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Progress of Theoretical Physics

Domain Structure in a Second-Neighbour Ising Chain

Norman E. Frankel, Dennis C. Rapaport

Published May 1, 1970• 6 citations
Publisher

Abstract

Two different methods of obtaining the partition function for a spin ½ open ended Ising chain with interactions between nearest and second nearest neighbour spins are presented. These are non-matrix methods, in that they do not require the diagonalization of a transfer matrix, and they provide a useful representation for studying the analytical structure of the partition function. The pair correlation functions are also derived, using the second of these methods. These correlation functions are utilized in a study of the ground state and the finite temperature spin configurations. An interesting application of these results to the problem of out-of-phase domain structure in binary alloys is discussed.

DOI
10.1143/ptp.43.1170

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Two different methods of obtaining the partition function for a spin ½ open ended Ising chain with interactions between nearest and second nearest neighbour spins are presented. These are non-matrix methods, in that they do not require the diagonalization of a transfer matrix, and they provide a useful representation for studying the analytical structure of the partition function. The pair correlation functions are also derived, using the second of these methods. These correlation functions are utilized in a study of the ground state and the finite temperature spin configurations. An interesting application of these results to the problem of out-of-phase domain structure in binary alloys is discussed.
A one-dimensional Ising chain in which nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) exchange interactions depend linearly on the distance between the spins is considered. The magnetic field and temperature dependence of the chain length is exactly calculated by use of a transfer matrix. It is shown that the chain length changes variously because of the competition among the n.n. and n.n.n. exchange interactions and the magnetic field. The zero-temperature phase diagram is also determined, which is composed of four spin states ↑, ↑↓, ↑↑↓ and ↑↑↓↓ (the smallest units of repetitions are shown). Dimerization and trimerization of the lattice are found in the ↑↑↓↓ and ↑↑↓ states respectively.
Transfer matrix methods are used to obtain the partition function and correlation length of Ising chains at low temperatures. For the infinite-spin chain, asymptotically exact eigenvalues and eigenvectors are obtained from an iteration scheme, and from a secular equation method. It is also shown how the Ising chain for general spin S (including S to infinity ) may be treated using a 2*2 transfer matrix. For low temperatures (K identical to J/K B T>>1) the inverse correlation length, in units of the spin spacing, is xi -1 =4S e -2K +O(e -4K ) for finite S, and xi -1 =8K e -2K (ln 4K+ gamma +O((lnK)/K)) for infinite spin where gamma is Euler's constant.
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- .
A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest-neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.