Advanced Monte Carlo Study of the Goldstone Mode Singularity in the 3D XY Model

Advanced Monte Carlo simulations of magnetisation and susceptibility in 3D XY model are performed at two different coupling constantsβ=0.55 andβ=0.5,completing our previous simulation results with additional data points and extending the range of the external field to twice as small values as previously reported(h≥0.00015625).The simulated maximal lattices sizes are also increased from L=384 to L=512.Our aim is an improved estimation of the exponentρ,describing the Goldstone mode singularity M(h)=M(+0)+chρat h→0,where M is the magnetisation.The data reveal some unexpected small oscillations.It makes the estimation by manyparameter fits of the magnetisation data unstable,and we are looking for an alternative method.Our best estimateρ=0.555(17)is extracted from the analysis of effective exponents determined from local fits of the susceptibility data.This method gives stable and consistent results for both values ofβ,taking into account the leading as well as the subleading correction to scaling.We report also the values of spontaneous magnetisation.

Scaling Regimes and the Singularity of Specific Heat in the 3D Ising Model

The singularity of specific heat CV of the three-dimensional Ising model is studied based on Monte Carlo data for lattice sizes L≤1536.Fits of two data sets,one corresponding to certain value of the Binder cumulant and the other-to the maximum of CV,provide consistent values of C0 in the ansatz CV(L)=C0+AL^(a/n) at large L,if a/n=0.196(6).However,a direct estimation from our Cmax V data suggests that a/n,most probably,has a smaller value(e.g.,a/n=0.113(30)).Thus,the conventional power-law scaling ansatz can be questioned because of this inconsistency.We have found that the data are well described by certain logarithmic ansatz.

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

Correlation functions in the 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 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_(⊥)(k)≈ak−λ_(⊥)and G_(||)(k)≈bk−λk,respectively.Here a and b are the amplitudes,k=|k|is the magnitude of the wave vector k.The exponentsλ_(⊥),λk 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_(⊥)(k)and G_(||)(k),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。