PHYSICAL REVIEW B VOLUME 35, NUMBER 10 1 APRIL 1987 Localization in a one-dimensional quasiperiodic Hamiltonian with o8'-diagonal disorder J. A. Verges and L. Brey Departamento de Fisica del Estado Solido, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain E. Louis Departamento de Fisica, Universidad de Alicante, 03080 Alicante, Spain C. Tejedor Departamento de Fisica del Estado Solido, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain (Received 28 July 1986) Localization in a one-dimensional quasiperiodic system with oH'-diagonal disorder was investi- gated by means of recursion relations similar to those already used by other authors to treat diag- onal disorder. When a part of the quasiperiodic chain is periodically repeated to fill the whole space, the bandwidth goes to zero as a function of the cell length with an exponent which depends upon the degree of disorder. The average transmission through finite parts of quasiperiodic chains was also calculated; our results indicate that it scales with size as the bandwidth does. In recent years great attention has been devoted to the study of the properties of electrons in one-dimensional quasiperiodic potentials. ' Quasiperiodic systems are in- termediate between disorder and periodic solids, and it is expected that their apparent regularity should be reflected in their properties. In particular, localization of electron states has been the subject of many theoretical studies. The results obtained by Kohmoto, Kadanoff', and Tang are particularly revealing, indicating that the properties of some quasiperiodic one-dimensional systems may follow universal scaling laws. Those authors arrived at this con- clusion by periodically repeating a part of the quasiperiod- ic chain and studying the scaling behavior of the total width of the allowed bands. The purpose of the present paper is twofold. First, to study whether that behavior is also found in Hamiltonians with ofl-diagonal disorder. Second, to search a similar scaling law for a physical mag- nitude, such as transmission through a finite chain. 5 The results indicate that both the total bandwidth and the aver- age transmission scale with the size of the chain following a power law as found by Kohmoto et al. for the total width of quasiperiodic chains with diagonal disorder. Although the techniques used in the present work are rather similar to those developed by Kohmoto et al. for diagonal disorder, several nontrivial points specific to the Hamiltonian herewith considered are worthy of comment. The quasiperiodic Hamiltonian discussed in this work con- tains two interactions between first-nearest neighbors lo- cated in the chain according to the Fibonacci sequence (V„Vb, V„V„Vb, V„Vb, V„... ). Then the relevant pa- rameter is the ratio between those two interactions a = Vb/V, (energy will be hereafter expressed in units of V, ). Atomic levels are assumed equal for all lattice sites, all the energies will be referred to this level. The sequence of transfer matrices contains three different matrices A, B, and C, being as follows: BCABCBCABCABCBCABCBC. . . where B joins bonds of type a and b, C does for b and a, Pn+i -Pngn, Qn+|-Pn, or equivalently, Pn+ i -PnPn- I, & — 1, as in the case of diagonal disorder. 2 The initial conditions are the following: — E E — 1 P]=I/a E I, PP 1 0 (3) The recursion relations for x„(trP„)/2 are straightfor- wardly obtained. The results coincide with that for the case of diagonal disorder, namely, xn+) 2Xnxn-] Xn — 2. n ~ 2 and the corresponding initial conditions are xp=E/2, x) - (E2 — a2 — I)/(2a), x2 E(E — a — 2)/(2a) . Now we use the device introduced by Kohmoto et al. and repeat periodically quasiperiodic sequences of increas- ing size. The period of the resulting periodic chain will be the Fibonacci number F„(F„1 for n 0). The corre- sponding bandwidth is directly obtained from x„as seen in Ref. 2. The results for the total width of the allowed ener- and A does for a and a. The recursion relation is then most easily obtained by defining two new transfer matrices as P =BC, Q— : A . Then a simple recursion relation is obtained which is simi- lar to that found by Kohmoto et al. for the case of diago- nal disorder, namely, 35 5270 1987 The American Physical Society