Critical behavior of nonlinear random conductance networks: Real-space renormalization group approach

Abstract

The real-space renormalization group theory is used to study the critical behavior of nonlinear random conductance networks in two and three dimensions, in which the components obey a current-voltage ( I – V) relation of the form I = χ μ V γ with γ > 0 and μ = i, m where i, m represent the inclusion of volume fraction p and the host medium of volume fraction q ( p + q = 1), respectively. Two important limits are worth studying: (1) normal conductor-insulator ( N/ I) mixture ( χ m = 0) and (2) superconductor-normal conductor ( S/ N) mixture ( χ m = ∞). As the percolation threshold p c (or q c ) is approached, the effective nonlinear response χ e is found to behave as χ e ≈ ( p − p c ) t in the N/ I limit while χ e ≈ ( q c − q) − s in the S/ N limit. We calculate the critical exponents t and s as a function of γ. A more general duality relation is found to obey in two dimensions. By a connection with the links-nodes-blobs picture, t and s can be related to critical exponents ζ R and ζ s , which describe the geometry of the percolating network. The results are compared with those of the series analysis and excellent agreements are found over a wide range of γ.