Avoiding Boundary Effects in Wang-Landau Sampling

A simple modification of the ``Wang-Landau sampling'' algorithm removes the systematic error that occurs at the boundary of the range of energy over which the random walk takes place in the original algorithm.

Monte Carlo renormalization group methods

An Ising ferromagnet with an antiferromagnetic surface layer: A simple model for magnetic surface reconstruction

Simple cubic Ising lattices are studied by Monte Carlo simulation, using a thin film geometry (usually 40 atomic layers thick), with nearest neighbour ferromagnetic exchange J in the bulk and nearest neighbour antiferromagnetic interaction Js between surface spins. Applying a technique of preferential sampling in the surface layers, we investigate the ordering for a variety of values of JsJ and for various temperatures. For JsAF < Js < − 0.25J (where JsAF ≈ − 2.01J) ferromagnetic ordering occurs at a higher temperature than the antiferromagnetic surface ordering, while for − 0.25J < Js no antiferromagnetic long range order is possible. For Js < JsAF the surface transition occurs at a higher…

Continuous Phase Transitions at Surfaces of CuAu Alloy Models — A Monte Carlo Study of Surface Induced Order and Disorder

The influence of surface on phase transitions has found significant attention in recent years, and a number of excellent reviews exists. [1, 2, 3] A variety of complex phenomena occur which are also related to the physics of adsorption and wetting. The scenario of wetting requires three distinct phases, for instance the vacuum, the bulk phase and a third phase intervening in between at equilibrium. In case of surface induced disorder (SID, a film of disordered layers at the surface “wets” the bulk phase as the temperature approaches the bulk transition temperature T c,b. The transition at the surface may be continuous (standard critical wetting phenomena), and, as theoretically investigated…

A Monte Carlo Simulation of the Stillinger-Weber Model for Si-Ge Alloys

ABSTRACTThe bulk phase behavior of silicon-germanium alloys is investigated by means of a constant pressure Monte Carlo simulation of the Stillinger-Weber potential in the semi-grand-canonical ensemble. At low temperatures, Si and Ge phase separate into a Si-rich phase and a Ge-rich phase. The two-phase region is terminated by a critical point whose nature is investigated thoroughly by the multihistogram method combined with finite size scaling analysis. These results showed that the critical behavior of the alloy belongs to the mean field universality class, presumably due to the elastic degrees of freedom. We have also studied the structural properties of the mixture and found that the li…

Interface Localization-Delocalization in a Double Wedge: A New Universality Class with Strong Fluctuations and Anisotropic Scaling

Using Monte Carlo simulations and finite-size scaling methods we study ``wetting'' in Ising systems in a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ pore with quadratic cross section. Antisymmetric surface fields ${H}_{s}$ act on the free $L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ surfaces of the opposing wedges, and periodic boundary conditions are applied along the $y$ direction. In the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, ${L}_{y}/{L}^{3}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, the system exhibits a new type of phase transition, which is the analog of the ``filling transition'' that occurs in a single wedge. It is charac…

Structural properties ofSi1−xGexalloys: A Monte Carlo simulation with the Stillinger-Weber potential

The structural properties of binary silicon-germanium alloys are investigated by means of large-scale constant-pressure Monte Carlo simulations of the Stillinger-Weber model. At low temperatures, the binary-mixture phase separates into Si-rich and Ge-rich phases. The two-phase coexistence region is terminated by a critical point that belongs to the mean-field universality class. We also studied the structural properties of pure Si and Ge as well as the binary mixture. In particular, we found that the linear thermal expansions for both Si and Ge are in agreement with experiments, and that V\'egard's law is valid at temperatures above the critical point. Finally, we compare the bond-length an…

The Ising model as a playground for the study of wetting and interface behavior

Computer simulations have played an important role in the elucidation of wetting and interface unbinding phenomena. In particular, use of the Ising-lattice-gas model in a film geometry and subject to diverse surface and bulk magnetic fields has permitted extensive Monte Carlo simulations to reveal new features of the phase diagrams associated with these phenomena and to provoke new theoretical studies. The status of our knowledge about the nature of wetting and interface-delocalization transitions which has resulted from these Ising model simulations will be summarized.

