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

We address the logarithmic finite-size scaling (FSS) of the O(n) symmetry at the upper critical dimensionality. 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 finite-size critical two-point correlation exhibits a two-length behavior, which is governed by Gaussian fixed point at shorter distance, and enters a plateau at larger distance whose height decreases with system size in a power law corrected by a logarithmic exponent. We carry out extensive Monte Carlo simulations for n-vector models, and obtain solid evidence supporting the conjectured scaling forms from the FSS of macroscopic quantities as well as the two-point correlation. According to the quantum-to-classical mapping, the three-dimensional quantum O(n) systems are at the upper critical dimensionality. Hence, the present study is of practical relevance to a large number of experimental systems including quantum magnetic materials, Josephson junction arrays, and ultracold atomic systems.
Ref.: J.-P. Lv, W. Xu, Y. Sun, K. Chen, Y. Deng, National Science Review, nwaa212, 2020 (https://doi.org/10.1093/nsr/nwaa212)