Cluster Variation Method Research Articles

Cluster Variation Method as a Theoretical Tool for the Study of Phase Transformation

Cluster variation method (CVM) has been widely employed to calculate alloy phase diagrams. The atomistic feature of the CVM is consistent with first-principles electronic structure calculations, and the combination of CVM with electronic structure calculation enables one to formulate free energy from the first-principles. CVM free energy conveys affluent information of a given system, and the second-order derivative traces the stability locus against configurational fluctuation. The kinetic extension of the CVM is the path probability method (PPM) which is utilized to calculate transformation and relaxation kinetics associated with the temperature change. Hence, the CVM and PPM are coherent methods to perform a synthetic study from initial non-equilibrium to final equilibrium states. By utilizing CVM free energy as a homogeneous free energy density term, one can calculate the time evolution of ordered domains within the phase field method. Finally, continuous displacement cluster variation method (CDCVM) is discussed as the recent development of CVM. CDCVM is capable of introducing the local lattice displacement into the free energy. Moreover, it is shown that CDCVM can be extended to study collective atomic displacements leading to displacive phase transformation.

B2-disorder phase boundary calculations in Fe rich region of Fe-Si binary system with tetrahedron approximation of CVM

The phase boundary between B2 ordered and disordered phases in the Fe-rich region of the Fe-Si binary system is calculated by Cluster Variation Method (CVM). The configurational entropy is formulated within the tetrahedron approximation of CVM, and the internal energy is derived by Cluster Expansion Method (CEM) operated on a set of total energies calculated by DFT. The Debye Gruneisen model is employed to introduce the vibrational effect. The second order transition for the B2 order-disorder transition is confirmed, which is in agreement with the published phase diagram data and the results of previous CALPHAD calculations. The calculated transition temperature in the present study is higher around the stoichiometric composition and lower in the Fe-rich region compared to the experimental transition temperature. One reason for this overestimation and underestimation of the transition temperature may stem from the facts that the local atomic displacement and wide range atomic correlations are not considered in the present study. The transition temperature is also determined using Thermo-Calc software with the SSOL4 database. The transition temperature obtained by Thermo-Calc calculations accurately reproduces the experimental results. Hence, it is considered that the interaction parameters and the ordering parameters of CALPHAD free energy implicitly include the contributions of short range ordering and local atomic displacement.

±J Blume–Capel model in the cluster variation method

The ± J Blume–Capel model is studied by using the cluster variation method (CVM) in its first approximation which yields results equal to the mean-field ones. The spins are assumed to be interacting either ferromagnetically or antiferromagnetically with the given probabilities p or 1−p, respectively. The phase diagrams of the model are obtained on the temperature (T)-probability (p) or crystal field (D) planes for given D and p, respectively. The model is examined by introducing the dipolar and quadrupolar order-parameters and, ferromagnetic (F), antiferromagnetic (AF) and paramagnetic (P) phase regions are explored. It is found that the model gives first- and second-order phase transitions and tricritical points.

Field-Tuned Order by Disorder in Frustrated Ising Magnets with Antiferromagnetic Interactions.

We demonstrate the appearance of thermal order by disorder in Ising pyrochlores with staggered antiferromagnetic order frustrated by an applied magnetic field. We use a mean-field cluster variational method, a low-temperature expansion, and MonteCarlo simulations to characterize the order-by-disorder transition. By direct evaluation of the density of states, we quantitatively show how a symmetry-broken state is selected by thermal excitations. We discuss the relevance of our results to experiments in 2D and 3D samples and evaluate how anomalous finite-size effects could be exploited to detect this phenomenon experimentally in two-dimensional artificial systems, or in antiferromagnetic all-in-all-out pyrochlores like Nd_{2}Hf_{2}O_{7} or Nd_{2}Zr_{2}O_{7}, for the first time.

