Study of One-DimensionalXYModel by the Transfer-Matrix Method

The Suzuki-Trotter formula has been used to get the $m$ th approximant to the classical representation of the partition function of the $N$-spin $S=\frac{1}{2}$ $\mathrm{XY}$ chain. The equivalent two-dimensional ($N\times 2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for $T>~0.05$ and $m<~8$. The limiting $m\to \infty$ behavior has been estimated from the extrapolation formula of the form $E\left(m\right)=E\left(\infty \right)+\frac{a}{m^2}$. The limiting value of the energy agrees with the exact solution of Katsura.