Computing the roughening transition of Ising and solid-on-solid models by BCSOS model matching

Thermodynamics of the two-dimensional XY model from functional renormalization

We solve the nonperturbative renormalization-group flow equations for the two-dimensional XY model at the truncation level of the (complete) second-order derivative expansion. We compute the thermodynamic properties in the high-temperature phase and compare the nonuniversal features specific to the XY model with results from Monte Carlo simulations. In particular, we study the position and magnitude of the specific-heat peak as a function of temperature. The obtained results compare well with Monte Carlo simulations. We additionally gauge the accuracy of simplified nonperturbative renormalization-group treatments relying on ϕ^{4}-type truncations. Our computation indicates that such an approximation is insufficient in the high-T phase and a correct analysis of the specific-heat profile requires account of an infinite number of interaction vertices.

Tensor renormalization group study of classical XY model on the square lattice

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.

Residual discrete symmetry of the five-state clock model

It is well known that the q-state clock model can exhibit a Kosterlitz-Thouless (KT) transition if q is equal to or greater than a certain threshold, which has been believed to be five. However, recent numerical studies indicate that helicity modulus does not vanish in the high-temperature phase of the five-state clock model as predicted by the KT scenario. By performing Monte Carlo calculations under the fluctuating twist boundary condition, we show that it is because the five-state clock model does not have the fully continuous U(1) symmetry even in the high-temperature phase while the six-state clock model does. We suggest that the upper transition of the five-state clock model is actually a weaker cousin of the KT transition so that it is q≥6 that exhibits the genuine KT behavior.

Monte Carlo tests of nucleation concepts in the lattice gas model

The conventional theory of homogeneous and heterogeneous nucleation in a supersaturated vapor is tested by Monte Carlo simulations of the lattice gas (Ising) model with nearest-neighbor attractive interactions on the simple cubic lattice. The theory considers the nucleation process as a slow (quasistatic) cluster (droplet) growth over a free energy barrier ΔF(*), constructed in terms of a balance of surface and bulk term of a critical droplet of radius R(*), implying that the rates of droplet growth and shrinking essentially balance each other for droplet radius R=R(*). For heterogeneous nucleation at surfaces, the barrier is reduced by a factor depending on the contact angle. Using the definition of physical clusters based on the Fortuin-Kasteleyn mapping, the time dependence of the cluster size distribution is studied for quenching experiments in the kinetic Ising model and the cluster size ℓ(*) where the cluster growth rate changes sign is estimated. These studies of nucleation kinetics are compared to studies where the relation between cluster size and supersaturation is estimated from equilibrium simulations of phase coexistence between droplet and vapor in the canonical ensemble. The chemical potential is estimated from a lattice version of the Widom particle insertion method. For large droplets it is shown that the physical clusters have a volume consistent with the estimates from the lever rule. Geometrical clusters (defined such that each site belonging to the cluster is occupied and has at least one occupied neighbor site) yield valid results only for temperatures less than 60% of the critical temperature, where the cluster shape is nonspherical. We show how the chemical potential can be used to numerically estimate ΔF(*) also for nonspherical cluster shapes.

Crossover in growth laws for phase-separating binary fluids: molecular dynamics simulations

Pattern and dynamics during phase separation in a symmetrical binary (A+B) Lennard-Jones fluid are studied via molecular dynamics simulations after quenching homogeneously mixed critical (50:50) systems to temperatures below the critical one. The morphology of the domains, rich in A or B particles, is observed to be bicontinuous. The early-time growth of the average domain size is found to be consistent with the Lifshitz-Slyozov law for diffusive domain coarsening. After a characteristic time, dependent on the temperature, we find a clear crossover to an extended viscous hydrodynamic regime where the domains grow linearly with time. Pattern formation in the present system is compared with that in solid binary mixtures, as a function of temperature. Important results for the finite-size and temperature effects on the small-wave-vector behavior of the scattering function are also presented.

Phase transitions in two-dimensional Z(N) vector models for N>4

We investigate both analytically and numerically the renormalization group equations in two-dimensional (2D) Z(N) vector models. The position of the critical points of the two phase transitions for N>4 is established and the critical index ν is computed. For N=7 and 17 the critical points are located by Monte Carlo simulations, and some of the corresponding critical indices are determined. The behavior of the helicity modulus is studied for N=5, 7, and 17. Using these and other available Monte Carlo data we discuss the scaling of the critical points with N and some other open theoretical problems.

Boson pairing and unusual criticality in a generalized XY model

We discuss the unusual critical behavior of a generalized XY model containing both 2π-periodic and π-periodic couplings between sites, allowing for ordinary vortices and half-vortices. The phase diagram of this system includes both single-particle condensate and pair-condensate phases. Using a field theoretic formulation and worm algorithm Monte Carlo simulations, we show that in two dimensions it is possible for the system to pass directly from the disordered (high temperature) phase to the single particle (quasi)condensate via an Ising transition, a situation reminiscent of the "deconfined criticality" scenario.

Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensional XY model

We study a percolation problem on a substrate formed by two-dimensional XY spin configurations using Monte Carlo methods. For a given spin configuration, we construct percolation clusters by randomly choosing a direction x in the spin vector space, and then placing a percolation bond between nearest-neighbor sites i and j with probability p(ij)=max(0,1-e(-2Ks(i)(x)s(j)(x))), where K>0 governs the percolation process. A line of percolation thresholds K(c)(J) is found in the low-temperature range J≥J(c), where J>0 is the XY coupling strength. Analysis of the correlation function g(p)(r), defined as the probability that two sites separated by a distance r belong to the same percolation cluster, yields algebraic decay for K≥K(c)(J), and the associated critical exponent depends on J and K. Along the threshold line K(c)(J), the scaling dimension for g(p) is, within numerical uncertainties, equal to 1/8. On this basis, we conjecture that the percolation transition along the K(c)(J) line is of the Berezinskii-Kosterlitz-Thouless type.

