Finite-size scaling for mean-field percolation

By studying transfer matrix eigenvalues, correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime. Near criticality, finite-size scaling behavior is derived and an analytic technique is provided for obtaining the finite-size scaling function. Our methods involve the generating function, matched asymptotic expansions, and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability.     

Corrections to scaling for diffusion exponents on three-dimensional percolation systems at criticality

Recent results of Monte Carlo simulations of the ant-in-the-labyrinth method in three-dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz, motivated by inconsistent results that have appeared in the literature. The results are observed to be sensitive to the form of the scaling correction terms. Using a single correction term, we estimate the valuek=0.197±0.004 for the anomalous diffusion exponent at criticality. When two correction terms are included,k=0.200±0.002 is obtained. These new estimates are consistent with known theoretical bounds, with recent series expansion results, and with numerical calculations of the conductance of random resistor networks above criticality.     

Finite-size scaling for correlations of quantum spin chains at criticality

We study the finite-size scaling behavior of two-point correlation functions of translationally invariant many-body systems at criticality. We propose an efficient method for calculating the two-point correlation functions in the thermodynamic limit from numerical data of finite systems. Our method is most effective when applied to a two-dimensional (classical) system which possesses a conformal invariance. By using this method with numerical data obtained from exact diagonalizations and Monte Carlo simulations, we study the spin-spin correlations of the quantum spin-1/2 and-3/2 antifierromagnetic chains. In particular, the logarithmic corrections to power-law decay of the correlation of the spin-1/2 isotropic Heisenberg antiferromagnetic chain are studied thoroughly. We clarify the cause of the discrepancy in previous calculations for the logarithmic corrections. Our result strongly supports the field-theoretic prediction based on the mappings to the Wess-Zumino-Witten nonlinear -model or the sine-Gordon model. We also treat logarithmic corrections and crossover phenomena in the spin-spin correlation of the spin-3/2 isotropic Heisenberg antiferromagnetic chain. Our results are consistent with the Affleck-Haldane prediction that the correlation of the spin-3/2 chain exhibits a crossover to the same asymptotic behavior as in the spin-1/2 chain.     

Logarithmic corrections to finite-size scaling in the four-state Potts model

The leading corrections to finite-size scaling predictions for eigenvalues of the quantum Hamiltonian limit of the critical four-state Potts model are calculated analytically from the Bethe ansatz equations for equivalent eigenstates of a modifiedXXZ chain. Scaled gaps are found to behave for large chain lengthL asx+dL+0[(lnL)^–1], wherex is the anomalous dimension of the associated primary scaling operator. For the gaps associated with the energy and magnetic operators, the values of the amplitudesd are in agreement with predictions of conformai invariance. The implications of these analytical results for the extrapolation of finite lattice data are discussed. Accurate estimates of x andd are found to be extremely difficult even with data available from large lattices,L500.     

Beyond Scaling and Locality in Turbulence

An analytic perturbation theory is suggested in order to find finite-size corrections to the scaling power laws. In the frame of this theory it is shown that the first order finite-size correction to the scaling power laws has following form , where η is a finite-size scale (in particular for turbulence, it can be the Kolmogorov dissipation scale). Using data of laboratory experiments and numerical simulations it is shown shown that a degenerate case with α ₀=0 can describe turbulence statistics in the near-dissipation range r > η, where the ordinary (power-law) scaling does not apply. For moderate Reynolds numbers the degenerate scaling range covers almost the entire range of scales of velocity structure functions (the log-corrections apply to finite Reynolds number). Interplay between local and non-local regimes has been considered as a possible hydrodynamic mechanism providing the basis for the degenerate scaling of structure functions and extended self-similarity. These results have been also expanded on passive scalar mixing in turbulence. Overlapping phenomenon between local and non-local regimes and a relation between position of maximum of the generalized energy input rate and the actual crossover scale between these regimes are briefly discussed. PACS: 47.27.-i, 47.27.Gs.     

Fluctuations and finite-size effect in the Bak-Sneppen model

The self-organized criticality in the nearest-neighbor version of the Bak-Sneppen model is investigated from the event-by-event fluctuations of the mean fitness. The finite-size effect on the evolution of the critical state is shown, and a scaling solution to the gap equation for an infinite one-dimensional lattice is given numerically for the first time. The mean lifetime of avalanches is presented as a function of the gap from the solution. The critical value of the gap and an exponent are calculated from the solution. Received 10 December 2000 and Received in final form 18 January 2001     

Computer simulation of the critical behavior of 3D disordered ising model

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.     

