Monopole Interaction Truncation

• Monopole-interaction-based truncation is a shell-model technique that ranks configuration partitions by their mean-field monopole energy to identify the most relevant low-energy states.
• It decomposes the effective Hamiltonian into a monopole (mean-field) part and a multipole residual, ensuring angular momentum conservation while reducing computational complexity.
• Various schemes—sharp cutoff, distribution-type, and weight-based—optimize the retention of key configurations, demonstrating rapid convergence in large-scale nuclear calculations.

Monopole-interaction-based truncation is a configuration-interaction shell-model strategy in which the monopole part of the effective Hamiltonian is used to rank occupation-defined subspaces and to retain only the energetically favored sectors of Hilbert space. In the modern shell-model literature, the method rests on the decomposition $H_A=H_m+H_M$, where $H_m$ is the monopole, or mean-field, part and $H_M$ is the multipole residual interaction; low-lying states are then approximated by configurations with the lowest monopole energies or centroids, while angular momentum conservation and rotational symmetry are preserved by construction (Choudhary et al., 15 Jul 2025, Johnson et al., 5 Nov 2025).

1. Formal definition in the shell-model Hamiltonian

In the shell model with a frozen core, the effective Hamiltonian is written as

$$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$

with $\alpha$ labeling single-particle orbitals, $\epsilon_\alpha$ the single-particle energies, and $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$ the antisymmetrized two-body matrix elements. Following the standard decomposition used in shell-model monopole analyses, one writes

$H_A=H_m+H_M,$

where $H_m$ is the monopole Hamiltonian and $H_M$ the multipole Hamiltonian (Choudhary et al., 15 Jul 2025, Qi et al., 2012).

The monopole matrix elements are the angular-momentum-weighted averages of the diagonal two-body matrix elements: $H_m$0 The monopole expectation value in a state $H_m$1 is

$H_m$2

Because this term depends only on orbital occupancies and not on detailed angular-momentum coupling, it defines the effective mean field, controls shell evolution, and sets the gross ordering of configurations; the multipole part is then responsible for pairing, quadrupole correlations, seniority splittings, and other spectroscopic fine structure (Qi et al., 2012, Kaneko et al., 2015).

This separation is the formal basis of monopole-interaction-based truncation. If $H_m$3 dominates the smooth part of the spectrum, then configurations with low monopole energy are natural candidates for retention, while high-monopole-energy configurations can be discarded with comparatively small impact on low-lying states. This suggests a mean-field-guided importance ordering rather than an a priori $H_m$4 or particle-hole counting.

2. Partitions, centroids, and occupation-space ordering

The practical object in monopole truncation is the partition, or configuration, defined by a fixed set of orbital occupation numbers. For a partition $H_m$5, with occupations $H_m$6, the monopole energy is

$H_m$7

Within a given partition, all basis states have the same monopole energy, because the monopole term depends only on occupations, not on the detailed coupling of single-particle angular momenta (Choudhary et al., 15 Jul 2025).

The same logic appears in centroid-based formulations. If a configuration subspace is labeled by occupations $H_m$8, its dimension is

$H_m$9

and its energy centroid is

$H_M$0

where $H_M$1 is the monopole potential obtained from the angular-momentum-weighted average of the diagonal two-body matrix elements. These centroids are traces over occupation-defined subspaces and can be computed without constructing the many-body Hamiltonian matrix (Johnson et al., 5 Nov 2025).

This centroid formulation makes the ordering principle explicit. One ranks configurations by $H_M$2 or $H_M$3, identifies the minimum monopole energy,

$H_M$4

and then retains only the subset of partitions lying sufficiently close to it. The resulting truncation is not arbitrary: it is tied to the average interaction energy of occupation patterns. In this sense, monopole-interaction-based truncation is an occupation-space implementation of a mean-field energy hierarchy.

A related development is the TRACER code, which computes monopole potentials, configuration dimensions, and configuration centroids rapidly from shell-model interaction files, and then uses those centroids to optimize energy-based truncation schemes for codes such as BIGSTICK (Johnson et al., 5 Nov 2025).

3. Truncation schemes and algorithmic realizations

The simplest scheme is the sharp cutoff at the partition level. A partition is retained if

$H_M$5

All basis states belonging to the retained partitions are kept; all others are discarded. As $H_M$6 increases, the truncated space approaches the full space monotonically (Choudhary et al., 15 Jul 2025).

A second scheme is the distribution-type cutoff truncation. Instead of a hard boundary, one defines

$H_M$7

with $H_M$8 a smoothing parameter. For each partition, one generates a random number $H_M$9; if $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$0, that partition is retained. Low-monopole-energy partitions therefore have $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$1, whereas higher-energy partitions are sampled only with some probability (Choudhary et al., 15 Jul 2025).

A third variant applies the same distribution function inside each retained partition at the $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$2-scheme level. If a partition contains $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$3 many $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$4-scheme basis states, the allowed number is set to

$$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$5

Low-energy partitions keep nearly all their basis states, while high-energy partitions are heavily thinned. This is a more aggressive reduction that directly targets the projected basis (Choudhary et al., 15 Jul 2025).

