Finite-size Dependence Research Articles

Finite-size dependence of the free energy in two-dimensional critical systems

We show that the free energy at criticality of a finite two-dimensional system of characteristic size L has in general a term which behaves like ln L as L → ∞. The coefficient of this term is universal and is proportional to the conformal anomaly number c. Furthermore, when the metric is non-singular and the boundaries are smooth, this coefficient depends only on the topology and is equal to − 1 6 c gX, where Ξ is the Euler characteristic. However, if there are conical singularities in the metric, or corners on the boundary, this is no longer true. For these cases, we give the correct result.

Monte Carlo study of dynamic universality in two-dimensional Potts models.

We have carried out extensive computer simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L q-state Potts models for q=2, 3, and 4. We fit the equilibrium relaxation functions for energy and order parameter to a sum of exponential decays and use the finite-size dependence at ${T}_{c}$, \ensuremath{\tau}\ensuremath{\propto}${L}^{z}$, to extract estimates for z. We find for the longest relaxation time for both energy and for order parameter that z=2.17\ifmmode\pm\else\textpm\fi{}0.04 describes all three values of q. Our results support the dynamic generalization of the ``weak universality'' hypothesis proposed by Suzuki.

Role of boundary conditions in the finite-size Ising model.

Boundary conditions monitor the finite-size dependence of scaling functions for the Ising model. We study the low-temperature phase for the extremely anisotropic limit, or quantum version of the 2D classical Ising model, by means of combined exact results and large-size numerical calculations. The mass gap (inverse of correlation length) is the suitable order parameter for the finite system, and its finite-size behavior is studied as a function of variable boundary conditions. We find that the well-known exponential convergence to zero of the mass gap is only valid in a limited range of parameters; it strikingly changes into a power law for antiperiodic boundary conditions. We suggest that this puzzling phenomenon is associated with topological excitations.

Universality of finite-size scaling: Role of the boundary conditions.

The influence of the boundary conditions on the finite-size dependence of the mass gap in the 1D Ising model with a transverse field is studied by means of combined exact results and large-size numerical calculations. The well-known exponential form (for transverse fields smaller than the critical one) is only recovered in a limited range of parameters. A power-law behavior for the mass gap is even found in the case of antiperiodic boundary conditions.

Direct calculation of absolute free energy for lattice systems by Monte Carlo sampling of finite-size dependence.

L'energie libre absolue d'un systeme de spins de reseau peut etre calculee directement par une nouvelle application de l'echantillonnage de Monte Carlo a variation de taille finie. On obtient des resultats pour les modeles d'Ising 2D et 3D a Tc. Ces resultats sont coherents avec de recentes predictions de theories d'echelle de taille finie

Finite-size effects in the three-state quantum asymmetric clock model

Abstract The one-dimensional quantum hamiltonian of the asymmetric three-state clock model is studied using finite-size scaling. Various boundary conditions are considered on chains containing up to eight sites. We calculate the boundary of the commensurate phase and the mass gap index. The model shows an interesting finite-size dependence in connexion with the presence of the incommensurate phase indicating that for the infinite system there is no Lifshitz point.

Finite-size behaviour of the two-dimensional ANNNI model

The two-dimensional axial next-nearest neighbour Ising (ANNNI) model of finite size with periodic boundary conditions is studied by the Monte Carlo method. The model shows an interesting finite size dependence in connection with its oscillatory correlations pretending for finite systems a Lifshitz point in one part of the phase diagram, while the infinite system appears to display one in another part of the phase diagram.