Searcharxiv⌕ Search

arXiv subjects

J. R. Shepard

Publications and source records attributed to J. R. Shepard.

At least 19 recordsLinked to original sources
arXiv2009

Separable Expansions of V_{low} for 2- and 3-Nucleon Systems

We present an alternative organizational scheme for developing effective theories of 2- and 3-body systems that is systematic, accurate, and efficient with controlled errors. To illustrate our approach we consider the bound state and scattering properties of the 2- and 3-nucleon systems. Our approach combines the computational benefits of using separable potentials with the improved convergence properties of potentials evolved with a renormalization group procedure. Long ago Harms showed that any potential can be expanded in a series of separable terms, but this fact is only useful if the expansion can be truncated at low order. The separable expansion provides an attractive organizational scheme that incorporates the two body bound state in the leading term while allowing for systematic corrections thereafter. We show that when applied to a renormalization group-evolved potential, the separable expansion converges rapidly, with accurate results for both 2- and 3-body scattering processes using only two separable terms.

nucl-th↗
arXiv2009

Scaling functions applied to three-body recombination of Cesium-133 atoms

We demonstrate the implications of Efimov physics in the recently measured recombination rate of Cesium-133 atoms. By employing previously calculated results for the energy dependence of the recombination rate of Helium-4 atoms, we obtain three independent scaling functions that are capable of describing the recombination rates over a large energy range for identical bosons with large scattering length. We benchmark these and previously obtained scaling functions by successfully comparing their predictions with full atom-dimer phase shift calculations with artificial Helium-4 potentials yielding large scattering lengths. Exploiting universality, we finally use these functions to determine the 3-body recombination rate of Cesium-133 atoms with large positive scattering length, compare our results to experimental data obtained by the Innsbruck group and find excellent agreement.

cond-mat.other↗
arXiv2009

Convergence of 2-Body Effective Range Expansions for $nd$ Quartet Scattering

We examine the convergence properties of the 2-nucleon Effective Range Expansion as used in Effective Theories (ET-ERE's) for 3-nucleon calculations. We accomplish this by accounting for the 2-body dynamics with a simple rank-1 separable 2-body potential where the finite range effects can be incorporated systematically in both the 2- and 3-body problems. We make our initial comparisons in the simple context of the $^3S_1$ 2-nucleon channel and the $^4S_{3/2}$ 3-nucleon channel. We find that convergence problems for some of the 3-nucleon scattering amplitudes using the ET-ERE can be traced to its poor account of finite range effects that soften the momentum dependence of the deuteron propagator in the Faddeev kernel. In contrast, our simple separable potential with dipole form factors works very well in all cases considered.

nucl-th↗
arXiv2007

New Calculations of Recombination Rates for Cold $^4$He Atoms and Determination of Universal Scaling Functions

Three-body recombination rates for cold $^4$He are calculated with a new method which exploits the simple relationship between the imaginary part of the atom-dimer elastic scattering phase shift and the $S$-matrix for recombination. The elastic phase shifts are computed above breakup threshold by solving a three-body Faddeev equation in momentum space with inputs based on a variety of modern atom-atom potentials. Recombination coefficients for the HFD-B3-FCII potential agree very well with the only previously published results. Since the elastic scattering and recombination processes for $^4$He are governed by "Efimov physics", they depend on universal functions of a scaling variable. The newly computed recombination coefficients for potentials other than HFD-B3-FCII make it possible to determine these universal functions for the first time.

cond-mat.other↗
arXiv2003

Low-energy operators in effective theories

Modern effective-theory techniques are applied to the nuclear many-body problem. A novel approach is proposed for the renormalization of operators in a manner consistent with the construction of the effective potential. To test this approach, a one-dimensional, yet realistic, nucleon-nucleon potential is introduced. An effective potential is then constructed by tuning its parameters to reproduce the exact effective-range expansion and a variety of bare operators are renormalized in a fashion compatible with this construction. Predictions for the expectation values of these effective operators in the ground state reproduce the results of the exact theory with remarkable accuracy (at the 0.5% level). This represents a marked improvement over a widely practiced approach that uses effective interactions but retains bare operators. Further, it is shown that this improvement is more impressive as the operator becomes more sensitive to the short-range structure of the potential. We illustrate the main ideas of this work using the elastic form factor of the deuteron as an example.

nucl-th↗
arXiv2001

Novel Methods for Determining Effective Interactions for the Nuclear Shell Model

The Contractor Renormalization (CORE) method is applied in combination with modern effective-theory techniques to the nuclear many-body problem. A one-dimensional--yet ``realistic''--nucleon-nucleon potential is introduced to test these novel ideas. It is found that the magnitude of ``model-space'' (CORE) corrections diminishes considerably when an effective potential that eliminates the hard-momentum components of the potential is first introduced. As a result, accurate predictions for the ground-state energy of the there-body system are made with relatively little computational effort when both techniques are used in a complementary fashion.

