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1 Numerical evidence for Goldstone-mode effects in the three-dimensional O(4)-model ∗

Jürgen Engels, Tereza Mendes A

Published Jan 1, 1999• 1 citation
OpenAlex

Abstract

We investigate the three-dimensional O(4)-model on 24(3)-96(3) lattices as a function of the magnetic field H. Below the critical point, in the broken phase, we confirm explicitly the H-1/2 dependence of the magnetization and the corresponding H-1/2 divergence of the longitudinal susceptibility, which are predicted from the existence of Goldstone modes. At the critical point the magnetization follows the expected finite-size-scaling behavior, with critical exponent delta = 4.87(1).

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Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G ⊥ (k) ≃ ak -λ ⊥ and G ∥ (k) ≃ bk -λ ∥ in the ordered phase at k → 0, is performed in the three-dimensional O(n) models with n = 2,4,10.Our aim is to test some challenging theoretical predictions, according to which the exponents λ ⊥ and λ ∥ are non-trivial (3/2 < λ ⊥ < 2 and 0 < λ ∥ < 1 in three dimensions) and the ratio bM 2 /a 2 (where M is a spontaneous magnetization) is universal.The trivial standardtheoretical values are λ ⊥ = 2 and λ ∥ = 1.Our earlier MC analysis gives λ ⊥ = 1.955 ± 0.020 and λ ∥ about 0.9 for the O(4) model.A recent MC estimation of λ ∥ , assuming corrections to scaling of the standard theory, yields λ ∥ = 0.69±0.10for the O(2) model.Currently, we have performed a similar MC estimation for the O(10) model, yielding λ ⊥ = 1.9723(90).We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by ∆λ ⊥ = 0.0121(52).It is consistent with the idea that 2 -λ ⊥ decreases for large n and tends to zero at n → ∞.We have also verified and confirmed the expected universality of bM 2 /a 2 for the O(4) model, where simulations at two different temperatures (couplings) have been performed.

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We investigate the three-dimensional O(4)-model on 24(3)-96(3) lattices as a function of the magnetic field H. Below the critical point, in the broken phase, we confirm explicitly the H-1/2 dependence of the magnetization and the corresponding H-1/2 divergence of the longitudinal susceptibility, which are predicted from the existence of Goldstone modes. At the critical point the magnetization follows the expected finite-size-scaling behavior, with critical exponent delta = 4.87(1).
We investigate numerically the transverse and longitudinal correlation lengths of the three-dimensional O(4) model as a function of the external field H. In the low-temperature phase we verify explicitly the H^{-1/2}-dependence of the transverse correlation length, which is expected due to the Goldstone modes of the model. On the critical line we find the universal amplitude ratio xi^c_T / xi^c_L = 1.99(1). From our data we derive the universal scaling function for the transverse correlation length. The H-dependencies of the correlation lengths in the high temperature phase are discussed and shown to be in accord with the scaling functions.
Critical phenomena and Goldstone mode effects in spin models with the O(n) rotational symmetry are considered. Starting with Goldstone mode singularities in the XY and O(4) models, we briefly review various theoretical concepts, as well as state-of-the-art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for lattices with linear sizes up to L = 1536, which are very large as compared to L ≤ 128 usually used in the finite-size scaling analysis. These results are obtained, using a parallel OpenMP implementation of the Wolff single-cluster algorithm. The finite-size scaling analysis of the critical exponent η, assuming the usually accepted correction-to-scaling exponent ω ≈ 0.8, shows that η is likely to be somewhat larger than the value 0.0335 ± 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: η = ω =1/8 and ν = 2/3, found within the GFD theory.
Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G ⊥ (k) ≃ ak -λ ⊥ and G ∥ (k) ≃ bk -λ ∥ in the ordered phase at k → 0, is performed in the three-dimensional O(n) models with n = 2,4,10.Our aim is to test some challenging theoretical predictions, according to which the exponents λ ⊥ and λ ∥ are non-trivial (3/2 < λ ⊥ < 2 and 0 < λ ∥ < 1 in three dimensions) and the ratio bM 2 /a 2 (where M is a spontaneous magnetization) is universal.The trivial standardtheoretical values are λ ⊥ = 2 and λ ∥ = 1.Our earlier MC analysis gives λ ⊥ = 1.955 ± 0.020 and λ ∥ about 0.9 for the O(4) model.A recent MC estimation of λ ∥ , assuming corrections to scaling of the standard theory, yields λ ∥ = 0.69±0.10for the O(2) model.Currently, we have performed a similar MC estimation for the O(10) model, yielding λ ⊥ = 1.9723(90).We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by ∆λ ⊥ = 0.0121(52).It is consistent with the idea that 2 -λ ⊥ decreases for large n and tends to zero at n → ∞.We have also verified and confirmed the expected universality of bM 2 /a 2 for the O(4) model, where simulations at two different temperatures (couplings) have been performed.
Critical phenomena and Goldstone mode effects in spin models with O(n) rotational symmetry are considered. Starting with the Goldstone mode singularities in the XY and O(4) models, we briefly review different theoretical concepts as well as state-of-the art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for very large lattices with linear sizes up to L=1536. These results are obtained, using a parallel OpenMP implementation of the Wolff single cluster algorithm. The finite-size scaling analysis of the critical exponent eta, assuming the usually accepted correction-to-scaling exponent omega=0.8, shows that eta is likely to be somewhat larger than the value 0.0335 +/- 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: eta=omega=1/8 and nu=2/3, found within the GFD theory.
Correlation functions in the O\mathcal{O}O(n)(n)(n) models below the critical temperature are considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the O\mathcal{O}O(4)(4)(4) model for lattice sizes about L=120L=120L=120 and small external fields hhh is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by certain non-Gaussian approximation. We have also tested larger lattice sizes, up to L=512L=512L=512. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit at k→0k→0k→0 and h=+0h=+0h=+0, i.e., G⊥G_⊥G⊥​(k)≃ak−λ⊥≃ak^{−λ_⊥}≃ak−λ⊥​ and G∣∣(G_{||}(G∣∣​(k)≃bk−λ∣∣)≃bk^{−λ_{||}})≃bk−λ∣∣​, respectively. Here aaa and bbb are the amplitudes, kkk=|k| is the magnitude of the wave vector k. The exponents λ⊥λ_⊥λ⊥​, λ∣∣λ_{||}λ∣∣​ and the ratio bM2/a2bM^2/a^2bM2/a2, where MMM is the spontaneous magnetization, are universal according to the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding bM2/a2bM^2/a^2bM2/a2=(n−1)/16(n−1)/16(n−1)/16. Our MC estimates of this ratio are 0.06±0.010.06±0.010.06±0.01 for n=2n=2n=2, 0.17±0.010.17±0.010.17±0.01 for n=4n=4n=4 and 0.498±0.0100.498±0.0100.498±0.010 for n=10n=10n=10. According to these and our earlier MC results, the asymptotic behavior and Goldstone mode singularities are not exactly described by the standard theory. This is expected from the GFD theory. We have found appropriate analytic approximations for G⊥G_⊥G⊥​(k) and G∣∣G_{||}G∣∣​(k), well fitting the simulation data for small kkk. We have used them to test the Patashinski-Pokrovski relation and have found that it holds approximately.