Conformal thermal tensor network and universal entropy on topological manifolds

Thermal quantum critical systems, with partition functions expressed as conformal tensor networks, are revealed to exhibit universal entropy corrections on nonorientable manifolds. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy ${S}_{\mathcal{K}}=lnk$ on the Klein bottle, where $k$ relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy $lng$ $(g$ reflects noninteger ground-state degeneracy), ${S}_{\mathcal{K}}$ has $no$ boundary dependence or surface energy terms accompanying it, and can be very conveniently extracted from thermal data. On the M\"obius-strip manifold, we uncover an entropy ${S}_{\mathcal{M}}=\frac{1}{2}(lng+lnk)$ in CFT, where $\frac{1}{2}lng$ is associated with the only open edge of the M\"obius strip and $\frac{1}{2}lnk$ is associated with the nonorientable topology. As a useful application, we employ the universal entropy to accurately pinpoint the quantum phase transitions, even for those without local order parameters.