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Canadian Journal of Physics

Calculation of the isothermal susceptibility by the Kubo formula

T. Morita, T. Horiguchi

Published Jul 15, 1976• 1 citation
Publisher

Abstract

The relation between the zero frequency limit of the frequency-dependent susceptibility and the isothermal susceptibility is made clearer by expressing them in terms of the spectral density function. The general formulas are illustrated for the perpendicular susceptibilities of the Ising model.
DOI
10.1139/p76-173

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The Kubo linear response theory is applied to the quantum mechanical Ising model to obtain a general expression which is valid for all regular lattices for the perpendicular response function in terms of spin correlations. For the honeycomb and the square lattices, exact expressions are found for the frequency-dependent initial perpendicular susceptibility, [Formula: see text] valid for all temperatures. For the honeycomb lattice, the present result for zero frequency agrees with Fisher's static calculation, whereas there is disagreement in the case of the square lattice. The discrepancy is resolved by using an extended calculation which explicitly illustrates the difference between isothermal and adiabatic susceptibilities.
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.
Bounds are presented relating zero-field static isothermal magnetic susceptibilities to the mean-square fluctuations of corresponding magnetization variables. The lower bounds contain the first frequency moment of a spectral density. When this moment $\overline{\ensuremath{\omega}}$ approaches zero, the upper and lower bounds merge, and the susceptibility is determined by the mean-square fluctuation. In particular, if the susceptibility diverges at a temperature Tc{T}_{c}Tc​, and if the expectation of the double commutator appearing in $\overline{\ensuremath{\omega}}$ is finite at and near Tc{T}_{c}Tc​, then the fluctuation and the susceptibility diverge in the same manner, and their critical exponents will be identical.