arXiv:1805.08607

Finite- Size Scaling of Correlation Function

Xin Zhang, Gaoke Hu, Yongwen Zhang, Xiaoteng Li, Xiaosong Chen

Published May 22, 2018• 1 citation

Abstract

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 $η$.

DOI: 10.48550/arxiv.1805.08607

Related papers

Finding critical points and correlation length exponents using finite size scaling of Gini index

Soumyaditya Das, Soumyajyoti Biswas, Anirban Chakraborti +1 more

arXiv:2307.01075Physical review. EFeb 20, 20242 citations

Finite-size scaling analysis of the anisotropic critical behavior of the two-dimensional Ising model under shear

D. Winter, Peter Virnau, Jürgen Horbach +1 more

Europhysics Letters (EPL)Sep 1, 201014 citations

Crossover Finite-Size Scaling Theory and Its Applications in Percolation

Ming Li, Sheng Fang, Jingfang Fan +1 more

arXiv:2412.06228Dec 9, 20241 citation

Fidelity approach to quantum phase transitions: finite-size scaling for the quantum Ising model in a transverse field

Huan-Qiang Zhou, Jian-Hui Zhao, Bo Li

arXiv:0704.2940Journal of Physics A Mathematical and TheoreticalOct 29, 200896 citations

Zeros of Partition Function and Critical Exponents of the 3D Diluted Ising Model

Andrey N. Vakilov

Materials science forumMar 1, 20160 citations

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

Jian-Ping Lv, Wanwan Xu, Yanan Sun +2 more

arXiv:1909.10347National Science ReviewAug 26, 202019 citations

arXiv:1805.08607

Finite- Size Scaling of Correlation Function

Xin Zhang, Gaoke Hu, Yongwen Zhang, Xiaoteng Li, Xiaosong Chen

Published May 22, 2018• 1 citation

arXiv HTML PDF

Abstract

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

DOI: 10.48550/arxiv.1805.08607

Related papers

Finding critical points and correlation length exponents using finite size scaling of Gini index

Soumyaditya Das, Soumyajyoti Biswas, Anirban Chakraborti +1 more

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.

arXiv:2307.01075Physical review. EFeb 20, 20242 citations

Finite-size scaling analysis of the anisotropic critical behavior of the two-dimensional Ising model under shear

D. Winter, Peter Virnau, Jürgen Horbach +1 more

The critical behavior of the two-dimensional Ising Model with non-conserved order parameter in steady-state shear is studied by large-scale Monte Carlo simulations. Studying the structure factor S ( q x , q y ) in the disordered phase, the ratio of correlation length exponents ν x /ν y in the two lattice directions ( x , y ) is estimated, and the critical temperature is determined as a function of the shear rate as T c ( ) − T c (0) ∝ ϕ with ϕ≈0.45. Critical exponents β≈0.37, γ≈1.1, ; ν ⊥ ≈0.46, ν ∥ ≈1.38 are roughly compatible with anisotropic hyperscaling.The critical behavior of the two-dimensional Ising Model with non-conserved order parameter in steady-state shear is studied by large-scale Monte Carlo simulations. Studying the structure factor S ( q x , q y ) in the disordered phase, the ratio of correlation length exponents ν x /ν y in the two lattice directions ( x , y ) is estimated, and the critical temperature is determined as a function of the shear rate as T c ( ) − T c (0) ∝ ϕ with ϕ≈0.45. Critical exponents β≈0.37, γ≈1.1, ; ν ⊥ ≈0.46, ν ∥ ≈1.38 are roughly compatible with anisotropic hyperscaling.

Europhysics Letters (EPL)Sep 1, 201014 citations

Crossover Finite-Size Scaling Theory and Its Applications in Percolation

Ming Li, Sheng Fang, Jingfang Fan +1 more

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

Fidelity approach to quantum phase transitions: finite-size scaling for the quantum Ising model in a transverse field

Huan-Qiang Zhou, Jian-Hui Zhao, Bo Li

We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.

arXiv:0704.2940Journal of Physics A Mathematical and TheoreticalOct 29, 200896 citations

Zeros of Partition Function and Critical Exponents of the 3D Diluted Ising Model

Andrey N. Vakilov

We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

Materials science forumMar 1, 20160 citations

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

Jian-Ping Lv, Wanwan Xu, Yanan Sun +2 more

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 $r^{2-d_c}$ governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances 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 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.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 $r^{2-d_c}$ governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances 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 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.

arXiv:1909.10347National Science ReviewAug 26, 202019 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

η

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.

r^{2-d_c}

governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances 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 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.