A different but closely related construction is the Approximately Cutoff-Energy-dependent truncation. Here one seeks a weight-based truncation of the type

$$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$6

where the orbital weights $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$7 are integers chosen so that the weight-based truncated space approximates a centroid cutoff $$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$8. The optimization is driven by the fitness function

$$H_A=\sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha} a_{\alpha} + \frac{1}{4} \sum_{\alpha\beta\gamma\delta JT} \langle j_{\alpha} j_{\beta} | V | j_{\gamma} j_{\delta} \rangle_{JT} A^{\dagger}_{JT; j_{\alpha} j_{\beta}}A_{JT; j_{\delta} j_{\gamma}},$$9

with $\alpha$0 the enveloping configuration space and $\alpha$1 the subset satisfying the weight criterion. TRACER performs a Monte Carlo search over $\alpha$2 and scans $\alpha$3 to minimize $\alpha$4, then writes a new single-particle-space file with the optimized weights for use in BIGSTICK (Johnson et al., 5 Nov 2025).

In the KTH version of NuShellX, monopole truncation is implemented by generating partitions, computing $\alpha$5, selecting important partitions, constructing the truncated $\alpha$6-scheme basis, and then restoring good total $\alpha$7 by projection. Because truncation is imposed at the level of partitions and projected basis states rather than by arbitrary removal of $\alpha$8-scheme determinants, angular momentum conservation and rotational symmetry are preserved (Choudhary et al., 15 Jul 2025).

4. Physical basis: shell evolution, ESPEs, and configuration hierarchies

The physical justification for the method comes from the empirical hierarchy between monopole and multipole effects. In the tin calculations based on a CD-Bonn-derived interaction, the monopole contribution dominates the absolute scale of energies and their smooth evolution with neutron number, whereas the multipole contribution is relatively small and peaks near mid-shell. The same work shows that light tin isotopes are dominated by $\alpha$9-$\epsilon_\alpha$0 configurations, that the $\epsilon_\alpha$1 orbit becomes half-filled around $\epsilon_\alpha$2–$\epsilon_\alpha$3, corresponding to $\epsilon_\alpha$4–$\epsilon_\alpha$5Sn, and that a truncation to the $\epsilon_\alpha$6 space is natural near $\epsilon_\alpha$7, with explicit $\epsilon_\alpha$8 inclusion required toward mid-shell (Qi et al., 2012).

This occupation-driven hierarchy is often expressed through effective single-particle energies. Standard practice is to write

$\epsilon_\alpha$9

so that monopole filling shifts shell gaps and determines which orbitals are near the Fermi surface. Although several of the cited works use this logic without always printing the formula explicitly, the practical consequence is consistent: orbits with low $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$0 and sizable occupancy define the active space, while orbits remaining high in energy can be removed or treated approximately through effective monopole shifts (Qi et al., 2012, Kaneko et al., 2015).

The PMMU framework in the $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$1 shell makes the same point in a broader region. Its Hamiltonian,

$$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$2

combines single-particle energies, pairing-plus-multipole terms, and a monopole interaction obtained from the monopole-based universal force and empirical fits. In Cu isotopes, the proton $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$3 effective single-particle energy is pulled down relative to $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$4 as neutrons fill $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$5, and for $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$6, $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$7 becomes the lowest proton orbit. In Ge at $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$8, attractive $$\langle j_\alpha j_\beta|V|j_\gamma j_\delta\rangle_{JT}$$9 monopole terms increase the $H_A=H_m+H_M,$0–$H_A=H_m+H_M,$1 neutron gap and suppress $H_A=H_m+H_M,$2 excitations in the $H_A=H_m+H_M,$3 state, whereas yrast states require those excitations. This suggests a state-dependent truncation strategy in which monopole-favored intruders are retained for collective yrast structures but can be more strongly restricted for near-closed-shell excitations (Kaneko et al., 2015).

The same lesson appears in the Ni region through the monopole tensor interaction. In $H_A=H_m+H_M,$4Ni, the Monte Carlo shell model with the A3DA-m interaction places the prolate $H_A=H_m+H_M,$5 state at $H_A=H_m+H_M,$6 keV, with a collective $H_A=H_m+H_M,$7 state at about $H_A=H_m+H_M,$8 keV. The prolate minimum is driven by neutron $H_A=H_m+H_M,$9 and $H_m$0 occupancy together with correlated proton cross-shell excitations; when the proton-neutron monopole part is frozen, the prolate minimum nearly vanishes. A plausible implication is that monopole truncation is not equivalent to simply minimizing basis size: it must retain precisely those intruder occupations that receive large monopole-tensor energy gains (Mărginean et al., 2020).

Cross-shell applications based on $H_m$1 extend this logic to truncated multi-$H_m$2 spaces. In boron, carbon, nitrogen, and oxygen isotopes, a single Hamiltonian is used in $H_m$3, $H_m$4, and $H_m$5 spaces without deriving a separate effective interaction for each truncation, and the inclusion of $H_m$6 excitations is important for drip lines, spectra, and electromagnetic properties. In the PMMU program for the $H_m$7 and $H_m$8 regions, limited excitations out of deeply bound orbitals are introduced only where needed computationally, while the monopole sector preserves the shell evolution and effective single-particle-energy hierarchy that justifies the restricted valence spaces (Yuan et al., 2012, Kaneko et al., 2013).

