PHYSICAL REVIEW B VOLUME 46, NUMBER 19 15 NOVEMBER 1992-I Localized electrons on a lattice with incommensurate magnetic flux Shmuel Fishman Department of Physics, Technion, Israeli Institute of Technology, 32000 Haifa, Israel Yonathan Shapir Department of Physics and Astronomy, Uniuersity of Rochester, Rochester, New York 14627 0011- Xiang-Rong Wang School of Physics and Astronomy, Uniuersi ty of Minnesota, Minneapolis, Minnesota 55455 and Department of Physics, The Hong Kong Uni uersity of Science and Technology, Clear 8'ater Bay, Komloon, Hong Kong (Received 27 May 1992) The magnetic-field effects on lattice wave functions of Hofstadter electrons strongly localized at boun- daries are studied analytically and numerically. The exponential decay of the wave function is modulat- ed by a field-dependent amplitude J(t) = g', u2 cos(trar), where a is the magnetic flux per plaquette (in units of a flux quantum) and t is the distance from the boundary (in units of the lattice spacing). The be- havior of ~ J(t) ~ is found to depend sensitively on the value of a. While for rational values a=p/q the envelope of J(t) increases as 2 ~, the behavior for a irrational (q ~ 00) is erratic with an aperiodic struc- ture which drastically changes with a. For algebraic a it is found that J(t) increases as a power law t@ ' while it grows faster (presumably as t@ ""') for transcendental a. This is very different from the growth rate J(t)-e ' that is typical for cosines with random phases. The theoretical analysis is extended to products of the type J(t)=g'„: 2ucos{ mar") with v&0. Different behavior of J"(t) is found in various regimes of v. It changes from periodic for small v to randomlike for large v. I. INTRODUCTION The properties of noninteracting electrons on a lattice subjected to an external magnetic field have attracted much attention since the early works of Hofstadter, ' Wannier, and Azbel. These works have focused pri- marily on the exotic spectral properties as a function of the electron's energy and the parameter a = //go, where P is the magnetic flux per plaquette and Po=R/e is the Aux quantum. The spectrum has special scaling proper- ties as a function of the commensurability q for a =p /q (rational) and becomes a Cantor set for incommensurate fluxes (a irrational). The wave function itself is extended in the q subbands for commensurate values of rational a. The Hamiltonian describing the lattice electron is g y, , =eP/A (1. 2) will do. More recently there has been growing interest in the effect of a magnetic field on localized electrons. The combined effects of lattice periodicity and magnetic Aux create a very complex behavior in the spatial variation of the wave functions even in the absence of disorder. This &=Kg a +aVg aa e "+c c i The phase y;, is associated to the link (ij ) between the nearest-neighbor sites in accordance with the "Peierls an- satz. " Any gauge such that the sum around a plaquette V„=A, cosa.an . (1. 4) As discussed below, this model was generalized ' to is the case when the electron's energy is deep inside a gap between the quasibands of the bulk eigenstates. ' The electron may be localized at inhomogeneities such as the edge of the lattice (i.e. , surface gap states) or at isolated impurities in an otherwise ordered bulk. Clearly these states decay exponentially going away from the inhomo- geneity into the bulk. This exponential decay and partic- ularly the "localization" length (associated with this ex- ponential decay) will be affected by the application of an external magnetic field to the system. In the present work we look at some basic aspects of this problem. While avoiding the full complexities of the related questions, we look at a simple (maybe the simpler) model in which the intricate salient features of this prob- lem are predominant and can be addressed. As is well known, ' the two-dimensional (2D) tight- binding lattice electron problem reduces (in the Landau gauge) to that of a 1D electron hopping in a potential which varies as cos(k~ j+2ma) where k~ are the trans- verse momenta in the y direction, and may be taken to be zero by shifting the origin in the x direction. ' The Schrodinger equation reduces then to the famous Harper equation u„+, +u„&+ V„u„=(E — W)u„, where the diagonal potential is 46 12 154 1992 The American Physical Society