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arXiv:2501.05194 · Physical Review A

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

Jiansen Zhang, Zipeng Wang, Shina Tan

Published May 22, 2025• 1 citation
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Abstract

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.

DOI
10.1103/physreva.111.053313

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We study the zero-energy collision of three fermions, two of which are in the spin-down (\ensuremath↓\ensuremath{\downarrow}\ensuremath↓) state and one of which is in the spin-up (\ensuremath↑\ensuremath{\uparrow}\ensuremath↑) state. Assuming that the two-body and three-body interactions have a finite range, we find a parameter DDD called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-\ensuremath↑\ensuremath{\uparrow}\ensuremath↑ (spin-\ensuremath↓\ensuremath{\downarrow}\ensuremath↓) 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 DDD appears in the coefficients of such expansions at the order of B\ensuremath−5{B}^{\ensuremath{-}5}B\ensuremath−5, where B=[(s12+s22+s32)/2]1/2B={[({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2})/2]}^{1/2}B=[(s12​+s22​+s32​)/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 s1,s2,s3{s}_{1}, {s}_{2}, {s}_{3}s1​,s2​,s3​ are the sides of the triangle. We compute the TTT-matrix element for three such fermions colliding at low energy in terms of DDD in the absence of two-body interactions. When the interactions are weak, we calculate DDD approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to DDD and generalize this result to the many-fermion system. DDD also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of DDD and the density and temperature of the gas.
We define the three-body scattering hypervolume DF{D}_{F}DF​ for identical spin-polarized fermions in two dimensions by considering the wave function of three such fermions colliding at zero energy and zero orbital angular momentum. We derive the asymptotic expansions of such a wave function when three fermions are far apart or one pair and the third fermion are far apart, and DF{D}_{F}DF​ appears in the coefficients of such expansions. For weak-interaction potentials, we derive an approximate formula of DF{D}_{F}DF​ by using the Born expansion. We then study the shift of energy of three such fermions in a large periodic area due to DF{D}_{F}DF​. This shift is proportional to DF{D}_{F}DF​ times the square of the area of the triangle formed by the momenta of the fermions. We also calculate the shifts of energy and of pressure of spin-polarized two-dimensional Fermi gases due to a nonzero DF{D}_{F}DF​ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases in two dimensions.
We study the zero-energy collision of three identical spin-polarized fermions with short-range interactions in one dimension. We derive the asymptotic expansions of the three-body wave function when the three fermions are far apart or one pair and the third fermion are far apart, and the three-body scattering hypervolume DF{D}_{F}DF​ appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, DF{D}_{F}DF​ acquires a negative imaginary part related to the amplitudes of the outgoing waves describing the departure of the resultant bound pair and the remaining free fermion. For weak interaction potentials, we derive an approximate formula of the hypervolume by using the Born expansion. For the square-barrier and the square-well potentials and the Gaussian potential, we solve the three-body Schr\"odinger equation to compute DF{D}_{F}DF​ numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero DF{D}_{F}DF​ and the three-body recombination rate in one dimension.
We analyze the collision of three particles with arbitrary mass ratio at zero collision energy, assuming arbitrary short-range potentials, and generalize the three-body scattering hypervolume DDD first defined for identical bosons in 2008. We solve the three-body Schr\"odinger equation asymptotically when the three particles are far apart or one pair and a third particle are far apart, deriving two asymptotic expansions of the wave function, and the parameter DDD appears at the order 1/B41/{B}^{4}1/B4, where BBB is the overall size of the triangle formed by the particles. We then analyze the ground state energy of three such particles with vanishing or negligible two-body scattering lengths in a large periodic volume of side length LLL, where the three-body parameter contributes a term of the order D/L6D/{L}^{6}D/L6. From this result we derive some properties of a two-component Bose gas with negligible two-body scattering lengths: its energy density at zero temperature, the corresponding generalized Gross-Pitaevskii equation, the conditions for the stability of the two-component mixture against collapse or phase separation, and the decay rates of particle densities due to three-body recombination.
We analyze the collision of three identical spin-polarized fermions at zero collision energy, assuming arbitrary finite-range potentials, and define the corresponding three-body scattering hypervolume DF{D}_{F}DF​. The scattering hypervolume DDD was first defined for identical bosons in 2008. It is the three-body analog of the two-body scattering length. We solve the three-body Schr\"odinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which DDD has the dimension of length to the fourth power, here the DF{D}_{F}DF​ we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\ensuremath{\hbar}{\mathbf{k}}_{1}$, $\ensuremath{\hbar}{\mathbf{k}}_{2}$, and $\ensuremath{\hbar}{\mathbf{k}}_{3}$ in a large periodic cubic box. The energy shift due to DF{D}_{F}DF​ is proportional to ${D}_{F}/{\mathrm{\ensuremath{\Omega}}}^{2}$, where $\mathrm{\ensuremath{\Omega}}$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero DF{D}_{F}DF​ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.
We study the zero-energy collision of three identical spin-polarized fermions with short-range interactions in one dimension. We derive the asymptotic expansions of the three-body wave function when the three fermions are far apart or one pair and the third fermion are far apart, and the three-body scattering hypervolume DFD_FDF​ appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, DFD_FDF​ acquires a negative imaginary part related to the amplitudes of the outgoing waves describing the departure of the resultant bound pair and the remaining free fermion. For weak interaction potentials, we derive an approximate formula of the hypervolume by using the Born expansion. For the square-barrier and the square-well potentials and the Gaussian potential, we solve the three-body Schrödinger equation to compute DFD_FDF​ numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero DFD_FDF​ and the three-body recombination rate in one dimension.
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