Distinguishing continuous and weakly first-order transitions via the Klein bottle ratio

Determine, using the Klein bottle ratio g, the distinctions between the two-dimensional ferromagnetic 4-state Potts model (which undergoes a continuous transition) and the 5-state Potts model (which exhibits a weakly first-order transition), and, more generally, characterize how the Klein bottle ratio g can be used to diagnose and differentiate continuous versus weakly first-order phase transitions.

Background

The Klein bottle ratio g is a universal quantity originally defined within conformal field theory and can be computed numerically using tensor-network methods. In two-dimensional Potts models, g has been observed to locate and characterize phase transitions: for q=4 the transition is continuous with central charge c≈1, while for q=5 the transition is weakly first-order with pseudo-critical scaling linked to complex CFT fixed points.

The authors find that g exhibits size-dependent behavior and even reasonable data collapse in regimes expected to be first-order (e.g., q=5 and q=6), suggesting that g encodes information beyond standard CFT expectations. This motivates a precise understanding of how g distinguishes the continuous q=4 transition from the weakly first-order q=5 case and, more generally, how g discriminates between continuous and weakly first-order transitions.