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arXiv:2507.11796 · Symmetry

Monopole and Seniority Truncations in the Large-Scale Configuration Interaction Shell Model Approach

Priyanka Choudhary, Chong Qi

Published Dec 19, 2024• 4 citations
arXivHTMLPDF

Abstract

This paper addresses the challenges of solving the quantum many-body problem, particularly within nuclear physics, through the configuration interaction (CI) method. Large-scale shell model calculations often become computationally infeasible for systems with a large number of valence particles, requiring truncation techniques. We propose truncation methods for the nuclear shell model, in which angular momentum is conserved and rotational symmetry is restored. We introduce the monopole-interaction-based truncation and seniority truncation strategies, designed to reduce the dimension of the calculations. These truncations can be established by considering certain partitions based on their importance and selecting physically meaningful states. We examine these truncations for Sn, Xe, and Pb isotopes, demonstrating their effectiveness in overcoming computational limits. These truncations work well for systems with either a single type of valence nucleon or with both types. With these truncations, we are able to achieve good convergence for the energy at a very small portion of the total dimension.

DOI
10.3390/sym16121685

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This paper addresses the challenges of solving the quantum many-body problem, particularly within nuclear physics, through the configuration interaction (CI) method. Large-scale shell model calculations often become computationally infeasible for systems with a large number of valence particles, requiring truncation techniques. We propose truncation methods for the nuclear shell model, in which angular momentum is conserved and rotational symmetry is restored. We introduce the monopole-interaction-based truncation and seniority truncation strategies, designed to reduce the dimension of the calculations. These truncations can be established by considering certain partitions based on their importance and selecting physically meaningful states. We examine these truncations for Sn, Xe, and Pb isotopes, demonstrating their effectiveness in overcoming computational limits. These truncations work well for systems with either a single type of valence nucleon or with both types. With these truncations, we are able to achieve good convergence for the energy at a very small portion of the total dimension.
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 nanbn_a n_bna​nb​ where nan_ana​ is the occupation of the single-particle orbital labeled by aaa. 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.
We summarize recent efforts to develop an angular-momentum-conserving variant of the Density Matrix Renormalization Group method into a practical truncation strategy for large-scale shell model calculations of atomic nuclei. Following a brief description of the key elements of the method, we report the results of test calculations for 48 Cr and 56 Ni . In both cases we consider nucleons limited to the 2p-1f shell and interacting via the KB3 interaction. Both calculations produce a high level of agreement with the exact shell-model results. Furthermore, and most importantly, the fraction of the complete space required to achieve this high level of agreement goes down rapidly as the size of the full space grows.
We propose an importance truncation scheme for no-core shell model or configuration interaction approaches, which enables converged calculations for nuclei well beyond the p shell. It is based on an a priori measure for the importance of individual basis states constructed by means of many-body perturbation theory. Only the physically relevant states of the no-core model space are considered, which leads to a dramatic reduction of the basis dimension. We analyze the validity and efficiency of this truncation scheme using different realistic nucleon-nucleon interactions and compare to conventional no-core shell model calculations for 4He and 16O. Then, we present first converged calculations for the ground state of 40Ca within no-core model spaces including up to 16 PlanckOmega excitations using realistic low-momentum interactions. The scheme is universal and can be easily applied to other quantum many-body problems.