Monte Carlo studies of finite-size effects at first-order transitions

Abstract First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussi…

Finite-size scaling for a first-order transition where a continuous symmetry is broken: The spin-flop transition in the three-dimensional XXZ Heisenberg antiferromagnet

Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological ``degeneracy'' factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, $XXZ$ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattice for the first-order ``spin-flop'' transition between the Ising-like antiferromagnetic state and the canted, $XY$-like state. Our theory predicts that for large linear dimension $L$ the field dependen…

Quantum Monte Carlo methods

Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenbe…

Monte Carlo Calculations on Phase Transitions in Adsorbed Layers

MONTE CARLO METHODS FOR FIRST ORDER PHASE TRANSITIONS: SOME RECENT PROGRESS

This brief review discusses methods to locate and characterize first order phase transitions, paying particular attention to finite size effects. In the first part, the order parameter probability distribution and its fourth-order cumulant is discussed for thermally driven first-order transitions (the 3-state Potts model in d=3 dimensions is treated as an example). First-order transitions are characterized by a minimum of the cumulant, which gets very deep for large enough systems. In the second part, we discuss how to locate first order phase boundaries ending in a critical point in a large parameter space. As an example, the study of the unmixing transition of asymmetric polymer mixtures…

Monte Carlo study of surface phase transitions in the three-dimensional Ising model.

We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling ${\mathit{J}}_{\mathit{s}}$. The surface-bulk multicritical point is located with improved precision, ${\mathit{J}}_{\mathit{s}}$/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}…

Simple sampling Monte Carlo methods

Wetting in fluid systems. Wetting and capillary condensation of lattice gases in thin film geometry

Monte Carlo studies of lattice gas models with attractive interactions between nearest neighbors on a simple cubic lattice are carried out for a L×L×D geometry with two hard walls of size L×L and periodic boundary conditions parallel to the wall. Two types of short-range forces at the walls are considered: (i) Both walls are of the same type and exert an attractive force of the same strength (in Ising model terminology, surface fields HD = H1 occur). (ii) The walls differ, one attracts and the other repels particles, again with the same strength (HD = −H1). In the first case, capillary condensation occurs at a chemical potential differing from its value for phase coexistence in the bulk, an…

Phase transition shifts in films

Abstract We present a Monte Carlo computer simulation study of phase transitions in a three-dimensional Ising/lattice gas model with nearest neighbor attractive coupling and confined to a slit-like capillary with absorbing walls. Data are generated for thicknesses D ⩽ 40 and are used to study the shift of the phase boundaries due to finite wall separation.

Critical behavior of the surface-layer magnetization at the extraordinary transition in the three-dimensional Ising model.

We have used a vectorized multispin-coding Monte Carlo method to determine the behavior of the surface-layer magnetization ${\mathit{m}}_{1}$ at the bulk transition in a simple-cubic Ising film with strongly enhanced surface coupling, i.e., at the extraordinary transition. In contrast to recent renormalization-group calculations we find no evidence for a discontinuous slope in the temperature dependence of ${\mathit{m}}_{1}$; the data are consistent with a free-energy-like (T-${\mathit{T}}_{\mathit{c}}$${)}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ behavior plus background terms.

Statistical and systematic errors in Monte Carlo sampling

We have studied the statistical and systematic errors which arise in Monte Carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used. We find that, depending on the degree of self-averaging exhibited by the quantities measured, the statistical errors can increase, decrease, or stay the same as the system size is increased. The systematic underestimation of response functions due to the finite number of measurements made is also studied. We develop a scaling formalism to describe the size dependence of these errors, as well as their dependence on the “bin length” (size of the statistical sample), both…

Critical wetting with short-range forces: Is mean-field theory valid?

Dynamical scaling of surface growth in simple lattice models

We present extensive simulations of the atomistic Edwards-Wilkinson (EW) and Restricted Edwards-Wilkinson (REW) models in 2+1 dimensions. Dynamic finite-size scaling analyses of the interfacial width and structure factor provide the estimates for the dynamic exponent z=1.65+/-0.05 for the EW model and z=2.0+/-0.1 for the REW model. The stochastic contribution to the interface velocity U due to the deposition and diffusion of particles is characterized for both the models using a blocking procedure. For the EW model the time-displaced temporal correlations in U show nonexponential decay, while the temporal correlations decay exponentially for the REW model. Dynamical scaling of the temporal …

Some necessary background

Monte Carlo studies of adsorbed monolayers: Lattice-gas models with translational degrees of freedom

