Nearest-neighbor Ising Model Research Articles

Interplay of phase segregation and chemical reaction: Crossover and effect on growth laws.

By combining the nonconserved spin-flip dynamics driving ferromagnetic ordering with the conserved Kawasaki-exchange dynamics driving phase segregation, we perform Monte Carlo simulations of the nearest-neighbor Ising model. This kind of mixed dynamics is found in a system consisting of a binary mixture of isomers, simultaneously undergoing a segregation and an interconversion reaction among themselves. Here, we study such a system following a quench from the high-temperature homogeneous phase to a temperature below the demixing transition. We monitor the growth of domains of both the winner; the isomer, which survives as the majority; and the loser, the isomer that perishes. Our results show a strong interplay of the two dynamics at early times, leading to a growth of the average domain size of both the winner and loser as ∼t^{1/7}, slower than a purely phase-segregating system. At later times, eventually the dynamics becomes reaction dominated and the winner exhibits a ∼t^{1/2} growth, expected for a system with purely nonconserved dynamics. On the other hand, the loser at first show a faster growth, albeit, slower than the winner, and then starts to decay before it almost vanishes. Further, we estimate the time τ_{s} marking the crossover from the early-time slow growth to the late-time reaction-dominated faster growth. As a function of the reaction probability p_{r}, we observe a power-law scaling τ_{s}∼p_{r}^{-x}, where x≈1.05, irrespective of the temperature. For a fixed value of p_{r} too, τ_{s} appears to be independent of the temperature.

On the origin of cooperativity effects in the formation of self-assembled molecular networks at the liquid/solid interface.

In this work we investigate the behaviour of molecules at the nanoscale using scanning tunnelling microscopy in order to explore the origin of the cooperativity in the formation of self-assembled molecular networks (SAMNs) at the liquid/solid interface. By studying concentration dependence of alkoxylated dimethylbenzene, a molecular analogue to 5-alkoxylated isophthalic derivatives, but without hydrogen bonding moieties, we show that the cooperativity effect can be experimentally evaluated even for low-interacting systems and that the cooperativity in SAMN formation is its fundamental trait. We conclude that cooperativity must be a local effect and use the nearest-neighbor Ising model to reproduce the coverage vs. concentration curves. The Ising model offers a direct link between statistical thermodynamics and experimental parameters, making it a valuable tool for assessing the thermodynamics of SAMN formation.

Perspectives on a Few Puzzles in Phase Transformations: When Should the Farthest Reach the Earliest?

We briefly review the facts concerning two important aspects of phase transitions, namely, critical and coarsening phenomena. A discussion of the universal features, highlighting the current challenges, is provided. Following this, we elaborate on a topic of much recent interest, viz., the Mpemba effect, a puzzle that found mention even in the works of Aristotle. After a description of the debated case of faster freezing of a hotter sample of liquid water, into ice, than a colder one, when quenched to the same subzero temperature, we discuss more modern interest. There one asks, should a hotter body of a material equilibrate faster than a colder one when quenched to a common lower temperature? Within this broad scenario, we focus on magnetic systems. A surprising observation of the effect during the para- to ferromagnetic transition, in a simple model system, viz., the nearest-neighbor Ising model, without any built-in frustration, is described. Some associated future directions are pointed out. A discussion is provided by considering the effect as a kinetic outcome in the background of critical phenomena. A picture is drawn by putting emphasis on the role of spatial correlations in the initial configurations alongside discussing the importance of frustration and metastability in evolution from one state to another. In connection with dynamical freezing, concerning metastability, we have introduced the complex Ginzburg-Landau equation that has relevance in phase transitions, chemical oscillations, and elsewhere. For this model and a few other cases also, we have described how a lack of order or correlation in certain parameters can lead to quicker evolution.

Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

We consider a class of non-integrable 2D Ising models whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature via a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest neighbor Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green’s function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.

