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

Abstract Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (dc = 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 n-vector model (n = 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 g(r, L), with L the linear size, exhibits a two-length behavior: follows \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$r^{2-d_c}$\end{document} governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$L^{-d_c/2}({\rm ln}L)^{\hat{p}}$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\hat{p}=1/2$\end{document} 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.

We elaborate the quantifications of the finite-size scaling (FSS) mentioned in the main text. Subsequently, we address the location of the critical temperatures T c , the FSS of the susceptibility χ 0 , and the FSS of the two-point correlation g(r, L) with r = L/2.

THE ESTIMATE OF Tc
For each of the four-dimensional Ising, XY, and Heisenberg models, the estimate of T c is achieved by fitting the finite-size Monte Carlo data of the Binder cumulant Q to the scaling ansatz where t ≡ T c − T , for t → 0. Q c is the critical dimensionless ratio and a, b, c are constants. In the fits, the mean-field thermal exponent is fixed as y t = 2 for reducing uncertainty. For the Ising model, as shown in Table I, we perform fits withŷ t being fixed to beŷ t = 4−n 2n+16 = 1/6 as predicated by renormalization-group calculations [1] or being free. After obtaining an estimate of T c , we also perform fits right at T c . Stable fits with χ 2 /DF 1 are achieved for all of these scenarios. As we let Q c free, it is found Q c = 0.45(1), which is close to Q c = 0.456 947 of the complete-graph model [2]. By these fits, we estimatep ≈ 0.5. The fits for Q of the XY and Heisenberg models are summarized in Tables II and III, where the estimates for Q c are again close to the results Q c ≈ 0.635 and 0.728 by Monte Carlo simulations of XY and Heisenberg models on complete graph [3], respectively. Meanwhile, we note that the amplitude of correction b ≈ 0.1 is sizeable for each of the four-dimensional models. Hence, the prediction of this study on the existence of the finite-size correction b(lnL) −p is further confirmed. This correction form is visualized by Fig. 1 for the Ising, XY, and Heisenberg models. The locating of T c is shown in Fig. 2.
The final estimates of T c are determined by comparing the fits, and are T c = 6.680 30(1), 3.314 437(6), and 2.198 79(2), for the Ising, XY, and Heisenberg models, respectively. These estimates can be examined independently using the quantities other than Q, e.g., the magnetization density m.

FSS OF THE SUSCEPTIBILITY χ0
We fit the Monte Carlo data of the susceptibility χ 0 at T c to the ansatz which is an inference drawn from the scaling formulae of free energy density and two-point correlation. The fits are summarized in Table IV   cludes both q 1 and q 2 terms and letp = 1/2 and 2y h − d = 2 fixed, stable fits with χ 2 /DF 1 can be achieved. If the mean-field exponent 2y h − d is free in the fits, we obtain 2y h − d ≈ 2.00 for each model, in perfect agreement with the exact value 2 within error bars. For the Ising and XY models (for which we have Monte Carlo data with L max = 96), we achieve stable fits withp being free, which yieldp ≈ 0.5, consistent with the predictionp = 1/2.

FSS OF THE TWO-POINT CORRELATION g(L/2, L)
We perform FSS for the large-distance plateau of two-point correlation g(r, L), by fitting the finite-size g(L/2, L) data to the ansatz (3)  for the critical XY model. We perform fits under the situations that 2y h − 2d = −2 andp = 1/2 are both fixed, and that only one of them is fixed. The fits are summarized by Table V, which demonstrates that the inclusion of both v 1 and v 2 terms correctly produces the mean-field exponent y h . As an instance, one of the fits yields 2y h − 2d = −1.99(1) with L min = 24 and χ 2 /DF ≈ 0.8, which is in good agreement with the exact 2y h − 2d = −2. By comparing the fits witĥ p = 1/2 being fixed, we estimate y h = 3.01(2). As a further verification, once the mean-field exponent 2y h − 2d = −2 is fixed, the estimatep = 0.