We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for $\uparrow$ and $\downarrow$ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for $\uparrow$ and $\downarrow$ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.

arXiv:2211.09765Nov 17, 20220 citations

Three-body contact for fermions. II. Non-degenerate limit

X. Leyronas, Félix Werner

A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact $C_3$ , which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute $C_3$ to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length $a$ . At $a=\infty$ , using a wavefunction approach, we obtain the analytical expression of $C_3$ , which has a remarkably slow $1/T^{0.22728}$ dependence on the temperature $T$ . In the Feynman diagram technique, which we use for $a<0$ , the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact $C_3$, which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute $C_3$ to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length $a$. At $a=\infty$, using a wavefunction approach, we obtain the analytical expression of $C_3$, which has a remarkably slow $1/T^{0.22728}$ dependence on the temperature $T$. In the Feynman diagram technique, which we use for $a<0$, the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.

arXiv:2608.15324Aug 15, 20260 citations

General relations for quantum gases in two and three dimensions. II. Bosons and mixtures

Félix Werner, Yvan Castin

We derive exact general relations between various observables for $N$ bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body $s$ -wave scattering length $a$ . Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by $a$ , but also by a three-body parameter ${R}_{t}$ . We then find additional relations which involve the derivative of the energy with respect to ${R}_{t}$ . In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.We derive exact general relations between various observables for $N$ bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body $s$-wave scattering length $a$. Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by $a$, but also by a three-body parameter ${R}_{t}$. We then find additional relations which involve the derivative of the energy with respect to ${R}_{t}$. In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.

arXiv:1210.1784Physical Review ANov 30, 2012147 citations

General relations for quantum gases in two and three dimensions: Two-component fermions

Félix Werner, Yvan Castin

We derive exact general relations between various observables for $N$ spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range ${r}_{e}$ in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading $1/{k}^{6}$ tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite- $1/a$ and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.We derive exact general relations between various observables for $N$ spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range ${r}_{e}$ in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading $1/{k}^{6}$ tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-$1/a$ and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.

arXiv:1204.3204Physical Review AJul 19, 2012201 citations

Three-body scattering area of identical bosons in two dimensions

Junjie Liang, Hongye Yu, Shina Tan

We study the wave function $φ^{(3)}$ of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length $a$ . We derive asymptotic expansions of $φ^{(3)}$ in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as $\ln^3(B/a)$ at large hyperradius $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$ . At order $B^{-2}\ln^{-3}(B/a)$ , we identify a three-body parameter $D$ with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, $D$ acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of $D$ in terms of $φ^{(3)}$ and the changes of the two-body and three-body potentials. We also study the effects of $D$ and $φ^{(3)}$ on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.We study the wave function $φ^{(3)}$ of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length $a$. We derive asymptotic expansions of $φ^{(3)}$ in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as $\ln^3(B/a)$ at large hyperradius $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$. At order $B^{-2}\ln^{-3}(B/a)$, we identify a three-body parameter $D$ with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, $D$ acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of $D$ in terms of $φ^{(3)}$ and the changes of the two-body and three-body potentials. We also study the effects of $D$ and $φ^{(3)}$ on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.

cond-mat.quant-gasCross-listedarXiv:2601.20509Jan 29, 20260 citations

Three-body scattering hypervolume of two-component fermions in three dimensions

Jiansen Zhang, Zipeng Wang, Shina Tan

We study the zero-energy collision of three fermions, two of which are in the spin-down ($\ensuremath{\downarrow}$) state and one of which is in the spin-up ($\ensuremath{\uparrow}$) state. Assuming that the two-body and three-body interactions have a finite range, we find a parameter $D$ called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\ensuremath{\uparrow}$ (spin-$\ensuremath{\downarrow}$) fermion and one pair, formed by the other two fermions, are far apart and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume $D$ appears in the coefficients of such expansions at the order of ${B}^{\ensuremath{-}5}$, where $B={[({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2})/2]}^{1/2}$ is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and ${s}_{1}, {s}_{2}, {s}_{3}$ are the sides of the triangle. We compute the $T$ -matrix element for three such fermions colliding at low energy in terms of $D$ in the absence of two-body interactions. When the interactions are weak, we calculate $D$ approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to $D$ and generalize this result to the many-fermion system. $D$ also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of $D$ and the density and temperature of the gas.We study the zero-energy collision of three fermions, two of which are in the spin-down ($\ensuremath{\downarrow}$) state and one of which is in the spin-up ($\ensuremath{\uparrow}$) state. Assuming that the two-body and three-body interactions have a finite range, we find a parameter $D$ called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\ensuremath{\uparrow}$ (spin-$\ensuremath{\downarrow}$) fermion and one pair, formed by the other two fermions, are far apart and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume $D$ appears in the coefficients of such expansions at the order of ${B}^{\ensuremath{-}5}$, where $B={[({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2})/2]}^{1/2}$ is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and ${s}_{1}, {s}_{2}, {s}_{3}$ are the sides of the triangle. We compute the $T$-matrix element for three such fermions colliding at low energy in terms of $D$ in the absence of two-body interactions. When the interactions are weak, we calculate $D$ approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to $D$ and generalize this result to the many-fermion system. $D$ also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of $D$ and the density and temperature of the gas.

arXiv:2501.05194Physical Review AMay 22, 20251 citation

\uparrow

and

\downarrow

particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.

A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact

C_3

, which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute

C_3

to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length

a

. At

a=\infty

, using a wavefunction approach, we obtain the analytical expression of

C_3

, which has a remarkably slow

1/T^{0.22728}

dependence on the temperature

T

. In the Feynman diagram technique, which we use for

a<0

, the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.

We derive exact general relations between various observables for

N

bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body

s

-wave scattering length

a

. Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by

a

, but also by a three-body parameter

{R}_{t}

. We then find additional relations which involve the derivative of the energy with respect to

{R}_{t}

. In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.

We derive exact general relations between various observables for

N

spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range

{r}_{e}

in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading

1/{k}^{6}

tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-

1/a

and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.

We study the wave function

φ^{(3)}

of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length

a

. We derive asymptotic expansions of

φ^{(3)}

in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as

\ln^3(B/a)

at large hyperradius

B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}

. At order

B^{-2}\ln^{-3}(B/a)

, we identify a three-body parameter

D

with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states,

D

acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of

D

in terms of

φ^{(3)}

and the changes of the two-body and three-body potentials. We also study the effects of

D

and

φ^{(3)}

on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.

D

called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\ensuremath{\uparrow}$ (spin-$\ensuremath{\downarrow}$) fermion and one pair, formed by the other two fermions, are far apart and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume

D

appears in the coefficients of such expansions at the order of ${B}^{\ensuremath{-}5}$, where

B={[({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2})/2]}^{1/2}

is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and

{s}_{1}, {s}_{2}, {s}_{3}

are the sides of the triangle. We compute the

T

-matrix element for three such fermions colliding at low energy in terms of

D

in the absence of two-body interactions. When the interactions are weak, we calculate

D

approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to

D

and generalize this result to the many-fermion system.

D

also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of

D

and the density and temperature of the gas.