Statistical Mechanics of LiquidHe4

The partition function proposed by Feynman for liquid ${\mathrm{He}}^4$, based on his path integral method, is evaluated for a simple cubic lattice considering long-range permutations as well as nearest-neighbor permutations (to which the previous analysis by one of the authors was restricted). The result indicates a second-order phase transition at the $\lambda$ point. The marked improvements over the previous treatment are: (1) the specific heat behaves as $T^3$ near absolute zero, (2) the specific heat peak is more pronounced at the $\lambda$ point, and (3) when triangles are added as possible finite polygons above $T_{\lambda }$ the specific heat just above $T_{\lambda }$ increases over the previous result, showing an improvement. Equating the theoretical $\lambda$ point with the experimental, a value for the effective mass of a helium atom about 1.6 times the normal mass is obtained.