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arXiv:1602.01571 · Journal of Physics A Mathematical and Theoretical

The existence of bound states in a system of three particles in an optical lattice

Saidakhmat Norzhigitovich Lakaev, Sh. S. Lakaev

Published Jul 21, 2017• 26 citations
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Abstract

Abstract We consider the hamiltonian H μ , μ ∈ R of a system of three-particles (two identical fermions and one different particle) moving on the lattice Z d , d = 1 , 2 interacting through repulsive ( μ > 0 ) or attractive ( μ < 0 ) zero-range pairwise potential μ v . We prove for any μ ≠ 0 the existence of bound states of the discrete three-particle Schrödinger operator H μ ( K ) , K ∈ T d being the three-particle quasi-momentum, associated to the hamiltonian H μ .

DOI
10.1088/1751-8121/aa7db8

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Abstract We consider the hamiltonian H μ , μ ∈ R of a system of three-particles (two identical fermions and one different particle) moving on the lattice Z d , d = 1 , 2 interacting through repulsive ( μ > 0 ) or attractive ( μ < 0 ) zero-range pairwise potential μ v . We prove for any μ ≠ 0 the existence of bound states of the discrete three-particle Schrödinger operator H μ ( K ) , K ∈ T d being the three-particle quasi-momentum, associated to the hamiltonian H μ .
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 prove the existence of two-and three-particle bound states of the Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians hμ\mathrm{h}_μhμ​ and Hμ\mathrm{H}_μHμ​ of a system of two and three identical bosons on the lattice Zd,d=1,2\Z^d, d=1,2Zd,d=1,2 interacting via pairwise zero-range attractive μ<0μ<0μ<0 or repulsive μ>0μ>0μ>0 potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.
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.