Showing papers on "Cluster expansion published in 2019"
TL;DR: The relationship between heating rates and the smooth function that characterizes the off-diagonal matrix elements of the drive operator in the eigenbasis of the static Hamiltonian is discussed and it is shown that such a function, in nonintegrable and (remarkably) integrable Hamiltonians, can be probed experimentally by studying heating rates as functions of theDrive frequency.
Abstract: We study heating rates in strongly interacting quantum lattice systems in the thermodynamic limit. Using a numerical linked cluster expansion, we calculate the energy as a function of the driving time and find a robust exponential regime. The heating rates are shown to be in excellent agreement with Fermi's golden rule. We discuss the relationship between heating rates and, within the eigenstate thermalization hypothesis, the smooth function that characterizes the off-diagonal matrix elements of the drive operator in the eigenbasis of the static Hamiltonian. We show that such a function, in nonintegrable and (remarkably) integrable Hamiltonians, can be probed experimentally by studying heating rates as functions of the drive frequency.
49 citations
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TL;DR: This work introduces machine-learned potentials for Ag-Pd to describe the energy of alloy configurations over a wide range of compositions and compares two different approaches to Moment tensor potentials, which are polynomial-like functions of interatomic distances and angles.
Abstract: We introduce machine-learned potentials for Ag-Pd to describe the energy of alloy configurations over a wide range of compositions. We compare two different approaches. Moment tensor potentials (MTP) are polynomial-like functions of interatomic distances and angles. The Gaussian Approximation Potential (GAP) framework uses kernel regression, and we use the Smooth Overlap of Atomic Positions (SOAP) representation of atomic neighbourhoods that consists of a complete set of rotational and permutational invariants provided by the power spectrum of the spherical Fourier transform of the neighbour density. Both types of potentials give excellent accuracy for a wide range of compositions and rival the accuracy of cluster expansion, a benchmark for this system. While both models are able to describe small deformations away from the lattice positions, SOAP-GAP excels at transferability as shown by sensible transformation paths between configurations, and MTP allows, due to its lower computational cost, the calculation of compositional phase diagrams. Given the fact that both methods perform as well as cluster expansion would but yield off-lattice models, we expect them to open new avenues in computational materials modeling for alloys.
42 citations
TL;DR: At high temperatures, it is shown that many-body cluster correlations still play an important role in understanding the configuration entropy before reaching the solid solution limit of high-entroy alloys (HEAs).
Abstract: Configuration entropy is believed to stabilize disordered solid solution phases in multicomponent systems at elevated temperatures over intermetallic compounds by lowering the Gibbs free energy. Traditionally, the increment of configuration entropy with temperature was computed by time-consuming thermodynamic integration methods. In this work, a new formalism based on a hybrid combination of the Cluster Expansion (CE) Hamiltonian and Monte Carlo simulations is developed to predict the configuration entropy as a function of temperature from multi-body cluster probability in a multi-component system with arbitrary average composition. The multi-body probabilities are worked out by explicit inversion and direct product of a matrix formulation within orthonomal sets of point functions in the clusters obtained from symmetry independent correlation functions. The matrix quantities are determined from semi canonical Monte Carlo simulations with Effective Cluster Interactions (ECIs) derived from Density Functional Theory (DFT) calculations. The formalism is applied to analyze the 4-body cluster probabilities for the quaternary system Cr-Fe-Mn-Ni as a function of temperature and alloy concentration. It is shown that, for two specific compositions (Cr 25Fe 25Mn 25Ni 25 and Cr 18Fe 27Mn 27Ni 28), the high value of probabilities for Cr-Fe-Fe-Fe and Mn-Mn-Ni-Ni are strongly correlated with the presence of the ordered phases L1 2 -CrFe 3 and L1 0-MnNi, respectively. These results are in an excellent agreement with predictions of these ground state structures by ab initio calculations. The general formalism is used to investigate the configuration entropy as a function of temperature and for 285 different alloy compositions. It is found that our matrix formulation of cluster probabilities provides an efficient tool to compute configuration entropy in multi-component alloys in a comparison with the result obtained by the thermodynamic integration method. At high temperatures, it is shown that many-body cluster correlations still play an important role in understanding the configuration entropy before reaching the solid solution limit of high-entroy alloys (HEAs).
