The similarity renormalization group for three-body interactions in one dimension

Abstract

We report on recent progress of the implementation of the similarity renormalization group (SRG) for three-body interactions in a one-dimensional, bosonic model system using the plane wave basis. We discuss our implementation of the flow equations and show results that confirm that results in the three-body sector remain unchanged by the transformation of the Hamiltonian. We also show how the SRG transformation decouples low-from high-momentum nodes in the three-body sector and therefore simplifies the numerical calculation of observables.Let us denote with H s the transformed (and initially unknown) Hamiltonian where s denotes the so-called flow arXiv:1107.3064v1 [nucl-th]

Universality in similarity renormalization group evolved potential matrix elements andT-matrix equivalence

“…We can confirm the binding energies of Ref. [30] for different choices of σ i and c i , using b = 0.5f m and N max = 120. These results also employ a 3-body potential of the form, Table 5.1: 3-body ground-state energies in fm −2 for different parameters chosen in Ref.…”

“…The HO-basis also simplifies bound states calculations; simply choose your favorite eigenvalue routine and input the Hamiltonian matrix; this is true even for many-body bound states which require Fadeev formalism in a plane-wave basis [30]. An extremely difficult aspect of an A-body calculation is the presence of spectator delta-functions that arise from the spectator particle when embedding fewer-body potentials into the A-body space (we will discuss this further in section 4.1.3).…”

“…These results also employ a 3-body potential of the form, Table 5.1: 3-body ground-state energies in fm −2 for different parameters chosen in Ref. [30]. α v2 = 1 means that c 1 = 12. fm −1 , c 2 = −12.…”

“…We began this chapter confirming ground-state energies for a previously used 1-D potential for spinless bosons [30,31]. We introduced a 2-body potential more beneficial to our study because its long-range interaction is not as confined as a Gaussian, it requires smaller basis size for accurate calculations, and it is easier to fit with our δ-shell procedure.…”

Similarity renormalization group evolution of three-nucleon forces in a hyperspherical momentum representation

“…In contrast, three-nucleon forces (3NF) have primarily been studied in the HO basis [8][9][10][11][12] with only recent work in momentum representation [13,14]. We have developed an alternative momentum representation SRG evolution that exploits hyperspherical harmonics (HH) to build a hybrid method combining the relative strengths of the previous HO SRG implementations and the recent momentum representation SRG implementations.…”