The general Wiedemann–Franz law derived by the structural-stability-based swallowtail catastrophe model

Abstract The Wiedemann–Franz (WF) law, a foundational principle in condensed matter physics, posits a direct proportionality between the electronic contributions to thermal and electrical conductivity. However, recent experimental observations, particularly in materials like vanadium dioxide (VO 2 ), have revealed significant violations of this law. Furthermore, a growing body of evidence indicates its inapplicability under cryogenic conditions, highlighting the need for a more comprehensive theoretical framework. This study addresses this discrepancy by deriving a generalized WF law. Beginning with the thermodynamic potential of a relativistic degenerate electron gas and electron density functions, we employ a swallowtail catastrophe model grounded in structural stability principles, in conjunction with non-dimensional analysis. The resulting formulation yields a general WF law that correctly satisfies the boundary conditions at both cryogenic and infinite-temperature limits. Our findings reveal that the relationship between thermal and electrical conductivity is fundamentally nonlinear, intrinsically dependent on the material’s relaxation time and chemical potential. To validate our framework, we have applied the derived law to calculate the thermal conductivity of VO 2 , achieving excellent agreement with previously reported experimental data. This generalized WF law provides a robust theoretical tool for analyzing electron transport phenomena and holds significant potential for investigating phase transitions in metals and superconductors.