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Physical Review A

Proof of the Third Law of Thermodynamics for Ising Ferromagnets

Harvey S. Leff

Published Dec 1, 1970• 16 citations
Publisher

Abstract

The third law of thermodynamics is proved for a large class of Ising models with generalized ferromagnetic many-body interactions. A sufficient condition for the third law to hold is that the model have nearest-neighbor couplings which are bounded from below by a positive constant. The proof is based on a spin-correlation inequality of Griffiths which implies a corresponding inequality for the bulk entropy per spin. Ground-state degeneracy considerations are completely avoided.

DOI
10.1103/physreva.2.2368

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The third law of thermodynamics is proved for a large class of Ising models with generalized ferromagnetic many-body interactions. A sufficient condition for the third law to hold is that the model have nearest-neighbor couplings which are bounded from below by a positive constant. The proof is based on a spin-correlation inequality of Griffiths which implies a corresponding inequality for the bulk entropy per spin. Ground-state degeneracy considerations are completely avoided.
A cumulant-like expansion for the entropy of an NNN-spin system is presented. The successive terms in the expansion relate to successively higher orders of statistical association among spins. It is proved that for any Ising system of general dimensionality with ferromagnetic interactions of arbitrary range, the first two terms in the entropy expansion provide a lower bound for the exact entropy. A corollary of the theorem is that the lower-bound property is also valid for any two-sublattice Ising system with antiferromagnetic interactions between sublattices. An example is given which illustrates the fact that the vanishing of the two-spin cumulant (correlation) does not necessarily imply that the spins are statistically independent. The sum of the first two terms in the expansion is compared numerically with the exact entropy of an NNN-spin chain (and also a ring) with nearest-neighbor ferromagnetic or antiferromagnetic Ising interactions. The comparison, which measures the validity of a Kirkwood-type truncation in this context, is favorable only at sufficiently high temperatures.
Antiferromagnetic ground-state energies for infinite Heisenberg linear chains with up to third-nearest-neighbor interactions have been determined by means of numerical extrapolations. On the other hand, new conditions to produce a ferromagnetic ground state have been established for finite chains. These conditions are expressed as lower bounds for the coupling constants. The particular case in which the coupling constant for third-nearest-neighbor interactions is zero is compared in two different ways with previous calculations by other authors, showing good correspondence.
The Ising model of an antiferromagnet exhibits an anomalous entropy peak at a critical magnetic field which remains even at the absolute zero of temperature, in violation of the third law of thermodynamics. By investigating numerically the behaviour of finite chains of spins interacting through the more general anisotropic Heisenberg coupling, described by the Hamiltonian provided it is shown that the zero point anomalous entropy is peculiar to the Ising model (γ = 0). If the anisotropy is sufficiently great, however, an appreciable entropy peak does persist down to low temperatures although in the anisotropic (pure Heisenberg) limit the entropy has only a broad maximum which falls steadily in height as T approaches zero.
For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.