Finite-size scaling in a microcanonical ensemble

The finite-size scaling technique is extended to a microcanonical ensemble. As an application, equilibrium magnetic properties of anL×L square lattice Ising model are computed using the microcanonical ensemble simulation technique of Creutz, and the results are analyzed using the microcanonical ensemble finite-size scaling. The computations were done on the multitransputer system of the Condensed Matter Theory Group at the University of Mainz.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results

We consider two single-species reaction-diffusion models on one-dimensional lattices of lengthL: the coagulation-decoagulation model and the annihilation model. For the coagulation model the system of differential equations describing the time evolution of the empty interval probabilities is derived for periodic as well as for open boundary conditions. This system of differential equations grows quadratically withL in the latter case. The equations are solved analytically and exact expressions for the concentration are derived. We investigate the finite-size behavior of the concentration and calculate the corresponding scaling functions and the leading corrections for both types of boundary conditions. We show that the scaling functions are independent of the initial conditions but do depend on the boundary conditions. A similarity transformation between the two models is derived and used to connect the corresponding scaling functions.     

Logarithmic finite-size corrections in the three-dimensional mean spherical model

The finite-size scaling prediction about logarithmic corrections in the free energy arising from corners in the geometry of the system is tested on the three-dimensional mean spherical model. The general case of boundary conditions which are periodic ind 0 dimensions and free or fixed in the remaining 3 –d dimensions is considered. Logarithmic and double-logarithmic size corrections stemming from corners, edges, and surfaces are obtained.     

Leading corrections to finite-size scaling for mixed-spin chains

The leading corrections to finite-size scaling relations for the correlation length and twist order parameter of three mixed-spin quantum spin chains for the critical feature that develops at ϑ = π, corresponding to a change in the topological realization of the ground states, are identified. The text was submitted by the authors in English.     

Continuum limits and exact finite-size-scaling functions for one-dimensionalO(N)-invariant spin models

We solve exactly the general one-dimensionalO(N)-invariant spin model taking values in the sphereS ^N–1, with nearest-neighbor interactions, in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discussed for a general one-parameter family of interactions and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated and no additional universality classes appear. In the appendices we give new formulae for the Clebsch-Gordan coefficients and 6–j symbols of theO(N) group, and some new generalizations of the Poisson summation formula; these may be of independent interest.     

Born effective charge reversal and metallic threshold state at a band insulator-Mott insulator transition

We study the quantum phase transition between a band (“ionic”) insulator and a Mott-Hubbard insulator, realized at a critical value in a bipartite Hubbard model with two inequivalent sites, whose on-site energies differ by an offset . The study is carried out both in D=1 and D=2 (square and honeycomb lattices), using exact Lanczos diagonalization, finite-size scaling, and Berry's phase calculations of the polarization. The Born effective charge jump from positive infinity to negative infinity previously discovered in D=1 by Resta and Sorella is confirmed to be directly connected with the transition from the band insulator to the Mott insulating state, in agreement with recent work of Ortiz et al. In addition, symmetry is analysed, and the transition is found to be associated with a reversal of inversion symmetry in the ground state, of magnetic origin. We also study the D=1 excitation spectrum by Lanczos diagonalization and finite-size scaling. Not only the spin gap closes at the transition, consistent with the magnetic nature of the Mott state, but also the charge gap closes, so that the intermediate state between the two insulators appears to be metallic. This finding, rationalized within Hartree-Fock as due to a sign change of the effective on-site energy offset for the minority spin electrons, underlines the profound difference between the two insulators. The band-to-Mott insulator transition is also studied and found in the same model in D=2. There too we find an associated, although weaker, polarization anomaly, with some differences between square and honeycomb lattices. The honeycomb lattice, which does not possess an inversion symmetry, is used to demonstrate the possibility of an inverted piezoelectric effect in this kind of ionic Mott insulator. Received 21 May 1999     

Analytic calculation of scaling dimensions: Tricritical hard squares and critical hard hexagons

The finite-size corrections, central chargesc, and scaling dimensionsx of tricritical hard squares and critical hard hexagons are calculated analytically. This is achieved by solving the special functional equation or inversion identity satisfied by the commuting row transfer matrices of these lattice models at criticality. The results are expressed in terms of Rogers dilogarithms. For tricritical hard squares we obtainc=7/10,x=3/40, 1/5, 7/8, 6/5 and for hard hexagons we obtainc=4/5,x=2/15, 4/5, 17/15, 4/3, 9/5, in accord with the predictions of conformal and modular invariance.     