Standard lattice-gas models for the description of the phase behavior of adsorbed monolayers are generalized to ``elastic lattice gases'' which allow for translational degrees of freedom of the adsorbate atoms but have the substrate lattice structure built into the adsorbate-adsorbate interaction. For such models, we derive a simple and efficient grand-canonical Monte Carlo algorithm, which treats the occupied and empty sites in precisely the same way. Using this method, we calculate the phase diagram of a simple model for the adsorption of hydrogen on palladium (100); this model includes only pairwise interactions and exhibits an ordered $c(2\ifmmode\times\else\texttimes\fi{}2)$ structure.…

Appendix: listing of programs mentioned in the text

Critical Wetting and Interface Localization—Delocalization Transition in a Double Wedge

Using Monte Carlo simulations and finite-size scaling methods we study “wetting” in Ising systems in a L x L x L y pore with quadratic cross section. Antisymmetric surface fields H s act on the free L x L y surfaces of the opposing wedges, and periodic boundary conditions are applied along the y-direction. Our results represent the first simulational observation of fluctuation effects in three dimensional wetting phenomena and corroborate recent predictions on wedge filling. In the limit L → ∞ L y /L 3 = const the system exhibits a new type of phase transition, which is the analog of the “filling transition” that occurs in a single wedge. It is characterized by critical exponents α = 3/4, β…

Thin Ising films with competing walls: A Monte Carlo study.

Ising magnets with a nearest neighbor ferromagnetic exchange interaction J on a simple cubic lattice are studied in a thin film geometry using extensive Monte Carlo simulations. The system has two large L\ifmmode\times\else\texttimes\fi{}L parallel free surfaces, a distance D apart from each other, at which competing surface fields act, i.e., ${\mathit{H}}_{\mathit{D}}$=-${\mathit{H}}_{1}$. In this geometry, the phase transition occurring in the bulk at a temperature ${\mathit{T}}_{\mathit{c}\mathit{b}}$ is suppressed, and instead one observes the gradual formation of an interface between coexisting phases stabilized by the surface fields. While this interface is located in the center of th…

More on importance sampling Monte Carlo methods for lattice systems

Probing predictions due to the nonlocal interface Hamiltonian: Monte Carlo simulations of interfacial fluctuations in Ising films

Extensive Monte Carlo simulations have been performed on an Ising ferromagnet under conditions that would lead to complete wetting in a semi-infinite system. We studied an L×L×D slab geometry with oppositely directed surface fields so that a single interface is formed and can undergo a localization-delocalization transition. Under the chosen conditions the interface position is, on average, in the middle of the slab, and its fluctuations allow a sensitive test of predictions that the effective interactions between the interface and the confining surfaces are nonlocal. The decay of distance dependent correlation functions are measured within the surface, in the middle of the slab, and betwee…

Off-lattice models

Wetting transitions near the bulk critical point: Monte Carlo simulations for the Ising model

Critical, tricritical, and first-order wetting transitions are studied near the bulk critical point of a simple cubic nearest-neighbor Ising model by extensive Monte Carlo simulations. The model applies an exchange J in the bulk and exchange ${J}_{s}$ in the surface planes, where surface fields ${H}_{1}$ also act in addition to a possible bulk field H. Lattices in a thin-film geometry L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D are used, with two free L\ifmmode\times\else\texttimes\fi{}L surfaces (with L up to 256) and film thickness D up to 160, applying a very fast fully vectorizing multispin coding program. Our results present the first quantitative evidence f…

Critical phenomena at surfaces

Abstract The presence of free surfaces adds a rich and interesting complexity to critical phenomena associated with phase transitions occurring in bulk materials. We shall review Monte Carlo computer simulation studies of surface critical behavior in simple cubic Ising- and XY-models with nearest-neighbor interactions J in the bulk and Js at the surface. These studies allow the identification of various critical exponents and critical amplitude ratios involving both the critical behavior of local quantities and of surface excess corrections to the bulk. We consider both the “ordinary” transition (surface criticality controlled by the bulk) and the “special transition” (a multicritical point…

Monte Carlo simulations of Ising models and polymer blends in double wedge geometry: Evidence for novel types of critical phenomena

Abstract Two-phase coexistence in systems with free surfaces is enforced by boundary fields requiring the presence of an interface. Varying the temperature or the surface field, one can observe new types of phase transitions where the interface essentially disappears (it becomes bound to a wall or a wedge or a corner of the system). These transitions are simulated with Monte Carlo for Ising ferromagnets and polymer blends, applying finite size scaling analysis. Anisotropic critical fluctuations may occur, and in the limit where the system becomes macroscopically large in all three directions the order parameter vanishes discontinuously (either because its exponent β = 0 , or its critical am…

Phase transitions in thin films with competing surface fields and gradients.