32 citations
TL;DR: In this paper, nonperturbative aspects of the quantum many-body problem are revisited, discussed and advanced in the equation of motion framework, and a new class of solutions for the response functions, which include explicitly beyond-mean-field correlations between up to six fermions.
Abstract: Nonperturbative aspects of the quantum many-body problem are revisited, discussed and advanced in the equation of motion framework. We compare the approach to the two-fermion response function truncated on the two-body level by the cluster expansion of the dynamical interaction kernel to the approach known as time blocking approximation. Such a comparison leads to an extended many-body theory with nonperturbative treatment of high-order configurations. The present implementation of the advanced theory introduces a new class of solutions for the response functions, which include explicitly beyond-mean-field correlations between up to six fermions. The novel approach, which includes configurations with two quasiparticles coupled to two phonons ($2\mathrm{q}\ensuremath{\bigotimes}2\mathrm{phonon}$), is discussed in detail for the particle-hole nuclear response and applied to medium-mass nuclei. The proposed developments are implemented numerically on the basis of the relativistic effective meson-nucleon Lagrangian and compared to the models confined by two-fermion and four-fermion configurations, which are considered as state-of-the-art for the response theory in nuclear structure calculations. The results obtained for the dipole response of $^{42,48}\mathrm{Ca}$ and $^{68}\mathrm{Ni}$ nuclei in comparison to available experimental data show that the higher configurations are necessary for a successful description of both gross and fine details of the spectra in both high-energy and low-energy sectors.
26 citations
TL;DR: In this article, the authors investigated a compromise where the quantum character of light is taken into account at modest computational cost by employing the cumulant or cluster expansion method to the Heisenberg equations of motion up to first, second and third order.
Abstract: Strong coupling of quantum emitters with confined electromagnetic modes of nanophotonic structures may be used to change optical, chemical and transport properties of materials, with significant theoretical effort invested towards a better understanding of this phenomenon. However, a full theoretical description of both matter and light is an extremely challenging task. Typical theoretical approaches simplify the description of the photonic environment by describing it as a single or few modes. While this approximation is accurate in some cases, it breaks down strongly in complex environments, such as within plasmonic nanocavities, and the electromagnetic environment must be fully taken into account. This requires the quantum description of a continuum of bosonic modes, a problem that is computationally hard. We here investigate a compromise where the quantum character of light is taken into account at modest computational cost. To do so, we focus on a quantum emitter that interacts with an arbitrary photonic spectral density and employ the cumulant or cluster expansion method to the Heisenberg equations of motion up to first, second and third order. We benchmark the method by comparing with exact solutions for specific situations and show that it can accurately represent dynamics for many parameter ranges.
24 citations
TL;DR: In this article, the QCD equation of state at finite baryon density is studied in the framework of a Cluster Expansion Model (CEM), which is based on the fugacity expansion of the net baryons density and uses the two leading Fourier coefficients, obtained from lattice simulations at imaginary μB, as the only model input and permits a closed analytic form.
15 citations
TL;DR: In this article, a multi-scale model for the prediction of precipitation kinetics of binary alloys and its application to the formation of Guinier-Preston zones (GPZ) in Al-rich binary Al-Cu alloys is presented.
15 citations
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TL;DR: An FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides of the bipartition is given.
Abstract: We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $\lambda=1$ (approximating the number of independent sets of $G$) and $\Delta_R \geq 7\Delta_L \log(\Delta_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.
14 citations
TL;DR: In this article, the authors developed a framework to express highly anharmonic first-principles potential energy surfaces as polynomials of collective cluster deformations and further adapted the approach to a nonlinear extension of the cluster expansion formalism through the use of an artificial neural net model.
