Logarithmic Rainbow Free Energy on a Topological Manifold.

In this work, a boundary matrix-product-state technique is employed to extract the universal data of the conformal field theory (CFT) on non-orientable surfaces. On the Klein bottle manifold, we demonstrate that each crosscap boundary contributes a free energy term $\frac{1}{2} \ln{k}$, where $k$ is a universal constant in two-dimensional CFT. On the real projective plane (cross-capped disk), we uncover that, besides the universal crosscap term, the rainbow boundary gives rise to a logarithmic free energy term $\frac{c}{4 \mathcal{A}} \ln{\beta}$, where $\beta$ is the width (or inverse temperature) of the infinite-size system, $c$ is the central charge, and $\mathcal{A}$ is a geometry-dependent coefficient. Our calculations reveal that $\mathcal{A} = \sin{\frac{2\pi}{3}}$ for statistical models on the hexagonal, triangular, and kagome lattices, and $\mathcal{A} = \sin{\frac{2\pi}{4}}$ (i.e., $1$) for the square lattice models as well as 1+1D quantum critical chains.