Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, $\mathbb{ RP}^2$). Through a cut-and-sew process, $\mathbb{ RP}^2$ is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term $\frac{ 1}{ 2} \ln{ k}$ related to nonorientable genus, with $k$ a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{ c}{ 4} \ln{ \beta}$, with $\beta$ the manifold size and $c$ the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.
Categories: cond-mat.str-el cond-mat.stat-mech math-ph math.MP