Critical phenomena: Corrections to scaling behaviour

In preceding chapters, while deriving the scaling behaviour of correlation functions, we have always kept only the leading term in the critical region. We examine now the different corrections to the leading behaviour. For instance, when we have solved the renormalizaton group (RG) equations, so far, we have neglected the small deviation of the effective coupling constant from its fixed-point value. Moreover, to establish RG equations, we have neglected corrections subleading by powers of the cut-off, and effects of other couplings of higher canonical dimensions. Subleading terms related to the value of the effective coupling constant which give the leading corrections, at least near four dimensions, can easily be derived from the solutions of the renormalization group (RG) equations and are discussed first. The situations below and at four dimensions (the upper-critical dimension) have to be examined separately. The second type of corrections involves additional considerations and is examined in the second part of the chapter. The last section is devoted to one physics application, provided by systems with strong dipolar forces, which have 3 as upper-critical dimension.

Corrections to finite-size scaling in the 3D Ising model based on nonperturbative approaches and Monte Carlo simulations

Corrections to scaling in the 3D Ising model are studied based on nonperturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes [Formula: see text]. Analytical arguments show the existence of corrections with the exponent [Formula: see text], the leading correction-to-scaling exponent being [Formula: see text]. A numerical estimation of [Formula: see text] from the susceptibility data within [Formula: see text] yields [Formula: see text], in agreement with this statement. We reconsider the MC estimation of [Formula: see text] from smaller lattice sizes, [Formula: see text], using different finite-size scaling methods, and show that these sizes are still too small, since no convergence to the same result is observed. In particular, estimates ranging from [Formula: see text] to [Formula: see text] are obtained, using MC data for thermodynamic average quantities, as well as for partition function zeros. However, a trend toward smaller [Formula: see text] values is observed in one of these cases in a refined estimation from extended data up to [Formula: see text]. We discuss the influence of [Formula: see text] on the estimation of critical exponents [Formula: see text] and [Formula: see text].

Corrections to finite-size scaling in the φ4 model on square lattices

Corrections to scaling in the two-dimensional (2D) scalar [Formula: see text] model are studied based on nonperturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L ([Formula: see text]) and different values of the [Formula: see text] coupling constant [Formula: see text], i.e. [Formula: see text], 1, 10. According to our analysis, amplitudes of the nontrivial correction terms with the correction–to–scaling exponents [Formula: see text] become small when approaching the Ising limit ([Formula: see text]), but such corrections generally exist in the 2D [Formula: see text] model. Analytical arguments show the existence of corrections with the exponent [Formula: see text]. The numerical analysis suggests that there exist also corrections with the exponent [Formula: see text] and, perhaps, also with the exponent about [Formula: see text], which are detectable at [Formula: see text]. The numerical tests provide an evidence that the structure of corrections to scaling in the 2D [Formula: see text] model differs from the usually expected one in the 2D Ising model.