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arXiv:1804.09857

BCS Pairing Gap in the Infrared Limit of the Similarity Renormalization Group

S. Szpigel, V. S. Timóteo, E. Ruiz Arriola

Published Apr 26, 2018• 0 citations
arXivHTMLPDF

Abstract

Effective interactions have been used to compute the pairing gap for nuclear and neutron matter in several schemes. In this work we analyze the impact of phase-shift equivalent interactions within the BCS theory on the $^1S_0$-channel pairing gap for a translational invariant many-fermion system such as nuclear and neutron matter. We solve the BCS pairing gap equation on a finite momentum grid for a toy model separable Gaussian potential in the $^1S_0$-channel explicitly evolved through the Similarity Renormalization Group (SRG) transformation and show that in the on-shell and continuum limits the pairing gap vanishes. For finite size systems the momentum is quantized and the on-shell limit is realized for SRG cutoffs comparable to the momentum resolution. In this case the pairing gap can be computed directly from the scattering phase-shifts by an energy-shift formula. While the momentum grid is usually used as an auxiliary way of solving the BCS pairing gap equation, we show that it actually encodes some relevant physical information, suggesting that in fact finite grids may represent the finite size of the system.

DOI
10.48550/arxiv.1804.09857

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Effective interactions have been used to compute the pairing gap for nuclear and neutron matter in several schemes. In this work we analyze the impact of phase-shift equivalent interactions within the BCS theory on the 1S0^1S_01S0​-channel pairing gap for a translational invariant many-fermion system such as nuclear and neutron matter. We solve the BCS pairing gap equation on a finite momentum grid for a toy model separable Gaussian potential in the 1S0^1S_01S0​-channel explicitly evolved through the Similarity Renormalization Group (SRG) transformation and show that in the on-shell and continuum limits the pairing gap vanishes. For finite size systems the momentum is quantized and the on-shell limit is realized for SRG cutoffs comparable to the momentum resolution. In this case the pairing gap can be computed directly from the scattering phase-shifts by an energy-shift formula. While the momentum grid is usually used as an auxiliary way of solving the BCS pairing gap equation, we show that it actually encodes some relevant physical information, suggesting that in fact finite grids may represent the finite size of the system.
Renormalization group methods can be applied to the nuclear many-body problem using the approach proposed by Shankar. We start with the two-body low momentum interaction V_{low k} and use the RG flow from the particle-hole channels to calculate the full scattering amplitude in the vicinity of the Fermi surface. This is a new straightforward approach to the many-body problem which is applicable also to condensed matter systems without long-ranged interactions, such as liquid 3He. We derive the one-loop renormalization group equations for the quasiparticle interaction and scattering amplitude at zero temperature. The RG presents an elegant method to maintain all momentum scales and preserve the antisymmetry of the scattering amplitude. As a first application we solve the RG equations for neutron matter. The resulting quasiparticle interaction includes effects due to the polarization of the medium, the so called induced interaction of Babu and Brown. We present results for the Fermi liquid parameters, the equation of state of neutron matter and the 1S0 superfluid pairing gap.
In this paper we consider a two-component gas of fermions on the BCS side of the BCS-BEC crossover at zero temperature. We use a momentum dependent interaction that reproduces the s-wave scattering phase shifts of a contact interaction up to a momentum cutoff that is scaled with the Fermi momentum. Using a diagrammatic formulation of Bogoliubov many-body perturbation theory, suitably augmented by self-consistency conditions, we obtain the Hartree shift and the pairing gap to third order. In the weak-coupling regime, our results are not only well-converged but also agree with the well-established Gor'kov-Melik-Barkhudarov corrections for the gap and the Galitskii result for the Hartree shift. Near the unitary regime, our results for the Nambu-Gor'kov self-energy are less converged, but there is still reasonable agreement with experiments as well as with quantum Monte-Carlo results. Perspectives for improvements and applications of this approach to neutron matter are discussed.
We report ab initio calculations of the S wave pairing gap in neutron matter calculated using realistic nuclear Hamiltonians that include two- and three-body interactions. We use a trial state, properly optimized to capture the essential pairing correlations, from which we extract ground state properties by means of auxiliary field diffusion Monte Carlo simulations. We extrapolate our results to the thermodynamic limit by studying the finite-size effects in the symmetry-restored projected Bardeen-Cooper-Schrieffer (PBCS) theory and compare our results to other ab initio studies done in the past. Our quantum Monte Carlo results for the pairing gap show a modest suppression with respect to the mean-field BCS values. These results can be connected to cold atom experiments, via the unitarity regime where fermionic superfluidity assumes a unified description, and they are important in the prediction of thermal properties and the cooling of neutron stars.
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We discuss relations and differences between two methods for the construction of unitarily transformed effective interactions, the Similarity Renormalization Group (SRG) and Unitary Correlation Operator Method (UCOM). The aim of both methods is to construct a soft phase-shift equivalent effective interaction which is well suited for many-body calculations in limited model spaces. After contrasting the two conceptual frameworks, we establish a formal connection between the initial SRG-generator and the static generators of the UCOM transformation. Furthermore we propose a mapping procedure to extract UCOM correlation functions from the SRG evolution. We compare the effective interactions resulting from the UCOM-transformation and the SRG-evolution on the level of matrix elements, in no-core shell model calculations of light nuclei, and in Hartree-Fock calculations up to 208Pb^{208}\mathrm{Pb}208Pb. Both interactions exhibit very similar convergence properties in light nuclei but show a different systematic behavior as function of particle number.
The similarity renormalization group (SRG), based on the simple one-body free harmonic oscillator Hamiltonian, is applied to various nucleon-nucleon realistic interactions to investigate the unitarity of the SRG transformations. Two-body and three-body contributions to the SRG-evolved Hamiltonian are studied in the framework of spectral distribution theory for reasonable SRG cutoffs and in multiparticle systems, with up through 28 particles considered. The outcome points to the first evidence for the overall importance of three-body SRG-induced interactions and especially, of its two-body effective content in multinucleon systems, without the need for large-scale shell model calculations for many light to heavier nuclei.