Random Walks through the Ensemble: Linking Spectral Statistics with Wave-Function Correlations in Disordered Metals

Abstract

We use a random walk in the ensemble of impurity configurations to generate a Brownian motion model for energy levels in disordered conductors. Treating arc-length along the random walk as fictitous time, the resulting Langevin equation relates spectral statistics to eigenfunction correlations. Solving this equation at energy scales large compared with the mean level spacing, we obtain the spectral form factor, and its parametric dependence. PACS numbers: 05.45+b, 71.25.-s, 72.15.Rn (Submitted to PRL 29 March 1996) Statistical properties of the spectra of finite quantum systems have been a focus for research in three successive contexts: nuclear physics [1], the semiclassical limit of quantum mechanics [2], and studies of mesoscopic conductors [3,4]. A unifying idea is that the energies of individual eigenstates are frequently neither calculable nor interesting: instead, the concern should be with eigenvalue correlations, which typically are independent of many details of the Hamiltonian.Disordered mesoscopic conductors bring to this field both new behaviour and a natural ensemble for a statistical description -the ensemble of impurity configurations. For weak disorder, new behaviour arises because there can be a broad window in time, and hence a corresponding energy interval, between the scale, t el , for electron scattering from impurities and that for diffusion across the system, t erg ∼ L 2 /D (where L and D are the system size and diffusion constant). And at the mobility edge, specific, critical spectral statistics are expected [5,6]. The ensemble average provides the departure point for established perturbative [4] and non-perturbative [3] calculations of spectral correlations in disordered metals.In this paper we describe an alternative approach, in which the energy level distribution is averaged along a random walk through the ensemble. There are several precedents for study of levels as a function of position in the space of Hamiltonians. Most recently, a number of authors [7], in particular Szafer, Simons and Altshuler [8], have investigated parametric statistics: level correlations between different points on a smooth path in this space. Earlier, in connection with the semiclassical limit, Pechukas [9] used motion along such a path, with coordinate λ, to generate a dynamics for the one-dimensional gas formed by levels on the energy axis. And originally, in the context of random matrix theory (RMT), Dyson [10], employed a random walk through the matrix ensemble as the foundation for Brownian dynamics of levels, with arc-length, τ , being the fictitious time. Pechukas' and Dyson's ideas are linked, as shown schematically in Fig. 1, by the usual relation [10] between the end-to-end distance and the length of a random walk, λ 2 ∼ τ . In previous work on level dynamics [9][10][11][12], eigenfunction correlations have played no role, either (in the semiclassical limit) by assumption, or (for random matrices) by construction, RMT having no preferred basis. In contrast, for disordered metals, the basis o...

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