The critical Ising model on a torus with a defect line

TL;DR

This paper provides the first exact solution for the critical 2D Ising model with a defect line on a torus, deriving eigenvalues, partition function, and finite-size corrections, and confirms universality of amplitude ratios.

Contribution

It offers the first exact analytical results for the Ising model with a defect line, including eigenvalues, partition function, and universal amplitude ratios.

Findings

Finite-size corrections follow specific power laws with integer exponents.

Amplitude ratios are universal and verified through conformal perturbation theory.

Exact solutions for eigenvalues and partition function with twisted boundary conditions.

Abstract

The critical Ising model in two dimensions with a defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference $L_x$. We find that finite-size corrections to scaling are of the form $a_k/L^{2k-1}_x$ for the free energy $f$ and $b_k(p)/L_x^{2k-1}$ and $c_k(p)/L_x^{2k-1}$ for inverse correlation lengths $\xi^{-1}_p$ and $\xi^{-1}_{L-p}$, respectively, with integer values of $k$. By exact evaluation we find that the amplitude ratios $b_k(p)/a_k$ and $c_k(p)/a_k$ are universal and verify this universal behavior using a perturbative conformal approach.