PHYSICAL REVIE%' 8 VOLUME 37, NUMBER 3 15 JANUARY 1988-II Hierarchical structure of a one-dimensional quasiperiodic model H. Eduardo Roman Fakultat fur Physik, Uniuersitat Konstanz, D 775-0 Konstanz i, Federa/ Republic of Germany (Received 24 August 1987) A hierarchical structure of a Fibonacci chain is obtained exactly. This is done by transforming the original chain into another one in which the ne~ hopping matrix elements and on-site energies are arranged hierarchically. An exact renormalization-group transformation is derived for the hierarchical chain. The problem of quasiperiodic structures has attracted a great deal of attention in the last years, in particular since the experimental observation of s possible qussi- crystal phase in rapidly quenched Al-Mn alloys. ' From a theoretical point of view, one-dimensional (1D) models play an important role in understanding the unusual physical behavior associated with quasiperiodic ordering. Recently, the experimental realization of a quasiperiodic superlsttice has been achieved and studied by x-ray scattering techniques. Several models have been proposed as 1D versions of qussicrystals. Here we are interested in two tight- binding variants of a Fibonacci chain; i. e. , one in which the potential V„at site n can take two values, a or b, with constant hopping matrix elements (diagonal model), and a second one which consists of two types of bonds, a and b, with constant V„(ofF-diagonal model). In both cases the elements a and b are arranged in a Fibonacci sequence Q„which can be obtained from the iterating equation Q„=Q„, Q„2, n &2 with Qti=b and Qi — — a. The number of elements in Q„ is given by the Fibonacci number F„=F„1+F„2 with I'o — — E1 — — 1. These models have been extensively stud- ied " and their characteristic features like the Cantor- set spectrum and "critical" wave functions are well un- derstood. The existence of s self-similar structure in the spectrum and wave functions has long been recognized in those works and in some cases it is possible to relate it to that of the underlying lattice. There exists, however, s basic property of a Fibonacci chain, that to our knowledge hss not been explicitly pointed out so far. This has to do with an exact hierarchical structure of s Fibonacci lattice that we aim to discuss in this Brief Re- port. To start with, let us rewrite (1) in the form Q„= Q„zg„3Q„2, n & 3 . If one writes the sequence for odd n (even n }, one recog- nizes a kind of self-similar structure in which even (odd) scqucilccs ai'c salidwlclicd by odd (cvcil) oilcs. This fact. suggests an underlying hierarchical structure. Recently, I have introduced a simple tight-binding model' in which the hopping matrix elements are arranged in a hierarchical fashion. It turns out that a Fibonacci chain can be viewed as a hierarchical model with both renor- malized (energy-dependent) hopping terms and on-site energies. Several qualitative features known about the above-mentioned models follow straightforwardly from the present considerations. We proceed with a more detailed discussion of (2) by considering first the diagonal model defined by to'IP i + e 'I/J + t pP + i = 0, where s„are the on-site en- ergies, to the constant hopping matrix element, and f„ the amplitude of the wave function at site n. In Fig. 1(a) the sequence for Q„ is plotted in such a way' as to make evident the self-similar structure of (2). Note that the first three barriers correspond to Qs, the first six to Qs, the first 12 to Q7, and so on. Next, the b sites (e„=b) displayed in Fig. 1(a) can be decimated from the lattice. This generates the chain shown in Fig. 1(b), in which new hopping matrix elements t1 and on-site ener- gies e are obtained. Finally, sites belonging to Q„, for even n, are decimated from the chain. The result of this operation [Fig. 1(c}] is that one has effective hopping terms t2„, n &1, which connect sites with effective on- site energies e'I"x ", arranged in a hierarchical way [Fig. 1(d)]. The problem is now reduced to finding closed ex- pressions for these efFective coefficients. Before doing this, a brief remark on the corresponding transformation for the o8'-diagonal model can be made. An inspection of Fig. 2 indicates that both models lead to the same efFective hierarchical chain (EHC), and are therefore qualitatively equivalent. To proceed further in our calculations it is convenient to find first a renormalization-group (RG) transforma- tion for the EHC, assuming the t2„'s and cL"z ", n &1, are known. The decimation procedure is described in Fig. 3. We just quote the final result which is written in terms of the following recursion relations: 2 t2n + i tzn t 2n — i /52n (n) (n — 1) (n — 1) +2 ~1+m ~1+m ~2'+~ +1» 2n — 1 s' ~2@ (~i e(~ — 1) e(» — 1 1t 2 + +m 2" + +m 2" +m 2n — 1 2n where (n — 1) (n — 1) t 2 ~2n 2tt ~(2tt+ 1) t 2n 37 1399 1988 The American Physical Society