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Journal of Mathematical Physics

The short-range expansion for multiple well scattering theory

Helge Holden, Raphael J. Høegh-Krohn, M. Mebkhout

Published Jan 1, 1985• 5 citations
Publisher

Abstract

We expand the scattering amplitude and the scattering matrix around the zero-range limit, i.e., with point interactions, in the case when the potential has a finite number of wells. The lower-order terms are largely independent of the shape of the potentials.
DOI
10.1063/1.526802

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Abstract The textbook effective-range expansion of scattering theory is useful in the analysis of low-energy scattering phenomenology when the scattering length ∣ a ∣ is much larger than the range R of the scattering potential: ∣ a ∣ ≫ R . Nevertheless, the same has been used for systems where the scattering length is much smaller than the range of the potential. We suggest and numerically study an effective-range expansion for ∣ a ∣ ≪ R as well as improved two-parameter effective-range expansions for the cases ∣ a ∣ > R and ∣ a ∣ < R . The improved effective-range expansions for ∣ a ∣ > R and ∣ a ∣ < R reduce to the expansions for ∣ a ∣ / R ≫ 1 and ∣ a ∣ / R ≪ 1 , respectively, in appropriate limits.
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In this article, we develop an exact (nonadiabatic, nonperturbative) density matrix scattering theory for a two component quantum liquid which interacts or scatters off from a generic spin-dependent quantum potential. The generic spin dependent quantum potential [Eq. (1)] is a matrix potential, hence, adiabaticity criterion is ill-defined. Therefore the full matrix potential should be treated nonadiabatically. We succeed in doing so using the notion of vectorial matrices which allows us to obtain an exact analytical expression for the scattered density matrix (SDM), ϱsc[Eq. (30)]. We find that the number or charge density in scattered fluid, Tr (ϱsc), expressions in Eqs. (32) depends on nontrivial quantum interference coefficients, [Formula: see text], which arises due to quantum interference between spin-independent and spin-dependent scattering amplitudes and among spin-dependent scattering amplitudes. Further it is shown that Tr (ϱsc) can be expressed in a compact form [Eq. (39)] where the effect of quantum interference coefficients can be included using a vector Qαβ, which allows us to define a vector order parameterQ. Since the number density is obtained using an exact scattered density matrix, therefore, we do not need to prove that Q is non-zero. However, for sake of completeness, we make detailed mathematical analysis for the conditions under which the vector order parameterQ would be zero or nonzero. We find that in presence of spin-dependent interaction the vector order parameterQ is necessarily nonzero and is related to the commutator and anti-commutator of scattering matrix S with its dagger S†[Eq. (78)]. It is further shown that Q≠0, implies four physically equivalent conditions,i.e., spin-orbital entanglement is nonzero, non-Abelian scattering phase, i.e., matrices, scattering matrix is nonunitary and the broken time reversal symmetry for SDM. This also implies that quasi particle excitation are anyonic in nature, hence, charge fractionalization is a natural consequence. This aspect has also been discussed from the perspective of number or charge density conservation, which implies i.e., Tr (ϱ}sc) = Tr (ϱin). On the other hand Q = 0 turns out to be a mathematically forced unphysical solution in presence of spin-dependent potential or scattering which is equivalent to Abelian hydrodynamics, unitary scattering matrix, absence of spin-space entanglement and preserved time reversal symmetry. We have formulated the theory using mesoscopic language, specifically, we have considered two terminal systems connected to spin-dependent scattering region, which is equivalent to having two potential wells separated by a generic spin-dependent potential barrier. The formulation using mesoscopic language is practically useful because it leads directly to the measured quantities such as conductance and spin-polarization density in the leads, however, the presented formulation is not limited to the mesoscopic system only, its generality has been stressed at various places in this article.