Surveying the Multicomponent Scattering Matrix: Unitarity and Symmetries

Abstract

Multicomponent-multiband fluxes of spim-charge carriers, whose components propagate mixed and synchronously, with a priori nonzero incoming amplitudes, do not obey the standard unitarity condition on the scattering matrix for an arbitrary basis set. For such cases, we have derived a robust theoretical procedure, which is fundamental in quantum-transport problems for unitarity preservation and we have named after structured unitarity condition. Our approach deals with (N × N ) interacting components (for N ≥ 2), within the envelope function approximation (EFA), and yet the standard unitary properties of the (N = 1) scattering matrix are recovered. Rather arbitrary conditions to the basis-set and/or to the output scattering coefficients, are not longer required, if the eigen-functions are orthonormalized in both the configuration and the spinorial spaces. We expect the present model to be workable, for different kind of multiband-multicomponent physical systems described by Hermitian Hamiltonians within the EFA, with small transformations if any. We foretell the interplay for the state-vector transfer matrix, together with the large values of its condition number, as a novel complementary tools for a more accurate definition of the threshold for tunnelling channels in a scattering experiment. Contents I. Introductory outline II. Flux Tunneling III. Structured unitarity of S A. EFA general case: N ≥ 2 B. EFA particular case: N = 4 C. Reduction from the structured case to the generalized one. IV. Convergence from EFA to EMA: flux and unitarity A. Limit of uncoupled N -component flux 1. Flux Convergence 2. Reduction of the structured-unitarity: N ≥ 2 3. Reduction of the generalized-unitarity: N = 4 B. Completely Orthonormalized Basis V. Symmetry Relations VI. Tunneling amplitudes A. Probabilities Flux Conservation 1.