5. Quantitative performance in Sn, Xe, and Pb

Recent large-scale tests establish the method as a practical truncation strategy rather than only a qualitative guide. In the Pb, Sn, and Xe studies, convergence is assessed by increasing $H_m$9 and monitoring energies, excitation energies, and selected $H_M$0 values (Choudhary et al., 15 Jul 2025).

System	Full-space scale	Representative truncated result
$H_M$1Pb	$H_M$2 dimension $H_M$3; full spaces up to $H_M$4	At $H_M$5 MeV, about $H_M$6 of the total basis gives $H_M$7 MeV versus full-space $H_M$8 MeV
$H_M$9Sn	large full space in the $H_m$00 space	At $H_m$01 MeV, about $H_m$02 of the full dimension gives ground-state energy $H_m$03 MeV versus full $H_m$04 MeV
$H_m$05Xe	ground-state $H_m$06-scheme dimension $H_m$07	At $H_m$08 MeV, about $H_m$09 of the model space gives $H_m$10, $H_m$11, $H_m$12 MeV
$H_m$13Xe	ground-state $H_m$14-scheme dimension $H_m$15	At $H_m$16 MeV, about $H_m$17 of the full dimension gives $H_m$18, $H_m$19, $H_m$20 MeV

For $H_m$21Pb, the distribution-type cutoff with $H_m$22 MeV converges faster than the sharp cutoff at low cutoff energies. At $H_m$23 MeV, the sharp cutoff gives a $H_m$24 energy of about $H_m$25 MeV, while the distribution-type cutoff gives a minimum $H_m$26 energy of about $H_m$27 MeV. The basis-level distribution scheme improves convergence further for excited states, and combining monopole and seniority truncation yields energies closer to the full-space result at a given basis fraction (Choudhary et al., 15 Jul 2025).

For $H_m$28Sn, the full-space calculation gives $H_m$29, smaller than the experimental $H_m$30, likely because of missing contributions from the $H_m$31 orbital below the $H_m$32 core. Monopole truncation converges rapidly to that full-space shell-model value, which indicates that the truncation error is smaller than the model-space error in this case (Choudhary et al., 15 Jul 2025).

For $H_m$33Xe, energies are essentially converged with less than $H_m$34 of the full $H_m$35-scheme basis, and the $H_m$36 value also stabilizes rapidly. For $H_m$37Xe, the excitation energies $H_m$38 and $H_m$39 converge faster than the absolute binding energy, and around $H_m$40 of the basis is sufficient for an essentially converged low-lying spectrum. These examples show that the method remains effective for systems with both protons and neutrons and for nuclei with enhanced collectivity (Choudhary et al., 15 Jul 2025).

6. Relation to seniority truncation, limitations, and current scope

Monopole truncation is frequently paired with seniority truncation, but the two are conceptually distinct. Monopole truncation ranks partitions by mean-field energy, whereas seniority truncation emphasizes the dominance of monopole pairing by favoring basis states with low quasi-seniority. In the recent Pb and Sn tests, adding seniority information on top of monopole selection improves convergence for ground and low-lying yrast states, although high-spin states require higher-seniority components and therefore converge more slowly under an overly restrictive seniority bias (Choudhary et al., 15 Jul 2025).

The method also has explicit limitations. Centroid-based ranking does not encode all multipole correlations. Collective intruder states can be built from configurations whose monopole centroids are not especially low, and group-theoretical truncations based on seniority, SU(3), or Sp(3,R) can capture collectivity more directly. The centroid paper notes that computing analogous centroids for such symmetry-based partitions is not currently practical; existing symmetry-guided schemes therefore rely on different selection criteria (Johnson et al., 5 Nov 2025).

Implementation details matter in mixed proton-neutron systems. In the current NuShellX realization, truncation is applied separately in the proton and neutron spaces, whereas KSHELL can use the full proton-neutron monopole Hamiltonian for truncation. The resulting differences in $H_m$41Xe and $H_m$42Xe show that monopole truncation is not purely abstract: the bookkeeping of partitions and the treatment of proton-neutron cross terms affect numerical performance (Choudhary et al., 15 Jul 2025).

At a broader methodological level, the PMMU and $H_m$43 programs suggest that monopole-interaction-based truncation is most effective when the monopole sector has already been calibrated to reproduce shell evolution and effective single-particle-energy trends. Realistic interactions often get the multipole part approximately right but require monopole corrections; once those corrections are imposed, the valence space becomes naturally stratified by monopole energy, and truncation by partitions or optimized weights becomes physically meaningful (Kaneko et al., 2015, Kaneko et al., 2013).

In current usage, monopole-interaction-based truncation is therefore best understood as a hierarchy-building tool. It defines which configurations are mean-field favored, which orbitals are active near the Fermi surface, and which sectors of a very large CI space should be retained exactly before more selective pairing-, seniority-, or collectivity-based refinements are applied.