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arXiv:2410.09730

In-depth Study of Spin- and Momentum-Dependent Interaction Potentials Between Two Spin-1/2 Fermions Mediated by Light Spin-0 Particles

Zhong, Yang, Zhi-Hui Guo, Haiqing Zhou

Published Oct 13, 2024• 0 citations
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Abstract

We present a calculation by including the relativistic and off-shell contributions to the interaction potentials between two spin-1/2 fermions mediated by the exchange of light spin-0 particles, in both momentum and coordinate spaces. Our calculation is based on the four-point Green function rather than the scattering amplitude. Among the sixteen potential components, eight that vanish in the non-relativistic limit are shown to acquire nonzero relativistic and off-shell corrections. In addition to providing relativistic and off-shell corrections to the operator basis commonly used in the literature, we introduce an alternative operator basis that facilitates the derivation of interaction potentials in coordinate space. Furthermore, we calculate both the long-range and short-range components of the potentials, which can be useful for future experimental analyses at both macroscopic and atomic scales.

DOI
10.48550/arxiv.2410.09730

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We present a calculation by including the relativistic and off-shell contributions to the interaction potentials between two spin-1/2 fermions mediated by the exchange of light spin-0 particles, in both momentum and coordinate spaces. Our calculation is based on the four-point Green function rather than the scattering amplitude. Among the sixteen potential components, eight that vanish in the non-relativistic limit are shown to acquire nonzero relativistic and off-shell corrections. In addition to providing relativistic and off-shell corrections to the operator basis commonly used in the literature, we introduce an alternative operator basis that facilitates the derivation of interaction potentials in coordinate space. Furthermore, we calculate both the long-range and short-range components of the potentials, which can be useful for future experimental analyses at both macroscopic and atomic scales.
This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift δ(k)=π/2δ(k)=π/2δ(k)=π/2 a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off ΛΛΛ such that δ(k)=0δ(k)=0δ(k)=0 for k>Λk>Λk>Λ. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass m∗=1m^{*}=1m∗=1. Using only the zero relative momentum component of this interaction the total energy is ξ=4/9ξ=4/9ξ=4/9 (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to ξ=0.24.ξ=0.24.ξ=0.24. This result is independent of density and of the cut-off ΛΛΛ if Λ>∼3kfΛ> \sim 3k_{f}Λ>∼3kf​. With m∗≠1m^{*}\neq 1m∗=1 there is however a strong dependence on this cut-off. Including hh-ladders estimates give ξ=0.4↔0.6ξ=0.4\leftrightarrow 0.6ξ=0.4↔0.6, but a reliable result would in this case require a Green's function calculation.
In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift π/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k')=-v(k)v(k') with v^{2}(k)=(4π)^{2}(Λ^{2}-k^{2})^{-1/2}, Λbeing the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave ξ=0.24 (in units of kinetic energy) independent of density and with Λ---->infinity. This value of ξis appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from ξ~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of ξ~0.4 ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.
We study spin-1/2 fermions in spin-dependent potentials under the spin-model approximation, in which interatomic collisions that change the total occupation of single-particle modes are ignored. The spin-model approximation maps the interacting fermion problem to an ensemble of lattice spin models in energy space, where spin-spin interactions are long range and spin anisotropic. We consider a Ramsey-type sequence in which quantum degenerate, initially spin-polarized fermions are rotated to a spin superposition, and the differences in potential for the two spin states induce sss-wave interactions. Applied to such a setup, we show that the spin-model approximation qualitatively captures the collective spin dynamics for weak interactions compared to the harmonic-oscillator frequency, and often captures quantitative features at timescales much longer than would be expected from perturbation theory. We explore corrections to the spin model, and the relative importance of corrections for both collective spin and fidelity metrics when realistic anharmonic potential corrections are taken into account, demonstrating that anharmonicity can significantly improve the fidelity of the spin-model approximation. Additionally, we present numerical techniques that are useful for analysis of spin models on an energy lattice, including enacting a change of single-particle basis on a many-body state as an effective time evolution, and fitting of spatially inhomogeneous long-range interactions with exponentials. This latter technique is useful for constructing matrix product operators for use in density-matrix renormalization group analyses, and may have broader applicability within the tensor network community.
New formulae for the 2(2s + 1)-component relativistic basis spinors useful for the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where s = 1/2, 3/2, 5/2, . . . . These spinors are reduced to independent sets of two-component spinors. The relations presented in this study can be useful in the linear combination of atomic orbital approximation for the solution of different problems arising in relativistic quantum mechanics when orthonormal basis sets of relativistic exponential-type spinor wave functions and Slater-type spinor orbitals in position, momentum and four-dimensional spaces are employed.
Starting from the non-relativistic field theory of spin- fermions interacting through the Abelian Chern-Simons term, we show that the quantized field theory leads, in the two-particle sector, to a two-particle Aharonov-Bohm-like Schrödinger equation with an antisymmetric (fermionic) wavefunction and without a delta function term. Calculating perturbatively the field-theoretic two-particle scattering amplitude up to one-loop order, we show that, in contrast to the scalar theory, the contribution of all the one-loop diagrams is finite and null, and that of the tree level ones coincides with the exact amplitude. Further, the Pauli matter-magnetic field interaction term is shown not to contribute to the amplitude to this order.
Four different spin-dependent scattering cross sections are calculated by semiclassical and by partial-wave analyses for each of the five alkali atoms, Li, Na, K, Rb, and Cs, for 124 different relative kinetic energies. These cross sections are finally averaged thermally. The results agree very well with experiments.In the calculation, the interaction potential between two alkali atoms is calculated by individually considering the interaction due to the two valence sss electrons and to the closed electronic shells of the two atoms. The two-sss-electron interaction is generated ad hoc phenomenologically from the known hydrogen-hydrogen interatomic potential and the closed-shell interaction from interatomic potentials of two inert atoms corresponding to the cores. The potentials so obtained are in agreement with those constructed from band spectra.