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arXiv:1905.05275 · Physical Review C

Extracting free-space observables from trapped interacting clusters

Xilin Zhang

Published May 14, 2020• 24 citations
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Abstract

The energy spectrum of two short-range interacting particles in a harmonic potential trap has previously been related to free-space scattering phase shifts. But the existing formula for systems with a nonzero interaction range is exact only in the limit of an infinitely shallow trap. Here I provide a systematically improved formula---describing the low-energy dynamics---that enables the use of finite traps. This paves the way for extracting nuclear scattering phase shifts from ab initio nuclear many-body structure calculations, a long-sought goal in nuclear physics. The derivation establishes effective field theory as a powerful framework for studying the connection between structure information of a trapped system (with two or more subclusters) and continuum physics in the fields of both nuclear and condensed-matter physics.

DOI
10.1103/physrevc.101.051602

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The energy spectrum of two short-range interacting particles in a harmonic potential trap has previously been related to free-space scattering phase shifts. But the existing formula for systems with a nonzero interaction range is exact only in the limit of an infinitely shallow trap. Here I provide a systematically improved formula---describing the low-energy dynamics---that enables the use of finite traps. This paves the way for extracting nuclear scattering phase shifts from ab initio nuclear many-body structure calculations, a long-sought goal in nuclear physics. The derivation establishes effective field theory as a powerful framework for studying the connection between structure information of a trapped system (with two or more subclusters) and continuum physics in the fields of both nuclear and condensed-matter physics.
We study systems of few two-component fermions interacting via short-range interactions within a harmonic-oscillator trap. The dominant interactions, which are two-body interactions, are organized according to the number of derivatives and defined in a two-body truncated model space made from a bound-state basis. Leading-order (LO) interactions are solved for exactly using the formalism of the no-core shell model, whereas corrections are treated as many-body perturbations. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. We present results at unitarity for three- and four-fermion systems, which show excellent agreement with the exact solution (for the three-body problem) and results obtained by other methods (in the four-body case). We also present results for finite scattering lengths and nonzero range of the interaction, including (at positive scattering length) observation of a change in the structure of the three-body ground state and extraction of the atom-dimer scattering length.
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.
In this work we derive a systematic short-range expansion of the many-body wave function. At leading order, the wave function is factorized to a zero-energy s -wave correlated pair and spectator particles, while terms that include energy derivatives and larger orbital angular momentum two-body functions appear at subleading orders. The validity of the expansion is tested for the two-body case, as well as the many-body case, where infinite neutron matter is considered. An accurate and consistent description of both coordinate-space two-body densities and the one-body momentum distribution is obtained. These results show the possibility to utilize such an expansion for describing different observables in strongly-interacting many-body systems, including nuclear, atomic and condensed-matter systems. This work also opens the path for a systematic description of large momentum transfer reactions in nuclear systems sensitive to short-range correlations, provides a link between such experiments and low-energy nuclear physics, and motivates measurement of new observables in these experiments.
We address the issue of determining an effective two-body interaction for mean-field calculations of the energies of many-body systems. We show that the effective interaction is proportional to the phase shift and demonstrate this result in the quasiclassical approximation when there is a trapping potential in addition to the short-range interaction between a pair of particles. We calculate numerically energy levels for the case of an interaction with a short-range square well and a harmonic trapping potential and show that the numerical results agree well with the analytical expression. We derive a generalized Gross-Pitaevskii equation which includes effective range corrections and discuss the form of the electron-atom effective interaction to be used in calculations of Rydberg atoms and molecules.
In present work, a relation that connects the integrated correlation function of a trapped two-particle system to infinite volume particles scattering phase shift is derived. It has the potential to provide an alternative approach for extracting two-particle scattering phase shift from integrated correlation function in lattice simulation at small Euclidean time region. Both (i) perturbation calculation of (1+11+11+1)-dimensional lattice Euclidean field theory model of fermions interacting with a contact interaction and (ii) Monte Carlo simulation of a 1D exactly solvable quantum mechanics model are carried out to test the proposed relation. In contrast to conventional two-step approach of extracting energy levels from temporal correlation function in lattice simulation at large Euclidean time first and then applying L\"uscher formula to convert energy levels into scattering phase shifts, we show that the difference of integrated correlation functions between interacting and noninteracting trapped systems converges rapidly to infinite volume limit that is given in terms of scattering phase shifts at small Euclidean time region.
In present work, a relation that connects the integrated correlation function of a trapped two-particle system to infinite volume particles scattering phase shift is derived. It has the potential to provide an alternative approach for extracting two-particle scattering phase shift from integrated correlation function in lattice simulation at small Euclidean time region. Both (i) perturbation calculation of 1+1 dimensional lattice Euclidean field theory model of fermions interacting with a contact interaction and (ii) Monte Carlo simulation of a 1D exactly solvable quantum mechanics model are carried out to test the proposed relation. In contrast to conventional two-step approach of extracting energy levels from temporal correlation function in lattice simulation at large Euclidean time first and then applying Lüscher formula to convert energy levels into scattering phase shifts, we show that the difference of integrated correlation functions between interacting and noninteracting trapped systems converges rapidly to infinite volume limit that is given in terms of scattering phase shifts at small Euclidean time region.