As a generic model for phase equilibria under confinement in a thin-film geometry in the presence of a gradient in the field conjugate to the order parameter, an Ising-lattice gas system is studied by both Monte Carlo simulations and a phenomenological theory. Choosing an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$ geometry with $L\ensuremath{\gg}D$ and periodic boundary conditions in the $x,y$ directions, we place competing surface fields on the two $L\ifmmode\times\else\texttimes\fi{}L$ surfaces. In addition, a field gradient $g$ is present in the $z$ direction across the film, in competition with the surface fields. At temperatures $T$ exceeding the critical…

Monte Carlo studies of anisotropic surface tension and interfacial roughening in the three-dimensional Ising model.

Extensive Monte Carlo simulations of the simple cubic Ising model with nearest-neighbor ferromagnetic interactions with a tilted interface are presented for a wide range of lattice size L, temperature T, and tilt angles \ensuremath{\theta}. The anisotropic interfacial tension is studied in detail. From the small-angle data, we obtain the step free energy density ${f}_{S}$(T,L). Finite-size scaling of the step free energy density is discussed and used to probe the predicted temperature dependence of the correlation length near and above the roughening transition. The square-root temperature dependence predicted by solid-on-solid model calculations is exhibited. Finite-size scaling implies th…

Character of the Phase Transition in Thin Ising Films with Competing Walls

By extensive Monte Carlo simulations of a lattice gas model we have studied the controversial nature of the gas-liquid transition of a fluid confined between two parallel plates that exert competing surface fields. We find that the transition is shifted to a temperature just below the wetting transition of a semi-infinite fluid but belongs to the two-dimensional Ising universality class. In between this new type of critical point and bulk criticality, a response function ${x}_{\mathrm{nn}}^{max}$ varying exponentially with $D$ is observed, $\frac{2 \mathrm{ln}{\ensuremath{\chi}}_{\mathrm{nn}}^{max}}{D}={\ensuremath{\ell}}^{\ensuremath{-}1}$, where $\ensuremath{\ell}$ is a new length charact…

Wedge filling and interface delocalization in finite Ising lattices with antisymmetric surface fields

Theoretical predictions by Parry et al. for wetting phenomena in a wedge geometry are tested by Monte Carlo simulations. Simple cubic $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ Ising lattices with nearest neighbor ferromagnetic exchange and four free $L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ surfaces, at which antisymmetric surface fields $\ifmmode\pm\else\textpm\fi{}{H}_{s}$ act, are studied for a wide range of linear dimensions $(4l~Ll~320,30l~{L}_{y}l~1000),$ in an attempt to clarify finite size effects on the wedge filling transition in this ``double-wedge'' geometry. Interpreting the Ising model as a lattice gas, the problem is equivalent to a li…

Monte Carlo study of surface critical behavior in the XY model.

We have used Monte Carlo simulations to study the behavior of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$ slabs containing classical spins which interact via nearest-neighbor $\mathrm{XY}$ coupling. The coupling constant ${J}_{S}$ for spins in the surface layer is fixed at $0.5J$. Finite-size scaling is used to analyze data for $D=59$ and to extract estimates for the surface critical exponents. We find that ${\ensuremath{\beta}}_{1}$ is in good agreement with theoretical predictions.

Wetting and layering in the nearest-neighbor simple-cubic Ising lattice: A Monte Carlo investigation.

Critical, tricritical, and first-order wetting transitions are studied in a simple-cubic nearest-neighbor Ising model, with exchange J in the bulk and exchange ${J}_{s}$ in the surface planes, by applying suitable bulk and surface fields H and ${H}_{1}$. Monte Carlo calculations are presented for systems of size L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D, in a thin film geometry with D=40 layers and two free L\ifmmode\times\else\texttimes\fi{}L surfaces, with L ranging from L=10 to L=50. In addition, evidence for prewetting transitions and for layering transitions (the latter occur for temperatures T less than the roughening temperature ${T}_{R}$) is presented. …