Exact solution of the critical Ising model with special toroidal boundary conditions

TL;DR

This paper derives exact solutions for the critical 2D Ising model with special toroidal boundary conditions involving a defect line, providing explicit eigenvalues, partition function, and universal finite-size correction ratios.

Contribution

It introduces duality twisted boundary conditions in the Ising model, deriving exact eigenvalues and partition functions, and analyzes universal finite-size correction ratios.

Findings

Exact eigenvalues and partition function for the Ising model with defect line boundary conditions.

Universal ratio of finite-size correction terms in free energy and correlation lengths.

Verification of universality via conformal perturbation theory.

Abstract

The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary condition, which we call duality twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along non-contractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of the transfer matrix for the critical ferromagnetic Ising model on the M x N square lattice wrapped on the torus with a specific defect line. As result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of perturbating conformal approach by calculating the universal structure constant Cn1n for descendent states generated by the operator product expansion (OPE) of the primary fields. For such states the calculations of the universal structure constants is difficult task, since its involve the knowledge of the four-point correlation function, which in general does not fix by conformal invariance except for some particular cases, including the Ising model.