Pair correlations and structure factor of theJ1-J2square lattice Ising model in an external field

We compute the structure factor of the $J_1$-$J_2$ Ising model in an external field on the square lattice within the Cluster Variation Method. We use a four point plaquette approximation, which is the minimal one able to capture phases with broken orientational order in real space, like the recently reported Ising-nematic phase in the model. The analysis of different local maxima in the structure factor allows us to track the different phases and phase transitions against temperature and external field. Although the nematic susceptibility is not directly related to the structure factor, we show that because of the close relationship between the nematic order parameter and the structure factor, the latter shows unambiguous signatures of the presence of a nematic phase, in agreement with results from direct minimization of a variational free energy. The disorder variety of the model is identified and the possibility that the CVM four point approximation be exact on the disorder variety is discussed.

The Cluster Variation Method: A Primer for Neuroscientists.

Effective Brain–Computer Interfaces (BCIs) require that the time-varying activation patterns of 2-D neural ensembles be modelled. The cluster variation method (CVM) offers a means for the characterization of 2-D local pattern distributions. This paper provides neuroscientists and BCI researchers with a CVM tutorial that will help them to understand how the CVM statistical thermodynamics formulation can model 2-D pattern distributions expressing structural and functional dynamics in the brain. The premise is that local-in-time free energy minimization works alongside neural connectivity adaptation, supporting the development and stabilization of consistent stimulus-specific responsive activation patterns. The equilibrium distribution of local patterns, or configuration variables, is defined in terms of a single interaction enthalpy parameter (h) for the case of an equiprobable distribution of bistate (neural/neural ensemble) units. Thus, either one enthalpy parameter (or two, for the case of non-equiprobable distribution) yields equilibrium configuration variable values. Modeling 2-D neural activation distribution patterns with the representational layer of a computational engine, we can thus correlate variational free energy minimization with specific configuration variable distributions. The CVM triplet configuration variables also map well to the notion of a M = 3 functional motif. This paper addresses the special case of an equiprobable unit distribution, for which an analytic solution can be found.

Cycle-Based Cluster Variational Method for Direct and Inverse Inference

We elaborate on the idea that loop corrections to belief propagation could be dealt with in a systematic way on pairwise Markov random fields, by using the elements of a cycle basis to define region in a generalized belief propagation setting. The region graph is specified in such a way as to avoid dual loops as much as possible, by discarding redundant Lagrange multipliers, in order to facilitate the convergence, while avoiding instabilities associated to minimal factor graph construction. We end up with a two-level algorithm, where a belief propagation algorithm is run alternatively at the level of each cycle and at the inter-region level. The inverse problem of finding the couplings of a Markov random field from empirical covariances can be addressed region wise. It turns out that this can be done efficiently in particular in the Ising context, where fixed point equations can be derived along with a one-parameter log likelihood function to minimize. Numerical experiments confirm the effectiveness of these considerations both for the direct and inverse MRF inference.

Approximate solutions to the cluster variation free energies by the variable basis cluster expansion

In this paper we obtain approximate solutions to the Cluster Variation free energy by carrying out a cluster expansion of the probabilities appearing in the free energy functional in terms of concentration-dependent basis functions, and by truncating the expansion at different cluster levels prior to minimization. We show that a significant improvement over the Bragg–Williams approximation can be achieved by truncating the expansion of the cluster probabilities at relatively small clusters, thus dramatically reducing the number of equations that need to be solved in order to minimize the free energy. Furthermore, the free energy functional in the Cluster Variation Method offers a well-controlled case study to infer the effects of truncating the expansion of the energy of alloy formation in the commonly used Cluster Expansion method, versus the effects of truncating the expansion when using a concentration-dependent basis. Examples of the approach are given for simple Ising models for fcc- and bcc-based prototype alloy systems.

