VOLUME 73, NUMBER 25 PHYSICAL REVIEW LETTERS 19 DECEMBER 1994 Multifractal Energy Spectra and Their Dynamical Implications Italo Guarneri* and Giorgio Mantica~ Universita di Milano, sede di Corno, via Lucini 3, 22100 Corno, Italy (Received 11 January 1994) We present a method for constructing lattice tridiagonal Hamiltonians having a preassigned multifractal measure as local spectrum. Using this construction we investigate how the fractal structure of the spectrum affects the motion of wave packets. We find that the quantum evolution is intermittent: The moments of particle s position on the lattice are characterized by a nontrivial scaling function, even when the spectrum is a one-scale, balanced Cantor set. Numerical data show that the minimum scaling exponent is always larger than the information dimension of the spectral measure, and qualitatively follows the behavior of this quantity, as the spectral measure is varied. PACS numbers: 05. 45.+b, 02.30. -f, 71. 30. +h, 71. 55. Jv Until recently, the traditional classification of energy spectra in pure point and absolutely continuous has proven quite sufficient to describe quantum evolution. The third element exhausting this classification, singular continuous energy spectra, has been usually considered by physicists as a mathematical curiosity and overlooked. However, this exotic kind of spectra is now attracting a growing attention: In fact, it is generic (in a mathematical sense) for one-body Schrodinger operators [1] and appears fairly frequently in theoretical studies on quasiperiodic systems [2,3], on incommensurate structures [4], and on the electron dynamics of crystals in magnetic fields [5]. The success of fractals in various areas of physics has also invested this field, and the techniques of the thermodynamical formalism [6] have been applied to the systems just mentioned [7 — 9], although more often in a descriptive than in a predictive approach. Quite on the contrary, these spectra lead to dynamical behaviors of deep physical relevance; for instance, it has been found [10,11] that, for a particle moving on a one-dimensional lattice, the time-averaged probability of staying at the starting site decays in time asymptotically as t ', D2 being the correlation dimension of the associated spectral measure. This is just one of the relations between dynamics and multifractal properties, which make the subject of this Letter. The analysis of the Harper system [7, 8] first brought into evidence a multifractal spectrum (with Hausdorff dimension close to 2) associated with a pseudodiffusive evolution of wave packets initially focused at the origin: the expectation value of the square of the position of such packets grows (approximately) linearly in time. On the grounds of heuristic arguments [12,13] it has been conjectured [13] that the exponent p2 ruling anomalous diffusion (x2)(t) — t2P' (the bar meaning time average), should coincide with the Hausdorff dimension DH of the support of the spectral measure of the initial state. Although numerical data consistent with this surmise have been presented [9,13], quite recently some doubts on its generality have been put forward on the strength of other heuristic observations [14]; in summary, no decisive evidence in either sense has been provided so far. A complete study of the scaling properties of wave- packet propagation can be centered around the behavior of the moments v (t):= (x )(t) for real a & 0. Under suitable circumstances, these moments behave like t P, thereby defining a scaling function P(n) [15]. An exact analysis shows that P(u) must be larger than the infor- mation dimension D& of the spectral measure for any u [16, 17]. The limit P(0) can be defined, and it represents the lowest scaling exponent. In the general case, more precise results seem difficult to derive; at the same time a numerical investigation of the relations between the mo- ment scaling function and the various fractal dimensions has to cope with the difficulties present in the accurate de- termination of both quantities. For these reasons, in this Letter we take a different approach: Instead of studying a given Hamiltonian, and computing spectral properties and dynamics, we start from a given spectral measure, p„, arbitrarily chosen in a wide multifractal class, for which all fractal dimensions are ex- plicitly computable. Then we show that a constructive procedure can be set to compute all matrix elements of a (tridiagonal) Hamiltonian H, which is the most natural operator processing p, as spectral measure. The related Schrodinger equation is then numerically solved via reliable procedures and P(a) extracted. We investi- gate various choices of the spectral measure and we find that the conjecture p(2) = Dp is at best approximate; that the wave-packet propagation exhibits multiscaling in time (i.e. , intermittency), in the sense that it is characterized by a nontrivial range of exponents P(a); and that inter- mittency is present even in the case of a homogeneous fractal measure, that is, multiscaling does not require multifractality. The first step in our approach requires the solution of an inverse problem of the sort "can we build a drum with a given spectrum. " The "drum" is, in our case, a tridiagonal Hamiltonian matrix H of the tight-binding type widely used to model the dynamics of electrons in 0031-9007/94/73(25)/3379(4)$06. 00 1994 The American Physical Society 3379