Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Three-Body Bound States on a Lattice

Daniel Charles Mattis, Sergey Rudin

Published Feb 27, 1984• 21 citations
Publisher

Abstract

The theory of three-body bound states for particles moving on a lattice and interacting with attractive two-body pointlike potentials is presented. The applications are to bosons, fermions (no three-body bound states are found), and magnons. When a three-body bound state forms in three dimensions, it does so discontinuously. Thus there is a maximum size for the three-body bound state, of approximately two lattice constants. Some of the various analyses are relevant to magnetism and superconductivity.

DOI
10.1103/physrevlett.52.755

Related papers

The few-body problem on a lattice

Daniel Charles Mattis

The author explores some of the inherent simplifications of "quantum lattice physics." He distinguishes between fermions and bosons and analyzes the $n$-body problem for each, with $n=1,2,3\dots{}$ typically a small number. With delta-function (zero-range) interactions, the three-body problem on a lattice is manageable, and some results can even be extrapolated to $n\ensuremath{\ge}4$. Such calculations are not limited to one dimension (where the well-known Bethe ansatz solves a number of $n$-body problems). On the contrary, studies cited are mainly in three dimensions and actually simplify with increasing dimensionality. For example, it is found that bound states of $n\ensuremath{\ge}3$ particles in $d\ensuremath{\ge}3$ dimensions are formed discontinuously as the strength of two-body attractive forces is increased, and are therefore always in the easily analyzed "strong coupling limit." In the Appendix, an exactly solved example from the theory of itinerant-electron magnetism illustrates how a rigorous solution to the few-body problem is capable of yielding information concerning the $N$-body problem.

Reviews of Modern Physics•Apr 1, 1986•302 citations

Absence of binding of three fermions on a two-dimensional lattice

S. Rudin

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.

Physical Review A•May 1, 1985•6 citations

Universal bound states of one-dimensional bosons with two- and three-body attractions

Yusuke Nishida

When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.

arXiv:1803.06759•Physical Review A•Jun 14, 2018•41 citations

Three-boson bound states in two dimensions

Tianhao Ren, I. L. Aleǐner

The presence of few-body bound states is an important issue in the study of interacting bosonic systems due to the instability it introduces into the condensation. The problem is more interesting in two dimensions because the system can host only a finite number of bound states, in contrast to the Efimov effect in three dimensions. The authors considered a two-component bosonic system in two dimensions, with repulsive intra- and attractive inter-species interactions. They found that there is only one three-particle bound state and the binding energies for various scattering lengths fall on an universal curve, where the microscopic details are of no relevance. Their conclusion is pertinent for various systems such as excitons in quantum well structures and bosonic dipoles in bilayer geometries. They developed an efficient renormalization group type method, capable of handling both short- and long- range physics and free of numerical instabilities. This method also makes use of the Hopf coordinates and running basis to reduce the proliferation of hyperspherical harmonics appearing in alternative approaches, yielding significant saving in numerical implementations.

arXiv:1607.05135•Physical review. B./Physical review. B•Jan 3, 2017•4 citations

Universal trimers of one-dimensional bosons with two- and three-body attractions

Yusuke Nishida

When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions and predict the existence of two three-body bound states when both interactions are attractive. Their binding energies are universal functions of the two-body and three-body scattering lengths and have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.

arXiv:1803.06759?context=cond-mat.quant-gas•Mar 18, 2018•0 citations

Existence of three-particle bound states in optical lattice

Saidakhmat Norzhigitovich Lakaev, Sh. I. Khamidov, Собир Сахибжанович Улашов

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 $$\mu_1\in\mathbb{R}$$ , while the interaction between a fermion and one other particle is via an on-site potential with strength $${\mu_2\in\mathbb{R}}$$ . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum $$K\in\mathbb{T}^1$$ . Furthermore, we show that both the bound state $$f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)$$ and its corresponding eigenvalue $$E_{\mu_1\mu_2}(K)$$ depend holomorphically on the quasimomentum.

Theoretical and Mathematical Physics•Dec 1, 2025•3 citations
The theory of three-body bound states for particles moving on a lattice and interacting with attractive two-body pointlike potentials is presented. The applications are to bosons, fermions (no three-body bound states are found), and magnons. When a three-body bound state forms in three dimensions, it does so discontinuously. Thus there is a maximum size for the three-body bound state, of approximately two lattice constants. Some of the various analyses are relevant to magnetism and superconductivity.
The author explores some of the inherent simplifications of "quantum lattice physics." He distinguishes between fermions and bosons and analyzes the nnn-body problem for each, with n=1,2,3…n=1,2,3\dots{}n=1,2,3… typically a small number. With delta-function (zero-range) interactions, the three-body problem on a lattice is manageable, and some results can even be extrapolated to $n\ensuremath{\ge}4$. Such calculations are not limited to one dimension (where the well-known Bethe ansatz solves a number of nnn-body problems). On the contrary, studies cited are mainly in three dimensions and actually simplify with increasing dimensionality. For example, it is found that bound states of $n\ensuremath{\ge}3$ particles in $d\ensuremath{\ge}3$ dimensions are formed discontinuously as the strength of two-body attractive forces is increased, and are therefore always in the easily analyzed "strong coupling limit." In the Appendix, an exactly solved example from the theory of itinerant-electron magnetism illustrates how a rigorous solution to the few-body problem is capable of yielding information concerning the NNN-body problem.
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.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
The presence of few-body bound states is an important issue in the study of interacting bosonic systems due to the instability it introduces into the condensation. The problem is more interesting in two dimensions because the system can host only a finite number of bound states, in contrast to the Efimov effect in three dimensions. The authors considered a two-component bosonic system in two dimensions, with repulsive intra- and attractive inter-species interactions. They found that there is only one three-particle bound state and the binding energies for various scattering lengths fall on an universal curve, where the microscopic details are of no relevance. Their conclusion is pertinent for various systems such as excitons in quantum well structures and bosonic dipoles in bilayer geometries. They developed an efficient renormalization group type method, capable of handling both short- and long- range physics and free of numerical instabilities. This method also makes use of the Hopf coordinates and running basis to reduce the proliferation of hyperspherical harmonics appearing in alternative approaches, yielding significant saving in numerical implementations.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions and predict the existence of two three-body bound states when both interactions are attractive. Their binding energies are universal functions of the two-body and three-body scattering lengths and have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
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.