Thermodynamic Crossovers in Supercritical Matter: One, No One, and One Hundred Thousand

Beyond a critical point, two thermodynamic phases become continuously connected, without being separated by any thermodynamic singularity. The paradigmatic example is the liquid-gas transition, where several criteria have been proposed to distinguish liquid-like and gas-like regimes within the supercritical region. Recently, a pair of thermodynamic crossover lines, $L^\pm$, was introduced to address a proposed incompatibility of previous definitions with the universal critical behavior of the Ising model. These lines divide the supercritical region into three regimes, meet only at the critical point, and obey the expected critical scaling laws. Here, we demonstrate that these requirements do not uniquely determine the crossover boundaries. Instead, infinitely many pairs of lines satisfy the same conditions, with different choices of thermodynamic paths selecting different crossover boundaries. We further show that the Lee-Yang formalism, despite providing an intrinsic characterization of critical singularities, does not resolve this ambiguity in defining real-axis crossover boundaries. Restricting to regular probing paths that respect the $Z_2$ symmetry of the critical point recovers the Widom and $L^\pm$ prescriptions asymptotically. This restriction, however, is a choice of protocol rather than a property of the system, and even then two inequivalent prescriptions survive. Our results demonstrate that universal critical behavior alone does not select privileged supercritical thermodynamic crossover boundaries. While such lines may identify physically meaningful changes in thermodynamic, structural, or transport properties, their precise locations and the widths of the regions they delimit remain dependent on the operational criteria used to define them.