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Logarithmic finite-size scaling of the O($n$) universality class at the uppercritical dimensionality ($d_c=4$) has a fundamental role in statistical andcondensed-matter physics and important applications in various experimentalsystems. Here, we address this long-standing problem in the context of the$n$-vector model ($n=1, 2, 3$) on periodic four-dimensional hypercubiclattices. We establish an explicit scaling form for the free energy density,which simultaneously consists of a scaling term for the Gaussian fixed pointand another term with multiplicative logarithmic corrections. In particular, weconjecture that the critical two-point correlation $g(r,L)$, with $L$ thelinear size, exhibits a two-length behavior: following the behavior $r^{2-d_c}$governed by Gaussian fixed point at shorter distance and entering a plateau atlarger distance whose height decays as $L^{-d_c/2}({\rm ln}L)^{\hat{p}}$ with$\hat{p}=1/2$ a logarithmic correction exponent. Using extensive Monte Carlosimulations, we provide complementary evidence for the predictions through thefinite-size scaling of observables including the two-point correlation, themagnetic fluctuations at zero and non-zero Fourier modes, and the Bindercumulant. Our work sheds light on the formulation of logarithmic finite-sizescaling and has practical applications in experimental systems.

Finite-size Scaling of O($n$) Systems at the Upper Critical Dimensionality