To determine the universality class of critical phenomena, we propose a method of statistical inference in the scaling analysis of critical phenomena. The method is based on Bayesian statistics, most specifically, the Gaussian process regression. It assumes only the smoothness of a scaling function, and it does not need a form. We demonstrate this method for the finite-size scaling analysis of the Ising models on square and triangular lattices. Near the critical point, the method is comparable in accuracy to the least-square method. In addition, it works well for data to which we cannot apply the least-square method with a polynomial of low degree. By comparing the data on triangular lattices with the scaling function inferred from the data on square lattices, we confirm the universality of the finite-size scaling function of the two-dimensional Ising model.To determine the universality class of critical phenomena, we propose a method of statistical inference in the scaling analysis of critical phenomena. The method is based on Bayesian statistics, most specifically, the Gaussian process regression. It assumes only the smoothness of a scaling function, and it does not need a form. We demonstrate this method for the finite-size scaling analysis of the Ising models on square and triangular lattices. Near the critical point, the method is comparable in accuracy to the least-square method. In addition, it works well for data to which we cannot apply the least-square method with a polynomial of low degree. By comparing the data on triangular lattices with the scaling function inferred from the data on square lattices, we confirm the universality of the finite-size scaling function of the two-dimensional Ising model.

Feb 21, 20110 citations

Finite-Size Scaling in Hamiltonian Field Theory

CJ Hamer, Michael N. Barbert

Current physics sources and commentsJan 1, 19880 citations

Bayesian inference in the scaling analysis of critical phenomena

Kenji Harada

arXiv:1102.4149Physical Review ENov 18, 201199 citations

Modular Invariance, Lattice Field Theories, and Finite Size Corrections

C. Nash, Denjoe O’Connor

arXiv:hep-th/9606137Annals of PhysicsApr 1, 199917 citations

Lattice versions of quantum field theory models in two dimensions

Anatoli G. Izergin, V. E. Korepin

Nuclear Physics BJun 1, 1982189 citations

The Phase Transition of the Spin-1/2 Heisenberg Model with a Spatially Staggered Anisotropy on the Square Lattice

Fu-Jiun Jiang

Puzzled by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice as suggested in \cite{Wenzel08}, we re-investigate the phase transition of this model induced by dimerization. We focus on studying the finite-size scaling of the observables $ρ_{s1} L$ and $ρ_{s2} L$ , where $L$ stands for the spatial box sizes used in the simulations and $ρ_{si}$ with $i \in \{1,2\}$ is the spin-stiffness in $i$ -direction. We find by performing finite-size scaling using the observable $ρ_{s2} L$ , which corresponds to the spatial direction with a fixed antiferromagnetic coupling, one would suffer a much less severe correction compared to that of using $ρ_{s1} L$ . Therefore $ρ_{s2} L$ is a better quantity than $ρ_{s1} L$ for finite-size scaling analysis concerning the limitation for the availability of large volumes data in our study. Remarkably, by employing the method of fixing the aspect-ratio of spatial winding numbers squared in the simulations, even from $ρ_{s1} L$ which receives the most serious correction among the observables considered in this study, we arrive at a value for the critical exponent $ν$ which is consistent with the expected O(3) value by using only up to $L = 64$ data points.Puzzled by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice as suggested in \cite{Wenzel08}, we re-investigate the phase transition of this model induced by dimerization. We focus on studying the finite-size scaling of the observables $ρ_{s1} L$ and $ρ_{s2} L$, where $L$ stands for the spatial box sizes used in the simulations and $ρ_{si}$ with $i \in \{1,2\}$ is the spin-stiffness in $i$-direction. We find by performing finite-size scaling using the observable $ρ_{s2} L$, which corresponds to the spatial direction with a fixed antiferromagnetic coupling, one would suffer a much less severe correction compared to that of using $ρ_{s1} L$. Therefore $ρ_{s2} L$ is a better quantity than $ρ_{s1} L$ for finite-size scaling analysis concerning the limitation for the availability of large volumes data in our study. Remarkably, by employing the method of fixing the aspect-ratio of spatial winding numbers squared in the simulations, even from $ρ_{s1} L$ which receives the most serious correction among the observables considered in this study, we arrive at a value for the critical exponent $ν$ which is consistent with the expected O(3) value by using only up to $L = 64$ data points.

arXiv:0911.0653Nov 3, 20090 citations

ρ_{s1} L

and

ρ_{s2} L

, where

L

stands for the spatial box sizes used in the simulations and

ρ_{si}

with

i \in \{1,2\}

is the spin-stiffness in

i

-direction. We find by performing finite-size scaling using the observable

ρ_{s2} L

, which corresponds to the spatial direction with a fixed antiferromagnetic coupling, one would suffer a much less severe correction compared to that of using

ρ_{s1} L

. Therefore

ρ_{s2} L

is a better quantity than

ρ_{s1} L

for finite-size scaling analysis concerning the limitation for the availability of large volumes data in our study. Remarkably, by employing the method of fixing the aspect-ratio of spatial winding numbers squared in the simulations, even from

ρ_{s1} L

which receives the most serious correction among the observables considered in this study, we arrive at a value for the critical exponent

ν

which is consistent with the expected O(3) value by using only up to

L = 64

data points.