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Physical Review A

Effective mass of three-body complexes for finite-range potentials on a lattice

S. Rudin

Published Feb 1, 1986• 6 citations
Publisher

Abstract

For the bound state of three identical lattice bosons interacting via attractive potentials (with strength U, range r, and repulsive core ${U}_{0}$) we investigate how its mass depends on U for large U. We find that the mobility of this bound complex depends strongly on r and that for a nearest-neighbor interaction (r=1) it may depend strongly on ${U}_{0}$.

DOI
10.1103/physreva.33.1402

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For the bound state of three identical lattice bosons interacting via attractive potentials (with strength U, range r, and repulsive core U0{U}_{0}U0​) we investigate how its mass depends on U for large U. We find that the mobility of this bound complex depends strongly on r and that for a nearest-neighbor interaction (r=1) it may depend strongly on U0{U}_{0}U0​.
We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength μ1∈R\mu_1\in\mathbb{R}μ1​∈R , while the interaction between a fermion and one other particle is via an on-site potential with strength μ2∈R{\mu_2\in\mathbb{R}}μ2​∈R . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum K∈T1K\in\mathbb{T}^1K∈T1 . Furthermore, we show that both the bound state fμ1μ2(K; ⋅ , ⋅ )f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)fμ1​μ2​​(K;⋅,⋅) and its corresponding eigenvalue Eμ1μ2(K)E_{\mu_1\mu_2}(K)Eμ1​μ2​​(K) depend holomorphically on the quasimomentum.
We investigate one-dimensional three-body systems composed of two identical bosons and one mass-imbalanced atom (impurity) with attractive two-body and three-body zero-range interactions. In the absence of three-body interaction, we give a complete phase diagram of the number of three-body bound states in the whole region of mass ratio and the ratio of intra- and intercomponent interaction strength via direct calculation of Skornyakov-Ter-Martirosyan equations. We demonstrate that other low-lying three-body bound states emerge when the mass of the impurity particle is different from other two identical particles. We obtain the binding energies together with the corresponding wave functions. When the mass of impurity atom is very large, there are at most three three-body bound states. In the presence of three-body zero-range interaction, we reveal that weak three-body interaction will not always induce one more three-body bound state. At some special parameter points, arbitrary small three-body interaction can generate one more three-body bound state. This corresponds to the transition of the number of three-body bound states induced only by two-body attractive interaction.
Three quantum particles with on-site repulsion and nearest-neighbour attraction on a one-dimensional lattice are considered. The three-body Schrödinger equation is reduced to a set of single-variable integral equations. Energies of three-particle bound complexes (trions) are found from self-consistency of the approximating matrix equation. In the case of spin- fermions, the ground-state trion energy, the excited-state energies, the trion spectra and stability regions are obtained for total spins S = 1/2 and S = 3/2. In the S = 1/2 sector, a narrow but finite parameter region is identified where the ground state consists of a stable fermion pair and an unbound fermion. Also presented is the reference case of spin-0 bosons.
We estimate the ratio R=g3/g2{R=g}_{3}{/g}_{2}R=g3​/g2​ of the critical coupling constants g2{g}_{2}g2​ and g3{g}_{3}g3​ that are required to achieve binding of two or three bosons, respectively, with a short-range interaction, and examine how this ratio depends on the shape of the potential. Simple monotonous potentials give $R\ensuremath{\simeq}0.8.$ A wide repulsive core pushes this ratio close to R=1.R=1.R=1. On the other hand, for an attractive well protected by an external repulsive barrier, the ratio approaches the rigorous lower bound R=2/3.R=2/3.R=2/3. We also present results for N=4N=4N=4 bosons, sketch the extension to N>4,N>4,N>4, and discuss various consequences.
Numerical calculations [D. Mattis and S. Rudin, Phys. Rev. Lett. 52, 755 (1984)] indicate the absence of bound states of three identical spin-(1/2) fermions moving on a two-dimensional lattice and interacting with attractive two-body zero-range potentials. Here, we give a proof of this result in the strong-coupling limit, and we also prove that, in the weak-coupling limit, the ground state of six fermions consists of three bound pairs. We discuss the relevance of this result to superconductivity.