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.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.

arXiv:1211.5557Ядерная физикаJan 1, 20144 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.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:1512.06786Europhysics Letters (EPL)Sep 13, 201615 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 $b$ .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:1904.04943Apr 9, 20191 citation

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.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:1107.3017Journal of Physics B Atomic Molecular and Optical PhysicsSep 26, 201129 citations

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.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, 200655 citations

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 mass m and a particle of mass m 1 with the zero-range two-body interaction in states of 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 dependences 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 ⩾ λ 1 ≈ 8.172 60 and m / m 1 ⩾ λ 2 ≈ 12.917 43, respectively. For m / m 1 ≲ λ 1 and m / m 1 ≲ λ 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 dependences of the cross sections is connected with the rise of the bound states.The universal three-body dynamics in ultra-cold binary Fermi and Fermi–Bose mixtures is studied. Two identical fermions of mass m and a particle of mass m 1 with the zero-range two-body interaction in states of 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 dependences 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 ⩾ λ 1 ≈ 8.172 60 and m / m 1 ⩾ λ 2 ≈ 12.917 43, respectively. For m / m 1 ≲ λ 1 and m / m 1 ≲ λ 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 dependences of the cross sections is connected with the rise of the bound states.

arXiv:physics/0610261Journal of Physics B Atomic Molecular and Optical PhysicsMar 15, 2007125 citations

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.

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

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.