Equation of state for neutron stars with hyperons using a variational method

We investigate the effects of the odd-state part of bare $\Lambda \Lambda$ interactions on the structure of neutron stars (NSs) by constructing equations of state (EOSs) for uniform nuclear matter containing $\Lambda$ and $\Sigma^-$ hyperons with use of the cluster variational method. The isoscalar part of the Argonne v18 two-nucleon potential and the Urbana IX three-nucleon potential are employed as the interactions between nucleons, whereas, as the bare $\Lambda N$ and even-state $\Lambda \Lambda$ interactions, two-body central potentials that are determined so as to reproduce the experimental data on single- and double-$\Lambda$ hypernuclei are adopted. In addition, the $\Sigma^- N$ interaction is constructed so as to reproduce the empirical single-particle potential of $\Sigma^-$ in symmetric nuclear matter. Since the odd-state part of the $\Lambda \Lambda$ interaction is not known owing to lack of experimental data, we construct four EOSs of hyperonic nuclear matter, each with a different odd-state part of the $\Lambda \Lambda$ interaction. The EOS obtained for NS matter becomes stiffer as the odd-state $\Lambda \Lambda$ interaction becomes more repulsive, and correspondingly the maximum mass of NSs increases. It is interesting that the onset density of $\Sigma^-$ depends strongly on the repulsion of the odd-state $\Lambda \Lambda$ interaction. Furthermore, we take into account the three-baryon repulsive force to obtain results that are consistent with observational data on heavy NSs.

Lattice Statistics and Dynamics within Cluster Variation Method

Lattice Statistics and Dynamics within Cluster Variation Method

Statistical analysis of loopy belief propagation in random fields.

Loopy belief propagation (LBP), which is equivalent to the Bethe approximation in statistical mechanics, is a message-passing-type inference method that is widely used to analyze systems based on Markov random fields (MRFs). In this paper, we propose a message-passing-type method to analytically evaluate the quenched average of LBP in random fields by using the replica cluster variation method. The proposed analytical method is applicable to general pairwise MRFs with random fields whose distributions differ from each other and can give the quenched averages of the Bethe free energies over random fields, which are consistent with numerical results. The order of its computational cost is equivalent to that of standard LBP. In the latter part of this paper, we describe the application of the proposed method to Bayesian image restoration, in which we observed that our theoretical results are in good agreement with the numerical results for natural images.

Entropy and Diffuse Scattering: Comparison of NbTiVZr and CrMoNbV

The chemical disorder intrinsic to high entropy alloys inevitably creates diffuse scattering in their x-ray or neutron diffraction patterns. Through first principles hybrid Monte Carlo/molecular dynamics simulations of two BCC high entropy alloy forming compounds, CrMoNbV and NbTiVZr, we identify the contributions of chemical disorder, atomic size and thermal fluctuations to the diffuse scattering. As a side benefit, we evaluate the reduction in entropy due to pair correlations within the framework of the cluster variation method. Finally, we note that the preference of Ti and Zr for HCP structures at low temperature leads to a mechanical instability reducing the local BCC character of NbTiVZr, while preserving global BCC symmetry.

Properties of a localized magnetic impurity in a superconducting host: The Anderson–Holstein–BCS model

A symmetric Anderson–Holstein model with a BCS interaction term is considered to investigate the effect of local electron–phonon interaction on a magnetic impurity in a superconductor. The Kikuchi–Morita Cluster variation (CV) method is used to calculate the local impurity magnetic moment and the binding energy between the impurity and the conduction electrons in the superconductor. The effect of electron–phonon interaction and the order parameter on the bound state is discussed.

Random field Ising model in two dimensions: Bethe approximation, cluster variational method and message passing algorithms

We study two free energy approximations (Bethe and plaquette-CVM) for the Random Field Ising Model in two dimensions. We compare results obtained by these two methods in single instances of the model on the square grid, showing the difficulties arising in defining a robust critical line. We also attempt average case calculations using a replica-symmetric ansatz, and compare the results with single instances. Both, Bethe and plaquette-CVM approximations present a similar panorama in the phase space, predicting long range order at low temperatures and fields. We show that plaquette-CVM is more precise, in the sense that predicts a lower critical line (the truth being no line at all). Furthermore, we give some insight on the non-trivial structure of the fixed points of different message passing algorithms. A study of the Monte Carlo dynamics for an arbitrary sample shows that GBP states are very well correlated with states that are attractors of the stochastic dynamics.

