Dynamics of the formation of two-dimensional ordered structures

The growth of ordered domains in lattice gas models, which occurs after the system is quenched from infinite temperature to a state below the critical temperatureT _c, is studied by Monte Carlo simulation. For a square lattice with repulsion between nearest and next-nearest neighbors, which in equilibrium exhibits fourfold degenerate (2×1) superstructures, the time-dependent energy E(t), domain size L(t), and structure functionS(q, t) are obtained, both for Glauber dynamics (no conservation law) and the case with conserved density (Kawasaki dynamics). At late times the energy excess and halfwidth of the structure factor decrease proportional tot ^–x, whileL(t) t ^x, where the exponent x=1/2 for Glauber dynamics and x1/3 for Kawasaki dynamics. In addition, the structure factor satisfies a scaling lawS(k,t)=t ^2xS(kt^x). The smaller exponent for the conserved density case is traced back to the excess density contained in the walls between ordered domains which must be redistributed during growth. Quenches toT>T _c, T=T_c (where we estimate dynamic critical exponents) andT=0 are also considered. In the latter case, the system becomes frozen in a glasslike domain pattern far from equilibrium when using Kawasaki dynamics. The generalization of our results to other lattices and structures also is briefly discussed.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

A monte Carlo study of the two-dimensional Heisenberg model with easy-plane symmetry

Monte Carlo estimates are reported for the (topological) critical temperatures, T_c, of classical Heisenberg spin models in two spatial dimensions with easy-plane exchange anisotropy. The variation of T_c with spin anisotropy is compared with theoretical predictions.     

Non-equilibrium phase transitions in the two-temperature Ising model with Kawasaki dynamics

Phase transitions of the two-finite temperature Ising model on a square lattice are investigated by using a position space renormalization group (PSRG) transformation. Different finite temperatures, T _x ?and?T _y , and also different time-scale constants, ?? _x and ?? _y for spin exchanges in the x and y directions define the dynamics of the non-equilibrium system. The critical surface of the system is determined by RG flows as a function of these exchange parameters. The Onsager critical point (when the two temperatures are equal) and the critical temperature for the limit when the other temperature is infinite, previously studied by the Monte Carlo method, are obtained. In addition, two steady-state fixed points which correspond to the non-equilibrium phase transition are presented. These fixed points yield the different universality class properties of the non-equilibrium phase transitions.     

Exploiting the flexibility of a family of models for taxation and redistribution

In this paper we present and discuss results of Monte Carlo numerical simulations of the two-dimensional Ising ferromagnet in contact with a heat bath that intrinsically has a thermal gradient. The extremes of the magnet are at temperatures T ₁?<?T _c ?<?T ₂, where T _c is the Onsager critical temperature. In this way one can observe a phase transition between an ordered phase (T?<?T _c ) and a disordered one (T?>?T _c ) by means of a single simulation. By starting the simulations with fully disordered initial configurations with magnetization m????0 corresponding to T?=???, which are then suddenly annealed to a preset thermal gradient, we study the short-time critical dynamic behavior of the system. Also, by setting a small initial magnetization m?=?m ₀, we study the critical initial increase of the order parameter. Furthermore, by starting the simulations from fully ordered configurations, which correspond to the ground state at T?=?0 and are subsequently quenched to a preset gradient, we study the critical relaxation dynamics of the system. Additionally, we perform stationary measurements (t??????) that are discussed in terms of the standard finite-size scaling theory. We conclude that our numerical simulation results of the Ising magnet in a thermal gradient, which are rationalized in terms of both dynamic and standard scaling arguments, are fully consistent with well established results obtained under equilibrium conditions.     

Monte Carlo investigation of magnetic properties of a two-dimensional plane-rotator model

The equilibrium properties of a simple quadratic lattice of plane rotators with nearest-neighbor isotropic interactions have been examined by the Monte Carlo method. It was found that the 2-d plane-rotator model shows much the same magnetic properties as those of the 2-d Heisenberg model if temperature is scaled in units of T_M, the transition temperature predicted by the mean-field theory. It is pointed out that the present results are closely related to the prediction of Zittartz of a phase transition of continuous order.     

Universality in the critical two-dimensionalφ 4-model

The equilibrium properties of the discreteφ ⁴-model ind=2 dimensions are investigated at critically using the hybrid Monte Carlo algorithm and the Ferrenberg-Swendsen extrapolation scheme. By combining the extrapolation scheme with a finite-size scaling analysis, the critical line of the model could be located very accurately. The universality of the model along the critical line is then explicity demonstrated both with respect to the universal nature of the probability distribution function of the global order parameter and by direct comparison with the critical two-dimensional Ising model.     

