Finite-size scaling (FSS) for a critical phase transition ( $t=0$ ) states that within a window of size $|t|\sim L^{-1/ν}$ , the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})$ , where $ν$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/ν}$ . We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with $t \to 0$ and $|x|\to\infty$ . This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-λ}$ with $λ<1/ν$ , the FSS becomes $λ$ -dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality $t=0$ can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.Finite-size scaling (FSS) for a critical phase transition ($t=0$) states that within a window of size $|t|\sim L^{-1/ν}$, the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})$, where $ν$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/ν}$. We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with $t \to 0$ and $|x|\to\infty$. This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-λ}$ with $λ<1/ν$, the FSS becomes $λ$-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality $t=0$ can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.

arXiv:2412.06228Dec 9, 20241 citation

Finite size effects at the critical point

J. D. Gunton

Physics Letters AMar 1, 196812 citations

Scaling Theory for Finite-Size Effects in the Critical Region

Michael Ellis Fisher, Michael N. Barber

Current physics sources and commentsJan 1, 19882 citations

Finite- Size Scaling of Correlation Function

Xin Zhang, Gaoke Hu, Yongwen Zhang +2 more

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 and Universality above the Upper Critical Dimensionality

Erik Luijten, Henk W. J. Blöte

Physical Review LettersMay 6, 199630 citations

Finite-size scaling for quantum criticality above the upper critical dimension: Superfluid–Mott-insulator transition in three dimensions

Yasuyuki Kato, Naoki Kawashima

The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is z=2. We consider the N -component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large- N limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of N=1.The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is z=2. We consider the N -component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large- N limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of N=1.

arXiv:0910.0082Physical Review EJan 19, 20107 citations

Finite-size scaling (FSS) for a critical phase transition (

t=0

) states that within a window of size

|t|\sim L^{-1/ν}

, the scaling behavior of any observable

Q

in a system of linear size

L

asymptotically follows a scaling form as

Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})

, where

ν

is the correlation-length exponent,

Y_Q

is an FSS exponent and

{\tilde Q}(x)

is a function of the scaled distance-to-criticality

x \equiv tL^{1/ν}

. We systematically study the asymptotic scaling behavior of

{\tilde Q}(|x|\to\infty)

for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with

t \to 0

and

|x|\to\infty

. This crossover FSS theory predicts that when the criticality is approached at a slower speed as

|t|\sim L^{-λ}

with

λ<1/ν

, the FSS becomes

λ

-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality

t=0

can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.

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