Ising Model with Curie–Weiss Perturbation

Consider the nearest-neighbor Ising model on $$\Lambda _n:=[-n,n]^d\cap {\mathbb {Z}}^d$$ at inverse temperature $$\beta \ge 0$$ with free boundary conditions, and let $$Y_n(\sigma ):=\sum _{u\in \Lambda _n}\sigma _u$$ be its total magnetization. Let $$X_n$$ be the total magnetization perturbed by a critical Curie–Weiss interaction, i.e., $$\begin{aligned} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp [x^2/\left( 2\langle Y_n^2 \rangle _{\Lambda _n,\beta }\right) ]}{\left\langle \exp [Y_n^2/\left( 2\langle Y_n^2\rangle _{\Lambda _n,\beta }\right) ]\right\rangle _{\Lambda _n,\beta }}, \end{aligned}$$ where $$F_{X_n}$$ and $$F_{Y_n}$$ are the distribution functions for $$X_n$$ and $$Y_n$$ respectively. We prove that for any $$d\ge 4$$ and $$\beta \in [0,\beta _c(d)]$$ where $$\beta _c(d)$$ is the critical inverse temperature, any subsequential limit (in distribution) of $$\{X_n/\sqrt{{\mathbb {E}}\left( X_n^2\right) }:n\in {\mathbb {N}}\}$$ has an analytic density (say, $$f_X$$ ) all of whose zeros are pure imaginary, and $$f_X$$ has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of $$Y_n$$ . We also prove that for any $$d\ge 1$$ and then for $$\beta $$ small, $$\begin{aligned} f_X(x)=K\exp (-C^4x^4), \end{aligned}$$ where $$C=\sqrt{\Gamma (3/4)/\Gamma (1/4)}$$ and $$K=\sqrt{\Gamma (3/4)}/(4\Gamma (5/4)^{3/2})$$ . Possible connections between $$f_X$$ and the high-dimensional critical Ising model with periodic boundary conditions are discussed.

Critical prewetting in the 2d Ising model

In this paper we develop a detailed analysis of critical prewetting in the context of the two-dimensional Ising model. Namely, we consider a two-dimensional nearest-neighbor Ising model in a $2N\times N$ rectangular box with a boundary condition inducing the coexistence of the $+$ phase in the bulk and a layer of $-$ phase along the bottom wall. The presence of an external magnetic field of intensity $h=\lambda/N$ (for some fixed $\lambda>0$) makes the layer of $-$ phase unstable. For any $\beta>\beta_{\rm c}$, we prove that, under a diffusing scaling by $N^{-2/3}$ horizontally and $N^{-1/3}$ vertically, the interface separating the layer of unstable phase from the bulk phase weakly converges to an explicit Ferrari-Spohn diffusion.

Dynamical phase transitions in the two-dimensional transverse-field Ising model

We investigate two separate notions of dynamical phase transitions in the two-dimensional nearest-neighbor transverse-field Ising model on a square lattice using matrix product states and a new \textit{hybrid} infinite time-evolving block decimation algorithm, where the model is implemented on an infinitely long cylinder with a finite diameter along which periodic boundary conditions are employed. Starting in an ordered initial state, our numerical results suggest that quenches below the dynamical critical point give rise to a ferromagnetic long-time steady state with the Loschmidt return rate exhibiting \textit{anomalous} cusps even when the order parameter never crosses zero. Within the accessible timescales of our numerics, quenches above the dynamical critical point suggest a paramagnetic long-time steady state with the return rate exhibiting \textit{regular} cusps connected to zero crossings of the order parameter. Additionally, our simulations indicate that quenching slightly above the dynamical critical point leads to a coexistence region where both anomalous and regular cusps appear in the return rate. Quenches from the disordered phase further confirm our main conclusions. Our work supports the recent finding that anomalous cusps arise only when local spin excitations are the energetically dominant quasiparticles. Our results are accessible in modern Rydberg experiments.

Zero-temperature coarsening in the two-dimensional long-range Ising model.

We investigate the nonequilibrium dynamics following a quench to zero temperature of the nonconserved Ising model with power-law decaying long-range interactions ∝1/r^{d+σ} in d=2 spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent α, the persistence exponent θ, and the fractal dimension d_{f}. It is found that the growth exponent α≈3/4 is independent of σ and different from α=1/2, as expected for nearest-neighbor models. In the large σ regime of the tunable interactions only the fractal dimension d_{f} of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponents θ this is a direct consequence of the different growth exponents α as can be understood from the relation d-d_{f}=θ/α; they just differ by the ratio of the growth exponents ≈3/2. This relation has been proposed for annihilation processes and later numerically tested for the d=2 nearest-neighbor Ising model. We confirm this relation for all σ studied, reinforcing its general validity.

