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 Möbius-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 Möbius 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.

• Received 17 August 2017
• Revised 1 November 2017

DOI:https://doi.org/10.1103/PhysRevB.96.174429