Energy gap of spin nanotube

Recently some quantum spin systems on tube lattices, so-called spin nanotubes, have been synthesized. They are expected to be interesting low-dimensional systems like the carbon nanotubes. As a first step of theoretical study on the spin nanotube, we investigate the three-leg spin tube, which is the simplest one, using numerical analyses of finite clusters and a finite-size scaling technique. The spin gap, which is one of the most interesting quantities reflecting the macroscopic quantum effect, was revealed to be open for any finite rung exchange couplings, in contrast to the three-leg spin ladder system which is gapless. We also found a quantum phase transition caused by an asymmetric rung interaction. When one of the three rung coupling constants is changed, the spin gap vanishes very rapidly.     

Crossover and finite-size effects in the (1+1)-dimensional Kardar-Parisi-Zhang equation

Crossover scaling of the surface width in the Kardar-Parisi-Zhang equation for surface growth is studied numerically. By means of a perturbative solution of the discretized equation and by comparison with the exact solution of the corresponding linear equation, the finite-size effects due to the spatial discretization are carefully analyzed. The dependence on the nonlinearity of both the finite-size and asymptotic scaling forms is then investigated.     

Universality Under Conditions of Self-tuning

We study systems with a continuous phase transition that tune their parameters to maximize a quantity that diverges solely at a unique critical point. Varying the size of these systems with dynamically adjusting parameters, the same finite-size scaling is observed as in systems where all relevant parameters are fixed at their critical values. This scheme is studied using a self-tuning variant of the Ising model. It is contrasted with a scheme where systems approach criticality through a target value for the order parameter that vanishes with increasing system size. In the former scheme, the universal exponents are observed in naïve finite-size scaling studies, whereas in the latter they are not.     

Nonlinear scaling and its application to ferromagnetic systems

Making use of renormalization-group ideas, a scaling equation of state applicable to ferromagnetic systems and involving the nonlinear scaling variables =ε/t and =h/t, instead of the usual linear scaling variables ε=(T-T_c)/Tc = t-1 and h=H (ordering field), has been derived. The magnetic equation of state so obtained is then generalized to take into account the effect of nonlinear relevant and irrelevant scaling fields. To facilitate a comparison with experiments, the analytic (non-analytic) corrections to the dominant singular behaviour of spontaneous magnetization (order parameter) M(T, 0), ‘zero-field’ susceptibility χ(T, 0), and specific heat in zero field that the nonlinear relevant (irrelevant) scaling fields give rise to are explicitly calculated up to third order in Due consideration is also given to the modifications in the Arrott-Noakes form of the scaling equation of state and the Kouvel-Fisher definition of the effective susceptibility exponent brought about by these scaling fields. A detailed analysis of the M(T, 0) and χ(T, 0) data for crystalline and amorphous ferromagnets in terms of the theoretical expressions derived in this work reveals that i) in conformity with the theoretical predictions, the “;non-analytic”; corrections to the singular behaviour dominate over the “;analytic”; ones for temperatures in the immediate vicinity of the critical point Tc, whereas reverse is the case for temperatures far away from Tc and (ii) the expression for χ(T, 0), based on the nonlinear scaling arguments, which includes the leading “;analytic”; correction, reproduces closely the observed variation of χ with T over a wide range of temperatures T_c ≤ T ≤ 1.5T_c (in some cases, up to 3T_c) for both ordered as well as quench-disordered ferromagnets.     

Spin coupling and magnetic field effects on the finite-size free energy and its non-extensivity for 1-D Ising model with nearest and next-nearest neighbor interactions in nanosystem

The effects of second-neighbor spin coupling interactions and a magnetic field are investigated on the free energies of a finite-size 1-D Ising model. For both ferromagnetic of nearest neighbor (NN) and next-nearest neighbor (NNN) spin coupling interactions, the finite-size free energy first increases and then approaches a constant value for any size of the spin chain. In contrast, when NNN and NN spin coupling interactions are antiferromagnetic and ferromagnetic, respectively, the finite-size free energy gradually decreases by increasing the competition factor and eventually vanishes for large values of it. When a magnetic field is applied, the finite-size free energy decreases with respect to the case of zero magnetic fields for both ferromagnetic and antiferromagnetic spin coupling interactions. Deviation of free energy per size for finite-size systems relative to the infinite system increases when the spin coupling interactions as well as the f parameter (the ratio of the magnetic field to NN spin coupling interaction) increase.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part II. Numerical methods

The scaling exponent and the scaling function for the 1D single-species coagulation model (A+AA) are shown to be universal, i.e., they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties of the concentration: Monte Carlo simulations and extrapolations of exact finite-lattice data. These methods are tested in a case where analytical results are available. To obtain reliable results from finite-size extrapolations, numerical data for lattices up to ten sites are sufficient.