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arXiv:2206.04393

Corrections to scaling in the 3D Ising model: a comparison between MC and MCRG results

J. Kaupužs, Roderick Melnik

Published Jun 9, 2022• 0 citations
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Abstract

Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to L=3456. Our estimated values of the correction-to-scaling exponent omega tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e., omega about 0.7 and omega = 0.75(5). We discuss the possibilities that omega is either really smaller than usually expected or these values of omega describe some transient behavior. We propose refining MCRG simulations and analysis to resolve this issue. In distinction from omega, our actual MC estimations of the critical exponents eta and nu provide stable values eta=0.03632(13) and nu=0.63017(31), which well agree with those of the conformal bootstrap method, i.e., eta=0.0362978(20) and nu=0.6299709(40).

DOI
10.48550/arxiv.2206.04393

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Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to L=3456. Our estimated values of the correction-to-scaling exponent omega tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e., omega about 0.7 and omega = 0.75(5). We discuss the possibilities that omega is either really smaller than usually expected or these values of omega describe some transient behavior. We propose refining MCRG simulations and analysis to resolve this issue. In distinction from omega, our actual MC estimations of the critical exponents eta and nu provide stable values eta=0.03632(13) and nu=0.63017(31), which well agree with those of the conformal bootstrap method, i.e., eta=0.0362978(20) and nu=0.6299709(40).
We present a status report on the ongoing analysis of the 3D Ising model with nearest-neighbor interactions using the Monte Carlo Renormalization Group (MCRG) and finite size scaling (FSS) methods on 643, 1283, and 2563 simple cubic lattices. Our MCRG estimates are [Formula: see text] and ν = 0.625(1). The FSS results for Kc are consistent with those from MCRG but the value of ν is not. Our best estimate η = 0.025(6) covers the spread in the MCRG and FSS values. A surprise of our calculation is the estimate ω ≈ 0.7 for the correction-to-scaling exponent. We also present results for the renormalized coupling gR along the MCRG flow and argue that the data support the validity of hyperscaling for the 3D Ising model.
We have tested the leading correction-to-scaling exponent omega in O(n)-symmetric models on a three-dimensional lattice by analysing the recent Monte Carlo (MC) data. We have found that the effective critical exponent, estimated at finite sizes of the system L and L/2, decreases remarkably within the range of the simulated L values. This shows the incorrectness of some claims that omega has a very accurate value 0.845(10) at n=1. A selfconsistent infinite volume extrapolation yields row estimates omega=0.547, omega=0.573, and omega=0.625 at n=1, 2, and 3, respectively, in approximate agreement with the corresponding exact values 1/2, 5/9, and 3/5 predicted by our recently developed GFD (grouping of Feynman diagrams) theory. We have fitted the MC data for the susceptibility of 3D Ising model at criticality showing that the effective critical exponent eta tends to increase well above the usually accepted values around 0.036. We have fitted the data within [L;8L], including several terms in the asymptotic expansion with fixed exponents, to obtain the effective amplitudes depending on L. This method clearly demonstrates that the critical exponents of GFD theory are correct (the amplitudes converge to certain asymptotic values at L tending to infinity), whereas those of the perturbative renormalization group (RG) theory are incorrect (the amplitudes diverge). A modification of the standard Ising model by introducing suitable "improved" action (Hamiltonian) does not solve the problem in favour of the perturbative RG theory.
We have tested the theoretical values of critical exponents, predicted for the three--dimensional Heisenberg model, based on the published Monte Carlo (MC) simulation data for the susceptibility. Two different sets of the critical exponents have been considered - one provided by the usual (perturbative) renormalization group (RG) theory, and another predicted by grouping of Feynman diagrams in phi^4 model (our theory). The test consists of two steps. First we determine the critical coupling by fitting the MC data to the theoretical expression, including both confluent and analytical corrections to scaling, the values of critical exponents being taken from theory. Then we use the obtained value of critical coupling to test the agreement between theory and MC data at criticality. As a result, we have found that predictions of our theory (gamma=19/14, eta=1/10, omega=3/5) are consistent, whereas those of the perturbative RG theory (gamma=1.3895, eta=0.0355, omega=0.782) are inconsistent with the MC data. The seemable agreement between the RG prediction for eta and MC results at criticality, reported in literature, appears due to slightly overestimated value of the critical coupling. Estimation of critical exponents of 3D Ising model from complex zeroth of the partition function is discussed. A refined analysis yields the best estimate 1/nu=1.518. We conclude that the recent MC data can be completely explained within our theory (providing 1/nu=1.5 and omega=0.5) rather than within the conventional RG theory.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We present results of a Monte Carlo renormalization-group study of the three-dimensional Ising model on 643{64}^{3}643 and 1283{128}^{3}1283 simple-cubic lattices. The eigenvalues of a linearized transformation matrix, constructed with the use of 53 even operators and 46 odd operators, are shown to be free of truncation errors. Our estimate of the critical coupling is 0.221 652\ifmmode\pm\else\textpm\fi{}0.000 003\ifmmode\pm\else\textpm\fi{}0.000 001 where the first error is statistical and the second due to the finite number of blocking steps. The results for the relevant exponents are \ensuremath{\nu}=0.624(2) and \ensuremath{\eta}=0.026(3). This estimate for \ensuremath{\nu} lies (2--3)\ensuremath{\sigma} below that obtained from other methods. The correction-to-scaling exponent is found to lie in the range \ensuremath{\omega}=0.8--0.85. We also find that the subleading magnetic exponent is relevant and present evidence that it corresponds to a redundant eigenvector of the majority-rule blocking transformation.