Finite-size corrections to the crosscap overlap in the two-dimensional Ising model

Abstract

We analyze the finite-size corrections to the crosscap overlap in the two-dimensional classical Ising model along its self-dual critical line. Using a fermionic formulation, we express the lattice crosscap overlap in terms of Bogoliubov angles and develop a contour-integral approach by analytically continuing the lattice momentum to the complex plane. This leads to a remarkably simple expression for the crosscap overlap, which demonstrates that the finite-size corrections decay exponentially with system size. We further derive an exact analytical formula for the corresponding decay constant and show that it is determined by the complex singularity structure of the Bogoliubov angle.

Figures (3)

• Figure 1: Left: schematic of the 2D classical Ising model with anisotropic couplings and periodic (crosscap) boundary conditions along the $x$ ($y$) direction. Right: geometric depiction of the cylinder with crosscap boundaries.
• Figure 2: Finite-size corrections to the crosscap overlap for the 2D Ising model along the self-dual line ($K_x=K_y^*$). (a) Logarithm of the deviation $\ln \Delta$ [Eq. \ref{['eq:crosscap-deviation']}] as a function of the system size $N$ for several values of the coupling $K_x$. (b) Decay constant $\alpha$ as a function of $K_x$: numerical estimates (blue dots) extracted from the slopes in (a), compared with the analytical result (red solid line) in Eq. \ref{['eq:decay-const']}.
• Figure 3: Contour integration for analyzing finite-size corrections. The contour $C$ is chosen to be a rectangle path extending infinitely along the imaginary axis, with small arcs around the branch points. It encloses all real-axis poles corresponding to the allowed $k>0$ momenta in the NS sector.