Exact Solution of a Model of Localization

The exact solution is presented for the "susceptibility," χ (the number of sites covered by the maximally extended eigenfunction), for the zero-energy solutions of a hopping model on a randomly dilute Cayley tree. If p is the concentration, then χ~(p*−p)−1 with p*~pce1/ξ1, where pc is the critical percolation concentration and ξ1 the one-dimensional localization length. This result is argued to hold for the dilute quantum Heisenberg antiferromagnet at zero temperature. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/313 VOLUME 49, NUMBER 4 PHYSICAL REVIE%' LETTERS 26 Jvav 1982 Exact Solution of a Model of Localization A. Brooks Harris DePartment of Physics, University of Pennsylvania, PhiladelPhia, Pennsylvania l9104 (Beceived ll May 1982) The exact solution is presented for the "susceptibility, " g (the number of sites covered by the maximally extended eigenfunction), for the zero-energy solutions of a hopping model on a randomly dilute Cayley tree. If p is the concentration, then y {p*-p) ' with p*-p, e ', where p, is the critical percolation concentration. and (~ the one-dimension~/& al localization length. This result is argued to hold for the dilute quantum Heisenberg antiferromagnet at zero temperature. PACS numbers: 71.55.Jv, 64.60.Cn Recently much interest has been shown in the relation between classical' and quantum' percolation. The latter phenomenon is a model for the transition from localized to extended states that may occur for excitations in a random potential. ' In these models bonds between nearest-neighboring sites are randomly present with probability p and absent with probability I -p. For both models we express the susceptibility, y, in terms of the susceptibility, y(I'), of the cluster I of sites connected by occupied bonds: where &(I") is the probability of occurrence of a cluster I which intersects the origin. For classical percolation, ' y(1) is simply the number of sites in the cluster I". To define y(I") for quantum percolation one considers the eigenfunctions of the hopping Hamiltonian which obey where t, , assumes the value t if the bond between sites i and j is present and o otherwise. I consider only lattices which can be decomposed into two sublattices a and b such that t;, is nonzero only when sites i and j are on different sublattices. Then, even in the presence of dilution, the density of states, p(E), is an even function of E and is nonzero for -zt &E &zt, where z is the coordination number of the lattice. As the concentration p is decreased from unity the states near the edge of the band become localized (See Fig. I). Thus there exist mobility edges at energies +E„such that states with ~E~ &E, are localized whereas for ~E ~ &E, extended states appear (possibly coexisting with special localized states'). The mobility edges move towards E =0 as p approaches a critical value, p*. Under the reasonable assumption that extended states first appear at E = 0, I locate p* by an exact analysis of the eigenfunctions for E =0. To do this I introduce a localization susceptibility which diverges when extended states begin to form. Following the concept of the participation ratio introduced by Thouless' I define y(I') by