Thermodynamics on the space-time Klein bottle——from the wonder trip in the flatland to the path integral of conformal quantum states

Due to the distinct topology of non-orientable surfaces (such as the Klein bottle and Möbius band), a 2D chiral creature changes its parity after travelling around the manifold. When we consider a space-time journey of quantum states in conformal field theory (CFT) on non-orientable world sheets, we find that the Euclidean path integral of CFT on non-orientable surfaces gives rise to intriguing universal thermal properties. For instance, on the Klein bottle there exists a universal entropy in the free energy which can be exploited to locate phase transition points and identify their corresponding CFTs. Universal thermal data of the Klein bottle, as well as other non-orientable surfaces, can play an important role in studying critical phenomena in condensed matter and statistical physics. We also anticipate their potential applications and far-reaching implications in other areas, such as holographic blackhole thermodynamics, and so forth.