Abstract: The temperature and pressure dependence of structural phase transitions determine the structure-functionality relationships in many technologically important materials. Harmonic Hamiltonians have proven successful in predicting the vibrational properties of many materials. However, they are inadequate for modeling structural phase transitions in crystals with potential energy surfaces that are either strongly anharmonic or nonconvex with respect to collective atomic displacements or homogeneous strains. In this paper we develop a framework to express highly anharmonic first-principles potential energy surfaces as polynomials of collective cluster deformations. We further adapt the approach to a nonlinear extension of the cluster expansion formalism through the use of an artificial neural net model. The machine learning models are trained on a large database of first-principles calculations and are shown to reproduce the potential energy surface with low error.
14 citations
TL;DR: The atomic order, electronic structure and thermodynamic stability of nickel aluminate, NiAl2O4, have been analyzed using periodic density functional theory and cluster expansion and it is found to be ferromagnetic and a semiconductor with an indirect band gap along the Γ → M symmetry points.
Abstract: The atomic order, electronic structure and thermodynamic stability of nickel aluminate, NiAl2O4, have been analyzed using periodic density functional theory and cluster expansion. NiAl2O4 forms a tetragonal structure with P4122 space group. At temperatures below 800 K, it is an inverse spinel, with Ni occupying the octahedral sites and Al occupying both the octahedral and the tetrahedral sites. Some Niocta + Altetra ⇌ Nitetra + Alocta exchange occurs above 800 K, but the structure remains largely inverse at high temperatures, with about 95% Niocta at 1500 K. Various functionals, with and without van der Waals corrections, were used to predict the experimental formation energy, lattice parameters and electronic structure. In all cases, the NiAl2O4 is found to be ferromagnetic and a semiconductor with an indirect band gap along the Γ → M symmetry points. NiAl2O4 is found to be thermodynamically stable at operating conditions of 900-1000 K and 1 atm relative to its competing oxide phases, NiO and Al2O3.
14 citations
TL;DR: In this article, via density functional theory (DFT), cluster expansion (CE), and Monte Carlo (MC) calculations, the authors investigated three binary MXene alloy systems of Ti2CO2 and M′2CO 2 (M′ = V, Nb, or Ta), where T...
Abstract: Herein, via density functional theory (DFT), cluster expansion (CE), and Monte Carlo (MC) calculations, we investigate 3 binary MXene alloy systems of Ti2CO2 and M′2CO2 (M′ = V, Nb, or Ta), where T...
TL;DR: In this article, the authors give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles, which correspond bijectively with perfect matching of the corresponding bipartite graph and minimal cuts of a quiver with potential.
Abstract: We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between perfect matchings of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.
TL;DR: In this article, tensor networks formulated via classic logic gates are used to treat electron-hole complexes inside the Brillouin zone of a single layer transition metal dichalcogenide MoS$_2$ like model system.
Abstract: Carriers such as electrons and holes inside the Brillouin zone of complex semiconducting materials can form bound states (excitons, biexcitons etc) For obtaining the corresponding eigenstates (eg through Wannier or Bethe Salpeter equation) and dynamics (eg cluster expansion) the number of involved electrons and holes as well as the accuracy is limited by the appearing high dimensional tensors (ie wavefunctions or correlations) These tensors can be efficiently represented and manipulated via tensor network methods We show how tensor networks formulated via classic logic gates can be used to treat electron-hole complexes inside the Brillouin zone The method is illustrated for the exciton and biexciton states of a single layer transition metal dichalcogenide MoS$_2$ like model system
TL;DR: An algorithm to compute the lattice energies of molecular crystals based on the many-body cluster expansion is presented and Coulomb-matrix descriptors from machine learning applications are found to be efficient in determining whether two N-mers are identical.
