The one-dimensional Ising model with general spin S and either open or periodic boundary conditions has been formulated as a difference equation of order 2S+1 with non-zero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=0, for the open chain, the order of the difference equation is reduced to [S+1]. It is shown that the correlation functions for the open chain satisfy similar difference equations. This result explicitly displays the lack of translational invariance of the open chain. For several cases it is shown that the partition function can be represented as the difference of two Chebyshev polynomials of the second kind.
The one-dimensional Ising model with general spin S and either open or periodic boundary conditions has been formulated as a difference equation of order 2S+1 with non-zero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=0, for the open chain, the order of the difference equation is reduced to [S+1]. It is shown that the correlation functions for the open chain satisfy similar difference equations. This result explicitly displays the lack of translational invariance of the open chain. For several cases it is shown that the partition function can be represented as the difference of two Chebyshev polynomials of the second kind.
The transfer-matrix method is employed to investigate a spin-1/2 Ising chain under open and periodic boundary conditions. It is demonstrated that finite-size Ising chains with antiferromagnetic coupling may exhibit significantly distinct magnetic behavior under open and periodic boundary conditions. While the open Ising chains display intriguing magnetic features regardless of the system size, mainly due to a specific contribution of boundary spins, the magnetic behavior of closed Ising chains depends basically on the number of spins. The closed Ising chains with an odd number of spins are subject to a geometric spin frustration leading to an additional plateau in the magnetization curve, which is naturally absent in the closed Ising chains with an even number of spins. Despite different microscopic origins, the magnetization curves of open and closed Ising chains with an odd number of spins exhibit an identical intermediate plateau, with only small quantitative differences appearing at moderate temperatures, which means that a geometric spin frustration of odd-membered rings is somewhat similar to the effect of open boundaries. The magnetization curves of the open Ising chains with an even number of spins differ drastically from those of the closed Ising chains due to the presence of an additional intermediate magnetization plateau. Furthermore, the initial susceptibility, inverse initial susceptibility, and susceptibility-temperature product are examined in detail as functions of temperature. These magnetic response functions demonstrate that the Curie constant and Weiss temperature represent fundamental characteristics of the magnetic system that are independent of the choice of boundary conditions.
For studying of the finite-size behavior of the Ising model under different boundary conditions we propose an alternative to the standard transfer matrix technique approach based on Abel\`{e}s theorem and Chebyshev polynomials. Using it we easily reproduce the known for periodic boundary conditions results for Lee-Yang zeros, the exact position space renormalization group transformation, etc., and extend them deriving new results for antiperiodic boundary conditions. Note that in the latter case one has a nontrivial order-parameter profile, which we also calculate, where the average value of a given spin depends on the distance from the seam with opposite bond in the system. It is interesting to stress that under both boundary conditions, the one-dimensional case exhibits Schottky anomaly.
An exact alternative solution method of 1D Ising model with Block-spin transformation at H=0
An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as 2=. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that =tanhK, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of {σiσi+1}={si} and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that {σ0σj}=(tanhK)j, here j assumes the values of j=2n, here n is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of N→∞ to =(tanhK)N/2, indicating that the second order phase transition is only possible in the limit of K→∞. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.
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.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.
We investigate a random field of mutually dependent random variables ("spins"), indexed by a finite one-dimensional lattice, called in physical sciences the one-dimensional Ising model, in which the random variables can take only ±1 values (see the text for a precise definition). One of the couplings, termed a "bond," that describes the mutual influence of two adjacent random variables is altered—it does not equal the others, thereby introducing a single "defect" bond. This defect bond represents a localised perturbation within an otherwise uniform system. Utilising the recurrence relations of Chebyshev polynomials and the bijective map between the number of spins and the polynomial index, we present a new method for calculations and systematically explore, using it, the system’s properties across different chain lengths and boundary conditions. As an application, we derive analytical expressions for the dependence of the average values of the random variables on their position within the chain, which we refer to as the "local magnetisation profile". From the findings related to the system with a defect bond, we present a novel result for this profile under free (Dirichlet) boundary conditions and re-derive the corresponding result for antiperiodic boundary conditions.