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Journal of Mathematical Physics

Monotonicity of correlation functions

Paul A. Pearce

Published May 1, 1976• 3 citations
Publisher

Abstract

Counterexamples to pair correlation monotonicity inequalities, analogous to Griffiths’ second inequality for the Ising model, are presented for the finite spin Heisenberg model, the spin−(1/2) X–Y model, the anisotropic planar classical Heinsberg model, and the spherical model.
DOI
10.1063/1.522972

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First, for a classical discrete approximation to the Heisenberg ferromagnet, in a Euclidean space of dimension 2k where k is a positive integer, the correlation is nonnegative and a monotonic increasing function of interactions. Second, for the quantum mechanical Heisenberg ferromagnet itself, the correlation is nonnegative but not necessarily a monotonic increasing function of interaction. Both results allow for long-range interactions. Third, a family of inequalities on the number of connected components in graphs is deduced.
We consider a new model Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}$ for interacting $\ensuremath{\nu}$-dimensional classical "spins"; ${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the Ising, planar, and Heisenberg models, respectively, for $\ensuremath{\nu}=1, 2, \mathrm{and} 3$. Certain critical properties of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found to be monotonic functions of $\ensuremath{\nu}$.
We propose a new method of constructing the quantum Griffiths inequality. From a viewpoint of operator inequalities, we first study the quantum rotor model. This viewpoint clarifies important connections between the reflection positivity and the Griffiths inequality. Next we apply our method to the spin-1/21/21/2 quantum Heisenberg model. Finally we point out a model-independent structure which governs the correlation inequalities.