PHYSICAL REVIE%' 8 VOLUME 29, NUMBER 3 1 FEBRUARY 1984 Renormalization-group analysis of the discrete quasiperiodic Schrodinger equation Stellan Ostlund Institute for Theoretical Physics, University of California, Santa Barbara, California 93I06 Rahul Pandit Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, ¹mFork l4853 (Received 11 July 1983} Recently developed scaling concepts in the theory of quasiperiodic dynamical systems are used to develop an exam» renorn alization group applicable to the discrete, quasiperiodic Schrodinger equa- tion. To illustrate the power of the method, me calculate the universal scaling properties of the states and eigenvalue spectrum at and below the localization transition for an energy which corre- sponds to an integrated density of states of 2. The modulating potential has a frequency T(l 5 — 1) relative to the underlying lattice for the example we work out in greatest detail. I. INTRODUCTION A. Overview Schrodinger equations with quasiperiodic potentials have been studied for many years. The motivation for this study is both physical and mathematical. Quasi- pcr1od1c potcnt1als ar1sc natUI'ally ln 1ncom mensurate structures, in nonstoichiometric intergrowth compounds, such as Hgs sASF6, in the calculation of band structures of periodic crystals in magnetic fields, 7 '6 and in super- conducting lattices in magnetic fields. ' Mathematically, quasiperiodic potentials aI'e interesting because Bloch's theorem is inapplicable. These potentials lead to rich spectra and wave functions because they are, in some scnsc, intermediate bctwccn periodic Rnd 1andoID. Period- ic potentials lead to absolutely continuous spectra' and extended c1gcnstatcs, whcrcas random potcnt181s lead to pure-point spectra and localized eigenstates in one dimen- sion. Though there are no rigorous proofs, the general be- lief is that generic quasiperiodic potentials lead to spectra that are Cantor sets, have both absolutely continuous and puI'c-po1nt co1Tlponcnts» Rnd, ln Rdd1tlon, 8 slngulal coQ- tlllUous colllpoiicIlt. Tllc wave fllIlctlolls CRIl bc ex- tended, localized, or "critical" in a sense that we will specify below. Quasiperiodic potentials pose mathematical problems that have a fundamental connection with the small-divisor problems that were investigated by Kolmogorov, Arnol'd, and Moser' (KAM) in classical mechanics. If the effect of 8 quasiperiodic potential is calculated within perturba- tloIl theory, * ' then high-order tcfIDs 1Q thc pcrtUIba- tion expansion (for, say, the wave function) have small denominators. However, D1QabUlg Rnd Sinai ' showed that for a sufficiently weak quasiperiodic potential, a large part of the spectrum is still absolutely continuous. These proofs use the ideas of the KAM theory and the results are the analog of the KAM theorem, which states that most of the invariant tori in the phase space of an inte- grable Hamiltonian system are not destroyed by a suffi- ciently weak nonintegrable perturbation. Recently, nonperturbative renormalization-group (RG) methods have been used to study scaling phenorDena in a variety of small-divisor problems. These include invariant circles of area-preserving maps, circle maps, ' and in- variant curves 111 nlapplllgs of tllc coIilplcx plaIic ollto It- self. Recent work has shown that similar RG methods can be used to study the scaling properties of the spectra and wave functions of a discrete, one-dimensional SchI'odlngcI' cquatlon with 8 spcc181 QOQRQalytlc~ quas1- periodic potential with two incommensurate frequencies. These ideas are extended to analytic, quasiperiodic poten- tials with two incommensurate frequencies in the present work. We believe scaling properties of the spectra and wave fuIlctlolls of quaslpcrlodlc Schrodinger opclatols cRI1 bc studied most conveniently by RG methods. The ex- ponents that characterize scaling behavior (which we de- fine in Sec. II) are simply related to the eigenvalues of the linearized RG transformation in the vicinity of its fixed points. The universality of these exponents follows naturally because different quasiperiodic Schrodinger operators that flow to the same fixed point under succes- sive iterations of the RG transformation share the same scaling behavior. Thus, even though we study a specific Uaslpcr1odlc potcnt181~ wc can prcdlct thc sca11ng propcI'- ties of the spectra and wave functions for a large class of quaslpcrlodic potclltlals. II1 addlt10n to tllcsc fundamental insights, the exact RG that we construct in Sec. IV gives a very efficient numerical algorithm for computing various properties of quasiperiodic operators. Thcrc have been soIIlc pl cvloUs attcIDpts to cxplaln properties of quasiperiodic operators using RG 1dcas. ' ' These 1Ilvcstigat1ons c1thcI' Usc approxima- tions or do not carry out a fixed-point analysis with which scaling could be investigated. By contrast, we analyze the fixed point of an exact renormalization group which governs the scaling properties of the states and spectra. One of the most powerful aspects of the present analysis is that the exact formulation can be implemented using a nu- OC1984 The American Physical Society