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Centroids of nuclear shell-model Hamiltonians, with optimization of energy-based truncation schemes

Calvin W. Johnson, Austin Keller

TL;DR

This work addresses the exponential growth of configuration-interaction shell-model basis spaces by leveraging energy centroids computed from monopole components of the Hamiltonian to characterize subspaces and guide truncations. It introduces TRACER, a Fortran90 tool that rapidly evaluates subspace centroids and implements an ACE truncation by Monte Carlo optimization of orbital weights to approximate a centroid-energy cutoff. The approach enables principled, efficient truncations that better capture low-energy configurations than traditional methods, demonstrated through examples such as $^{40}$Ar and $^{60}$Fe and accompanied by practical guidance for implementation and use with BIGSTICK-compatible inputs. The work thus provides a scalable pathway to more accurate, truncated CI calculations with potential extensions to include broader truncation schemes and future three-body forces in no-core contexts.

Abstract

The configuration-interaction shell model is an effective and widely-used approach to the nuclear many-body problem, whose main drawback is the exponential growth of the basis dimension. An useful way to character nuclear shell-model Hamiltonians is through traces, including traces in subspaces defined by orbital occupations. Such traces, or energy centroids, can be easily and efficiently computed through the monopole components of the nuclear interaction, that is, terms that go like $n_a n_b$ where $n_a$ is the occupation of the single-particle orbital labeled by $a$. These calculations can be carried out very quickly for both empirical (valence space) and no-core shell model spaces and interactions. In fact, they can be carried out so fast, one can use this to optimize an efficient, if approximate, many-body truncation scheme used in available nuclear shell-model codes such as BIGSTICK. To carry out both the traces and the optimization, we present the TRACER code, written in Fortran90 and described and available here. We give example results as well as discuss performance.

Centroids of nuclear shell-model Hamiltonians, with optimization of energy-based truncation schemes

TL;DR

This work addresses the exponential growth of configuration-interaction shell-model basis spaces by leveraging energy centroids computed from monopole components of the Hamiltonian to characterize subspaces and guide truncations. It introduces TRACER, a Fortran90 tool that rapidly evaluates subspace centroids and implements an ACE truncation by Monte Carlo optimization of orbital weights to approximate a centroid-energy cutoff. The approach enables principled, efficient truncations that better capture low-energy configurations than traditional methods, demonstrated through examples such as Ar and Fe and accompanied by practical guidance for implementation and use with BIGSTICK-compatible inputs. The work thus provides a scalable pathway to more accurate, truncated CI calculations with potential extensions to include broader truncation schemes and future three-body forces in no-core contexts.

Abstract

The configuration-interaction shell model is an effective and widely-used approach to the nuclear many-body problem, whose main drawback is the exponential growth of the basis dimension. An useful way to character nuclear shell-model Hamiltonians is through traces, including traces in subspaces defined by orbital occupations. Such traces, or energy centroids, can be easily and efficiently computed through the monopole components of the nuclear interaction, that is, terms that go like where is the occupation of the single-particle orbital labeled by . These calculations can be carried out very quickly for both empirical (valence space) and no-core shell model spaces and interactions. In fact, they can be carried out so fast, one can use this to optimize an efficient, if approximate, many-body truncation scheme used in available nuclear shell-model codes such as BIGSTICK. To carry out both the traces and the optimization, we present the TRACER code, written in Fortran90 and described and available here. We give example results as well as discuss performance.

Paper Structure

This paper contains 21 sections, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Nuclear shell-model Hamiltonians
  3. Monopole terms and traces
  4. Many-body truncations
  5. Approximately cutoff-energy dependent (ACE) truncation
  6. Compiling and running the code
  7. Example results: optimization of ACE truncations
  8. Conclusion
  9. Acknowledgements
  10. Format of input files
  11. Single-particle space
  12. How to handle 'different' proton-neutron spaces
  13. Interaction files
  14. Scaling and autoscaling
  15. Proton-neutron and other isospin-breaking formats
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Distribution of number of states (not levels) in configurations versus configuration centroids for $^{60}$Fe computed in the $1p0f$ with the GX1A interaction. Data binned in 1-MeV bins.
  • Figure 2: Average probabilities of configurations as a function of configuration energy centroid, for the ground state of $^{49}$Cr computed in the $1p0f$ shell using the GX1A interaction PhysRevC.65.061301PhysRevC.69.034335honma2005shell. The energy centroids, Eq. (\ref{['eq:centroid']}), are defined relative to the lowest centroid. Because each occupation configuration, e.g. $(0f_{7/2})^9$ contains multiple basis states, and we give only one probability per configuration, the probabilities here do not sum to one.
  • Figure 3: Distribution of number of states (not levels) in configurations versus configuration centroids for $^{40}$Ar computed in the $sd$-$pf$ space, for different orbital weights and truncations. FCI is the full space. The $N_\mathrm{max}$ denotes the number of particles out of the $sd$ shell into the $pf$ shell, while $W$ is the relative excitation using an optimized set of orbital weights. Data binned in 1-MeV bins.