nucl-th↗
arXiv1999

Lambda-Sigma^0 mixing in finite nuclei

Properties of hypernuclei are studied in the context of a chiral Lagrangian which successfully describes ordinary nuclei. Lambda-Sigma^0 mixing arises from nondiagonal vertices in flavor space induced by the vector mesons and by the electromagnetic field. The set of Dirac equations for the coupled hyperon system is discussed. Results are presented for energy spectra and electromagnetic properties of hyperons in the nuclear environment. Simple estimates suggest that flavor mixing can lead to sizable changes in the lifetimes of Lambda hypernuclei.

nucl-th↗
arXiv1998

Mean-Field Theory for Spin Ladders Using Angular-Momentum Coupled Bases

We study properties of two-leg Heisenberg spin ladders in a mean-field approximation using a variety of angular-momentum coupled bases. The mean-field theory proposed by Gopalan, Rice, and Sigrist, which uses a rung basis, assumes that the mean-field ground state consists of a condensate of spin-singlets along the rungs of the ladder. We generalize this approach to larger angular-momentum coupled bases which incorporate---by their mere definition---a substantial fraction of the important short-range structure of these materials. In these bases the mean-field ground-state remains a condensate of spin singlet---but now with each involving a larger fraction of the spins in the ladder. As expected, the ``purity'' of the ground-state, as judged by the condensate fraction, increases with the size of the elementary block defining the basis. Moreover, the coupling to quasiparticle excitations becomes weaker as the size of the elementary block increases. Thus, the weak-coupling limit of the theory becomes an accurate representation of the underlying mean-field dynamics. We illustrate the method by computing static and dynamic properties of two-leg ladders in the various angular-momentum coupled bases.

cond-mat.str-el↗
arXiv1998

Perturbation Theory for Spin Ladders Using Angular-Momentum Coupled Bases

We compute bulk properties of Heisenberg spin-1/2 ladders using Rayleigh-Schrödinger perturbation theory in the rung and plaquette bases. We formulate a method to extract high-order perturbative coefficients in the bulk limit from solutions for relatively small finite clusters. For example, a perturbative calculation for an isotropic $2\times 12$ ladder yields an eleventh-order estimate of the ground-state energy per site that is within 0.02% of the density-matrix-renormalization-group (DMRG) value. Moreover, the method also enables a reliable estimate of the radius of convergence of the perturbative expansion. We find that for the rung basis the radius of convergence is $λ_c\simeq 0.8$, with $λ$ defining the ratio between the coupling along the chain relative to the coupling across the chain. In contrast, for the plaquette basis we estimate a radius of convergence of $λ_c\simeq 1.25$. Thus, we conclude that the plaquette basis offers the only currently available perturbative approach which can provide a reliable treatment of the physically interesting case of isotropic $(λ=1)$ spin ladders. We illustrate our methods by computing perturbative coefficients for the ground-state energy per site, the gap, and the one-magnon dispersion relation.

cond-mat.str-el↗
arXiv1997

Renormalization Group Flow Equations for the Sigma Model

We present a nonperturbative renormalization group solution of the Gell-Mann--Levy $σ$-model which was originally proposed as a phenomenological description of the dynamics of nucleons and mesons. In our version of the model the fermions are interpreted as quarks which interact via the $σ$ and $π$ mesons. We derive and numerically solve renormalization group (RG) flow equations to leading order in a derivative expansion to study the behavior of the model as it evolves from high to low momentum scales. We develop an expansion in chiral-symmetry-breaking which enables us to track this symmetry breaking with the evolution of the scale. We use infrared observables to constrain the phenomenology allowing predictions of other quantities such as $π-π$ scattering lengths. The results show improvement over the tree level calculation and are consistent with experiment and the results of alternate theoretical approaches such as chiral perturbation theory and lattice gauge theory.

hep-ph↗
arXiv1997

Dynamic Spin Response for Heisenberg Ladders

We employ the recently proposed plaquette basis to investigate static and dynamic properties of isotropic 2-leg Heisenberg spin ladders. Simple non-interacting multi-plaquette states provide a remarkably accurate picture of the energy/site and dynamic spin response of these systems. Insights afforded by this simple picture suggest a very efficient truncation scheme for more precise calculations. When the small truncation errors are accounted for using recently developed Contractor Renormalization techniques, very accurate results requiring a small fraction of the computational effort of exact calculations are obtained. These methods allow us to determine the energy/site, gap, and spin response of 2x16 ladders. The former two values are in good agreement with density matrix renormalization group results. The spin response calculations show that nearly all the strength is concentrated in the lowest triplet level and that coherent many-body effects enhance the response/site by nearly a factor of 1.6 over that found for 2x2 systems.

