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Effective Mass and Machine Learning in Strongly Interacting Neutron Matter

Nawar Ismail

Published Sep 1, 2020• 0 citations
OpenAlex

Abstract

This thesis contains two topics related to strongly interacting neutron matter. The first considers the effective mass, described in Landau Fermi liquid theory, which aims to capture many-body physics in terms of a single parameter. Monte Carlo algorithms were used to determine energies for finite particle simulations. Studying the deviations from the thermodynamic limit for the non-interacting gas allows for claims about the macroscopic scale. Following this systematic investigation, the density dependence was determined to be less than one for the considered densities. The second topic focuses on machine learning algorithms which have proliferated into many fields, with an emerging popularity in nuclear physics. These algorithms work by learning patterns found in datasets to develop their predictive power. After identifying and resolving issues introduced by a small dataset, they were used to extrapolate finite calculations to zero effective-range in the thermodynamic limit, which best approximates neutron matter.

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This thesis contains two topics related to strongly interacting neutron matter. The first considers the effective mass, described in Landau Fermi liquid theory, which aims to capture many-body physics in terms of a single parameter. Monte Carlo algorithms were used to determine energies for finite particle simulations. Studying the deviations from the thermodynamic limit for the non-interacting gas allows for claims about the macroscopic scale. Following this systematic investigation, the density dependence was determined to be less than one for the considered densities. The second topic focuses on machine learning algorithms which have proliferated into many fields, with an emerging popularity in nuclear physics. These algorithms work by learning patterns found in datasets to develop their predictive power. After identifying and resolving issues introduced by a small dataset, they were used to extrapolate finite calculations to zero effective-range in the thermodynamic limit, which best approximates neutron matter.
We investigate the applicability of machine learning techniques in studying the finite-size effects associated with many-body physics. These techniques have an emerging presence in many-body theory as they have been used for interpolations, extrapolations, and in modeling wave functions. We will resolve several issues associated with machine learning and many-body calculations such as small datasets, outliers, and discontinuities, for the purpose of extrapolating finite calculations to macroscopic scales. We carry out a systematic investigation of two related systems by developing metrics that aim to avoid spurious effects and capture desired features. This work uses neural networks to extrapolate the unitary gas to the thermodynamic limit at zero range, which is otherwise difficult to reach. The effective mass of strongly interacting neutron matter is also studied and makes use of the noninteracting problem to resolve discontinuous predictions. For this investigation, we also carried out new auxiliary field diffusion Monte Carlo (AFDMC) calculations for a variety of densities and particle numbers. Ultimately, we demonstrate an effective utility for neural networks in this context.
We investigate the applicability of machine-learning techniques in studying the finite-size effects associated with many-body physics. These techniques have an emerging presence in many-body theory as they have been used for interpolations, extrapolations, and in modeling wavefunctions. We carry out a systematic investigation of two related systems by developing metrics that aim to avoid spurious effects and capture desired features. This work uses neural networks to extrapolate the Unitary Gas to the thermodynamic limit at zero-range, which is otherwise difficult to reach. The effective mass of strongly interacting neutron matter is also studied and makes use of the non-interacting problem to resolve discontinuous predictions. For this investigation, we also carried out new Auxiliary Field Diffusion Monte Carlo (AFDMC) calculations for a variety of densities and particle numbers. Ultimately, we demonstrate an effective utility for neural networks in this context.
We carry out nonperturbative calculations of the single-particle excitation spectrum in strongly interacting neutron matter. These are microscopic quantum Monte Carlo computations of many-neutron energies at different densities as well as several distinct excited states. As input, we employ both phenomenological and chiral two- and three-nucleon interactions. We use the single-particle spectrum to extract the effective mass in neutron matter. With a view to systematizing the error involved in this extraction, we carefully assess the impact of finite-size effects on the quasiparticle dispersion relation. We find an effective-mass ratio that drops from 1 as the density is increased. We conclude by connecting our results with the physics of ultracold gases as well as with energy-density functional theories of nuclei and neutron-star matter.
Uniform neutron matter is approximated by a cubical box containing a finite number of neutrons, with periodic boundary conditions. We report variational and Green's function Monte Carlo calculations of the ground state of fourteen neutrons in a periodic box using the Argonne ${v8}^{\ensuremath{'}}$ two-nucleon interaction at densities up to one and half times the nuclear matter density. The effects of the finite box size are estimated using variational wave functions together with cluster expansion and chain summation techniques. They are small at subnuclear densities. We discuss the expansion of the energy of low-density neutron gas in powers of its Fermi momentum. This expansion is strongly modified by the large nn\mathrm{nn}nn scattering length, and does not begin with the Fermi-gas kinetic energy, as assumed in both Skyrme and relativistic mean field theories. The leading term of neutron gas energy is approximately half the Fermi-gas kinetic energy. The quantum Monte Carlo results are also used to calibrate the accuracy of variational calculations employing Fermi hypernetted and single operator chain summation methods to study nucleon matter over a larger density range, with more realistic Hamiltonians including three-nucleon interactions.
In the first part of the thesis we consider the constraints of causality and unitarity for particles interacting via strictly finite-range interactions. We generalize Wigner's causality bound to the case of non-vanishing partial-wave mixing. Specifically we analyze the system of the low-energy interactions between protons and neutrons. We also analyze low-energy scattering for systems with arbitrary short-range interactions plus an attractive 1/rα1/r^α1/rα tail for α≥2α\geq2α≥2. In particular, we focus on the case of α=6α=6α=6 and we derive the constraints of causality and unitarity also for these systems and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive 1/r61/r^{6}1/r6 tail. We argue that a similar universality class exists for any attractive potential 1/rα1/r^α1/rα for α≥2α\geq2α≥2. In the second part of the thesis we present lattice Monte Carlo calculations of fermion-dimer scattering in the limit of zero-range interactions using the adiabatic projection method. The adiabatic projection method uses a set of initial cluster states and Euclidean time projection to give a systematically improvable description of the low-lying scattering cluster states in a finite volume. We use Lüscher's finite-volume relations to determine the sss-wave, ppp-wave, and ddd-wave phase shifts. For comparison, we also compute exact lattice results using Lanczos iteration and continuum results using the Skorniakov-Ter-Martirosian equation. For our Monte Carlo calculations we use a new lattice algorithm called impurity lattice Monte Carlo. This algorithm can be viewed as a hybrid technique which incorporates elements of both worldline and auxiliary-field Monte Carlo simulations.
Lattice field theory methods, usually associated with non-perturbative studies of quantum chromodynamics, are becoming increasingly common in the calculation of ground-state and thermal properties of strongly interacting non-relativistic few- and many-body systems, blurring the interfaces between condensed matter, atomic and low-energy nuclear physics. While some of these techniques have been in use in the area of condensed matter physics for a long time, others, such as hybrid Monte Carlo and improved effective actions, have only recently found their way across areas. With this topical review, we aim to provide a modest overview and a status update on a few notable recent developments. For the sake of brevity we focus on zero-temperature, non-relativistic problems. After a short introduction, we lay out some general considerations and proceed to discuss sampling algorithms, observables, and systematic effects. We show selected results on ground- and excited-state properties of fermions in the limit of unitarity. The appendix contains details on group theory on the lattice.