In this regime, the energy increase is governed by expressions such as symbolic notation Thus, the contribution is a measure of the Bloch-phase auto-correlation of individual levels. These differences notwithstanding, the Aharonov—Bohm analogy will be a useful guiding principle in the discussion given below of the system. The fields are matrices in the Hilbert space of angular momentum states n. The leading semiclassical theory, on which Gutzwiller's trace formula is based, breaks down in the vicinity of classical bifurcations. Physically, this means that for fixed values of the phases θ ± time reversal invariance is effectively broken the symmetry gets restored only upon integration over all phases , and Cooperon-like fluctuations fluctuations off-diagonal in T-space are strongly damped.
This analysis is conceptually straightforward yet tedious in practice. In this regime, periodicity effects relating to the q-commensurability of Planck's constant play no role and the system can be regarded as infinitely extended in angular momentum space. At these values, long-lived angular correlations form, and we are not able to get the massive mode fluctuations under control. Close to fundamental resonance The sensitivity of the rotor to the algebraic properties of manifests itself not only on resonances but also in the close vicinity of low-order resonant values. These are issues which are addressed especially in F.
For later reference, we finally note cf appendix that the Q-matrix representation of the density correlation function K ω n 1, n 2 is given by In this section, we reduce the functional integral formulation or, equivalently, equation to a more manageable field theory of localization in angular momentum space. We introduce a new class of eigenfunctions which live in the hierarchical region of the chaotic part of phase space. Formally, these are the field components diagonal and off-diagonal in T-space, respectively. As such, it can be expressed in a Fourier representation,. Starting from , the same action can be obtained by the straightforward if more tedious fluctuation expansion around arbitrary Z. This book discusses the following topics: spectral statistics and their semiclassical interpretation in terms of the Gutzwiller trace formula, quantum chaos and its application in mesoscopic physics, spectral statistics and conductance fluctuations and quantum chaos in systems with many degrees of freedom. The effects of bifurcations on various spectral statistics were discussed in M.
Specifically, for small values of time, the finite extension of the unit cell is not yet felt, and angular momentum will diffuse as in an infinite system. To understand the meaning of this limitation, recall the interpretation of Z n 1 n 2 as representative of a bilinear n 1 n 2. We now turn to the quantitative formulation of our theory. Because of obvious limitations, the school could not give due exposure to all the topics which were actively pursued in the past decade. The integration over all values of the flux restores time reversal invariance. We have obtained quantitative results for the rotor's kinetic energy , an observable that carries detailed information on its time-dependent transport characteristics. Secondly, we need to explore the role of massive fluctuations by way of an a posteriori justification of our identification of the above soft modes.
The rapid progress of the research field of quantum chaos and its application called for a book that keeps students abreast of the new developments and at the same time provides a solid basis in subjects which form the canon of the field. Rather, its effective stochasticity is rooted in mechanisms of incommensurability. For example, the action essentially measures the overlap of the one-step evolved state, , with the unevolved configuration. Therefore, the localization length scales as. To keep the notation simple, we denote the unitarily transformed operator again by. This scaling is indicative of a long time dynamics that is neither localized nor diffusive.
For a discussion of the meaning of this coefficient and of the correction terms in , we refer to section. To discuss the localization behavior of this system, we consider the density—density correlation function defined in. This generates a leading ~ t dependence in , which is purely classical. Throughout this paper, we consider the system prepared in a pure initial state with zero angular momentum, i. To leading order, this leads to a renormalization of the diffusion constant, saturates at a constant value.
Describing the auto-correlation of individual levels or 'bands' , j θ , this term is of 'deep quantum origin'. Nominally, we are still dealing with a q-periodic quantum system whose spectrum can be organized in Bloch bands. The motion of a quantum particle in a periodic, quasi-periodic, or disordered potential is the central problem of Localization Theory, which was historically the first meeting point between Quantum Chaos and Solid State Physics. In section , we introduce the functional integral approach to the problem, which will be reduced to an effective semiclassical action in section. Retracing their origin, we note that the T-indices have been introduced to account for the symmetry of the operator under transposition—the physical T c-symmetry. In doing so, we need to pay attention to fluctuations inhomogeneous in angular momentum space.
However, for periodicity intervals q much larger than the intrinsic localization length of the system, the wavefunction amplitudes at the boundaries of the 'unit cell' are exponentially small, which means that the system behaves as effectively localized. Wavefunctions and spectrum of the periodically extended system. We summarize the paper in section. However, in the absence of sources such as those specified in , the integration over this matrix field gives unity,. There, one usually considers correlations in the density of states of a system at slightly different values of some control parameter. In the present context, the role of that parameter is taken by the Bloch phases θ ±.