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arXiv:2006.09758 · Physical Review A

Few bosons to many bosons inside the unitary window: A transition between universal and nonuniversal behavior

Alejandro Kievsky, A. Polls, Bruno Juliá-Díaz, N. K. Timofeyuk, M. Gattobigio

Published Dec 17, 2020• 18 citations
arXivHTMLPDF

Abstract

Universal behavior in few-boson systems close to the unitary limit, where two bosons become unbound, has been intensively investigated in recent years both experimentally and theoretically. In this particular region, called the unitary window, details of the interparticle interactions are not important and observables, such as binding energies, can be characterized by a few parameters. With an increasing number of particles, the short-range repulsion, present in all atomic, molecular, and nuclear interactions, gradually induces deviations from the universal behavior. In the present paper, we discuss a simple way of incorporating nonuniversal behavior through one specific parameter which controls the smooth transition of the system from a universal to nonuniversal regime. Using a system of $N$ helium atoms as an example, we calculate their ground-state energies as trajectories within the unitary window and also show that the control parameters can be used to determine the energy per particle in homogeneous systems when $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physreva.102.063320

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Universal behavior in few-boson systems close to the unitary limit, where two bosons become unbound, has been intensively investigated in recent years both experimentally and theoretically. In this particular region, called the unitary window, details of the interparticle interactions are not important and observables, such as binding energies, can be characterized by a few parameters. With an increasing number of particles, the short-range repulsion, present in all atomic, molecular, and nuclear interactions, gradually induces deviations from the universal behavior. In the present paper, we discuss a simple way of incorporating nonuniversal behavior through one specific parameter which controls the smooth transition of the system from a universal to nonuniversal regime. Using a system of NNN helium atoms as an example, we calculate their ground-state energies as trajectories within the unitary window and also show that the control parameters can be used to determine the energy per particle in homogeneous systems when N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞.
Universal behavior has been found inside the window of Efimov physics for systems with N=4,5,6N=4,5,6N=4,5,6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\ensuremath{\le}6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having 1/21/21/2 spin-isospin symmetry. We perform a Gaussian characterization of the window using a Gaussian potential to define trajectories for low-energy quantities as binding energies and phase shifts. On top of this trajectories experimental values are placed or, when not available, quantities calculated using realistic potentials that are known to reproduce experimental values. The intention is to show that the Gaussian characterization of the window, thought as a contact interaction plus range corrections, captures the main low-energy properties of real systems as for example three helium atoms or three nucleons. The mapping of real systems on the Gaussian trajectories is taken as indication of universal behavior. The trajectories continuously link the physical points to the unitary limit allowing for the explanation of strong correlations between observables appearing in real systems and which are known to exist in that limit. In the present study we focus on low-energy bound, scattering, and virtual states.
Gaussian characterization of the unitary window for N=3N=3N=3: bound, scattering and virtual states
The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having 1/21/21/2 spin-isospin symmetry. We perform a gaussian characterization of the window using a gaussian potential to define trajectories for low-energy quantities as binding energies and phase shifts. On top of this trajectories experimental values are placed or, when not available, quantities calculated using realistic potentials that are known to reproduce experimental values. The intention is to show that the gaussian characterization of the window, thought as a contact interaction plus range corrections, captures the main low-energy properties of real systems as for example three helium atoms or three nucleons. The mapping of real systems on the gaussian trajectories is taken as indication of universal behavior. The trajectories continuously link the physical points to the unitary limit allowing for the explanation of strong correlations between observables appearing in real systems and which are known to exist in that limit. In the present study we focus on low-energy bound, scattering and virtual states.
Saturation properties are directly linked to the short-range scale of the two-body interaction of the particles. The case of helium is special; on one hand, the two-body potential has a strong repulsion at short distances. On the other hand, the extremely weak binding of the helium dimer locates this system very close to the unitary limit allowing for a description based on an effective theory. At leading order of this theory a two- and a three-body term appear, each one characterized by a low-energy constant. In a potential model this description corresponds to a soft potential model with a two-body term purely attractive plus a three-body term purely repulsive constructed to describe the dimer and trimer binding energies. Here we analyze the capability of this model to describe the saturation properties making a direct link between the low-energy scale and the short-range correlations. We will show that the energy per particle, EN/N{E}_{N}/NEN​/N, can be obtained with reasonable accuracy at leading order extending the validity of this approximation, characterizing universal behavior in few-boson systems close to the unitary limit, to the many-body system.
The description of unitary few-boson systems is conceptually simple: only one parameter – the three-body binding energy – is required to predict the binding energies of clusters with an arbitrary number of bosons. Whether this correlation between the three- and many-boson systems still holds for two species of bosons for which only the inter-species interaction is resonant depends on how many particles of each species are in the system. For few-body clusters with species A and B and a resonant AB interaction, it is known that the emergent AAB and ABB three-body scales are correlated to the ground-state binding energies of the AAAB and ABBB systems, respectively. We find that this link between three and four bodies is broken for the AABB tetramer whose binding energy is neither constrained by the AAB nor by the ABB trimer. From this de-correlation, we predict the existence of a scale unique to the AABB tetramer. In our explanation of this phenomenon, we understand the AABB and AAAB/ABBB tetramers as representatives of two different universal classes of N-body systems with distinct renormalization-group and discrete-scaling properties.
Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by continuous scale invariance, which is broken into discrete scale invariance when three or more particles come into play. This symmetry induces a universal behavior for different systems that is independent of the details of the underlying interaction and rooted in the smallness of the ratio ℓ/ a B ≪ 1, where the length a B is associated with the binding energy of the two-body system [Formula: see text], and ℓ is the natural length given by the interaction range. Efimov physics refers to this universal behavior, which is often hidden by the onset of system-specific nonuniversal effects. In this review, we identify universal properties by providing an explicit link of physical systems to their unitary limit, in which a B → ∞, and we show that nuclear systems belong to this class of universality.