Abstract: We present an algorithm to compute the lattice energies of molecular crystals based on the many-body cluster expansion. The required computations on dimers, trimers, etc., within the crystal are independent of each other, leading to a naturally parallel approach. The algorithm exploits the long-range three-dimensional periodic order of crystals to automatically detect and avoid redundant or unnecessary computations. For this purpose, Coulomb-matrix descriptors from machine learning applications are found to be efficient in determining whether two N-mers are identical. The algorithm is implemented as an open-source Python program, CrystaLattE, that uses some of the features of the Quantum Chemistry Common Driver and Databases library. CrystaLattE is initially interfaced with the quantum chemistry package Psi4. With CrystaLattE, we have applied the fast, dispersion-corrected Hartree–Fock method HF-3c to the lattice energy of crystalline benzene. Including all 73 symmetry-unique dimers and 7130 symmetry-unique trimers that can be formed from molecules within a 15 A cutoff from a central reference monomer, HF-3c plus an Axilrod-Teller-Muto estimate of three-body dispersion exhibits an error of only −1.0 kJ mol−1 vs the estimated 0 K experimental lattice energy of −55.3 ± 2.2 kJ mol−1. The convergence of the HF-3c two- and three-body contributions to the lattice energy as a function of intermonomer distance is examined.
TL;DR: In this article, the short-range order of the O/N distribution of perovskite oxynitrides was investigated based on first-principles calculations and the cluster expansion model based Monte Carlo simulations.
Abstract: Perovskite-type metal oxynitrides are emerging functional materials with tunable photocatalytic, dielectric and magnetic properties that may depend not only on the compositions but also on the distribution of oxygen and nitrogen ions. In this paper we use BaTaO2N as a representative to investigate the short-range order of the O/N distribution based on first-principles calculations and the cluster expansion model based Monte Carlo simulations. The impact of lattice vibration on short-range order is also considered by using the cluster expansion of the free energy. Strong short-range order is found to exist in BaTaO2N based on calculated cluster correlation functions, suggesting a strongly preferred cis configuration of the octahedron of Ta-(O4N2) with nearly random connection between octahedra. Special quasi-ordered structure is then constructed based on such anion ordering as a representative structure of BaTaO2N to perform further first-principles calculations of electronic and dielectric properties. Our work showcases a general approach to investigate chemical ordering of O/N distribution and its influences on physical properties of perovskite oxynitrides based on first-principles calculations.
TL;DR: The possibility of bypassing fitting models will lead to investigation of disordered systems where cluster expansion is known to perform badly, for example, systems with large lattice deformation due to defects, or systems where long-range interactions dominate such as electrochemical interfaces.
Abstract: We demonstrate the feasibility of performing sufficient configurational sampling of disordered oxides directly from first-principles without resorting to the use of fitted models such as cluster expansion. This is achieved by harnessing the power of modern-day cluster supercomputers using the replica exchange Monte Carlo method coupled directly with structural relaxation and energy calculation performed by density functional codes. The idea is applied successfully to the calculation of the temperature-dependence of the degree of inversion in the cation sublattice of MgAl2O4 spinel oxide. The possibility of bypassing fitting models will lead to investigation of disordered systems where cluster expansion is known to perform badly, for example, systems with large lattice deformation due to defects, or systems where long-range interactions dominate such as electrochemical interfaces.
TL;DR: In this paper, a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions), is provided.
Abstract: We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernandez and Procacci (2007), the proof is based on the Kirkwood–Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernandez, and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.
TL;DR: In this article, the authors give analogs of the cluster expansion formula of Musiker and Schiffler for cluster algebras of type A with coefficients arising from boundary arcs of the corresponding triangulated polygon.
Abstract: The aim of this paper is to give analogs of the cluster expansion formula of Musiker and Schiffler for cluster algebras of type A with coefficients arising from boundary arcs of the corresponding triangulated polygon. Indeed, we give three cluster expansion formulas by perfect matchings of angles in triangulated polygon, by discrete subsets of arrows of the corresponding ice quiver and by minimal cuts of the corresponding quiver with potential.
TL;DR: It is shown that a partial diagonalization of the largest clusters in the expansion using the Lanczos algorithm can be as useful as full diagonalization for the method while mitigating some of the time and memory issues.