Predicting nucleation using machine learning in the Ising model.

We use a convolutional neural network (CNN) and two logistic regression models to predict the probability of nucleation in the two-dimensional Ising model. The three methods successfully predict the probability for the nearest-neighbor Ising model for which classical nucleation is observed. The CNN outperforms the logistic regression models near the spinodal of the long-range Ising model, but the accuracy of its predictions decreases as the quenches approach the spinodal. An occlusion analysis suggests that this decrease is due to the vanishing difference between the density of the nucleating droplet and the background. Our results are consistent with the general conclusion that predictability decreases near a critical point.

Analysis of Tandem Repeat Protein Folding Using Nearest-Neighbor Models.

Cooperativity is a hallmark of protein folding, but the thermodynamic origins of cooperativity are difficult to quantify. Tandem repeat proteins provide a unique experimental system to quantify cooperativity due to their internal symmetry and their tolerance of deletion, extension, and in some cases fragmentation into single repeats. Analysis of repeat proteins of different lengths with nearest-neighbor Ising models provides values for repeat folding ([Formula: see text]) and inter-repeat coupling (ΔGi-1,i). In this article, we review the architecture of repeat proteins and classify them in terms of ΔGi and ΔGi-1,i; this classification scheme groups repeat proteins according to their degree of cooperativity. We then present various statistical thermodynamic models, based on the 1D-Ising model, for analysis of different classes of repeat proteins. We use these models to analyze data for highly and moderately cooperative and noncooperative repeat proteins and relate their fitted parameters to overall structural features.

Aging exponents for nonequilibrium dynamics following quenches from critical points.

Via Monte Carlo simulations we study nonequilibrium dynamics in the nearest-neighbor Ising model, following quenches to points inside the ordered region of the phase diagram. With the broad objective of quantifying the nonequilibrium universality classes corresponding to spatially correlated and uncorrelated initial configurations, in this paper we present results for the decay of the order-parameter autocorrelation function for quenches from the critical point. This autocorrelation is an important probe for the aging dynamics in far-from-equilibrium systems and typically exhibits power-law scaling. From the state-of-the-art analysis of the simulation results, we quantify the corresponding exponents (λ) for both conserved and nonconserved (order-parameter) dynamics of the model in space dimension d=3. Via structural analysis we demonstrate that the exponents satisfy a bound. We also revisit the d=2 case to obtain more accurate results. It appears that irrespective of the dimension, λ is approximately the same for both conserved and nonconserved dynamics.

When a defect is a pathway to improve stability: a case study of the L12 Co3TM superlattice intrinsic stacking fault

Effect of solutes of transition metals (TM = Cr, Fe, Hf, Mn, Mo, Nb, Ni, Pt, Rh, Ru, Re, Ta, Ti, V, W, Y and Zr) on the local phase transition between the L12 and D019 structures in superlattice intrinsic stacking fault (SISF) of Co3TM has been investigated. All the models employed herein, i.e. (1) the SISF-containing supercell, (2) the axial nearest-neighbor Ising (ANNI) model, and (3) both the L12- and D019-containing (L12 + D019) supercell, yield the same result regarding the stability of SISF in L12-type Co3TM. In the view of bonding charge density, the atomic and electronic basis of local D019 phase transition in the SISF fault layers of Co3TM are revealed. Especially, the negative SISF energy predicted by the L12 + D019 model suggests that both the SISF fault layers (i.e. the local D019 structure) and the L12 phase of Co3TM can be stabilized through a coupling interaction between the fault layers and the solutes, paving a pathway to stabilize Co-base superalloys via Co3TM precipitate. Moreover, the consist results of ESISF via the ANNI model with the classical SISF-supercell method utilized in first-principles calculations supports the approach to efficiently distinguish various planar faults and predict their corresponding energies, such as SISF, superlattice intrinsic stacking fault, anti-phase boundaries, and so on.