Laterally driven interfaces in the three-dimensional Ising lattice gas

We study the steady state of a phase-separated driven Ising lattice gas in three dimensions using computer simulations with Kawasaki dynamics. An external force field F(z) acts in the x direction parallel to the interface, creating a lateral order parameter current j^{x}(z) which varies with distance z from the interface. Above the roughening temperature, our data for "shearlike" linear variation of F(z) are in agreement with the picture wherein shear acts as effective confinement in this system, thus suppressing the interfacial capillary-wave fluctuations. We find sharper magnetization profiles and reduced interfacial width as compared to equilibrium. Pair correlations are more suppressed in the vorticity direction y than in the driving direction; the opposite holds for the structure factor. Lateral transport of capillary waves occurs for those forms of F(z) for which the current j^{x}(z) is an odd function of z , for example the shearlike drive, and a "steplike" driving field. For a V-shaped driving force no such motion occurs, but capillary waves are suppressed more strongly than for the shearlike drive. These findings are in agreement with our previous simulation studies in two dimensions. Near and below the (equilibrium) roughening temperature the effective-confinement picture ceases to work, but the lateral motion of the interface persists.

True and quasi-long-range order in the generalized q-state clock model

From consideration of the order-parameter distribution, we propose an observable which makes a clear distinction between true and quasi-long-range orders in the two-dimensional generalized q-state clock model. Measuring this quantity by Monte Carlo simulations for q=8, we construct a phase diagram and identify critical properties across the phase-separation lines among the true long-range order, quasi-long-range order, and disorder. Our result supports the theoretical prediction that there appears a discontinuous order-disorder transition as soon as the two phase-separation lines merge.

High-temperature expansion of the magnetic susceptibility and higher moments of the correlation function for the two-dimensional XY model

We calculate the high-temperature series of the magnetic susceptibility and the second and fourth moments of the correlation function for the XY model on the square lattice to order beta;{33} by applying the improved algorithm of the finite lattice method. The long series allow us to estimate the inverse critical temperature as beta_{c}=1.1200(1) , which is consistent with the most precise value given previously by the Monte Carlo simulation. The critical exponent for the multiplicative logarithmic correction is evaluated to be theta=0.054(10) , which is consistent with the renormalization group prediction of theta=1/16 .

Dissipation-driven phase transition in two-dimensional Josephson arrays

We analyze the interplay of dissipative and quantum effects in the proximity of a quantum phase transition. The prototypical system is a resistively shunted two-dimensional Josephson junction array, studied by means of an advanced Fourier path-integral Monte Carlo algorithm. The reentrant superconducting-to-normal phase transition driven by quantum fluctuations, recently discovered in the limit of infinite shunt resistance, persists for moderate dissipation strength but disappears in the limit of small resistance. For large quantum coupling our numerical results show that, beyond a critical dissipation strength, the superconducting phase is always stabilized at sufficiently low temperature. Our phase diagram explains recent experimental findings.

First-order interface localization-delocalization transition in thin Ising films using Wang-Landau sampling

Using extensive Monte Carlo simulations, we study the interface localization-delocalization transition of a thin Ising film with antisymmetric competing walls for a set of parameters where the transition is strongly first order. This is achieved by estimating the density of states (DOS) of the model by means of Wang-Landau sampling (WLS) in the space of energy, using both single-spin-flip as well as N-fold way updates. From the DOS we calculate canonical averages related to the configurational energy, like the internal energy and the specific heat, as well as the free energy and the entropy. By sampling micro-canonical averages during simulations we also compute thermodynamic quantities related to magnetization like the reduced fourth-order cumulant of the order parameter. We estimate the triple temperatures of infinitely large systems for three different film thicknesses via finite size scaling of the positions of the maxima of the specific heat, the minima of the cumulant, and the equal weight criterion for the energy probability distribution. The wetting temperature of the semi-infinite system is computed with help of the Young equation. In the limit of large film thicknesses the triple temperatures are seen to converge toward the wetting temperature of the corresponding semi-infinite Ising model in accordance with standard capillary wave theory. We discuss the slowing down of WLS in energy space as observed for the larger film thicknesses and lateral linear dimensions. In the case of WLS in the space of total magnetization we find evidence that the slowing down is reduced and can be attributed to persisting free energy barriers due to shape transitions.

Reentrant behavior of the phase stiffness in Josephson junction arrays

The phase diagram of a 2D Josephson junction array with large substrate resistance, described by a quantum XY model, is studied by means of Fourier path-integral Monte Carlo. A genuine Berezinskii-Kosterlitz-Thouless transition is found up to a threshold value g( small star, filled ) of the quantum coupling, beyond which no phase coherence is established. Slightly below g( small star, filled ) the phase stiffness shows a reentrant behavior with temperature, in connection with a low-temperature disappearance of the superconducting phase, driven by strong nonlinear quantum fluctuations.

Finite size scaling in the two-dimensional XY model and generalized universality

In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.

Invaded cluster simulations of the XY model in two and three dimensions

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000(2) and 120(3), respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K(c)=0.454 12(2) for the three-dimensional XY model and eta=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are K(c)=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.