Thermodynamic quantum phase transition by the structural-stability-based catastrophe theory

The thermodynamic quantum phase transition process is investigated quantitatively by the structural-stability-based catastrophe theory. For a canonical ensemble composed of <i>N</i> identical particles, the cusp catastrophe model is adopted to express the average free energy, and further the general quantum state equation about the pressure is obtained to describe the phase transition process from quantum scale to macro scale by using the dimensionless analysis. Subsequently, the ensemble free energy with considering the interaction potential energy among the particles can be gotten exactly, as well as the canonical partition function and the specific heat capacity of the system. Finally, we take the liquid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup><mml:msub><mml:mi>H</mml:mi> <mml:mi>e</mml:mi></mml:msub> <mml:mn>4</mml:mn></mml:msup> </mml:mrow> </mml:math> as an example to illustrate quantitatively its superfluid phase transition process, and the good agreement between the theoretical results and the classical experimental results verifies this theory, which could also explore quantitatively the quantum many-body problem.