Logarithmic coarsening in the Coulomb glass.

We present numerical results from a comprehensive Monte Carlo study in two dimensions (d=2) of coarsening kinetics in the Coulomb glass (CG) model at half-filling. The CG model is characterized by spin-spin interactions which are long-range Coulombic and antiferromagnetic. For the nonequilibrium properties studied by us (spatial correlation functions and domain growth laws), we find that domain growth in the CG is analogous to that in the nearest-neighbor random-field Ising model. The domain length scale L(t) shows a crossover from a regime of "power-law growth with a disorder-dependent exponent" [L(t)∼t^{1/z[over ¯]}] to a regime of "logarithmic growth with a universal exponent" [L(t)∼(lnt)^{1/ψ}].

Shape-preserving machining produces gradient nanolaminate medium entropy alloys with high strain hardening capability

A high density of grain boundaries can potentially increase structural materials' strength, but at the expense of losing the materials' strain hardening ability at high flow stress levels. However, endowing materials with grain size gradients and a high density of internal interfaces can simultaneously increase the strength and strain hardening ability. This applies particularly for through-thickness gradients of nanoscale interface structures. Here we apply a machining method that produces metals with nanoscale interface gradients. Conventional bulk plastic deformation such as rolling, a process applied annually to about 2 billion tons of material, aims to reduce the metal thickness. We have modified this process by introducing severe strain path changes, realized by leading the sheet through a U-turn while preserving its shape, an approach known as ‘hard turning’. We applied this process at both room temperature and 77 K to a NiCrCo medium entropy alloy. Micropillar compression was conducted to evaluate the mechanical response. After hard turning at room temperature, the surface microstructure obtained a ∼50% increase in yield stress (0.9 GPa) over the original state with homogeneous grain size (0.4 GPa), but the initial strain hardening rate did not show significant improvement. However, after hard turning at 77 k, the gradient nanolaminate structure tripled in yield stress and more than doubled its initial strain hardening rate. The improvements were achieved by introducing a specific microstructure that consists of gradient nanolaminates in the form of nanospaced twins and martensite in the face center cubic (fcc) phase. This microstructure was formed only at cryogenic temperature. It was found after turning at room temperature that only nanospaced twins were present in the fcc phase inside nanolaminates that had formed at the surface. The origin of the enhanced strain hardening mechanism was studied. Joint density functional theory (DFT) and axial next nearest neighbor Ising (ANNNI) models were used to explain the temperature-dependent phase formation of the NiCrCo nanolaminate at the surface of the hard-turned material.

