Crosscap states and duality of Ising field theory in two dimensions

Abstract

We propose two distinct crosscap states for the two-dimensional (2D) Ising field theory. These two crosscap states, identifying Ising spins or dual spins (domain walls) at antipodal points, are shown to be related via the Kramers-Wannier duality transformation. We derive their Majorana free field representations and extend bosonization techniques to calculate correlation functions of the 2D Ising conformal field theory (CFT) with different crosscap boundaries. Away from criticality, we develop a conformal perturbation theory to calculate the Klein bottle entropy (norm-square of the crosscap overlap) as a universal scaling function [Phys. Rev. Lett. 130, 151602 (2023)]. For the Ising field theory, our analytical results support the conjectured monotonicity of the Klein bottle entropy under relevant perturbations. The formalism provides a general framework for studying perturbed 2D CFTs on non-orientable manifolds.

Figures (3)

• Figure 1: Schematics of (a) typical configurations from lattice crosscap states $|\mathcal{C}^+_{\mathrm{latt}}\rangle$ (left panel) and $|\mathcal{C}^-_{\mathrm{latt}}\rangle$ (right panel) in the quantum Ising chain and (b) the conformal transformation from the semi-infinite cylinder with a crosscap boundary to the real projective plane ($\mathbb{RP}^2$).
• Figure S1: Crosscap overlap of the transverse field Ising chain with a longitudinal field compared with the leading-order (2nd order) conformal perturbation prediction. The numerical data with three different chain lengths ($N=20,40,60$), shown with different colored symbols, clearly collapse on the same line, corresponding to the universal scaling function. For three data sets obtained with different chain lengths, parabola fit gives (almost) the same numerical value $\mathcal{C}^{\mathrm{num}}_2 \approx -1.59$.
• Figure S2: Crosscap overlap of the three-state clock model compared with the leading-order (1st order) conformal perturbation prediction. Numerical results with three different chain lengths ($N=20,30,40$) are shown with different colored symbols. For all three data sets, fitting their slopes gives (almost) the same numerical value $\mathcal{C}^{\mathrm{num}}_1 \approx 0.55$.