Abstract: Numerical linked-cluster expansions allow one to calculate finite-temperature properties of quantum lattice models directly in the thermodynamic limit through exact solutions of small clusters. However, full diagonalization is often the limiting factor for these calculations. Here we show that a partial diagonalization of the largest clusters in the expansion using the Lanczos algorithm can be as useful as full diagonalization for the method while mitigating some of the time and memory issues. As test cases, we consider the frustrated Heisenberg model on the checkerboard lattice and the Fermi-Hubbard model on the square lattice. We find that our approach can surpass state of the art in performance in a parallel environment.
TL;DR: In this paper, a Hamiltonian Monte Carlo framework was developed to sample the high-dimensional spin space of magnetic materials, which is suitable for simulations based on electronic-structure methods.
Abstract: Atomistic simulations of the thermodynamic properties of magnetic materials rely on an accurate modeling of magnetic interactions and an efficient sampling of the high-dimensional spin space. Recent years have seen significant progress with a clear trend from model systems to material-specific simulations that are usually based on electronic-structure methods. Here we develop a Hamiltonian Monte Carlo framework that makes use of auxiliary spin dynamics and an auxiliary effective model, the temperature-dependent spin-cluster expansion, in order to efficiently sample the spin space. Our method does not require a specific form of the model and is suitable for simulations based on electronic-structure methods. We demonstrate fast warm-up and a reasonably small dynamical critical exponent of our sampler for the classical Heisenberg model. We further present an application of our method to the magnetic phase transition in bcc iron using magnetic bond-order potentials.
TL;DR: In this paper, it is shown that truncating the cluster expansion of the energy of alloys gives rise to renormalized effective cluster interactions that are explicit functions of the configurational variables.
Abstract: It is shown that truncating the cluster expansion of the energy of alloys gives rise to renormalized effective cluster interactions that are explicit functions of the configurational variables. Such a dependence of the renormalized cluster interactions is in addition to their dependence on the volume of the alloy and on other structural parameters. The physical picture that emerges is different from the commonly used representation of the configurational energy by means of a generalized Ising-like model, which follows from the assumption that the contributions of the effective cluster interactions can be neglected beyond a relatively small cluster size. The physical picture is one in which the sum of the effective interactions contributes over long distances but the expected ``near-sightedness'' of the energy is preserved by the renormalized interactions. Furthermore, the cluster expansion is implemented by simultaneously fitting the volume- and configuration-dependent energy function to the zero-pressure values of the energies of formation, volumes, bulk moduli, and pressure derivatives of the bulk modulus of a set of ordered compounds. As an example of this formulation of the cluster expansion, we apply the methodology to the Cu-Au system for different types of cell-internal and cell-external relaxations.
TL;DR: In this article, the authors considered the thermodynamic properties of systems with repulsive interactions and showed that the singular part of net baryonic density can to leading order be universally expressed in terms of polylogarithms.
Abstract: Thermodynamic properties of systems with repulsive interactions are considered in the grand-canonical ensemble. The analytic structure of the excluded-volume model in the complex plane of the system chemical potential (fugacity) is elaborated, based on the fact that the pressure function can be given in terms of the Lambert $W$ function. Even though the excluded-volume model has no phase transitions at real values of the chemical potential, it does exhibit a branch cut singularity in the complex plane, thus limiting the convergence range of the Taylor expansion in the chemical potential. Close similarities to analytic properties of the other models with repulsive interactions, such as a cluster expansion model, the mean-field model, and the ideal Fermi gas model, are pointed out. As an example, repulsive baryonic interactions in a hadron gas, with a focus on the fugacity/virial and Taylor expansion methods used in lattice QCD, are presented. The asymptotic behavior of the Fourier expansion coefficients in these various models suggests that the singular part of net baryonic density can to leading order be universally expressed in terms of polylogarithms.
TL;DR: This work demonstrates that this can be accurately and efficiently addressed by the combined Cluster-Expansion method and Wang–Landau algorithm, and is readily applicable to the study of similar second-order phase transitions in other binary and multi-component systems.
