TY - JOUR
T1 - Correlation Functions, Universal Ratios and Goldstone Mode Singularities in $n$-Vector Models
AU - Kaupužs , J.
AU - V. N. Melnik , R.
AU - Rimšāns , J.
JO - Communications in Computational Physics
VL - 5
SP - 1407
EP - 1430
PY - 2014
DA - 2014/05
SN - 15
DO - http://doi.org/10.4208/cicp.260613.301013a
UR - https://global-sci.org/intro/article_detail/cicp/7143.html
KW - 
AB - <p style="text-align: justify;">Correlation functions in the $\mathcal{O}$$(n)$ models below the critical temperature are
considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the $\mathcal{O}$$(4)$ model for
lattice sizes about $L=120$ and small external fields $h$ 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 $L=512$. The Fourier-transformed transverse and longitudinal two-point correlation
functions have Goldstone mode singularities in the thermodynamic limit at $k→0$ and $h=+0$, i.e., $G_⊥$(<strong>k</strong>)$≃ak^{−λ_⊥}$ and&nbsp;$G_{||}($<strong>k</strong>$)≃bk^{−λ_{||}}$, respectively. Here $a$ and $b$ are the amplitudes, $k$=|<strong>k</strong>| is the magnitude of the wave vector <strong>k</strong>. The exponents $λ_⊥$, $λ_{||}$ and the
ratio $bM^2/a^2$, where $M$ 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 $bM^2/a^2$=$(n−1)/16$. Our
MC estimates of this ratio are $0.06±0.01$ for $n=2$, $0.17±0.01$ for $n=4$ and $0.498±0.010$ for $n=10$. 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 $G_⊥$(<strong>k</strong>) and $G_{||}$(<strong>k</strong>), well fitting the simulation data for small $k$. We have used them
to test the Patashinski-Pokrovski relation and have found that it holds approximately.</p>