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arXiv:2205.01018 · Physical Review A

Binding of heavy fermions by a single light atom in one dimension

Andrea Tononi, Jules Givois, Dmitry S. Petrov

Published Jul 11, 2022• 16 citations
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Abstract

We consider the problem of $N$ identical fermions interacting via a zero-range attractive potential with a lighter atom in one dimension. Using the few-body approach based on the Skorniakov and Ter-Martirosian equation, we determine the energies and the critical mass ratios for the emergence of the tetramer, pentamer, and hexamer. For large $N$, we solve the problem analytically by using the mean-field theory based on the Thomas-Fermi approximation. The system becomes bound when the heavy-to-light mass ratio exceeds a critical value, which grows as ${N}^{3}$ at large $N$. We also employ a more sophisticated Hartree-Fock approach, which turns out to be equivalent to the Thomas-Fermi approximation for determining the energies, but provides a better description of the microscopic structure of the clusters.

DOI
10.1103/physreva.106.l011302

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We consider the problem of NNN identical fermions interacting via a zero-range attractive potential with a lighter atom in one dimension. Using the few-body approach based on the Skorniakov and Ter-Martirosian equation, we determine the energies and the critical mass ratios for the emergence of the tetramer, pentamer, and hexamer. For large NNN, we solve the problem analytically by using the mean-field theory based on the Thomas-Fermi approximation. The system becomes bound when the heavy-to-light mass ratio exceeds a critical value, which grows as N3{N}^{3}N3 at large NNN. We also employ a more sophisticated Hartree-Fock approach, which turns out to be equivalent to the Thomas-Fermi approximation for determining the energies, but provides a better description of the microscopic structure of the clusters.
Heavy-light N+1N+1N+1 clusters of two-dimensional fermions
We study binding of N N identical heavy fermions by a light atom in two dimensions assuming zero-range attractive heavy-light interactions. By using the mean-field theory valid for large N N we show that the N+1 N + 1 cluster is bound when the mass ratio exceeds 1.074N^2 1.074 N 2 . The mean-field theory, being scale invariant in two dimensions, predicts only the shapes of the clusters leaving their sizes and energies undefined. By taking into account beyond-mean-field effects we find closed-form expressions for these quantities. We also discuss differences between the Thomas-Fermi and Hartree-Fock approaches for treating the heavy fermions.
We study the emergence of universal tetramer and pentamer bound states in the two-dimensional (N+1) system, which consists of N identical heavy fermions interacting with a light atom. We show that the critical heavy-light mass ratio to support a (3+1) tetramer below the trimer threshold is 3.38, and to support a (4+1) pentamer below the tetramer threshold is 5.14. While the ground state tetramer and pentamer are both with zero total angular momentum, they exhibit very different density distributions and correlations in momentum space, due to their distinct angular momentum decompositions in the dimer-fermion frame. These universal bound states can be accessible by a number of Fermi-Fermi mixtures now realized in cold atoms laboratories, which also suggest novel few-body correlations dominant in their corresponding many-body systems.
For sufficiently large mass ratios the attractive exchange force caused by a single light atom interacting with a few heavy identical fermions can overcome their Fermi degeneracy pressure and bind them into an N+1 N + 1 cluster. Here, by using a mean-field approach valid for large N N , we find that N+1 N + 1 clusters can attract each other and form a self-bound charge density wave, the properties of which we fully characterize. Our work shows that there are no fundamental obstacles for having self-bound states in fermionic mixtures with zero-range interactions.
We study a heavy-heavy-light three-body system confined to one space dimension. Both binding energies and corresponding wave functions are obtained for (i) the zero-range and (ii) two finite-range attractive heavy-light interaction potentials. In the case of the zero-range potential, we apply the method of Skorniakov and Ter-Martirosian to explore the accuracy of the Born-Oppenheimer approach. For the finite-range potentials, we solve the Schr\"odinger equation numerically using a pseudospectral method. We demonstrate that when the two-body ground-state energy approaches zero, the three-body bound states display a universal behavior, independent of the shape of the interaction potential.
We show that four heavy fermions interacting resonantly with a lighter atom (4+1 system) become Efimovian at mass ratio 13.279(2), which is smaller than the corresponding 2+1 and 3+1 thresholds. We thus predict the five-body Efimov effect for this system in the regime where any of its subsystem is non-Efimovian. For smaller mass ratios we show the existence and calculate the energy of a universal 4+1 pentamer state, which continues the series of the 2+1 trimer predicted by Kartavtsev and Malykh and 3+1 tetramer discovered by Blume. We also show that the effective-range correction for the light-heavy interaction has a strong effect on all these states and larger effective ranges increase their tendency to bind.
We study a three-body system, formed by a light particle and two identical heavy dipoles, in two dimensions in the Born-Oppenheimer approximation. We present the analytic light-particle wave function resulting from an attractive zero-range potential between the light and each of the heavy particles. It expresses the large-distance universal properties which must be reproduced by all realistic short-range interactions. We calculate the three-body spectrum for zero heavy-heavy interaction as a function of light to heavy mass ratio. We discuss the relatively small deviations from Coulomb estimates and the degeneracies related to radial nodes and angular momentum quantum numbers. We include a repulsive dipole-dipole interaction and investigate the three-body solutions as functions of strength and dipole direction. Avoided crossings occur between levels localized in the emerging small and large-distance minima, respectively. The characteristic exchange of properties such as mean-square radii are calculated. Simulation of quantum information transfer is suggested. For large heavy-heavy-particle repulsion all bound states have disappeared into the continuum. The corresponding critical strength is inversely proportional to the square of the mass ratio, far from the linear dependence from the Landau criterion.