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 SK=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), SK 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 SM=12(lng+lnk) in CFT, where 12lng is associated with the only open edge of the Möbius strip and 12lnk 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.

Phys. Rev. B