cond-mat.str-el↗
arXiv1996

A Plaquette Basis for the Study of Heisenberg Ladders

We employ a plaquette basis-generated by coupling the four spins in a $2\times2$ lattice to a well-defined total angular momentum-for the study of Heisenberg ladders with antiferromagnetic coupling. Matrix elements of the Hamiltonian in this basis are evaluated using standard techniques in angular-momentum (Racah) algebra. We show by exact diagonalization of small ($2\times4$ and $2\times6$) systems that in excess of 90% of the ground-state probability is contained in a very small number of basis states. These few basis states can be used to define a severely truncated basis which we use to approximate low-lying exact eigenstates. We show how, in this low-energy basis, the isotropic spin-1/2 Heisenberg ladder can be mapped onto an anisotropic spin-1 ladder for which the coupling along the rungs is much stronger than the coupling between the rungs. The mapping thereby generates two distinct energy scales which greatly facilitates understanding the dynamics of the original spin-1/2 ladder. Moreover, we use these insights to define an effective low-energy Hamiltonian in accordance to the newly developed COntractor REnormalization group (CORE) method. We show how a simple range-2 CORE approximation to the effective Hamiltonian to be used with our truncated basis reproduces the low-energy spectrum of the exact $2\times6$ theory at the $\alt 1%$ level.

cond-mat↗
arXiv1994

Monte Carlo and Renormalization Group Effective Potentials in Scalar Field Theories

We study constraint effective potentials for various strongly interacting $ϕ^4$ theories. Renormalization group (RG) equations for these quantities are discussed and a heuristic development of a commonly used RG approximation is presented which stresses the relationships among the loop expansion, the Schwinger-Dyson method and the renormalization group approach. We extend the standard RG treatment to account explicitly for finite lattice effects. Constraint effective potentials are then evaluated using Monte Carlo (MC) techniques and careful comparisons are made with RG calculations. Explicit treatment of finite lattice effects is found to be essential in achieving quantitative agreement with the MC effective potentials. Excellent agreement is demonstrated for $d=3$ and $d=4$, O(1) and O(2) cases in both symmetric and broken phases.

hep-lat↗
arXiv1994

Comparison of $K^+$ and $e^-$ Quasielastic Scattering

We formulate $K^+$-nucleus quasielastic scattering in a manner which closely parallels standard treatments of $e^-$-nucleus quasielastic scattering. For $K^+$ scattering, new responses involving scalar contributions appear in addition to the Coulomb (or longitudinal) and transverse $(e,e')$ responses which are of vector character. We compute these responses using both nuclear matter and finite nucleus versions of the Relativistic Hartree Approximation to Quantum Hadrodynamics including RPA correlations. Overall agreement with measured $(e,e')$ responses and new $K^+$ quasielastic scattering data for $^{40}$Ca at $|\qs|=500$ MeV/c is good. Strong RPA quenching is essential for agreement with the Coulomb response. This quenching is notably less for the $K^+$ cross section even though the new scalar contributions are even more strongly quenched than the vector contributions. We show that this ``differential quenching'' alters sensitive cancellations in the expression for the $K^+$ cross section so that it is reduced much less than the individual responses. We emphasize the role of the purely relativistic distinction between vector and scalar contributions in obtaining an accurate and consistent description of the $(e,e')$ and $K^+$ data within the framework of our nuclear structure model.

nucl-th↗
arXiv1993

K$^+$-nucleus quasielastic scattering

K$^+$--nucleus quasielastic cross sections measured for a laboratory kaon beam momentum of 705 MeV/$c$ are presented for 3--momentum transfers of 300 and 500 MeV/$c$. The measured differential cross sections for C, Ca and Pb at 500 MeV/$c$ are used to deduce the effective number of nucleons participating in the scattering, which are compared with estimates based on the eikonal approximation. The long mean free path expected for K$^+$ mesons in nuclei is found. Double differential cross sections for C and Ca are compared to relativistic nuclear structure calculations.

nucl-th↗
arXiv1992

Relativistic Nuclear Matter with Self-Consistent Correlation Energy

We study relativistic nuclear matter in the $σ- ω$ model including the ring-sum correlation energy. The model parameters are adjusted self-consistently to give the canonical saturation density and binding energy per nucleon with the ring energy included. Two models are considered, mean-field-theory where we neglect vacuum effects, and the relativistic Hartree approximation where such effects are included but in an approximate way. In both cases we find self-consistent solutions and present equations of state. In the mean-field case the ring energy completely dominates the attractive part of the energy density and the elegant saturation mechanism of the standard approach is lost, namely relativistic quenching of the scalar attraction. In the relativistic Hartree approach the vacuum effects are included in an approximate manner using vertex form factors with a cutoff of 1 - 2 GeV, the range expected from QCD. Due to the cutoff, the ring energy for this case is significantlysmaller, and we obtain self-consistent solutions which preserve the basic saturation mechanism of the standard relativistic approach.

nucl-th↗