Finite-size scaling of O() systems at the upper critical dimensionality.

Logarithmic finite-size scaling of the O() universality class at the upper critical dimensionality ( = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the -vector model ( = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation (, ), with the linear size, exhibits a two-length behavior: follows [Formula: see text] governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as [Formula: see text] with [Formula: see text] a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.