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Physical review. B, Condensed matter

Real-space renormalization-group analysis of quantum percolation

Takashi Odagaki, K. C. Chang

Published Aug 1, 1984• 56 citations
Publisher

Abstract

Quantum percolation problems are studied with the use of a real-space renormalization-group method. A quantum effect is taken into account by calculating the quantum-mechanical efficiency of the Kadanoff cell in opening a channel for an electron in the cell. The quantum percolation threshold and the critical index for the correlation length are obtained for both the site and bond problems in the square and simplecubic lattices.

DOI
10.1103/physrevb.30.1612

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Quantum percolation problems are studied with the use of a real-space renormalization-group method. A quantum effect is taken into account by calculating the quantum-mechanical efficiency of the Kadanoff cell in opening a channel for an electron in the cell. The quantum percolation threshold and the critical index for the correlation length are obtained for both the site and bond problems in the square and simplecubic lattices.
A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.
A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.
A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration ppp of active first-neighbor bonds, and the concentration qqq of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.