Conformal Thermal Tensor Network and Universal Entropy on Topological Manifolds

Partition functions of quantum critical systems, expressed as conformal thermal tensor networks, are defined on various manifolds which can give rise to universal entropy corrections. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy $S_{\mathcal{K}} = \ln{k}$ 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 $\ln{g}$ ($g$ reflects non-integer groundstate degeneracy), $S_{\mathcal{K}}$ has \textit{no} boundary dependence or surface energy terms accompanied, and can be very conveniently extracted from thermal data. On the M\"obius-strip manifold, we uncover an entropy $S_{\mathcal{M}} = \frac{1}{2} (\ln{g} + \ln{k})$ in CFT, where $\frac{1}{2} \ln{g}$ is associated with the only open edge of the M\"obius strip, and $\frac{1}{2} \ln{k}$ with the non-orientable topology. We employ $S_{\mathcal{K}}$ to accurately pinpoint the quantum phase transitions, even for those without local order parameters.