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Journal of Physics A Mathematical and General

Test of finite-size scaling predictions in three dimensions

J.G. Brankov, V. B. Priezzhev

Published Aug 21, 1992• 8 citations
Publisher

Abstract

A constrained monomer-dimer (CMD) model on decorated hypercubic lattices, which is exactly solvable in any dimensionality d, is studied. The model is critical in the limit of maximal packing of the dimer configurations allowed by the constraint. For d=2 it is equivalent to the rooted-tree model on the square lattice considered by Duplantier and David (1988). The validity of two finite-size scaling predictions in higher dimensionalities is studied: (1) the existence of logarithmic corrections in the free energy due to corners, and (ii) the amplitude-exponent relation for the pair correlation function. The logarithmic finite-size corrections are obtained for arbitrary d, for boundary conditions which are periodic for d'>or=0 dimensions and free in the remaining d-d' dimensions. Dimer-dimer correlation functions are studied at d=3 for a system finite in two dimensions and infinite in the third. It is shown that the amplitude-exponent relation for the CMD model holds for d=2 and breaks down for d=3.

DOI
10.1088/0305-4470/25/16/008

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A constrained monomer-dimer (CMD) model on decorated hypercubic lattices, which is exactly solvable in any dimensionality d, is studied. The model is critical in the limit of maximal packing of the dimer configurations allowed by the constraint. For d=2 it is equivalent to the rooted-tree model on the square lattice considered by Duplantier and David (1988). The validity of two finite-size scaling predictions in higher dimensionalities is studied: (1) the existence of logarithmic corrections in the free energy due to corners, and (ii) the amplitude-exponent relation for the pair correlation function. The logarithmic finite-size corrections are obtained for arbitrary d, for boundary conditions which are periodic for d'>or=0 dimensions and free in the remaining d-d' dimensions. Dimer-dimer correlation functions are studied at d=3 for a system finite in two dimensions and infinite in the third. It is shown that the amplitude-exponent relation for the CMD model holds for d=2 and breaks down for d=3.
The validity of the finite-size scaling prediction about the existence of logarithmic corrections in the free energy due to corners is studied by the example of the mean spherical model. The general case of a hypercubic lattice of arbitrary dimensionality d>2, under boundary conditions which are periodic in d'>or=0 dimensions and free in the remaining d-d' dimensions is considered. The critical regime, as the size of the system L to infinity , is specified by the asymptotic behaviour of the ratio L/ xi L , where xi L is the correlation length fo the finite system. New results are the double-logarithmic corrections due to corners and logarithmic corrections due to one-dimensional edges in the regime L/ xi L varies as ln L which takes place at the bulk critical point.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
Abstract Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (dc = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows r2−dcr^{2-d_c}r2−dc​ governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as L−dc/2(lnL)p^L^{-d_c/2}({\rm ln}L)^{\hat{p}}L−dc​/2(lnL)p^​ with p^=1/2\hat{p}=1/2p^​=1/2 a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.
In this present work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model ıt via the so-called height mapping which will be extensively used to interpret exact results. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predict particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.
Lattice models are useful for understanding behaviors of interacting complex many-body systems. The lattice dimer model has been proposed to study the adsorption of diatomic molecules on a substrate. Here we analyze the partition function of the dimer model on a 2M×2N checkerboard lattice wrapped on a torus and derive the exact asymptotic expansion of the logarithm of the partition function. We find that the internal energy at the critical point is equal to zero. We also derive the exact finite-size corrections for the free energy, the internal energy, and the specific heat. Using the exact partition function and finite-size corrections for the dimer model on a finite checkerboard lattice, we obtain finite-size scaling functions for the free energy, the internal energy, and the specific heat of the dimer model. We investigate the properties of the specific heat near the critical point and find that the specific-heat pseudocritical point coincides with the critical point of the thermodynamic limit, which means that the specific-heat shift exponent λ is equal to ∞. We have also considered the limit N→∞ for which we obtain the expansion of the free energy for the dimer model on the infinitely long cylinder. From a finite-size analysis we have found that two conformal field theories with the central charges c=1 for the height function description and c=-2 for the construction using a mapping of spanning trees can be used to describe the dimer model on the checkerboard lattice.