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20
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84927320052
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An earlier attempt to apply semiclassical considerations to this problem is Y. Imry, in Quantum Coherence in Mesoscopic Systems, Vol. 254 of NATO Advanced Study Institute Series B: Physics, edited by B. Kramer (Plenum, New York, 1991), p. 221.
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22
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84927320051
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Note that the numerical coefficient in our Eq. (23) is larger by a factor of 3 than the result of this reference (the assumption of three dimensions is implicit there), due to the omission of ``two-ladder'' diagrams;
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29
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84927320049
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Note the error in the sign of Eq. (6) in the last reference;
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30
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84927320048
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the physical discussion following is based on the correct sign, however.
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46
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84927320044
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In this reference is suggested that tSCapp hbar-1/3 for the accuracy of the propagator, while in Ref. 23 is shown that, at least in some cases, tSC> tH for the accuracy of the spectral form factor.
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47
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84927320043
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It is worthwhile to note the expected magnitude of the fluctuation of K tilde ( E , t ) of a specific system from the ensemble-averaged value. The standard deviation ( langleK̃ >2 - langleK̃rangle2)1/2 may be estimated for t << tH, by assuming the absence of correlations between the actions in the semiclassical expression for K tilde. It depends on the amount of self-averaging included in K tilde, which is specified by the product Δ E Δ t, and turns out to be small if Δ E Δ t >> hbar. A simple demonstration can be given by use of the definitions in Ref. 27, when Δ E Δ t = hbar / 2. In this case the factor F ( t1, t2; t ) factorizes and K tilde ( E , t ) may be written as a product of an integral and its complex conjugate. This integral may be written semiclassically as a sum of terms which are assumed to be uncorrelated, and therefore its value is distributed according to a Gaussian distribution around the origin in the complex plane. This means that K tilde ( E , t ), which is proportional to the absolute value squared of the integral, will have a Poissonian distribution, with standard deviation equal to the mean. A similar type of argument also works for late times t >> tH, where the correlations between the energy eigenvalues may be neglected.
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48
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0002094837
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edited by, M. J. Giannoni, A. Voros, J. Zinn-Justin, North-Holland, Amsterdam
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(1991)
Chaos and Quantum Physics, Proceedings of the Les Houches Summer School in Theoretical Physics, Session LII, 1989
, pp. 251
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Berry, M.V.1
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53
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84927320041
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In order to obtain the exponentially small term within the semiclassical approach, one must keep track of the contributions of the few stable orbits which may still exist in a disordered sample.
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54
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84927320040
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A. Altland, S. Iida, A. Müller-Groeling, and H. A. Weidenmüller, Europhys. Lett. (to be published); Ann. Phys. (to be published).
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56
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84927320039
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see also O. Bohigas, in Chaos and Quantum Physics (Ref. 29), p. 87, Sec. 6, and references therein.
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58
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0001086836
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A detailed explanation of the relationship between the gauge and the boundary conditions may be found in Ref. 32.
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(1981)
JETP Lett.
, vol.33
, pp. 499
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59
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84927320038
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Note that although the averaging over the symmetry-breaking phases is identical here and in the case of weak localization, there are also some differences, e.g., the necessity for a special treatment of the end points of the paths (Appendix D of Ref. 11) as opposed to our case of periodic orbits without ends. Another important difference is in the way the spins are summed over.
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62
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84927320037
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Ya. G. Sinai, Proceedings of the Sixth International Conference on Mathematical Physics, Berlin, 1981 (Springer-Verlag, Berlin, 1982), p. 12.
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63
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84927320036
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The standard reference is A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, New Jersey, 1963); A modern textbook is J. W. Negele and H. Orland, Quantum Many-Body Systems (Addison-Wesley, New York, 1988).
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68
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84927320035
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Even after different pseudopotentials are chosen for the two approximations, the semiclassical and the perturbative formulas for the amplitude of a specific orbit in a specific member of the impurity averaging ensemble should not be expected to coincide. This is due to the different ways in which shadowing (which is necessary to preserve unitarity) is treated: in the semiclassical approach, an orbit connecting a specific set of scatterers may (or may not) be completely screened by an intervening impurity; in the perturbative approach there is a gradual reduction of the amplitude due to forward scattering on intervening impurities (which in that scheme is the only process that involves second-order perturbation scattering, and gives exactly the correct reduction due to the optical theorem). Note that in the ``semiclassical'' approach of Ref. 11 the reduction is also gradual, as in the perturbative approach.
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69
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84927320034
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In fact, there exists numerical evidence supporting the generality of the semiclassically derived result, Eq. (20), even when is applied to the region in which ħ is not small. One example is the kicked rotor mentioned in the next section (another is Ref. 10). The results (for the case with φl= τ l2/ 2 ) do not depend sensitively on ħ, and simulations with relatively large values of ħ are sometimes used to demonstrate semiclassically derived relationships. This can also be done for our case, and results very similar to Figs. 4 and 5 are obtained (simulations with small values of ħ were not attempted because they require a very large computation effort).
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70
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84927320033
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For a modern exposé see, e.g., H. Kamimura and A. Aoki, The Physics of Interacting Electrons in Disordered Systems (Clarendon, Oxford, 1989), pp. 97 – 102.
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73
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84927320032
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and Ref. 8.
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