Goldstone mode singularities in O(n) models

Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G_{\perp}(k) \simeq ak^{-\lambda_{\perp}} and G_{\parallel}(k) \simeq bk^{-\lambda_{\parallel}} 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 \lambda_{\perp} and \lambda_{\parallel} are non-trivial (3/2<\lambda_{\perp}<2 and 0<\lambda_{\parallel}<1 in three dimensions) and the ratio bM^2/a^2 (where M is a spontaneous magnetization) is universal. The trivial standard-theoretical values are \lambda_{\perp}=2 and \lambda_{\parallel}=1. Our earlier MC analysis gives \lambda_{\perp}=1.955 \pm 0.020 and \lambda_{\parallel} about 0.9 for the O(4) model. A recent MC estimation of \lambda_{\parallel}, assuming corrections to scaling of the standard theory, yields \lambda_{\parallel} = 0.69 \pm 0.10 for the O(2) model. Currently, we have performed a similar MC estimation for the O(10) model, yielding \lambda_{\perp} = 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 \Delta \lambda_{\perp} = 0.0121(52). It is consistent with the idea that 2-\lambda_{\perp} decreases for large $n$ and tends to zero at n ->\infty. 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.


Introduction
Our work is devoted to the Monte Carlo (MC) investigation of the Goldstone mode effects in n-component vector-spin models (O(n) models), which have O(n) global rotational symmetry at zero external field h. The Hamiltonian H is given by where T is temperature, s i ≡ s(x i ) is the n-component vector of unit length, i. e., the spin variable of the i -th lattice site with coordinate x i , and β is the coupling constant. The summation takes place over all nearest neighbors in the lattice with periodic boundary conditions. The Fourier-transformed longitudinal and transverse correlation functions are whereG ∥ (x) andG ⊥ (x) are the corresponding two-point correlation functions in the coordinate space.
There exist different theoretical predictions for the values of the exponents in these expressions. In a series of theoretical works (e. g., [1][2][3][4][5][6][7]), it has been claimed that these exponents are exactly ρ = 1/2 at d = 3, λ ⊥ = 2 and λ ∥ = 4− d. Here, d is the spatial dimensionality 2 < d < 4. These theoretical approaches are further referred to as the standard theory. More non-trivial universal values are expected according to [8], such that hold for 2 < d < 4. These relations were obtained in [8] by analyzing self-consistent diagram equations for correlation functions without cutting the perturbation series. As introduced in [9, 10], we will call this approach the GFD (grouping of Feynman diagrams) theory. Apart from the mathematical analysis, certain physical arguments were also provided [8] to show that λ ⊥ = 2 could not be the correct result for the X Y model (n = 2) within 2 < d < 4.
Several MC simulations were performed in the past [11][12][13][14] to verify the compatibility of MC data with some standard-theoretical expressions, where the exponents are fixed. In recent years, we performed a series of accurate MC simulations [10,15,16] for remarkably larger lattices than previously with an aim to reexamine the theoretical predictions by evaluating the exponents in (1.7)-(1.9). In particular, lattices of the linear sizes L 512 for n = 2 and L 350 for n = 4 were simulated in our papers [10,15] and [16], respectively. These L values remarkably exceed the largest sizes simulated by other authors, i. e., L = 160 for n = 2 in [13] and L = 120 for n = 4 in [12,14]. In the current work, the O(10) model is simulated up to L = 384.
The relations (1.7) and (1.8) are consistent with MC simulation results for the 3D O(4) model [16], where an estimate λ ⊥ = 1.955 ± 0.020 was found. It was also stated that the behavior of the longitudinal correlation function is well consistent with λ ∥ about 0.9 rather than with the standard-theoretical value λ ∥ = 1. According to (1.9), we have 1/2 < ρ < 1 in three dimensions. It is consistent with the MC estimate ρ = 0.555 (17) for the 3D X Y model [15], which corresponds to λ ⊥ = 1.929(21) according to (1.9). A clear MC evidence that the behavior of G ∥ (k) is not quite consistent with the standard-theoretical predictions has been recently provided [10], where an estimate λ ∥ = 0.69 ± 0.10 has been obtained for the 3D X Y (i. e., 3D O(2)) model (at β = 0.55), assuming corrections to scaling of the standard theory.
In the actual study, we have extended our MC simulations and analysis to include the n = 10 case and to test the n-dependence of the exponents. Apart from the exponents, we have performed here an extended analysis of the O(4) model to verify the expected universality of the ratio bM 2 /a 2 [8], where M ≡ M(+0) is the spontaneous magnetization, a and b are the amplitudes in (1.5) and (1.6).

Simulation results
We simulated the 3D O(10) model by a modified Wolff cluster algorithm, used also in [15,16], and evaluated the Fourier-transformed correlation functions by techniques described in [16]. The standard Wolff cluster algorithm [17] was modified to enable simulations at nonzero external field h. Simple cubic lattices of the linear size up to L = 384 were simulated at β = 3 and h =| h |= h min , 2h min , 4h min , where h min = 0.00021875. The coupling constant β = 3 corresponds to the ordered phase, since the spontaneous magnetization M(+0) is about 0.467 in this case -see section 5 for details. This value of M(+0) is comparable with those for the O(2) and O(4) models in our previous MC simulations [15,16]. The simulation