Nematic phase in theJ1−J2square-lattice Ising model in an external field

The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.

Effect of electron-phonon interaction and external magnetic field on the bound state in the Anderson-Holstein model: an improved variational treatment

The single-impurity Anderson-Holstein model is investigated in the presence of a magnetic field by an improved variational method. The phonon degrees of freedom are first eliminated by a modified Lang-Firsov transformation followed by a zero-phonon averaging. The resulting Hamiltonian is then treated by a cluster variational method to study the effects of the electron-phonon interaction and the magnetic field on the ground state energy, local magnetic moment and the binding energy between the magnetic impurity and the conduction electrons.

Analysis of Non-Monotonic Temperature Behavior of the Coefficient of Thermal Expansion in Fe-Ni Alloys Studied by First-Principles Cluster Variation Method

Analysis of Non-Monotonic Temperature Behavior of the Coefficient of Thermal Expansion in Fe-Ni Alloys Studied by First-Principles Cluster Variation Method

Equation of state for nuclear matter in core-collapse supernovae by the variational method

We construct a new nuclear equation of state (EOS) for core-collapse supernova (SN) simulations using the variational many-body theory. For uniform nuclear matter, the EOS is constructed with the cluster variational method starting from the realistic nuclear Hamiltonian composed of the Argonne v18 two-body potential and the Urbana IX three-body potential. The masses and radii of neutron stars calculated with the obtained EOS at zero temperature are consistent with recent observational data. For non-uniform nuclear matter, we construct the EOS in the Thomas-Fermi approximation. In this approximation, we assume a functional form of the density distributions of protons, neutrons, and alpha-particles, and minimize the free energy density in a Wigner-Seitz cell with respect to the parameters included in the assumed density distribution functions. The phase diagram of hot nuclear matter at a typical temperature is reasonable as compared with that of the Shen EOS.

Message passing and Monte Carlo algorithms: Connecting fixed points with metastable states

Mean field-like approximations (including naive mean-field, Bethe and Kikuchi and more general cluster variational methods) are known to stabilize ordered phases at temperatures higher than the thermodynamical transition. For example, in the Edwards-Anderson model in 2 dimensions these approximations predict a spin glass transition at finite T. Here we show that the spin glass solutions of the Cluster Variational Method (CVM) at plaquette level do describe well the actual metastable states of the system. Moreover, we prove that these states can be used to predict non-trivial statistical quantities, like the distribution of the overlap between two replicas. Our results support the idea that message passing algorithms can be helpful to accelerate Monte Carlo simulations in finite-dimensional systems.

The γ-Fe[N] and γ′-Fe4N1−x phase boundaries in high-nitrogen steels: The cube cluster approximation and the effect of vibrational energy contributions

Density functional theory (DFT) calculations and the cluster variation method (CVM) are combined to describe the γ-Fe[N] and γ′-Fe4N1−x phase boundaries. A new cluster approximation, the cube cluster, is used in the CVM. The advantage of using cube clusters is that they comprise both the metal and the interstitial sublattices and that a relatively small number of ordered structures needs to be considered. The cube energies are obtained by DFT calculations applying projector augmented wave pseudo-potentials wherein the electronic structure calculations are carried out in the generalized gradient approximation. The zero-point energy (ZPE), computed using the direct method, changes only slightly the total energy but it increases the cell volumes and decreases the bulk modulus of the cubes. Finite temperature effects are included through the Debye–Grüneisen model. Including the effects of ZPE and finite temperature effects, are necessary to predict the γ-Fe[N] and γ′-Fe4N1−x phase boundaries well.