Series and Monte Carlo study of high-dimensional Ising models

Ising models in high dimensions are used to compare high-temperature series expansions with Monte Carlo simulations. Simulations of the magnetization on four-, six-, and seven-dimensional hypercubic lattices give consistent values of the critical temperature from both equilibrium and nonequilibrium data ford=6 and 7. We tabulate 15 terms of series expansions for the susceptibility for generald and giveJ/k _B T _c =0.092295 (3) and 0.077706 (2) ford=6 and 7. In contrast to five dimensions, where earlier series found nonanalytic scaling corrections, for d=6 and 7 the leading scaling correction may be analytic inT-T _c . In most cases these expansions gave more accurate results than these simulations.     

Nonequilibrium model on Archimedean lattices

On (4, 6, 12) and (4, 8²) Archimedean lattices, the critical properties of the majority-vote model are considered and studied using the Glauber transition rate proposed by Kwak et al. [Kwak et al., Phys. Rev. E, 75, 061110 (2007)] rather than the traditional majority-vote with noise [Oliveira, J. Stat. Phys. 66, 273 (1992)]. We obtain T _c and the critical exponents for this Glauber rate from extensive Monte Carlo studies and finite size scaling. The calculated values of the critical temperatures and Binder cumulant are T _c = 0.651(3) and U ₄ ^* = 0.612(5), and T _c = 0.667(2) and U ₄ ^* = 0.613(5), for (4, 6, 12) and (4, 8²) lattices respectively, while the exponent (ratios) β/ν, γ/ν and 1/ν are respectively: 0.105(8), 1.48(11) and 1.16(5) for (4, 6, 12); and 0.113(2), 1.60(4) and 0.84(6) for (4, 8²) lattices. The usual Ising model and the majority-vote model on previously studied regular lattices or complex networks differ from our new results.     

Finite size scaling analysis of ising model block distribution functions

The distribution functionP _L (s) of the local order parameters in finite blocks of linear dimensionL is studied for Ising lattices of dimensionalityd=2, 3 and 4. Apart from the case where the block is a subsystem of an infinite lattice, also the distribution in finite systems with free [P _L ^(f) (s)] and periodic [P _L ^(p)(s) ] boundary conditions is treated. Above the critical pointT _c , these distributions tend for largeL towards the same gaussian distribution centered around zero block magnetization, while belowT _c these distributions tend towards two gaussians centered at ±M, whereM is the spontaneous magnetization appearing in the infinite systems. However, belowT _c the wings of the distribution at small |s| are distinctly nongaussian, reflecting two-phase coexistence. Hence the distribution functions can be used to obtain the interface tension between ordered phases.At criticality, the distribution functions tend for largeL towards scaled universal forms, though dependent on the boundary conditions. These scaling functions are estimated from Monte Carlo simulations. For subsystem-blocks, good agreement with previous renormalization group work of Bruce is obtained.As an application, it is shown that Monte Carlo studies of critical phenomena can be improved in several ways using these distribution functions:(i) standard estimates of order parameter, susceptibility, interface tension are improved(ii) T _c can be estimated independent of critical exponent estimates(iii) A Monte Carlo renormalization group similar to Nightingale's phenomenological renormalization is proposed, which yields fairly accurate exponent estimates with rather moderate effort(iv) Information on coarse-grained hamiltonians can be gained, which is particularly interesting if the method is extended to more general Hamiltonians.     

A fixed point for quantum gravity

Using the 1/N expansion a fixed point of the renormalization group is found for quantized gravitational theories which is non-trivial in all dimensions, d, including four. Using the fixed point it is shown how Einstein's theory can be renormalized for 3<d<4. In four dimensions the pure Einstein theory does not exist, but the R + C_μναβ² theory does. It is shown how gravitational theories whose quantum lagrangians are scale invariant may be renormalized such that the scale invariance is broken only by the choice of the critical renormalization group trajectory. A comparison is made with the renormalization of four-fermion and Yukawa theories in 4?? dimensions which suggests that quantum gravity might exist in four dimensions even if those theories do not.     

Monte Carlo investigation of magnetic properties of a two-dimensional classical Heisenberg magnet

The equilibrium properties of a simple quadratic lattice of classical spins with nearest-neighbor Heisenberg interactions have been examined by a Monte Carlo Method. The susceptibility was found to have a singular temperature dependence χ ∝ exp (const/T²)/T above the Stanley-Kaplan transition temperature (T_SK). A plausible argument has been presented to explain peculiar properties of the 2?d Heisenberg magnet on the basis of the observed singular behavior of the susceptibility.     

Damage spreading on the 3–12 lattice with competing Glauber and Kawasaki dynamics

The damage spreading of the Ising model on the 3–12 lattice with competing Glauber and Kawasaki dynamics is studied. The difference between the two kinds of nearest-neighboring spin interactions (interaction between two 12-gons, or interaction between a 12-gon and a triangle) are considered in the Hamiltonian. It is shown that the ratio of the interaction strengthF between the two kinds of interactions plays an important role in determining the critical temperature T_d of phase transition from frozen to chaotic. Two methods are used to introduce the bond dilution on the Ising model on the 3–12 lattice: regular and random. The maximum of the average damage spreading 〈D〉_max can approach values lower than 0.5 in both cases and the reason can be attributed to the ’survivors’ among the spins. We have also, for the first time, presented the phase diagram of the mixed G-K dynamics in the 3–12 lattice which shows what happens when going from pure Glauber to pure Kawasaki     