TL;DR: In this paper, the effective cluster interactions of O vacancies are mapped out by combining first-principles total energy calculation with the automated structure inversion method, and it is shown that the ordered O vacancies slice the CuO2 plane into not only 1D chains and but also two-leg ladders.
Abstract: A recently discovered high-Tc cuprate superconductor Ba2CuO$_{4-\delta}$ exhibits exceptional Jahn-Teller distortion, wherein the CuO6 octahedrons are compressed along the c axis. As a consequence, the O vacancies prefer to reside in the CuO2 plane, but the exact structure is not known. By combining first-principles total energy calculation with the automated structure inversion method, the effective cluster interactions of O vacancies are mapped out. Around $\delta$=0.8, where the 73K superconductivity was observed experimentally, we predict that the ordered O vacancies slice the CuO2 plane into not only 1D chains and but also two-leg ladders. A Monte Carlo simulation is performed based on the effective cluster interaction model, showing that such an ordering pattern is stable up to ~900 K. Our results put forth a concrete structural basis to discuss the underlying superconducting mechanism.
TL;DR: In this article, a cluster expansion framework is presented to describe the energetics of disordered mixed ionic/electronic conductors (MIECs) within the full solid solution composition space.
Abstract: Several mixed ionic/electronic conductors (MIECs) used as fuel or electrolysis cell electrodes may be thought of as solid solutions of perovskite oxides and ordered oxygen vacancy compounds. For example, the model MIEC SrTi1–xFexO3–x/2+δ (STF) can be described as a mixture of the perovskite SrTiO3 and the brownmillerite Sr2Fe2O5 that can accommodate some degree of oxygen off-stoichiometry δ. The large configurational space for these nondilute, disordered mixtures has hindered atomic scale modeling, limiting in-depth understanding and predictive analysis. We present a cluster expansion framework to describe the energetics of the disordered STF system within the full solid solution composition space Sr(Ti1–xFex)O3–x/2, 0 < x < 1, δ = 0. Cluster expansion Monte Carlo (CEMC) simulations are performed to identify low-energy configurations and to investigate the origin and degree of lattice disorder. Using realistic configurations obtained from CEMC, the electronic structure, band gap, and optical properties of...
TL;DR: A tensor optimized Fermi sphere (TOFS) method was developed to treat the nuclear matter using a bare interaction among nucleons in this paper, where the correlated nuclear matter wave function is taken to be a power series type, Ψ N = [ ∑ n = 0 N ( 1 ∕ n! ) F n ] Φ 0 and an exponential type, where the correlation operator F can induce central, spin-isospin, tensor etc.
TL;DR: In this paper, the binding of oxygen in the entire compositional range of Fe-Cr at minima and saddles was investigated using a cluster expansion model and a convex hull diagram was recorded.
Abstract: The behavior of oxygen in Fe-Cr binary alloys is important to understand for a variety of applications including fuel cell interconnects and nuclear energy systems. In this work, we performed an ab initio investigation of the binding of oxygen in the entire compositional range of Fe-Cr at minima and saddles. This database was subsequently used to parametrize concentration-dependent local cluster expansions which were utilized to examine the kinetics of oxygen transport in the alloy and the thermodynamics of the system. The behavior of oxygen in the alloy system was investigated using our cluster expansion model and a convex hull diagram was recorded. The most favorable composition for the incorporation of oxygen was found to occur at 70% Cr. The dependence of the oxygen solution energy on the Cr occupation of the nearest neighbor shells was carefully characterized. It was determined that the third nearest neighbor Cr occupation was unfavorable and that, generally speaking, the first and second nearest neighbor shell Cr occupation was favored with a few exceptions. Kinetic Monte Carlo models at dilute oxygen levels were performed using two models [a kinetically resolved activation (kRA) model and a cluster expansion model for the saddle points]. The models qualitatively agreed with the observed trends in the literature which reported a decrease in diffusivity at dilute Cr levels, but the models predicted different behaviors beyond the dilute Cr limit. Finally, using grand canonical Monte Carlo, we further examined the thermodynamics of oxygen incorporation into the alloy. Together, our results offer new insight into the behavior of oxygen in Fe-Cr alloys that have ramifications on the early stages of the corrosive behavior of the material.
