Monte Carlo estimation of transverse and longitudinal correlation functions in the model

Correlation Functions, Universal Ratios and Goldstone Mode Singularities inn-Vector Models

“…Indeed, at the smallest fields, the estimates of λ ⊥ for O(4) and O(10) models in Tab. 4 are consistent with the corresponding values 1.955±0.020 and 1.960(10) for n = 4 and 1.9723(90) for n = 10 reported in [13,15]. Moreover, in this case the longitudinal exponent λ = 2λ ⊥ −d, calculated from these asymptotic estimates, is consistent with λ for relatively small k values (k max = 0.28) at the smallest fields h in Tab.…”

“…Here we only note that it is consistent with the Gaussian form atã = m 2 and λ ⊥ = 2, as well as with the general power-law asymptotic ak −λ ⊥ at h → 0 under an appropriate choice ofã =ã(h). We have consideredã as the only fit parameter at a fixed exponent λ ⊥ = 1.955, consistent with the estimation for n = 4 in [13]. The χ 2 /d.o.f.…”

“…Note that this condition is automatically satisfied in (12) and (13) according to G ⊥ (0) = χ ⊥ , since all terms cancel each other after the summation over τ , except only those with k = 0. It is clear that (15) is not exactly consistent with (13), as it can be easily checked by writing down all terms in (13), e. g., at L = 2 (where only terms with ℓ = 0 and ℓ = 1 appear). However, the difference appears to be rather small for large L and small m. In Fig.…”

“…However, our fits are less perfect for larger sizes (longer curves). In Table 1: The estimates of transverse susceptibility χ ⊥ and χ * ⊥ (see text) and the fit parameter m (mass) in (13) for the O(2) model at β = 0.55 and h = 0.00021875 depending on the lattice size L. The values of χ 2 /d.o.f. of the fit are given in the last column.…”

“…However, a disadvantage is that no corrections to scaling are included in (32)-(33). Therefore, the values reported in [12][13][14][15] are preferable as asymptotic estimates. The exponents in Tabs.…”