Role of initial correlation in coarsening of a ferromagnet

We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of theIsing model, employing the Glauber spin-flip mechanism, in space dimensionsd = 2 and3, on square and simplecubic lattices. Results for the persistence probability and the domain growth arediscussed for quenches to various temperatures (T _f ) below the criticalone (T _c ), from differentinitial temperatures T _i ≥T _c . In long timelimit, for T _i >T _c ,the persistence probability exhibits power-law decay with exponents θ ? 0.22 and? 0.18 in d = 2 and 3, respectively. For finite T _i , the early timebehavior is a different power-law whose life-time diverges and exponent decreases asT _i →T _c . The two steps areconnected via power-law as a function of domain length and the crossover to the secondstep occurs when this characteristic length exceeds the equilibrium correlation length atT =T _i . T _i =T _c is expected toprovide a new universality class for which we obtain θ ≡θ _c ? 0.035 ind = 2 and?0.105 in d = 3. The time dependenceof the average domain size ?, however, is observed to be rather insensitive tothe choice of T _i .     

An analytical study of the Curie temperature in the D-vector model

The successive analytical expressions up to fifth order for the Curie temperature, T_c(D,d), of the D-vector model on d-dimensional hypercubic lattices are evaluated by the Ginzburg-Landau theory based on cumulant expansion. The results by substituting d = 3 and D = 1, 2, 3 into the expressions approach order by order those of the Ising, XY, and Heisenberg models by Monte Carlo simulation (MC). The differences between ours and the other models are about 5.52, 6.86, and 7.94 percent, respectively.     

Inequivalence of dynamical ensembles in a generalized driven diffusive lattice gas

We generalize the driven diffusive lattice gas model by using a combination of Kawasaki and Glauber dynamics. We find via Monte Carlo simulations and perturbation studies that the simplest possible generalization of the equivalence of the canonical and grand-canonical ensembles, which holds in equilibrium, does not apply for this class of nonequilibrium systems.     

Non-equilibrium behavior at a liquid-gas critical point

Second-order phase transitions in a non-equilibrium liquid-gas model with reversible mode couplings, i.e., model H for binary-fluid critical dynamics, are studied using dynamic field theory and the renormalization group. The system is driven out of equilibrium either by considering different values for the noise strengths in the Langevin equations describing the evolution of the dynamic variables (effectively placing these at different temperatures), or more generally by allowing for anisotropic noise strengths, i.e., by constraining the dynamics to be at different temperatures in d _|| - and d _⊥-dimensional subspaces, respectively. In the first, isotropic case, we find one infrared-stable and one unstable renormalization group fixed point. At the stable fixed point, detailed balance is dynamically restored, with the two noise strengths becoming asymptotically equal. The ensuing critical behavior is that of the standard equilibrium model H. At the novel unstable fixed point, the temperature ratio for the dynamic variables is renormalized to infinity, resulting in an effective decoupling between the two modes. We compute the critical exponents at this new fixed point to one-loop order. For model H with spatially anisotropic noise, we observe a critical softening only in the d _⊥-dimensional sector in wave vector space with lower noise temperature. The ensuing effective two-temperature model H does not have any stable fixed point in any physical dimension, at least to one-loop order. We obtain formal expressions for the novel critical exponents in a double expansion about the upper critical dimension d _c = 4 - d _|| and with respect to d _|| , i.e., about the equilibrium theory. Received 4 April 2002 Published online 13 August 2002     

Phase transition in a two-dimensional Ising ferromagnet based on the generalized zero-temperature Glauber dynamics

At zero temperature, based on the Ising model, the phase transition in a two-dimensional square lattice is studied using the generalized zero-temperature Glauber dynamics. Using Monte Carlo （MC） renormalization group methods, the static critical exponents and the dynamic exponent are studied; the type of phase transition is found to be of the first order.     

Modified action glueballs

The mass of the 0⁺ glueball in 4-dimensional lattice gauge theory with a mixed SU(2)-SO(3) action is obtained via Monte Carlo. We work in a region far from the critical end point in the phase diagram, with an action partly motivated by renormalization group flows in the Migdal-Kadanoff approximation. A large-N resummation of perturbation theory is used to show that the mass gap scales as predicted by the perturbative renormalization group. Independent of this, our results show that the ratio of the glueball mass to the square root of the string tension, obtained from a previous Monte Carlo, is a renormalization group invariant.     

Differential real-space renormalization of thed-Dimensional Gaussian model

With the aid of the differential real-space method we derive exact renormalization group (RG) equations for the Gaussian model ind dimensions. The equations involved + 1 spatially dependent nearest-neighbor interactions. We locate a critical fixed point and obtain the exact thermal critical indexy _T = 2. A special trajectory of the full nonlinear RG transformation is found and the free energy of the corresponding initial state calculated.Supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 130 Ferroelektrika.