Self-Consistent Spectral Potential Mean-Field (SC-SPMF) Method for First-Order Phase Transitions in Condensed Matter

We present a theoretical and numerical study of the Self-Consistent Spectral Potential Mean-Field (SC-SPMF) method for the description of structure and stability of phase transitions in condensed matter. The method extends the original Spectral Potential Mean-Field (SPMF) approach by introducing a fully self-consistent cycle between the Kirkwood relation and the Fisher equation. The SC-SPMF method determines the coordination shell in condensed matter, providing a genuinely predictive framework for the study without requiring approximate functionals or closure relations. The method is applied to the solid-to-liquid and liquid-to-solid phase transitions of argon, a prototypical monatomic system with well-characterized experimental and simulation data. The temperature evolution of key parameters of radial distribution function and potential of mean force clearly identifies the phase transition at approximately 86 K, in agreement with the experimental melting temperature of argon (83.8 K). A central result is the formulation of a spectral criterion for stability: the transition is marked by a change in the curvature of the first eigenvalue of the Fischer equation μ1(T), specifically by the temperature at which the second derivative d2μ1/dT2 becomes positive. This criterion does not rely on the absolute value of μ1, but on a qualitative change in the spectral response of the system, providing a system-independent fingerprint of the onset of the liquid phase. The method captures hysteresis and metastability between heating and cooling paths, with a hysteresis width of approximately 1.6 K, consistent with the first-order nature of the transition. The critical exponent β is found to differ between the two paths: β≈0.36 for the heating (close to the 3D Ising model value of 0.326) and β≈0.51 for the cooling (close to the mean-field value of 0.5). This difference reflects the role of short-range or long-range interactions in determining the effective critical behavior along melting or crystallization paths, respectively. This spectral interpretation offers a new perspective on phase stability, hysteresis, and critical phenomena, making the SC-SPMF method a powerful and original tool for the study of condensed matter.