The boundary disorder correlation for the Ising model on a cylinder
Rafael Leon Greenblatt
TL;DR
This work analyzes the boundary disorder correlation for the critical Ising model on a cylinder by exploiting the exact McCoy–Wu solution and a detailed finite-size scaling analysis. The correlation function $\langle \mu_{\text{top}} \mu_{\text{bottom}} \rangle_{\pm}$ is obtained from the ratio $Z_{\mp}/Z_{\pm}$ of partition functions, and its leading finite-size dependence is encoded in the constant term $z_{\pm}(\zeta)$ of $\log Z_{\pm}$, with $\zeta$ the rescaled aspect ratio and $\Xi$ the anisotropy factor. The main result expresses $\langle \mu_{\text{top}} \mu_{\text{bottom}} \rangle_{\pm}$ in terms of Jacobi theta-functions as $\langle \mu_{\text{top}} \mu_{\text{bottom}} \rangle_{\pm} \sim \exp( z_{\mp}(\zeta) - z_{\pm}(\zeta) )$, where $z_{\pm}(\zeta)$ are given by explicit theta-quotients and depend on $\zeta = \Xi \mathcal{M}/\mathcal{N}$. The paper also demonstrates consistency with conformal-field-theory predictions (Cardy) for boundary conditions, including the isotropic and anisotropic cases, with the anisotropy captured by the scaling factor $\Xi$. This provides an explicit lattice-to-CFT bridge for the boundary disorder correlation on the cylinder.
Abstract
I give an expression for the correlation function of disorder insertions on the edges of the critical Ising model on a cylinder as a function of the aspect ratio (rescaled in the case of anisotropic couplings). This is obtained from an expression for the finite size scaling term in the free energy on a cylinder in periodic and antiperiodic boundary conditions in terms of Jacobi theta functions.