The exact solution of quantum mechanical many-body problems is only possible for few particles. Therefore, numerical methods were developed in the fields of quantum physics and quantum chemistry for larger particle numbers. Configuration Interaction (CI) methods or the No-Core Shell Model (NCSM) allow ab initio calculations for light and intermediate-mass nuclei, without resorting to phenomenology. An extension of the NCSM is the Importance-Truncated No-Core Shell Model, which uses an a priori selection of the most important basis states. The importance truncation was first developed and applied in quantum chemistry in the 1970s and latter successfully applied to models of light and intermediate mass nuclei. Other numerical methods for calculations for ultra-cold fermionic many-body systems are the Fixed-Node Diffusion Monte Carlo method (FN-DMC) and the stochastic variational approach with Correlated Gaussian basis functions (CG). There are also such method as the Coupled-Cluster method, Green’s Function Monte Carlo (GFMC) method, et cetera, used for calculation of many-body systems. In this thesis, we adopt the IT-NCSM for the calculation of ultra-cold Fermi gases at unitarity. Ultracold gases are dilute, strongly correlated systems, in which the average interparticle distance is much larger than the range of the interaction. Therefore, the detailed radial dependence of the potential is not resolved, and the potential can be replaced by an effective contact interaction. At low energy, s-wave scattering dominates and the interaction can be described by the s-wave scattering length. If the scattering length is small and negative, Cooper-pairs are formed in the Bardeen-Cooper-Schrieffer (BCS) regime. If the scattering length is small and positive, these Cooper-pairs become strongly bound molecules in a Bose-Einstein-Condensate (BEC). In between (for large scattering lengths) is the unitary limit with universal properties. Calculations of the energy spectra (ground-state and first excited-state) have so far only been performed for up to five particles using CI or NCSM methods or for up to six particles using the CG method. Calculations with larger particle numbers have only been performed with Monte Carlo methods and only for the ground state of up to 30 particles. We extend ab initio calculations of the energy spectra of ultra-cold Fermi gases at unitarity for up to 20 particle using the IT-NCSM. For our calculations we use different interactions: an effective interaction introduced by Alhassid, Bertsch and Fang and an interaction constructed using an effective field theory (EFT) approach. Furthermore, we use a Gauss-shaped potential as it is also used for the calculations of ultra-cold Fermi gases at unitarity using CG or FN-DMC methods. Although more effort must be invested to make the Gauss-shaped potential suitable for IT-NCSM calculations of ultra-cold Fermi gases, calculations with the other two interactions yield ground-state energies which agree excellently with the results obtained using the FN-DMC method. The IT-NCSM extends the range of the NCSM and permits the calculation of energy spectra (ground-state and excited-state energies) of fermionic systems with large particle numbers for which previously only the ground states could be calculated using Monte Carlo methods.
We summarize our recent efforts to develop the Density Matrix Renormalization Group (DMRG) method into a practical truncation strategy for large-scale nuclear shell model calculations. Following an overview of the essential features of the DMRG, we discuss the changes we have implemented for its use in nuclei. In particular, we have found it useful to develop an angular-momentum conserving variant of the method (the JDMRG). We then summarize the principal results we have obtained to date, first reporting test results for 48^{48}48Cr and then more recent test results for 56^{56}56Ni. In both cases we consider nucleons limited to the 2p-1f shell. While both calculations produce a high level of agreement with the exact shell model results, the fraction of the complete space required to achieve this high level of agreement is found to go down rapidly as the size of the full space grows.
We study mode-entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority ν=0,  1,  2ν= 0,\;1,\;2ν=0,1,2, and ν=3ν=3ν=3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. We carry out numerical calculations for the lightest calcium isotopes and for 94Ru^{94}{\rm Ru}94Ru nucleus, and investigate the structure of their ground and low energy yrast states. We investigate the fulfillment of several predictions of the seniority model regarding the behavior of one-mode entropies, which we compare with the results of configuration interaction (CI) and density matrix renormalization group (DMRG) computations. For 94Ru^{94}{\rm Ru}94Ru, the seniority model accounts for the 0g9/20g_{9/2}0g9/2​ mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the 0f5/20f_{5/2}0f5/2​, 1p3/21p_{3/2}1p3/2​ and 1p1/21p_{1/2}1p1/2​ shells, and amount in finite mode entropies on these shells, too, clearly outside the scope of the simple (0g9/2)4(0g_{9/2})^4(0g9/2​)4 seniority model. The 0f7/20f_{7/2}0f7/2​ shell based seniority model is more accurate for the light Ca{\rm Ca}Ca isotopes, but seniority mixing is substantial for some 44Ca^{44}{\rm Ca}44Ca yrast states, too.