We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance ${\bf r}$ in the finite system with size $L$ , the correlation function can be written as the product of $|{\bf r}|^{-(d-2+η)}$ and its finite-size scaling function of variables ${\bf r}/L$ and $tL^{1/ν}$ , where $t=(T-T_c)/T_c$ . The directional dependence of correlation function is nonnegligible only when $|{\bf r}|$ becomes compariable with $L$ . This finite-size scaling of correlation function has been confirmed by correlation functions of the Ising model and the bond percolation in two-diemnional lattices, which are calculated by Monte Carlo simulation. We can use the finite-size scaling of correlation function to determine the critical point and the critical exponent $η$ .We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance ${\bf r}$ in the finite system with size $L$, the correlation function can be written as the product of $|{\bf r}|^{-(d-2+η)}$ and its finite-size scaling function of variables ${\bf r}/L$ and $tL^{1/ν}$, where $t=(T-T_c)/T_c$. The directional dependence of correlation function is nonnegligible only when $|{\bf r}|$ becomes compariable with $L$. This finite-size scaling of correlation function has been confirmed by correlation functions of the Ising model and the bond percolation in two-diemnional lattices, which are calculated by Monte Carlo simulation. We can use the finite-size scaling of correlation function to determine the critical point and the critical exponent $η$.

arXiv:1805.08607May 22, 20181 citation

Finite-Size Scaling at Criticality

WORLD SCIENTIFIC eBooksAug 1, 20000 citations

Finite-size scaling of correlation functions in finite systems

Xin Zhang, Gaoke Hu, Yongwen Zhang +2 more

Science China Physics Mechanics and AstronomyOct 8, 201815 citations

Finite-size scaling of the correlation length above the upper critical dimension in the five-dimensional Ising model

Jeff L. Jones, A. P. Young

We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size $L$ as ${L}^{5∕4}$ , rather than proportional to $L$ , as in standard finite-size scaling (FSS) theory. Our results confirm a hypothesis that FSS expressions in dimension $d$ greater than the upper critical dimension of 4 should have $L$ replaced by ${L}^{d∕4}$ for cubic samples with periodic boundary conditions. We also investigate numerically the logarithmic corrections to FSS in $d=4$ .We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size $L$ as ${L}^{5∕4}$, rather than proportional to $L$, as in standard finite-size scaling (FSS) theory. Our results confirm a hypothesis that FSS expressions in dimension $d$ greater than the upper critical dimension of 4 should have $L$ replaced by ${L}^{d∕4}$ for cubic samples with periodic boundary conditions. We also investigate numerically the logarithmic corrections to FSS in $d=4$.

arXiv:cond-mat/0412150Physical Review BMay 31, 200551 citations

When correlations exceed system size: ﬁnite-size scaling in free boundary conditions above the upper critical dimension

Yulian Honchar, Bertrand Berche, Yurij V. Holovatch +1 more

We progress finite-size scaling in systems with free boundary conditions above their upper critical dimension, where in the thermodynamic limit critical scaling is described by mean-field theory. Recent works show that the correlation length is not bound by the system's physical size, a belief that long held sway. Instead, two scaling regimes can be observed — at the critical and pseudo-critical temperatures. We demonstrate that both are manifest for free boundaries. We use numerical simulations of the d = 5 Ising model to analyse the magnetization, susceptibility, magnetization Fourier modes and the partition function zeros. While some of the response functions hide the dual finite-size scaling, the precision enabled by the analysis of Lee–Yang zeros allows this be brought to the fore. In particular, finite-size scaling of leading zeros at the pseudo-critical point confirms recent predictions coming from correlations exceeding the system size. This paper is dedicated to Jaroslav Ilnytskyi on the occasion of his 60th birthday.We progress finite-size scaling in systems with free boundary conditions above their upper critical dimension, where in the thermodynamic limit critical scaling is described by mean-field theory. Recent works show that the correlation length is not bound by the system's physical size, a belief that long held sway. Instead, two scaling regimes can be observed — at the critical and pseudo-critical temperatures. We demonstrate that both are manifest for free boundaries. We use numerical simulations of the d = 5 Ising model to analyse the magnetization, susceptibility, magnetization Fourier modes and the partition function zeros. While some of the response functions hide the dual finite-size scaling, the precision enabled by the analysis of Lee–Yang zeros allows this be brought to the fore. In particular, finite-size scaling of leading zeros at the pseudo-critical point confirms recent predictions coming from correlations exceeding the system size. This paper is dedicated to Jaroslav Ilnytskyi on the occasion of his 60th birthday.

arXiv:2311.11721Condensed Matter PhysicsMar 28, 20242 citations

We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance

{\bf r}

in the finite system with size

L

, the correlation function can be written as the product of

|{\bf r}|^{-(d-2+η)}

and its finite-size scaling function of variables

{\bf r}/L

and

tL^{1/ν}

, where

t=(T-T_c)/T_c

. The directional dependence of correlation function is nonnegligible only when

|{\bf r}|

becomes compariable with

L

. This finite-size scaling of correlation function has been confirmed by correlation functions of the Ising model and the bond percolation in two-diemnional lattices, which are calculated by Monte Carlo simulation. We can use the finite-size scaling of correlation function to determine the critical point and the critical exponent

η

We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size

L

{L}^{5∕4}

, rather than proportional to

L

, as in standard finite-size scaling (FSS) theory. Our results confirm a hypothesis that FSS expressions in dimension

d

greater than the upper critical dimension of 4 should have

L

replaced by

{L}^{d∕4}

for cubic samples with periodic boundary conditions. We also investigate numerically the logarithmic corrections to FSS in

d=4