An Interpretation for Defect-Induced Structural Transformation in SiC

Silicon carbide (SiC) has been considered as one of the alternative power semiconductor materials due to its excellent properties such as wide band gap, high break down field, and high thermal conductivity. Although SiC with 3C (zinc blende) is suitable for high-frequency power devices from its high electron mobility and high electron-saturation-velocity, SiC is well known as having dozens of polytypes including 4H and 6H in addition to 2H (wurtzite) and 3C that consist of different ordered stacking sequences of SiC atomic double layers. The 3C-SiC is believed to be more stable than the hexagonal structure at low temperatures. In contrast, 2H-SiC, which has the simplest stacking sequence, is rarely observed at high temperatures. Moreover, point defects also affect the stable polytype in SiC. Heavily Nitrogen (N)-doped single crystals of 6H-SiC were completely transformed into 3C-SiC by annealing in vacuum at presence of Si vapor at 2080 K and 2180 K (hereafter Case I). The 3C-SiC-like bands was also observed in 4H-SiC epilayers with heavily N-doping after thermal annealing (hereafter Case II). Presuming that vacancy formation is related to temperature, point defects such as vacancy and N in SiC are crucial factor for the structural transformation between hexagonal and cubic structures. In this study, the structural transformation in SiC is simply interpreted by considering point defect formation with the aid of ab initio calculations. Ab initio total energy calculations in a √3×√3×12 unit cell imply that stable stacking for SiC is 6H with Si-vacancy (VSi), 4H with C-vacancy (VC) and N substituting for Si (NSi), and 3C with N substituting for C (NC), whereas 2H-stacking does not appear as a stable stacking in these systems. Furthermore, the formation energy calculations reveal that vacancy formation depends on C chemical potential m C, where VSi and VC formations are favorable at large m C (corresponding to C-rich) and small m C (Si-rich), respectively. On the other hand, NC formation is more favorable than NSi formation over the entire range of m C and independent of conditions because of very small difference in covalent radius between N and C. Comparing these calculated results with the experimental finding in the Case I for transformation from 6H to 3C, thermal treatment with presence of Si vapor at high temperatures suppresses VSi formation favoring 6H-stacking while it enhances VC formation to be occupied by N (NC) in addition to N substitution for C (NC) that transform 6H to 3C. The Case II can be also interpreted by considering NC. Thermal annealing for N-doped 4H-SiC favored by VC easily produces NC with N occupying VC and substituting for C to realize 3C-SiC bands at high N-doped region in 4H-SiC epilayers. These results suggest that NC is crucial for the transformation from hexagonal stacking such as 6H and 4H to cubic stacking 3C. In order to give physical interpretation for the transformation to 3C-SiC with NC, the interaction energies Jn between nth neighbor SiC double layers are estimated in the range of n ≤ 3 using the axial next nearest neighbor Ising (ANNNI) model and calculated total energies for SiC polytypes obtained in this study. The calculated results elucidate that the absolute value of positive J 1 (51.70 meV) with J 3 (9.92 meV) favoring 3C-stacking is much larger than that of negative J 2 (-16.66 meV) inducing hexagonal stacking. It should be noted that the Jn values are one order of magnitude greater than those of perfect SiC such as J 1 (2.33 meV), J 2 (-1.41 meV), and J 3 (0.51 meV). This implies that the layer interaction in SiC becomes strong and long-range due to N-doping. Considering coordinate of (|J 2|/J 1, J 3/J 1) in the ANNNI phase diagram, it is also found that (0.31, 0.19) for SiC with NC is located in the stable region of 3C and is more distant from the multi-phase degeneracy point (0.5, 0) inducing polytype and phase boundary between 3C and 4H than (0.61, 0.21) for perfect SiC. This clarifies that the large J 1 in SiC with NC stabilizes 3C-stacking to dominate the structural transformation to 3C due to N-doping. Furthermore, atomistic interaction between the nearest neighbor layers corresponding to J 1 is also qualitatively discussed in terms of electrostatic interaction between bond charges located at the center of interatomic bonds and that between ionic charges located at the lattice sites on the basis of a simple energy description and charge distribution obtained by ab initio calculations.

An Interpretation for Defect-Induced Structural Transformation in SiC

The structural transformation in SiC is simply interpreted by considering point defect formation with the aid of ab initio calculations and axial nearest neighbor Ising model. Ab initio calculations reveal that stable stacking for SiC is 6H with Si-vacancy (VSi) under C-rich condition, 4H with C-vacancy (VC) under Si-rich condition and N substituting for Si (NSi), and 3C with N substituting for C (NC). On the basis of these results, thermal treatment with Si vapor is expected to suppress VSi formation favoring 6H-stacking while it enhances VC formation to be occupied by NC that transforms 6H to 3C. Furthermore, thermal annealing for N-doped 4H-SiC favored by VC easily produces NC to realize 3C-SiC. These results are consistent with experimental findings. It is also found that the large nearest layer interaction energy in SiC with NC stabilizes 3C-stacking to dominate the structural transformation to 3C due to N-doping.

Stability of multiple Shockley type basal plane stacking faults in heavily nitrogen-doped 4H-SiC crystals

The stability of various types of multiple Shockley type basal plane stacking fault in heavily nitrogen-doped 4H-SiC crystals was theoretically investigated on the basis of the quantum well action (QWA) mechanism. The energy levels of confined electrons within multiple Shockley stacking faults (SSFs) were calculated using the simple quantum well model and compared to the experimental values estimated from photoluminescence measurements. The formation energies of the stacking faults were also calculated on the basis of the axial next nearest neighbor Ising (ANNNI) model for SiC polytypes. Based on these physical and electronic parameters, the stability of multiple SSFs in heavily nitrogen-doped 4H-SiC crystals was discussed, and it was revealed that the stacking sequence of multiple SSFs largely affects their stability; SSFs having a cubic stacking sequence of medium thickness (five to six Si-C bilayer thickness) are stable in heavily nitrogen-doped 4H-SiC crystals at the typical growth and device processing temperatures of SiC, i.e., 1500–2600 K.

