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Physical Review A

Mathematically rigorous formulation of the Fermi pseudopotential for higher-partial-wave scattering in arbitrary dimension

Florian Stampfer, Peter Wagner

Published May 28, 2010• 7 citations
Publisher

Abstract

The Huang-Yang multipolar pseudopotential [see Huang and Yang, Phys. Rev. 105, 767 (1957); later revised by Huang, Statistical Mechanics (Wiley, New York, 1963)] is derived in a mathematically rigorous way using distribution theory in arbitrary dimension. For higher partial waves (i.e., all except $s$ waves), the original pseudopotential contained wrong numerical factors in dimension 3, which were corrected only recently [see Derevianko, Phys. Rev. A 72, 044701 (2005); Stampfer and Wagner, J. Math. Anal. Appl. 342, 202 (2008)]. Here, we generalize the expression for the pseudopotential, in particular to even dimension, enabling the study of scattering processes in a large class of systems, especially in dimension 2.

DOI
10.1103/physreva.81.052710

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The Huang-Yang multipolar pseudopotential [see Huang and Yang, Phys. Rev. 105, 767 (1957); later revised by Huang, Statistical Mechanics (Wiley, New York, 1963)] is derived in a mathematically rigorous way using distribution theory in arbitrary dimension. For higher partial waves (i.e., all except sss waves), the original pseudopotential contained wrong numerical factors in dimension 3, which were corrected only recently [see Derevianko, Phys. Rev. A 72, 044701 (2005); Stampfer and Wagner, J. Math. Anal. Appl. 342, 202 (2008)]. Here, we generalize the expression for the pseudopotential, in particular to even dimension, enabling the study of scattering processes in a large class of systems, especially in dimension 2.
Abstract Introduced eighty years ago, the Fermi pseudopotential has been a powerful concept in multiple fields of physics. It replaces the detailed shape of a potential by a delta-function operator multiplied by a parameter giving the strength of the potential. For Cartesian dimensions d > 1, a regularization operator is necessary to remove singularities in the wave function. In this study, we develop a Fermi pseudopotential generalized to d dimensions (including non-integer) and to non-zero wavenumber, k . Our new fractional calculus approach has the advantage of circumventing singularities that occur in the wave function at integer values of d while being valid arbitrarily close to integer d . In the limit of integer dimension, we show that our generalized pseudopotential is equivalent to previously derived s -wave pseudopotentials. Our pseudopotential generalizes the operator to non-integer dimension, includes energy ( k ) dependence, and simplifies the dimension-dependent coupling constant expression derived from a Green’s function approach. We apply this pseudopotential to the problem of two cold atoms ( k → 0) in a harmonic trap and extend the energy expression to arbitrary dimension.
The long-standing question of finding the momentum representation for the sss-wave zero-range interaction in three spatial dimensions is here solved. This is done by expressing a certain distribution, introduced in a formal way in [S. Tan, Ann. Phys. (NY) 323, 2952 (2008)], explicitly. The resulting form of the Fourier-transformed pseudopotential remains very simple. Operator forms for the so-called Tan's selectors, which, together with Fermi-Huang pseudopotential, largely simplify the derivation of Tan's universal relations for the Fermi gas, are here derived and are also very simple. A momentum cutoff version of the pseudopotential is also provided, and with this no apparent contradiction to the notion of integrals in Tan's methods is left. The equivalence, even at the intermediate-step level, between the pseudopotential approach and momentum-space renormalization of the bare Dirac delta interaction is then shown by using the explicit form of the cutoff pseudopotential.
Motivated by the recent progress in cooling and trapping polar molecules, we present a simplified version of the rigorous contact pseudopotential for anisotropically-interacting polarized particles [A. Derevianko, Phys. Rev. A 67, 033607 (2003); Phys. Rev. A 72, 039901(E) (2005)]. The simplifications are carried out for a practically important case of harmonically confined particles described by sufficiently smooth wavefunctions. The resulting contact pseudo-potential depends on the K-matrix of the underlying scattering process and is represented as a sum over pairs of partial waves coupled by the collision. The contribution of each pair of the coupled waves (l and l') involves a tensor product of derivatives of orders l and l'. The asymmetry in the derivatives reflects the anisotropy of the original interaction potential: there is a preferential appearance of the derivatives along the polarizing field.
We derive an analytical expression for the scattering amplitude of two ultracold atoms of arbitrary spin and with general spin-orbit (SO) coupling, on the basis of our recent work [Phys. Rev. A 86, 053608 (2012)]. As an application, we demonstrate that SO coupling can induce scattering resonance in the case with finite scattering length. The same approach can be applied to calculate the two-body bound state of SO-coupled ultracold atoms. For the ultracold spin-1/21/21/2 Fermi gases in three- or two- dimensional systems with SO coupling, we also obtain the renormalization relation of effective contact interaction with momentum cutoff, as well as the applicability of Huang-Yang pseudopotential.