states requires a short range three-body interaction, with a coupling that is governed by a precisely mapped limit cycle of the renormalization group. Additional three-body irrelevant interactions must be determined to control subleading dependence on the cutoff and this control is essential for an effective field theory since the continuum limit is not likely to match physical systems ({\it e.g.}, few-nucleon bound and scattering states at low energy). Leading order calculations precise to 11-12 digits allow clear identification of subleading corrections, but these corrections have not been computed.
Variational calculations of the deuteron and the triton illustrate that simple wave function ansätze become more effective after evolving the nucleon-nucleon potential to lower momentum (``$V_{low k}$''). This is consistent with many-body wave functions becoming much less correlated at lower cutoffs, as shown by two-particle wave functions and pair-distribution functions in nuclear matter. These results motivate a program to explore variational many-body calculations of binding energies and other low-energy nuclear properties using low-momentum potentials.
The application of density functional theory to nuclear structure is discussed, highlighting the current status of the effective action approach using effective field theory, and outlining future challenges.
An effective action approach to Kohn-Sham density functional theory is used to illustrate how the exact Green's function can be calculated in terms of the Kohn-Sham Green's function. An example based on Skyrme energy functionals shows that single-particle Kohn-Sham spectra can be improved by adding sources used to construct the energy functional.
This work continues a program to systematically generalize the Skyrme Hartree-Fock method for medium and heavy nuclei by applying effective field theory (EFT) methods to Kohn-Sham density functional theory (DFT). When conventional Kohn-Sham DFT for Coulomb systems is extended beyond the local density approximation, the kinetic energy density tau is sometimes included in energy functionals in addition to the fermion density. However, a local (semi-classical) expansion of tau is used to write the energy as a functional of the density alone, in contrast to the Skyrme approach. The difference is manifested in different single-particle equations, which in the Skyrme case include a spatially varying effective mass. Here we show how to generalize the EFT framework for DFT derived previously to reconcile these approaches. A dilute gas of fermions with short-range interactions confined by an external potential serves as a model system for comparisons and for testing power-counting estimates of new contributions to the energy functional.
The current generation of covariant mean-field models has had many successes in calculations of bulk observables for medium to heavy nuclei, but there remain many open questions. New challenges are confronted when trying to systematically extend these models to reliably address nuclear structure physics away from the line of stability. In this lecture, we discuss a framework for the next generation of relativistic models that can address these questions and challenges. We interpret nuclear mean-field approaches as approximate implementations of Kohn-Sham density functional theory (DFT), which is widely used in condensed matter and quantum chemistry applications. We look to effective field theory (EFT) for a systematic approach to low-energy nuclear physics that can provide the framework for nuclear DFT.
This contribution is an advertisement for applying effective field theory (EFT) to many-body problems, including nuclei and cold atomic gases. Examples involving three-body interactions are used to illustrate how EFT's quantify and systematically eliminate model dependence, and how they make many-body calculations simpler and more powerful.
Effective field theory (EFT) methods are applied to density functional theory (DFT) as part of a program to systematically go beyond mean-field approaches to medium and heavy nuclei. A system of fermions with short-range, natural interactions and an external confining potential (e.g., fermionic atoms in an optical trap) serves as a laboratory for studying DFT/EFT. An effective action formalism leads to a Kohn-Sham DFT by applying an inversion method order-by-order in the EFT expansion parameter. Representative results showing the convergence of Kohn-Sham calculations at zero temperature in the local density approximation (LDA) are compared to Thomas-Fermi calculations and to power-counting estimates.
A system of fermions with short-range interactions at finite density is studied using the framework of effective field theory. The effective action formalism for fermions with auxiliary fields leads to a loop expansion in which particle-hole bubbles are resummed to all orders. For spin-independent interactions, the loop expansion is equivalent to a systematic expansion in 1/N, where "N" is the spin-isospin degeneracy g. Numerical results at next-to-leading order are presented and the connection to the Bose limit of this system is elucidated.
In an effective hadronic theory constructed to describe long-range nuclear physics, the dynamics of the vacuum can be expanded in terms with zero or a finite number of derivatives acting on the fields. Thus vacuum dynamics can always be absorbed in the (infinite number of) counterterm parameters necessarily present in the effective lagrangian. These finite parameters, which at present must be fitted to data, encode the empirical vacuum physics as well as other short-range dynamics into the effective lagrangian; in practice, only a small number of parameters must be fitted. The strength of the effective field theory (EFT) framework is that there is no need to make a concrete picture of the vacuum dynamics, as one does in a renormalizable hadronic theory. At the one-loop level, the most convenient renormalization scheme requires explicit sums over long-range (``valence'') nucleon orbitals only, thus explaining the so-called ``no-sea approximation'' used in successful covariant mean-field theory (MFT) calculations of static ground states. When excited states are studied in the random-phase approximation (RPA), the same EFT scheme dictates the inclusion of both familiar particle-hole pairs and contributions that mix valence and negative-energy single-particle Dirac wave functions. The modern EFT strategy therefore justifies and explains the omission of some explicit contributions from the negative-energy Dirac sea of nucleons, as was done to maintain conservation laws in earlier pragmatic calculations of the nuclear linear response.