TL;DR: In this paper, a combined cluster expansion and atomic displacement expansion is proposed to fit first-principles energies, forces, and stresses to calculate thermodynamic quantities at nearly first-parameter levels of accuracy.
Abstract: Finite-temperature disordered solid solutions and magnetic materials are difficult to study directly using first-principles calculations, due to the large unit cells and many independent samples that are required. In this work, we develop a combined cluster expansion and atomic displacement expansion, which we fit to first-principles energies, forces, and stresses. We then use the expansion to calculate thermodynamic quantities at nearly first-principles levels of accuracy. Our model naturally includes both configurational and vibrational entropy, including anharmonic contributions and structural phase transitions. In addition, we can treat coupling between atomic displacement and chemical or magnetic degrees of freedom. As examples, we use our expansion to calculate properties of ${\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$, magnetic MnO, Al with vacancies, and ${\mathrm{Ba}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{TiO}}_{3}$. Finally, we demonstrate that by treating all the relevant degrees of freedom explicitly, we can in some cases achieve improved convergence of fitting parameters versus distance as compared to a pure cluster expansion.
TL;DR: This work uses numerical linked cluster expansions (NLCEs) to study the site-diluted transverse-field Ising model and investigates the Griffiths-McCoy singularities, finding that the magnetization develops nonlinearities in the GriffithS phase with exponents that vary continuously with h.
Abstract: We use numerical linked cluster expansions (NLCEs) to study the site-diluted transverse-field Ising model on the square lattice at T=0. NLCE with a self-consistent mean field on the boundary of the clusters is used to obtain the ground-state magnetization, susceptibility, and structure factor as a function of transverse field h and exchange constant J. Adding site dilution to the model turns NLCE into a series expansion in the dilution parameter p. Studying the divergence of the structure factor allows us to establish the phase diagram in the h/J and p plane. By studying the magnetization of the system in a longitudinal field, we investigate the Griffiths-McCoy singularities. We find that the magnetization develops nonlinearities in the Griffiths phase with exponents that vary continuously with h. Additionally, the probability distribution of the local susceptibility develops long tails in the Griffiths phase, which is studied in terms of its moments.
TL;DR: The scheme presented here provides a solid basis for electronic structure calculations for the ground state of solids and also matrix product states, which have been applied to one-dimensional systems with results of high precision.
Abstract: Wavefunctions for large electron numbers N are plagued by the Exponential Wall Problem (EWP), i.e., an exponential increase in the dimensions of Hilbert space with N. Therefore, they lose their meaning for macroscopic systems, a point stressed, in particular, by Kohn. The EWP has to be resolved in order to provide a solid basis for wavefunction based electronic structure calculations of macroscopic systems, e.g., solids. The origin of the EWP is the multiplicative property of wavefunctions when independent subsystems are considered. Therefore, it can only be avoided when wavefunctions are formulated so that they are additive instead, in particular, when matrix elements involving them are calculated. We describe how this is done for the ground state of a macroscopic electron system. Going over from a multiplicative to an additive quantity requires taking a logarithm. Here it implies going over from Hilbert space to the operator- or Liouville space with a metric based on cumulants. The operators which define the ground-state wavefunction generate fluctuations from a mean-field state. The latter does not suffer from an EWP and therefore may serve as a vacuum state. The fluctuations have to be connected like the ones caused by pair interactions in a classical gas when the free energy is calculated (Meyer’s cluster expansion). This fixes the metric in Liouville space. The scheme presented here provides a solid basis for electronic structure calculations for the ground state of solids. In fact, its applicability has already been proven. We discuss also matrix product states, which have been applied to one-dimensional systems with results of high precision. Although these states are formulated in Hilbert space, they are processed by using operators in Liouville space. We show that they fit into the general formalism described above.