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

arXiv:2404.00275 · Few-Body Systems

Discrete Scale Invariance and U(2) Family of Two-Body Contact Interactions in One Dimension

Satoshi Ohya

Published Mar 29, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U (2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U (2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.

DOI
10.1007/s00601-024-01904-6

Related papers

Scale Invariance Breaking and Discrete Phase Invariance in Few-Body Problems

Satoshi Ohya

Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.

quant-pharXiv:2601.09266•Few-Body Systems•Jan 15, 2026•0 citations

Efimov effect for two particles on a semi-infinite line

Satoshi Ohya

The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.

arXiv:2201.10869•American Journal of Physics•Sep 22, 2022•2 citations

Discrete scale-invariant boson-fermion duality in one dimension

Satoshi Ohya

We introduce models of one-dimensional $n(\ensuremath{\ge}3)$-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr\"odinger equation exactly; we obtain the exact $n$-body bound-state spectrum as well as the exact $n$-body $S$-matrix elements for arbitrary $n\ensuremath{\ge}3$, all of which enjoy discrete scale invariance or log periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of $n=3$; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.

arXiv:2110.09723•Physical Review A•Mar 21, 2022•10 citations

Topologically nontrivial three-body contact interaction in one dimension

Satoshi Ohya

Abstract It is known that three-body contact interactions in one-dimensional (1D) n(≥3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/3!-parameter family of n-body Hamiltonians that corresponds to one particular 1D unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely thin magnetic fluxes in the n-body configuration space.

arXiv:2310.16576•Progress of Theoretical and Experimental Physics•Dec 7, 2023•2 citations

Contact parameters in two dimensions for general three-body systems

F F Bellotti, T. Frederico, M T Yamashita +3 more

We study the two-dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive the analytic relations between the three different Faddeev components for the three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to the examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all the subsystems are bound. As an example, we present the results for mixtures of lithium with two cesium or two potassium atoms, which are systems of current experimental interest.

arXiv:1307.6813•New Journal of Physics•Jan 30, 2014•5 citations

Three two-component fermions with contact interactions: correct formulation and energy spectrum

O. I. Kartavtsev, A. V. Malykh

Properties of two identical particles of mass $m$ and a distinct particle of mass $m_1$ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum $L$ and parity $P$. For the unambiguous formulation of the problem in the interval $μ_r(L^P) < m/m_1 \le μ_c(L^P)$ ($μ_r(1^-) \approx 8.619$ and $μ_c(1^-) \approx 13.607$, $μ_r(2^+) \approx 32.948$ and $μ_c(2^+) \approx 38.630$,~etc.) in each $L^P$ sector an additional parameter $b$ determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on $m/m_1$ and $b$ in the sector of angular momentum and parity $L^P$ is calculated for $L \le 5$ and analysed with the aid of a simple model. A number of the bound states for each $L^P$ sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters $m/m_1$ and $b$.

arXiv:1904.04943•Apr 9, 2019•1 citation
Abstract Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U (2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U (2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.
Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.
The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.
We introduce models of one-dimensional $n(\ensuremath{\ge}3)$-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr\"odinger equation exactly; we obtain the exact nnn-body bound-state spectrum as well as the exact nnn-body SSS-matrix elements for arbitrary $n\ensuremath{\ge}3$, all of which enjoy discrete scale invariance or log periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of n=3n=3n=3; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.
Abstract It is known that three-body contact interactions in one-dimensional (1D) n(≥3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/3!-parameter family of n-body Hamiltonians that corresponds to one particular 1D unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely thin magnetic fluxes in the n-body configuration space.
We study the two-dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive the analytic relations between the three different Faddeev components for the three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to the examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all the subsystems are bound. As an example, we present the results for mixtures of lithium with two cesium or two potassium atoms, which are systems of current experimental interest.
Properties of two identical particles of mass mmm and a distinct particle of mass m1m_1m1​ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum LLL and parity PPP. For the unambiguous formulation of the problem in the interval μr(LP)<m/m1≤μc(LP)μ_r(L^P) < m/m_1 \le μ_c(L^P)μr​(LP)<m/m1​≤μc​(LP) (μr(1−)≈8.619μ_r(1^-) \approx 8.619μr​(1−)≈8.619 and μc(1−)≈13.607μ_c(1^-) \approx 13.607μc​(1−)≈13.607, μr(2+)≈32.948μ_r(2^+) \approx 32.948μr​(2+)≈32.948 and μc(2+)≈38.630μ_c(2^+) \approx 38.630μc​(2+)≈38.630,~etc.) in each LPL^PLP sector an additional parameter bbb determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on m/m1m/m_1m/m1​ and bbb in the sector of angular momentum and parity LPL^PLP is calculated for L≤5L \le 5L≤5 and analysed with the aid of a simple model. A number of the bound states for each LPL^PLP sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters m/m1m/m_1m/m1​ and bbb.