We present a microscopic derivation of the generalized Boltzmann and Eilenberger equations in the presence of non-Abelian gauges for the case of a nonrelativistic disordered Fermi gas. A unified and symmetric treatment of the charge ͓U͑1͔͒ and spin ͓SU͑2͔͒ degrees of freedom is achieved. Within this framework, just as the U͑1͒ Lorentz force generates the Hall effect, so does its SU͑2͒ counterpart gives rise to the spin Hall effect. Considering elastic and spin-independent disorder we obtain diffusion equations for charge and spin densities and show how the interplay between an in-plane magnetic field and a time-dependent Rashba term generates in-plane charge currents.
In many cases of interest such as a multilayer mesoscopic structure (e.g. a superlattice) or the grain boundaries network in high-T c 's, one deals with the situation where electrons traverses many partially transparent interfaces without loosing coherence. Then, the consequent reflection/transmission events require a simultaneous consideration. Theoretically, even an isolated interface poses certain difficulties: Since abrupt changes violate the quasiclassical condition, the theory of superconductivity in terms of the quasiclassical matrix Green's functionĝ R [1] is invalid at interfaces. The interface is included via the boundary condition derived by Zaitsev [3] -a cubic matrix relation. In case of many interfaces, one comes to a system of nonlinear matrix equations solution of which is far not simple if possible. Moreover, some authors argue [2] that the normalization condition is violated in the multiple interface case so that the quasiclassical scheme fails. The purpose of this paper is to suggest a new method which allows one to study systems with multiple partially transparent interfaces SIS ′ or SIN in the framework of the quasiclassical theory of superconductivity. Motivated by the paramagnetic effect problem (see references in [6]), the response to the vector potential (the superfluid density) of a ISIS ′ I sandwich is calculated.A convenient starting point is the formulation [4] of the quasiclassical technique in terms of the 2point trajectory Green's functionĝ R (x 1 , x 2 ) where x 1,2 are the coordinates along the trajectory characterised its direction n and position R ⊥ n. The equation of motion readsThe key point is the understanding that in the presence of partially reflective interfaces a typical trajectory remains extended and simply connected. In other words, the trajectories typically do not have loops and are not split to interfering paths so that there is only one path connecting any two points on the trajectory. Then, the coordinate along the trajectory is well defined and the method of 2-point Green's function is applicable. Due to reflection/transmission at the interfaces, the typical path becomes tree-like with "knots", i.e. the points where ballistic pieces of the trajectory cross. In between knots, the quasiclassical equation Eq. (1) is valid, whereas the knots are included via the boundary matching condition.The latter is formulated for the "wave functions" (2) and normalized,φ − φ + = 1. A general interface is partially diffusive, mixing states within a continuum of momenta. As a model, one allows mixing a finite number of states, N incoming and N outgoing, at a knot. The knot value of the wave functions φ i (φ k ′ ) in the in-coming (outgoing) channels, i, k ′ = 1, . . . , N are related to each other by the unitary scattering matrix [5]The energy independent matrix S, the normal state property, is considered as a given input.In this scheme, one finds the "wave function" φ by solving Eq.(2) on the pieces of the trajectory in between the knots and uses Eq.(3) to tailor the pieces. Th...
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