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arXiv:1603.05878 · Physical Review A

Matching universal behavior with potential models

R. Álvarez-Rodríguez, Arnoldas Deltuva, M. Gattobigio, Alejandro Kievsky

Published Jun 3, 2016• 22 citations
arXivHTMLPDF

Abstract

Two-, three-, and four-boson systems are studied close to the unitary limit using potential models constructed to reproduce the minimal information given by the two-body scattering length $a$ and the two-body binding energy or virtual-state energy ${E}_{2}$. The particular path used to reach the unitary limit is given by varying the potential strength. In this way the energy spectrum in the three- and four-boson systems is computed. The lowest-energy states show finite-range effects absorbed in the construction of level functions that can be used to study real systems. Higher-energy levels are free from finite-range effects, therefore the corresponding level functions tend to the zero-range universal function. Using this property, a zero-range equation for the four-boson system is proposed and the four-boson universal function is computed.

DOI
10.1103/physreva.93.062701

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Two-, three-, and four-boson systems are studied close to the unitary limit using potential models constructed to reproduce the minimal information given by the two-body scattering length aaa and the two-body binding energy or virtual-state energy E2{E}_{2}E2​. The particular path used to reach the unitary limit is given by varying the potential strength. In this way the energy spectrum in the three- and four-boson systems is computed. The lowest-energy states show finite-range effects absorbed in the construction of level functions that can be used to study real systems. Higher-energy levels are free from finite-range effects, therefore the corresponding level functions tend to the zero-range universal function. Using this property, a zero-range equation for the four-boson system is proposed and the four-boson universal function is computed.
We present an analysis of the NNN-boson spectrum computed using a soft two-body potential, the strength of which has been varied in order to cover an extended range of positive and negative values of the two-body scattering length aaa close to the unitary limit. The spectrum shows a tree structure of two states, one shallow and one deep, attached to the ground state of the system with one less particle. It is governed by a unique universal function $\ensuremath{\Delta}(\ensuremath{\xi})$, already known in the case of three bosons. In the three-particle system the angle $\ensuremath{\xi}$, determined by the ratio of the two- and three-body binding energies ${E}_{3}/{E}_{2}={tan}^{2}\ensuremath{\xi}$, characterizes the discrete scale invariance of the system. Extending the definition of the angle to the NNN-body system as ${E}_{N}/{E}_{2}={tan}^{2}\ensuremath{\xi}$, we study the NNN-boson spectrum in terms of this variable. The analysis of the results, obtained for up to N=16N=16N=16 bosons, allows us to extract a general formula for the energy levels of the system close to the unitary limit. Interestingly, a linear dependence of the universal function as a function of NNN is observed at fixed values of aaa. We show that the finite-range nature of the calculations results in the range corrections that generate a shift of the linear relation between the scattering length aaa and a particular form of the universal function. We also comment on the limits of applicability of the universal relations.
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Three- and four-boson systems expanded around the unitarity limit: Application to 4^44He molecules
The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of 4^44He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of 4^44He atomic clusters is governed by only small deviations from discrete scale invariance.
The occurrence of a new limit cycle in few-body physics, expressing a universal scaling function relating the binding energies of two successive tetramer states, is revealed by considering a renormalized zero-range two-body interaction in bound state of four identical bosons. The tetramer energy spectrum is obtained by adding a boson to an Efimov bound state with energy B3 in the unitary limit (for zero two-body binding energy or infinite two-body scattering length). Each excited Nth tetramer energy B(N)4 is shown to slide along a scaling function as a short-range four-body scale is changed, emerging from the 3+1 threshold for a universal ratio B4(N)/B3 = 4.6, which does not depend on N. The new scale can also be revealed by a resonance in the atom-trimer recombination process.
We apply a functional renormalization group to systems of four bosonic atoms close to the unitary limit. We work with a local effective action that includes a dynamical trimer field, and we use this field to eliminate structures that do not correspond to the Faddeev-Yakubovsky equations. In the physical limit, we find three four-body bound states below the shallowest three-body state. The values of the scattering lengths at which two of these states become bound are in good agreement with exact solutions of the four-body equations and experimental observations. The third state is extremely shallow. During the evolution we find an infinite number of four-body states based on each three-body state which follow a double-exponential pattern in the running scale. None of the four-body states shows any evidence of dependence on a four-body parameter.
We calculate energy levels of two and three bosons trapped in a harmonic oscillator potential with oscillator length aosc{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}aosc​. The atoms are assumed to interact through a short-range potential with a scattering length aaa, and the short-distance behavior of the three-body wave function is characterized by a parameter $\ensuremath{\theta}$. For large positive a/aosca/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}a/aosc​, the energies of states that, in the absence of the trap, correspond to three free atoms approach values independent of aaa and $\ensuremath{\theta}$. For other states, the $\ensuremath{\theta}$ dependence of the energy is strong, but the energy is independent of aaa for $|a/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}|\ensuremath{\gg}1$.