Spinor Analysis for the Partition Function and the Spin Correlations of the Ising Model on the Helical Lattice

Yoshiaki Tanaka

For the Ising model on the two-dimensional lattice with M × N lattice sites ( M is a finite number; N →∞), a new method is presented by which the two-spin correlations at arbitrary distances are evaluated exactly. We make use of spinor analysis, which was employed by Kaufman and Onsager. The zero-field magnetic susceptibility χ 0 that is equal to the sum of all the two-spin correlations, is calculated.

Barry M. McCoy, Craig A. Tracy, Tai Tsun Wu

N. Sh. Izmailian, Chin‐Kun Hu

We analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is lambda=1 for all values of the helicity factor d . We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.

Hidenori Sonoda

R. Z. Bariev, M.P. Zhelifonov

T. Mei

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.