Variational Theory and Parquet Diagrams for Nuclear Systems: A Comprehensive Study of Neutron Matter

TL;DR

This work presents a comprehensive framework that merges the Jastrow-Feenberg variational method with local parquet-diagram theory to address neutron matter with state-dependent nuclear interactions. By incorporating commutator (non-parquet) corrections and systematically summing rings, ladders, and exchange diagrams in the presence of operator-valued nucleon-nucleon forces, the authors obtain self-consistent estimates of energetics, structure, dynamic response, self-energy, and pairing. They demonstrate that non-parquet contributions can dominate short-range physics and that many-body correlations strongly suppress spin-orbit effects, altering $^3P_2$-$^3F_2$ and $^3P_0$ pairing behavior in neutron matter. The results highlight the importance of beyond-mean-field treatments for accurately predicting the equation of state, response functions, and superfluid gaps, with implications for neutron star physics and dense nuclear systems. Overall, the study provides a technically sophisticated, internally consistent bridge between variational wave functions and Green's-function based many-body theory for realistic nuclear interactions.

Abstract

To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow-Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, ``commutator diagrams'' can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow-Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called $v_8$ version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the $^3P_2$ and $^3P_2$-$^3F_2$ gaps. We also find pairing in $^3P_0$ states; the strength of the pairing gap depends sensitively on the potential model employed.

Figures (34)

• Figure 1: The figure shows the schematic equation of state of a self-bound Fermi liquid, such as nuclear matter or $^{3}$He, at zero temperature as described in the text. The left gray-shaded area is the density regime where no homogeneous system exists, and the right gray-shaded area is the density regime where nucleons might dissolve or where $^{3}$He becomes solid. The left scale and the red curve depict the energy per particle, the black curves and the right scale depict the isothermal speed of sound. From Ref. fullbcs.
• Figure 2: (color online) The left figure shows the central interactions in the spin-singlet (curves with markers) and the spin-triplet case for the above four potentials studied here. The right figure shows the tensor (curves with markers) and spin-orbit interactions for the same four interactions.
• Figure 3: The figure shows the diagrammatic representation of the lowest order exchange corrections $V_{\rm ee}(r)$ containing exactly one correlation line. We use the conventions of correlated wave functions Johnreview: Dots represent particle coordinates, black dots imply a density factor and the integration over the coordinate. Lines connecting represent correlations or interactions: For the interaction correction $V_{\rm ee}(r)$, the red wavy line is to be interpreted as the effective interaction $W(r_{ij})$. In the correlation correction $X_{\rm ee}(r)$, the wavy red line represents the function $\Gamma_{\rm dd}(r)$. The oriented solid lines represent exchange function $\ell(r_{ij}k_{\rm F})$, Eq. \ref{['eq:eldef']}.
• Figure 4: The figure shows the simplest second-order ladder diagrams including a "twisted chain" correction. The left diagram is the ordinary two-body ladder that is summed by the Bethe-Goldstone equation. The middle diagram is where one of the bare interactions is replaced by $\tilde{w}_I(q)$, and the right one is the simplest contribution to the totally irreducible interaction.
• Figure 5: (color online) Examples of the diagrams summed by the integral equation \ref{['eq:vi']}. The magenta wavy line represents the combination $\hat{v}(r)+\hat{w}_{\rm I}(r)$ and the blue line represents the induced interaction $\hat{w}_{\rm I}(r)$. The magenta rungs can all be summed to the $\hat{G}(r)$-matrix whereas the blue rungs sum to $\hat{G}_w(r)$