Modified Admittance, Perpendicular Susceptibilities, and Transformation of Correlations of the Ising Model

Abstract

A modified admittance is introduced to give Kubo's admittance at nonzero frequencies and to give the isothermal static admittance at zero frequency within the scope of the Kubo linear-response theory. The method is demonstrated by exact calculations of the frequency-dependent perpendicular susceptibility at zero field and its modified susceptibility of the regular Ising model. The results appear as linear combinations of the equilibrium spin-spin correlation functions of the lattice. The results are valid for all dimensions and all frequencies and temperatures. A (q + 1) × (q + 1) matrix a(q) describes the linear combinations explicitly, where q is the coordination number of the lattice. The properties of this matrix are extensively discussed as a special case of a matrix A(q)(ξ), which satisfies a simple quadratic equation of the form [A(q)(ξ)]2 = (1 + ξ)q for arbitrary values of ξ. Fisher's algebraic transformation of the spin-spin correlation functions for the regular Ising lattice is derived from the linear relation which holds between the perpendicular susceptibility and the corresponding modified susceptibility. By means of the product rules of the matrix A(q)(ξ), the higher-order spin-spin correlation functions are expressed in terms of the lower-order ones in complete generality.