Jan 19, 1998 - In a recent Letter [1], Altland and Zirnbauer claim that they rigorously proved the complete analogy between. (chaotic in the classical limit) ...
VOLUME 80, NUMBER 3
PHYSICAL REVIEW LETTERS
Comment on “Dynamical Theory of Quantum Chaos or a Hidden Random Matrix Ensemble?” In a recent Letter [1], Altland and Zirnbauer claim that they rigorously proved the complete analogy between (chaotic in the classical limit) dynamics of the quantum kicked rotor (QKR) and that of disordered solids. The main purpose of this Comment is to show that, in fact, the theory [1] does not represent at least some characteristic dynamical features of QKR. The phenomenon of quantum suppression of classically chaotic diffusion (see [2] and references therein) is well known to be in close analogy with Anderson localization in the 1D random solid model. Nevertheless, in spite of all mathematical efforts, a rigorous proof of quantum localization in the QKR is still lacking. The main mathematical difficulty lies in the fact that, for a rigorous proof of localization, the theoretic number properties of the period t are crucial. Actually, rigorous statements can be made only in the opposite direction. It was analytically shown that for rational values of the parameter T ty4p (quantum resonances), no localization, strictly speaking, takes place. Moreover, it was rigorously proven [3] that localization is absent also for a nonempty set of irrational values of T . Even for a typical irrational T , when the whole dense set of quantum resonances may be still insufficient to prevent localization, the eigenfunctions may be essentially modified if the detuning from resonance is small enough (see [3]). The measure of such T is typically small but finite. In our opinion, the crucial dynamical features mentioned above are lost in the approximate calculations following the exact functional (6) in Ref. [1]. The following arguments demonstrate our main point. By direct averaging over the parameter v0 , one easily finds kkl1 jG 1 sv1 d j l2 l kl2 jG 2 sv2 d j l1 llv0 ` X expfinsvt 1 i0dg j kl1jU n jl2 lj2 . (1)
19 JANUARY 1998
uncleared. Most likely, the very structures of the principal configurations in (6) must be quite different depending on the particular choice of the parameter T . Actually, the authors of [1] estimate the functional (6) by putting the system on an angular momentum torus with a period L ¿ 1 and angular lattice spacing 2pyL and restricting, therefore, to rational periods T MyL. This, however, does not circumvent the main difficulty mentioned above. Indeed, although localization on the scale of quasiperiodicity L ¿ k 2 may appear if the large integer numbers M, L are relatively prime, there is no localization when the ratio can be reduced to T ryq with r , q ø L [2]. An additional remark concerns the dependence of localization length j on the “symmetry breaking” parameter a sm 1 ad2pyL; a [ f0, 1g. Simple transformations lead to µ ∂ sl 2 1 l22 dt kl1 jUsadjl2 l expfisl1 2 l2 dag exp 2i 1 4 ` X 3 s2idfqL2sl1 2l2 dg JfqL2sl1 2l2 dg skd q2`
3 exps22ipqad .
(2)
It is now easily seen that under the condition L ¿ k 2 when localization does take place, the nontrivial dependence on a is exponentially weak and can hardly influence the value of j. In conclusion, the theory [1] in its present form may be expected to be valid only when a sort of band random matrix approach can be used ad hoc. However, it fails to take into account dynamical features which go beyond the random matrix theory description. A careful investigation of the conditions of validity of the suggested theory is needed. We are indebted to B. Chirikov for his numerous and deep remarks. We also appreciate interesting discussions with Ya. V. Fyodorov, A. D. Mirlin, and O. V. Zhirov.
n0
This exact expression contains matrix elements of the dynamical Floquet operator U so that all dynamical peculiarities of QKR survive the v0 averaging. Let us now consider, for example, the simplest quantum resonance with T 1. Then kl1 jU n jl2 l kl1 j expf2isnk cos udg jl2 l and, contrary to the localization regime, in the infrared limit v ! 0, the right-hand side in (1) does not depend on the distance jl1 2 l2 j. We reckon that Eq. (6) in Ref. [1] gives an integral representation of the quantity (1) and, therefore, should carry the same full dynamical information. Nevertheless, the delocalized resonance regimes similar to that mentioned above are not reproduced by the calculation reported in [1]. This leads us to suspect that some hidden additional statistical assumption has been made in [1] which remains 640
0031-9007y98y80(3)y640(1)$15.00
Giulio Casati International Center for the Study of Dynamical Systems, University of Milano at Como, Via Castelnuovo, 7, 22100 Como, Italy F. M. Izrailev and V. V. Sokolov Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia Received 11 March 1997 [S0031-9007(97)04912-0] PACS numbers: 05.45.+ b, 03.65.Sq, 72.15.Rn [1] A. Altland and M. Zirnbauer, Phys. Rev. Lett. 77, 4536 (1996). [2] F. M. Izrailev, Phys. Rep. 196, 300 (1990). [3] G. Casati and I. Guarneri, Commun. Math. Phys. 95, 121 (1984); G. Casati et al., Phys. Rev. A 34, 1413 (1986).
© 1998 The American Physical Society
