Abstract
We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation functions of the model with periodic-periodic boundary conditions on a finite lattice. The corresponding forms in the thermodynamic limit are presented, which show the short-range order. Then, we present the exact expression of the correlation function of the two farthest pair of spins in a column of the model with periodic-free boundary conditions on a finite lattice. Again, the corresponding form in the thermodynamic limit is discussed, from which the long-range order clearly emerges as the temperature decreases.
Highlights
Since the exact solution of the partition function in the absence of a magnetic field of the two-dimensional rectangular Ising model with periodic-periodic boundary conditions is obtained in the thermodynamic limit [1] and in finite-size systems [2], many authors have contributed to the knowledge of various aspects of this model, such as different boundary conditions, the arrangement modes of the spin lattice, surfaces, or mathematical methods, etc. [3,4,5,6,7]
Besides the partition function of the model, the calculations of spin-spin correlation functions are an important subject in the research of the two-dimensional Ising model
We present some exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice by employing the spinor analysis method [2]
Summary
Since the exact solution of the partition function in the absence of a magnetic field of the two-dimensional rectangular Ising model with periodic-periodic boundary conditions is obtained in the thermodynamic limit [1] and in finite-size systems [2], many authors have contributed to the knowledge of various aspects of this model, such as different boundary conditions, the arrangement modes of the spin lattice, surfaces, or mathematical methods, etc. [3,4,5,6,7]. Since RSPT is irregular, the approximate method is incapable when Q is very larger By this approach we can only obtain exact expressions of correlation functions when Q is a small number, for example, that of hσ , σ1 , 2 i , hσ , σ1 , 3 i , hσ , σ1 , 4 i , · · · , etc., which belong to the short-range order. After obtaining the exact expression of σl , 1 σl , N , we discuss the properties of the expression of σl , 1 σl , N in the thermodynamic limit, from which the long-range order emerges as the temperature decreases, as shown clearly
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