Critical phenomena in the bifurcation line's perspective

Abstract: On the phase diagram of a system undergoing a continuous phase transition of the second order, three lines, hyper-surfaces, convergent into the critical point feature prominently: the ordered and disordered phases in the thermodynamic limit, and a third line, extending into a domain of finite-size systems, defined by the bifurcation of the distribution of the order parameter. Unlike critical phenomena in the thermodynamic limit devoid of known thermodynamic potential and described rather by the conformal symmetry, in finite-size systems near the bifurcation line an explicit Hamiltonian for the zero-mode of the order parameter is found. It bears the impress of the universality class of the critical point in terms of the two critical exponents: $\beta$ and $\nu$.