Estimation of the exponents
Here we estimate the exponents λ ⊥ and λ ∥ , describing the behavior of the correlation functions in the limit k → 0, h → 0, L → ∞, taking the limit L → ∞ at first, followed by h → 0. For this purpose, first we find good approximations of the effective exponents at h → 0, L → ∞, and then fit these k-dependent effective exponents to evaluate their asymptotic values at k → 0. By comparing the simulation results for different L and h, we conclude that the largest-L and smallest-h data for k > k * with a good enough accuracy correspond to the thermodynamic limit at h = +0, i. e., h → 0, L → ∞. We have tested this precisely by looking how the estimates of the effective exponents depend on L and h. This method of analysis was applied in [10,16]. The effective transverse exponent λ eff (k) for the O(4) model was evaluated in [16] from the slope of the lnG ⊥ (k) vs ln k plot within [k, 4k]. Here we use a wider interval -[k, 6k], because we have found that the λ eff (k) data in this case can be perfectly fit by a parabola

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Goldstone mode singularities in O(n) models Another possible source of systematical errors is the existence of non-trivial corrections to scaling, which are not included in (3.1). These are corrections to scaling of the GFD theory [8], represented by an expansion in powers of k 2−λ ⊥ , k λ ⊥ −λ ∥ and k λ ∥ . Nevertheless, the actual estimation, where only the standard-theoretical corrections have been included, is well justified as a test of consistency of the standard theory. The existence of a small correction-to-scaling exponent 2 − λ ⊥ can make the extrapolation of the λ eff (k) plots unreliable. However, since the λ eff (k) data are really well described by a parabola, it might be true that the amplitude of such a correction term is small and the estimate λ ⊥ = 1.9723 (90) is quite reasonable. In any case, this estimation shows a small deviation from the standard-theoretical picture, where (3.1) should hold at small enough k with λ ⊥ = 2. This deviation can be indeed small at n = 10, since λ ⊥ → 2 is expected in the limit n → ∞, corresponding to the known behavior of the spherical model [18].
We have also attempted to evaluate the longitudinal exponent λ ∥ from the G ∥ (k) data within k > k * , where k * is indicated in figure 2 by a vertical dashed line. We have found that the longitudinal effective exponent, extracted from the data within [k, 4k], can be perfectly approximated by a parabola. It leads to an estimate λ ∥ = 0.85 ± 0.06. The error bars indicated here include a statistical standard error as well as a systematical error due to finite-h effects. However, due to a rather large extrapolation gap (from ≈ 1.17 to ≈ 0.85), we consider this estimation as a preliminary one.  We have reexamined the largest-L (L = 350) and smallest-h (h = 0.0003125) data of the O(4) model [16] at β = 1.1 with an aim to evaluate the transverse effective exponent in the same way as for the O(10) model. Like in the n = 10 case, we have verified that the thermodynamic limit at h = +0 is practically reached in this estimation. Besides, we have found a surprising similarity of the λ eff (k) plots, where the effective exponent in both cases was evaluated by fitting the G ⊥ (k) data within [k, 6k] (fits within [k, 4k] were used in [16]). As we can see from figure figure 4 and this estimation, it is quite plausible that a transverse exponent λ ⊥ of the O(10) model is somewhat larger than that of the O(4) model. It is consistent with the idea that 2−λ ⊥ decreases for large n and tends to zero at n → ∞. This behavior is fully consistent with the predictions of [8], but not so well consistent with the standard theory, according to which λ ⊥ is always 2 and, therefore, ∆λ ⊥ = 0 is expected. According to our estimates (λ ⊥ ) n=10 = 1.9723(90) and ∆λ ⊥ = 0.0121(52), we have λ ⊥ = 1.960(10) for n = 4. It perfectly agrees with our earlier estimate λ ⊥ = 1.955 ± 0.020 [16].

The ratio universality test
We have extended the MC analysis of our earlier data [16] for the O(4) model at two different couplings, β = 1.1 and β = 1.2, to test the expected (according to [8]) universality of the ratio bM 2 /a 2 , discussed already in the end of section 1. According to (1.5), (1.6) and (1.8), the universality of bM 2 /a 2 implies that the quantity

Spontaneous magnetization
We estimated the spontaneous magnetization of the 3D O(10) model at β = 3 based on our magnetization data M(h, L) depending on h and L. We observed a rather fast convergence to the thermodynamic limit, e.

Conclusions
In the actual work, the previous MC studies [10,16] of the transverse and longitudinal correlation functions in the 3D O(n) models with n = 2 and n = 4 have been extended, including the n = 10 case (sections 2 and 3). It gives us an important information about the behavior of the exponent λ ⊥ at large n. According to our MC analysis, a self-consistent (within the standard theory) estimation of λ ⊥ for n = 10 shows a small deviation from the standard-theoretical prediction λ ⊥ = 2, yielding λ ⊥ = 1.9723(90) (section 3). The fact that this deviation is quite small can be well understood, since λ ⊥ → 2 is expected at n → ∞ according to the known results for the spherical model, corresponding to this limit. Comparing the plots of the effective transverse exponent at n = 10 and n = 4, it has been stated that these plots are surprisingly similar, i. e., only slightly shifted with respect to each other. The estimation of this shift suggests that the transverse exponent for n = 10 is larger than that for n = 4 by an amount of ∆λ ⊥ = 0.0121(52) (section 3). 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 the ratio bM 2 /a 2 for the O(4) model by analyzing the correlation functions at two different couplings, i. e., β = 1.1 and β = 1.2 (section 4).
The actual MC results are fully consistent with the predictions of the GFD theory [8] (see section 1) and not so well consistent with the standard theory, according to which λ ⊥ is always 2.