Narrow Results

Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to $L=3456$. Our estimated values of the correction-to-scaling exponent $\omega$ tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes, i.e. $L<L_{\mathrm{\text{min}}}$ with $L_{\mathrm{\text{min}}}\in [6,64]$, are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e. $\omega \approx 0.7$ and $\omega =0.75(5)$. We discuss the possibilities that $\omega$ is either really smaller than usually expected or these values of $\omega$ describe some transient behavior which, eventually, turns into the correct asymptotic behavior at $L_{\mathrm{\text{min}}}>64$. We propose refining MCRG simulations and analysis to resolve this issue. Our actual MC estimations of the critical exponents $\eta$ and $\nu$ provide stable values $\eta =0.03632(13)$ and $\nu =0.63017(31)$, which well agree with those of the conformal bootstrap method, i.e. $\eta =0.0362978(20)$ and $\nu =0.6299709(40)$.

To determine anomalous dynamic scaling of continuum growth equations, López¹² proposed an analytical approach, which is based on the scaling analysis introduced by Hentschel and Family.¹⁵ In this work, we generalize this scaling analysis to the (d+1)-dimensional molecular-beam epitaxy equations to determine their anomalous dynamic scaling. The growth equations studied here include the linear molecular-beam epitaxy (LMBE) and Lai–Das Sarma–Villain (LDV). We find that both the LMBE and LDV equations, when the substrate dimension d>2, correspond to a standard Family–Vicsek scaling, however, when d<2, exhibit anomalous dynamic roughening of the local fluctuations of the growth height. When the growth equations exhibit anomalous dynamic scaling, we obtain the local roughness exponents by using scaling relation α_loc=α-zκ, which are consistent with the corresponding numerical results.