Localization and Quantum Percolation

Electronic wave functions are studied on dilute lattices, at dimensionalities 1⩽d⩽8. Generalized average inverse participation ratios are expanded in powers of the bond concentration, p. Dlog Padé approximants indicate that these ratios diverge as (pq−p)-γq, signaling the appearance of extended states for p>pq. These Anderson transitions occur above classical percolation. No divergence is detected at d=2. These results are consistent with the existence of localized states at the center of the band. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/481 VGLUMK 49, NUMBER 7 PHYSICAL REVIEW LETTERS 16 AUGUsr 1982 Localization and Quantum Percolation Yonathan Shapir"' and Amnon Aharony DePartment of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel A. Brooks Harris Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19174 (Received 14 May 1982) Electronic wave functions are studied on dilute lattices, at dimensionalities 1&d ~ 8. Generalized average inverse participation ratios are expanded in powers of the bond concentration, p. Dlog Pade approximants indicate that these ratios diverge as (p, -p) signaling the appearance of extended states for p&p, . These Anderson transitions occur above classical percolation. No divergence is detected at d = 2. These results are consistent with the existence of localized states at the center of the band. PACS numbers: 71.30.+h, 71.55.Jv The nature of the Anderson transition, "between localized and extended single-electron states in disordered materials, has been the subject of much recent research. A recent scaling approach' indicated that all states are localized at and below two dimensions of space, d ~ 2. For higher dimensionalities, d &2, all the states are localized only for a large amount of disorder. As the amount of disorder is decreased, extended states appear (via the "Anderson transition") at the center of the allowed energy band, bounded by two mobility edges. " Recent field-theoretical models yielded expansions in e =d —2 of the behavior near these mobility edges. ~ Related models gave a "mean-field theory, " and indicated that the critical dimensionality above which it describes extended states is d = 8.' Although the predictions at d =2 are confirmed by several calculations, ' they are still disputed by others. " Very recently it was suggested' that at d =2 the "extended" states decay algebraically, similarly to the spin correlations in the XF model. All these developments make it highly desirable to have a systematic study of localization at general dimensionalities. The above-quoted studies mostly considered tight-binding Hamiltonians with diagonal disorder. It has recently been suggested" that purely off-diagonal disorder yields different results, i.e., no Anderson localization at the band center. The simplest model with off-diagonal disorder is that of quantum percolation. ""Bonds on the lattice are present, with probability p, or absent, with probability 1 —p. The tight-binding Hamiltonian is then written as K= P t, , (a, ~a,. +c.c.), where a,. creates the electron at site i, the sum is over nearest-neighbor bonds (i,j), and t, , is equal to 1 (or 0) if the bond (ij) is present (or absent). Classica/ percolation' occurs at a concentration p„below which all the connected clusters are finite. An infinite cluster first appears at p„and grows to cover the whole lattice as p approaches unity. Clearly, all the quantum states of Eq. (1) are localized (i.e. , limited to a finite region of space) for p &p, . However, Kirkpatrick and Eggarter" showed that localized states always exist on the infinite cluster, i.e. , for Bll p & 1. Since all states are extended at p = 1 and no states are extended for p &p„one expects an Anderson transition from localized to extended states at some concentration p„with p, & p, & 1, such that no extended states occur for p & p, . Such a transition was indeed found by Kirkpatrick and Eggarter, ' and recently confirmed by Odagaki, Ogita, and Matsuda. ' An interesting result of Kirkpatrick Bnd Eggarter" is that there is Bn infinite sequence of discrete energies at which localized states can be formed on the infinite cluster. These states require a symmetric subcluster and also seem to be specific to the quantum percolation model. Using the commonly accepted argument excluding the coexistence in energy of localized and extended states (due to mixing of these degenerate states by some coupling), "we would then conclude that the density of extended states has as infinite number of zeros throughout the band. Presumably if this were true, the usual concept of mobility edges would have to be discarded. However, a specific counterexample presented below shows that localized and extended states can indeed coexist in energy, Bnd remain orthogonal to each other. 486 1982 The American Physical Society VQLUME 49, NUMBER 7 P HYSICAL REVIEW LETTERS 16 AUGUST 1982 Similarly, the degenerate localized states do not mix to create extended states. These peculiar features are special to the present model because of its high symmetry. The disagreement between various results at d =2, the general interest in the d dependence of localization, the question of universality (diagonal versus off-diagonal disorder), the curious role of the special localized states, and the relations of quantum to classical percolation" all motivated us to try a new approach to the model (1). In this communication we report on a series expansion of an appropriate measure of localization, )(, in powers of p, at general d. Our measure, related to the inverse participation ratio, ' is expected to diverge when extended states first appear, i.e. , as p —p, , in the form (p, —p) &. . Indeed, analysis of our series indicates divergences at p, = 0.32, 0.20, 0.15, 0.12, 0.10, 0.086, with y, -—1.7, 1.1, 0.96, 0.90, 0.87, 0.85, for d = 3, 4, 5, 6, 7, 8. Comparing to""p, = 0.247, 0.161, 0.118, 0.094, 0.079, 0.068, we see that indeed p, & p, . At d =2 the different approximants do not give any consistent results, probably indicating that p, = 1. An attempt to fit )( with exp[C(p, —p) &~ ], as predicted in Ref. 9, also failed. After we finished the present work we learned about recent work by Raghavan and Mattis, "who studied the model (1) on the extended-states side (p & p, ), using the tridiagonalization method. They found p, =1, 0.37, and 0.23 at d =2, 3, and 4, in rough agreement with our estimates. Like Ref. 18, most of the earlier quantitative calculations were performed on the side of the extended states. Recently, Schafer and Wegner" generated the first few terms in a perturbation expansion about the localized instanton states. However, our calculation contains many more terms and is the first to yield accurate quantitative results on the side of the localized states. Our calculation is based on a generalization of the inverse participation ratio. If we denote the gs degenerate orthonormal eigenfunctions of (1) on the finite cluster F by {Ps,(i)j, this ratio is defined via X(p) =Z „Y(F)p"' (1 —p)", (3) where N, and N~ are the numbers of bonds inside and adjacent to F. If Y'(I') is replaced by Ns' then