Chain Ising

In the case of antiferromagnetic Ising chains, there is a great difference between the parallel and perpendicular components of the susceptibility. The susceptibility curves of a ferromagnetic and an antiferromagnetic interaction are illustrated in Figs. 3 and 4. 

The zero-field susceptibihties for the S = /2 Ising chain have been derived by Fisher. They have the same form for both antiferromagnetic and ferromagnetic exchange, only the sign of J is different. 

An exact solution has also been obtained for the Ising chain model, for which the spins are assumed to be aligned along a given z axis, hi this limit, the spins possess an infinite uni-axial magnetic anisotropy with an easy z axis. The corresponding Hamiltonian reads ... 

Therefore, at low temperature and in the ferromagnetic case, the correlation function is obtained by T 1 - 2exp(4j6/S ) and the parallel magnetic susceptibility (x//) of the uniform Ising chain diverges exponentially as ... 

Using theoretical methods specifically adapted to ID problems, hke the matrix transfer method, the field dependence of the magnetization can also be determined [16,17]. In particular, a compact expression is obtained for the infinite Ising chain ... 

Fig.1 Schematic view of a magnetic Ising chain in zero applied magnetic field with large oriented domains of average length separated by sharp domain walls...

In the following sections, we will see that even the simplest presently known SCM system can not be considered as a regular ferromagnetic Ising chain. Therefore, some improvements of the previous models are essential to describe experimental magnetic properties. 

Fig. 11.19. Parallel and perpendicular product functions for an Si = 1/2 Ising chain left—J/k = 10 K right— J/k = 10 K (solid), J/k = 1000K (dashed).

When the Ising chain (SY spins) interacts substantially with another one (5, spins), forming thus a ladder-type structure, the model Hamiltonian is extended for an interchain coupling parameter. In the simplest case of the S) = Si = 1/2 system the representation of the transfer matrix has the dimension k = 4 x 4. Its eigenvalues are then obtained only numerically and thus the magnetisation and the magnetic susceptibility cannot be represented by an analytic function. However, numerical solution is accessible using computers [17]. 

Since the action for each spin path is complex valued, a stochastic Monte Carlo evaluation of the resulting isomorphic Ising chain has to deal with the inevitable dynamical sign problem. Relying on the strategy outlined in Section II, we wish to block configurations together in an optimal way. 

Lacombe, R. H., and Simha, R., Detailed balancing approach to disordered copolymeric Ising chains, J. Chem. Phys., 58, 1043-1053 (1973a). 

To obtain a more quantitative picture of the temperature dependence of tr and R IR, specimens of TPBIO were dissolved in a proprietary mixture of low-molar-mass nematics, designated E5, which has a broad nematic temperature range [Chiang et al., 2000]. Experimental results for two TPBIO specimens having DP= 18 and 63 are shown in Eigure 1.10. Eirst, the temperature dependence of R and R may be considered in the context of pertinent statistical theories. An analysis by Halperin and William [1992] of the conformation of main-chain LCPs using an Ising chain model led to the conclusion that the chain dimensions, and R , each exhibit exponential temperature dependence ... 

Halperin, A., and Williams, D. R. M., Liquid crystalline polymers as ising chains stretching and swelling, Eurvphys. Lett., 20, 601-606 (1992). 

Renormalization Group theory The partition function Q is the central quantity of statistical mechanics and many thermodynamic functions can be derived from it. The partition function of the one-dimensional Ising chain is... 

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