Exact thermodynamic quantities are derived for a chain of classical spins of arbitrary dimension interacting via an arbitrary isotropic nearest-neighbor exchange in both the annealed and quenched limit, and for both bond and site randomness.
An exact calculation of the thermodynamic properties of a linear chain of classical spins with nearest-neighbor bilinear and biquadratic isotropic random exchange interactions is presented. Both open and closed chains are discussed. Only bond disorder is being considered here. It is shown that in contrast to the uniform system, "dipolar disorder points" as well as "quadrupolar disorder points" with qualitative and quantitative differences from the uniform case occur in this system. This is because the competing dipolar and quadrupolar interactions are further enriched by the random signs of these interactions. It is interesting to point out that much of the analysis can be made by means of a study of the eigenvalues of the "transfer kernel." The method can also be applied to obtain exact results for a linear chain of spins of arbitrary dimensionality interacting through an arbitrary isotropic nearest-neighbor Hamiltonian. In the case of the open chain, the Edwards-Anderson expedient is found to be exact in all the cases considered.
The problem of a system of classical spins of arbitrary dimensionality interacting through an arbitrary, isotropic nearest-neighbor-exchange Hamiltonian is solved exactly for the closed linear chain. The 2m-uple spin-correlation function for spins a distance k apart is shown to be the kth power of the ratio of the mth eigenvalue to the largest one. It is proposed that experimental data be used to find the general functional form of the exchange interaction.
An Ising model with an arbitrary distribution P(J) of exchange interactions J is solved exactly for the annealed case in which the system is allowed to reach complete thermal equilibrium at each temperature. The solution is expressed in terms of an Ising model on the same lattice with a single exchange parameter, allowing an exact solution to be obtained in one and two dimensions. Some feeling for the effect of annealing can be found by examining correlations between exchange interactions on neighboring bonds. Some special distributions P(J) are examined in detail.
Consider a closed, linear chain of N localized spins (each s=21) with nearest-neighbor Ising interaction. Such a chain, exposed to a uniform external magnetic field, will be called the pure host. Replace a single spin in the pure host by an impurity spin which is also subjected to nearest-neighbor Ising interaction and to the external field. The magnitude of the impurity spin, the magnitude of its magnetic moment, and the magnitude and sign of its interaction with the host are allowed to differ from the corresponding values characterizing the host. For the bulk system ($N\ensuremath{\rightarrow}\ensuremath{\infty}$, for constant linear density of spins), the thermodynamic properties, such as the impurity magnetization, the (position-dependent) magnetization of the impurity-host system and the impurity-host spin correlation functions, are obtained exactly in terms of conventional, tabulated functions. Numerical results are presented for impurity spins of magnitude S=23,25.
We study the exact physical quantities of a competing spin chain which contains many interesting and meaningful couplings including the nearest neighbor, next nearest neighbor, chiral three spins, Dzyloshinsky-Moriya interactions and unparallel boundary magnetic fields in the thermodynamic limit. We obtain the density of zero roots, surface energies and elementary excitations in different regimes of model parameters. Due to the competition of various interactions, the surface energy and excited spectrum show many different pictures from those of the Heisenberg spin chain.