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Physical review. B, Solid state

Spin Correlation and Entropy

Harold Falk, Masuo Suzuki

Published Apr 1, 1970• 7 citations
Publisher

Abstract

A cumulant-like expansion for the entropy of an $N$-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 $N$-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.

DOI
10.1103/physrevb.1.3051

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The Kirkwood expansion for spin systems involving a priori averages within states of a given long-range order in the cumulant expansion of the free energy is examined. By defining a generating function for the averages the inconvenient averages can be replaced by a suitable canonical average in the thermodynamic limit. This modified Kirkwood expansion is generalized to the case of arbitrary spin. Simple applications to the Ising model and the spin one model of 3He-4He mixtures are illustrated. A two-spin cluster approximation for the Ising model gives estimates of the critical temperature in good agreement with the Bethe-Peierls approximation.
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Abstract An even-site spins- linear chain of cyclic in the nearest-neighbour and next-nearest-neighbour interactions has necessarily a three-ring structure. In this study, the interactions are considered to be short-range through the coupling constants J and K , respectively. For an arbitrary even number of spins- , the exact classical solution of the partition function and the pair correlations between the nearest- and next-nearest-neighbours are calculated in terms of the constants J and K , without using the energy levels and their degeneracy factors. Then, this partition function is used to verify the validity of an ansatz on the allowed energies of the chain and their corresponding degeneracy factors. The influence of next-nearest-neighbour interactions K on the specific heat capacity per spin at the low-temperatures is analysed for the ferromagnetic and antiferromagnetic chains. We especially show that specific heat capacity exhibits the spin frustration when the ferromagnetic and antiferromagnetic orders are demanded via the coupling constants J and K , respectively. Also, the maximal frustration is obtained analytically by two distinct ground levels for even number of spins- .
Recently Rojas et al. [J. Math. Phys. 43, 1390 (2002)] obtained a closed analytical expression for the coefficients of arbitrary order in the cumulant expansion of a one-dimensional periodic chain model with nearest-neighbor interaction and spatial translation invariance; that approach can be applied equally well to both nonintegrable and exact integrable models. Here, we obtain the exact analytic expressions for the six lowest-order terms of the high-temperature expansion of the Helmholtz free energy per site of the nonintegrable one-dimensional spin-111 XXZ\mathrm{XXZ}XXZ Heisenberg model. Our analytical results for the specific heat and the static magnetic susceptibility are compared, up to order ${\ensuremath{\beta}}^{6}$ for the ferromagnetic and antiferromagnetic phases, with the respective numerical results of a periodic chain with ten sites (N=10)(N=10)(N=10) and numerical results in the literature [Bl\"ote, Physica B 79, 427 (1975)]. A very good agreement is obtained for both phases in the high-temperature regime.