Extreme stability in de novo-designed repeat arrays is determined by unusually stable short-range interactions

Designed helical repeats (DHRs) are modular helix-loop-helix-loop protein structures that are tandemly repeated to form a superhelical array. Structures combining tandem DHRs demonstrate a wide range of molecular geometries, many of which are not observed in nature. Understanding cooperativity of DHR proteins provides insight into the molecular origins of Rosetta-based protein design hyperstability and facilitates comparison of energy distributions in artificial and naturally occurring protein folds. Here, we use a nearest-neighbor Ising model to quantify the intrinsic and interfacial free energies of four different DHRs. We measure the folding free energies of constructs with varying numbers of internal and terminal capping repeats for four different DHR folds, using guanidine-HCl and glycerol as destabilizing and solubilizing cosolvents. One-dimensional Ising analysis of these series reveals that, although interrepeat coupling energies are within the range seen for naturally occurring repeat proteins, the individual repeats of DHR proteins are intrinsically stable. This favorable intrinsic stability, which has not been observed for naturally occurring repeat proteins, adds to stabilizing interfaces, resulting in extraordinarily high stability. Stable repeats also impart a downhill shape to the energy landscape for DHR folding. These intrinsic stability differences suggest that part of the success of Rosetta-based design results from capturing favorable local interactions.

The critical 1-arm exponent for the ferromagnetic Ising model on the Bethe lattice

We consider the ferromagnetic nearest-neighbor Ising model on regular trees (Bethe lattice), which is well-known to undergo a phase transition in the absence of an external magnetic field. The behavior of the model at critical temperature can be described in terms of various critical exponents; one of them is the critical 1-arm exponent ρ which characterizes the rate of decay of the (root) magnetization as a function of the distance to the boundary. The crucial quantity we analyze in this work is the thermal expectation of the root spin on a finite subtree, where the expected value is taken with respect to a probability measure related to the corresponding finite-volume Hamiltonian with a fixed boundary condition. The spontaneous magnetization, which is the limit of this thermal expectation in the distance between the root and the boundary (i.e., in the height of the subtree), is known to vanish at criticality. We are interested in a quantitative analysis of the rate of this convergence in terms of the critical 1-arm exponent ρ. Therefore, we rigorously prove that ⟨σ0⟩n+, the thermal expectation of the root spin at the critical temperature and in the presence of the positive boundary condition, decays as ⟨σ0⟩n+≈n−12 (in a rather sharp sense), where n is the height of the tree. This establishes the 1-arm critical exponent for the Ising model on regular trees (ρ=12).

Stein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process

We provide non-asymptotic $L^1$ bounds to the normal for four well-known models in statistical physics and particle systems in $\mathbb{Z}^d$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^1$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for the total number of points in a finite region belonging to an infinite cluster in dimensions $d \ge 2$, in the voter model for the occupation time of the origin in dimensions $d \ge 7$, and for finite time integrals of non-constant increasing cylindrical functions evaluated on the one dimensional supercritical contact process started in its unique invariant distribution. The tool developed for these purposes is a version of Stein's method adapted to positively associated random variables. In one dimension, letting $\boldsymbol{\xi}=(\xi_1,\ldots,\xi_m)$ be a positively associated mean zero random vector with components that obey the bound $|\xi_i| \le B, i=1,\ldots,m$, and whose sum $W = \sum_{i=1}^m \xi_i$ has variance 1, it holds that $$ d_1 \left(\mathcal{L}(W),\mathcal{L}(Z) \right) \leq 5B + \sqrt{\frac{8}{\pi}}\sum_{i \neq j} \mathbb{E}[\xi_i \xi_j] $$ where $Z$ has the standard normal distribution and $d_1(\cdot,\cdot)$ is the $L^1$ metric. Our methods apply in the multidimensional case with the $L^1$ metric replaced by a smooth function metric.