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There the depression in the density of states was short ranged, and disappeared at energies a few level spacings above the Fermi level, because the system considered was not perfectly time-reversal symmetric. In contrast, here the density of states remains depressed through a range in energies of the order of the correlation energy, which is much larger than the level spacing, as explained in Sec. II
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We postpone a detailed discussion of the density of states and its relationship with reduced conductances to Ref. 16 (this reduction of the density of states is also obtained in the context of the Usadel equations, as mentioned, e.g., in Ref. 7). Note that it is much stronger (corresponding to a "leading-order" correction) than the reduction due to the "edge" of the spectrum, discussed, e.g., in A. Altland and M.R. Zirnbauer, Phys. Rev. Lett. 76, 3420 (1996). There the depression in the density of states was short ranged, and disappeared at energies a few level spacings above the Fermi level, because the system considered was not perfectly time-reversal symmetric. In contrast, here the density of states remains depressed through a range in energies of the order of the correlation energy, which is much larger than the level spacing, as explained in Sec. II.
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Although the majority of recent work uses the transmission matrix, there are also exceptions in which the scattering matrix was used directly, e.g., K. Frahm and J.-L. Pichard, J. Phys. (France) I 5, 847 (1995).
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this context, the semiclassical approximation should be understood as implying propagation along distinct classical orbits, with definite expressions for the contribution of each orbit (and an underlying chaotic classical dynamics for each specific realization of the impurity potential or the shape of the cavity). Unfortunately, confusion may arise because the word "semiclassical" is often used to imply other ℏ→0 approximations, e.g., those in which the impurity averaging has been performed before the ℏ→0 limit is taken
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See, e.g., R. Blumel and U. Smilansky, Phys. Rev. Lett. 60, 477 (1988). In this context, the semiclassical approximation should be understood as implying propagation along distinct classical orbits, with definite expressions for the contribution of each orbit (and an underlying chaotic classical dynamics for each specific realization of the impurity potential or the shape of the cavity). Unfortunately, confusion may arise because the word "semiclassical" is often used to imply other ℏ→0 approximations, e.g., those in which the impurity averaging has been performed before the ℏ→0 limit is taken.
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Note that, depending on the convention, an extra minus sign may appear in all of the Andreev reflection amplitudes; this is of no importance as interference can only occur between amplitudes differing by an even number of Andreev reflections
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For a detailed exposition, see K. Slevin, J.-L. Pichard and P.A. Mello, J. Phys. (France) I 6, 529 (1996). Note that, depending on the convention, an extra minus sign may appear in all of the Andreev reflection amplitudes; this is of no importance as interference can only occur between amplitudes differing by an even number of Andreev reflections.
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J. Phys. (France) I
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G. 't Hooft, Nucl. Phys. B72, 461 (1974); in The Large N Expansion in Quantum Field Theory and Statistical Physics: From spin systems to 2-dimensional gravity, edited by E. Brezin and S. R. Wadia (World Scientific, Singapore, 1993).
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Note that here the times considered, or the lengths of the corresponding orbits, are of the order of the escape time (Formula presented) and much shorter than the Heisenberg time ℏ/Δ. The fact that Δ, the single-particle level spacing in the cavity, is much smaller than ℏ/(Formula presented), i.e., that the cavity has a continuous rather than a discrete spectrum, follows from the fact that the total number of modes, N, is large
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See, e.g., M.V. Berry, Proc. R. Soc. London, A 400, 229 (1985). Note that here the times considered, or the lengths of the corresponding orbits, are of the order of the escape time (Formula presented) and much shorter than the Heisenberg time ℏ/Δ. The fact that Δ, the single-particle level spacing in the cavity, is much smaller than ℏ/(Formula presented), i.e., that the cavity has a continuous rather than a discrete spectrum, follows from the fact that the total number of modes, N, is large.
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See also Ref. 26, and especially Chap. 8 in M.J. Creutz, Quarks, Gluons and Lattices (Cambridge University Press, Cambridge, England, 1983). Although our results are certainly contained in those of the large-N QCD literature, we are not aware of an analog in the literature to our simple expression for the couplings defined below, Eq. (17)
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S. Samuel, J. Math. Phys. 21, 2695 (1980); See also Ref. 26, and especially Chap. 8 in M.J. Creutz, Quarks, Gluons and Lattices (Cambridge University Press, Cambridge, England, 1983). Although our results are certainly contained in those of the large-N QCD literature, we are not aware of an analog in the literature to our simple expression for the couplings defined below, Eq. (17).
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Samuel, S.1
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See, e.g., N. Taniguchi, A. Hashimoto, B.D. Simons and B.L. Altshuler, Europhys. Lett. 27, 335 (1994).
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