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:1211.5557 · Ядерная физика

Recent Advances in Description of Few Two-Component Fermions

O. I. Kartavtsev, A. V. Malykh

Published Jan 1, 2014• 4 citations
arXivHTMLPDF

Abstract

Overview of the recent advances in description of the few two-component fermions is presented. The model of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass $m$ and a distinct particle of mass $m_1$: it turns out that two $L^P = 1^-$ three-body bound states emerge if mass ratio $m/m_1$ increases up to the critical value $μ_c \approx 13.607$, above which the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, low-dimensional results, the four-body energy spectrum, crossover of the energy spectra for $m/m_1$ near $μ_c $, and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.

DOI
10.7868/s0044002714030131

Related papers

Recent advances in description of few two-component fermions

O. I. Kartavtsev, A. V. Malykh

Overview of the recent advances in description of the few two-component fermions is presented. Themodel of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass m and a distinct particle of mass m 1: it turns out that two L P = 1− three-body bound states emerge if mass ratio m/m 1 increases up to the critical value µ c ≈ 13.607, abovewhich the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, lowdimensional results, the four-body energy spectrum, crossover of the energy spectra for m/m 1 near µ c , and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.

Physics of Atomic Nuclei•Apr 1, 2014•9 citations

Universal description of three two-component fermions

O. I. Kartavtsev, A. V. Malykh

A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m 1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval μ r < m / m 1 ≤ μ c ( μ r ≈ 8.619 and μ c ≈ 13.607 ) 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. The dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P = 1 − is calculated and analysed with the aid of a simple model.

arXiv:1512.06786•Europhysics Letters (EPL)•Sep 13, 2016•15 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

Low-energy three-body dynamics in binary quantum gases

O. I. Kartavtsev, A. V. Malykh

The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass $m$ and a particle of the mass $m_1$ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio $m/m_1$ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at $m/m_1 \\ge \\lambda_1 \\approx 8.17260$ and $m/m_1 \\ge \\lambda_2 \\approx 12.91743$, respectively. For $m/m_1 \\alt \\lambda_1$ and $m/m_1 \\alt \\lambda_2$, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.

•Oct 28, 2006•55 citations

Scaling and universality in two dimensions: three-body bound states with short-ranged interactions

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

The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.

arXiv:1107.3017•Journal of Physics B Atomic Molecular and Optical Physics•Sep 26, 2011•29 citations

Absence of a four-body Efimov effect in the 2+2 fermionic problem

Shimpei Endo, Yvan Castin

In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant $s$-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this $2+2$ fermionic system can exhibit (as the $3+1$ fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where $s$ is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of $s$ are such that $M(s)$ has a zero eigenvalue; here the operator $M(s)$, that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of $M(s)$, analytical and for an arbitrary imaginary $s$ for the continuous spectrum and numerical and limited to $s=0$ and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of $M(0)$ crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of $M(s)$ has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the $2+2$ fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-$1/2$ Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary $s$ axis of $s$ times the logarithmic derivative of the determinant of $M(s)$ summed over all angular momenta. The conjectured value is in contradiction with the experimental results.

arXiv:1507.05580•Physical Review A•Nov 24, 2015•30 citations
Overview of the recent advances in description of the few two-component fermions is presented. The model of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass mmm and a distinct particle of mass m1m_1m1​: it turns out that two LP=1−L^P = 1^-LP=1− three-body bound states emerge if mass ratio m/m1m/m_1m/m1​ increases up to the critical value μc≈13.607μ_c \approx 13.607μc​≈13.607, above which the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, low-dimensional results, the four-body energy spectrum, crossover of the energy spectra for m/m1m/m_1m/m1​ near μcμ_c μc​, and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.
Overview of the recent advances in description of the few two-component fermions is presented. Themodel of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass m and a distinct particle of mass m 1: it turns out that two L P = 1− three-body bound states emerge if mass ratio m/m 1 increases up to the critical value µ c ≈ 13.607, abovewhich the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, lowdimensional results, the four-body energy spectrum, crossover of the energy spectra for m/m 1 near µ c , and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.
A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m 1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval μ r < m / m 1 ≤ μ c ( μ r ≈ 8.619 and μ c ≈ 13.607 ) 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. The dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P = 1 − is calculated and analysed with the aid of a simple model.
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.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass mmm and a particle of the mass m1m_1m1​ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio m/m1m/m_1m/m1​ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m/m1gelambda1approx8.17260m/m_1 \\ge \\lambda_1 \\approx 8.17260m/m1​gelambda1​approx8.17260 and m/m1gelambda2approx12.91743m/m_1 \\ge \\lambda_2 \\approx 12.91743m/m1​gelambda2​approx12.91743, respectively. For m/m1altlambda1m/m_1 \\alt \\lambda_1m/m1​altlambda1​ and m/m1altlambda2m/m_1 \\alt \\lambda_2m/m1​altlambda2​, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.
The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.
In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant sss-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this 2+22+22+2 fermionic system can exhibit (as the 3+13+13+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where sss is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of sss are such that M(s)M(s)M(s) has a zero eigenvalue; here the operator M(s)M(s)M(s), that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of M(s)M(s)M(s), analytical and for an arbitrary imaginary sss for the continuous spectrum and numerical and limited to s=0s=0s=0 and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of M(0)M(0)M(0) crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of M(s)M(s)M(s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the 2+22+22+2 fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-1/21/21/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary sss axis of sss times the logarithmic derivative of the determinant of M(s)M(s)M(s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results.