Field theory of finite-size effects in Ising-like systems

We propose a new perturbation approach to finitesize effects within the ⁴ field theory for a one-component order parameter with periodic boundary conditions. Our approach is applicable both above and belowT _c . Renormalization-group calculations of finite-size scaling functions in three dimensions are compared with new Monte Carlo data for theL×L×L Ising model withL=8, 16, 32. The field-theoretic predictions are in good overall agreement with the Monte Carlo data.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

Spin-pair correlations of the classical Heisenberg ferromagnet above and below the critical point: Results of self-consistent Monte Carlo calculations

The spin-pair correlation functions of the classical Heisenberg ferromagnet with simple cubic lattice have been calculated in the paramagnetic and ferromagnetic temperature range using the self-consistent Monte Carlo method. The results agree with high-temperature series expansions aboveT _c , for low temperatures with spin-wave theory. By two different approaches the divergence of the ferromagnetic homogeneous susceptibility in zero field throughout the ferromagnetic temperature range could be verified. The functional dependence of the static susceptibilityχ _T (k) upon the inverse correlation lengthκ ₁ is discussed above and belowT _c and a Fourier transform for the explicit dependence of the spin correlations upon correlation length belowT _c is given. According to these results the scaling assumptionv=v′ for the exponents of the correlation length in the critical region is consistent with a divergent ferromagnetic susceptibility.     

The Effect of the Number of Simulations on the Exponents Obtained by Finite-Size Scaling Relations of the Order Parameter and the Magnetic Susceptibility for the Four-Dimensional Ising Model on the Creutz Cellular Automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton using finite-size lattices with linear dimension 4≤L≤8. The exponents in the finite-size scaling relations for the order parameter and the magnetic susceptibility at the finite-lattice critical temperature are computed to be β=0.49(7), β=0.49(5), β=0.50(1) and γ=1.04(4), γ=1.03(4), γ=1.02(4) for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are consistent with the renormalization group predictions of β=0.5 and γ=1. The values for the critical temperature of the infinite lattice T _c (∞)=6.6788(65), T _c (∞)=6.6798(69), T _c (∞)=6.6802(70) are obtained from the straight-line fit of the magnetic susceptibility maxima using 4≤L≤8 for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are in very good agreement with the series expansion results of T _c (∞)=6.6817(15), T _c (∞)=6.6802(2), the dynamic Monte Carlo result of T _c (∞)=6.6803(1), the cluster Monte Carlo result of T _c (∞)=6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T _c (∞)=6.6802632±5×10⁻⁵.     

The Ising square lattice in aL×M geometry: A model for the effect of surface steps on phase transitions in adsorbed monolayers

Critical phenomena in adsorbed monolayers on surfaces are influenced by limited substrate homogeneity, such as surface steps. We consider the resulting finite-size and boundary effects in the framework of a lattice gas system with nearest neighbor attraction in aL×M geometry, with two free boundaries of lengthML, and periodic boundary conditions in the other direction (along the direction of the steps). This geometry thus models a terrace of the stepped surface, and adatoms adsorbed on neighboring terraces are assumed to be non-interacting. Also the effect of boundary fields is considered (describing the effects of missing neighbors and changed binding energy to the substrate near the boundary). Extensive Monte Carlo calculations on this model performed on a multi-transputer system are presented and analyzed in terms of phenomenological finite size scaling concepts. The fact that two scaling variables occur (/L,L/M, with being the correlation length in the bulk) is demonstrated explicitly. In the absence of boundary fields, the system forML orders nearT _c in a domain state, with domain walls running across the terrace, while at some temperature belowT _c a transition to a monodomain state occurs. This domain state slightly belowT _c is suppressed, however, by rather weak boundary fields. These results are interpreted in terms of exact theoretical predictions.     

Singlet-groundstate magnetism in TbP

Elastic neutron scattering and magnetic susceptibility data are reported for temperatures around the Neél-point,T _N=7.3 K, and for zero magnetic field. AboveT _N, the temperature dependence of the magnetic central peak intensity can adequately be described within the RPA assuming isotropic exchange between nearest and next-nearest neighbours as the only parameters. This two-parameter model is quantitatively confirmed by the susceptibility data. AtT _N, magnetic Bragg-intensities arise almost discontinuously (reaching 70% of the saturation within 0.1 K) accompanied by thermal hysteresis. For all temperatures belowT _N the sublattice magnetic moment is explained by solutions of meanfield equations, if an effective quadrupole-quadrupole interaction is included. The quadrupolar coupling parameter is fully consistent with a value determined recently from the softening of thec ₄₄-mode. These results show, that earlier suggestions of renormalization group results upon the origin of the first order phase transition in TbP are not needed.A project of the Sonderforschungsbereich 65 Festkörperspektroskopie Darmstadt-Frankfurt, supported by the Deutsche Forschungsgemeinschaft     

Interfacial Adsorption in Two-Dimensional Potts Models

The interfacial adsorption W at the first-order transition in two-dimensional q-state Potts models is studied. In particular, findings of Monte Carlo simulations and of density-matrix renormalization group calculations at q=16 are consistent with the analytic result, obtained in the Hamiltonian limit at large values of q, that Wt ^–1/3 on approach to the bulk critical temperature T _c, t=|T _c–T|/T _c. In addition, the numerical findings allow to estimate corrections to scaling. Our study supports and quantifies a previous conclusion by Bricmont and Lebowitz based on low temperature expansions.     

Dimensional crossover and finite-size scaling belowT c

Using the formalism developed in earlier work, dimensional crossover on ad-dimensional layered Ising-type system satisfying periodic boundary conditions and of sizeL is considered belowT _c (L), T _c (L) being the critical temperature of the finite-size system. Effective critical exponents _eff and _eff are shown explicitly to crossover between theird- and (d–1)-dimensional values for _L in the limitsL/ _L andL/ _L 0, respectively, _L , being the correlation length in the layers. Using anL-dependent renormalization group, the effective exponents are shown to satisfy natural generalizations of the standard scaling laws. In addition,L-dependent global scaling fields which span the entire crossover are defined and a scaling form of the equation of state in terms of them derived. All the above assertions are verified explicitly to one loop in perturbation theory, in particular effective exponents and a universal crossover equation of state are obtained and shown in the above asymptotic limits to be in good agreement with known results.     

The effect of a constant number of particles on the pair correlation function and the density profile in a one-dimensional lattice gas

A one-dimensional lattice gas (Ising model) of lengthL and with nearest-neighbor couplingJ is considered in a canonical ensemble with fixed number of particlesN=L/2. Exact expressions and asymptotic forms for largeL are derived for the density-density correlation function, using periodic boundary conditions, and for the density (magnetization) profile, using antisymmetric boundary conditions. The density-density correlation function,g, assumes for temperaturesT> T, withT = 2J(_BlnL)^–1 and forL large, the formg(x) =g ^gc(x) +BL ^–1 +a(x)L ^–1 +O(L^–2) wherex is a distance between considered lattice sites,B is known from earlier work of Lebowitz and Percus,(1b) anda(x) decays exponentially forx . For TT, the correlation function and the density profile behave differently, the latter exhibiting a step in the middle of the interface.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.