Hard-core Model Research Articles

Coupling with the stationary distribution and improved sampling for colorings and independent sets

We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for k/Δ>1.764, the Glauber dynamics on k-colorings of a graph on n vertices with maximum degree Δ converges in O(nlog n) steps, assuming Δ=Ω(log n) and that the graph is triangle-free. Previously, girth ≥5 was needed. As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at fugacity λ<(1−ɛ)e/Δ, on a regular graph G on n vertices of degree Δ=Ω(log n) and girth ≥6. The best known algorithm for general graphs currently assumes λ<2/(Δ−2).

Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries

Using exact computations we study the classical hard-core monomer-dimer models on m x n plane lattice strips with free boundaries. For an arbitrary number v of monomers (or vacancies), we found a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term depends on the number of monomers present (v) and the parity of the width n of the lattice strip: the coefficient equals to v when n is odd, and v/2 when n is even. The results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers. We also study the finite-size correction in the low dimer density limit, where the number of dimers d is fixed. In this case the coefficient of the logarithmic correction term equals to d, for both odd and even n.

Mott Transition in Lattice Boson Models

We use mathematically rigorous perturbation theory to study the transition between the Mott insulator and the conjectured Bose-Einstein condensate in a hard-core Bose-Hubbard model. The critical line is established to lowest order in the tunneling amplitude.

Correlations and confinement in nonplanar two-dimensional dimer models

We study classical hard-core dimer models on the square lattice with links extending beyond nearest-neighbors. Numerically, using a directed-loop Monte Carlo algorithm, we find that, in the presence of longer dimers preserving the bipartite graph structure, algebraic correlations persist. While the confinement exponent for monomers drifts, the leading decay of dimer correlations remains 1/r^2, although the logarithmic peaks present in the dimer structure factor of the nearest-neighbour model vanish. By contrast, an arbitrarily small fraction of next-nearest-neighbor dimers leads to the onset of exponential dimer correlations and deconfinement. We discuss these results in the framework of effective theories, and provide an approximate but accurate analytical expression for the dimer correlations.

Calculation of equilibrium and transport properties using modified hard-core potential models

We present in this paper a new method of modifying hard-core potentials to correctly predict the equilibrium and non-equilibrium properties of real fluids at low densities (specifically when ρ ⩽ B/ C where ρ is density, and B and C are the second and third virial coefficients) only by one set of potential parameters. Because the molecular diameter becomes smaller when temperature increases, we introduced a new expression for the variation of molecular diameter with temperature that incorporates this effect. The temperature dependence of the diameter was used in both Sutherland (ST) and square-well (SW) potential models to modify the second virial coefficient. Then we have shown that the experimental second virial coefficient fits into the modified Sutherland (MST) and modified square-well (MSW) quite well for the entire temperature range for which experimental data are reported, including the inversion temperature. Such fittings give the parameters of the modified potential models, which are used to calculate the non-equilibrium properties of He, Ne, N 2, O 2, CO, and NO, including viscosity, thermal conductivity, and self-diffusion coefficients. In comparison with experimental data, such modifications of the potential models give much better results than those obtained from the original ST and SW models, especially at high temperatures at which the deviations are reduced significantly. Therefore, there is no need to use two sets of potential parameters; one for the equilibrium and another for the non-equilibrium properties of real fluids, if we use the modified hard-core potential models.

Numerical study of magnetic field induced ordering inBaCuSi2O6and related systems

Thermodynamics of spin dimer system $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_{2}{\mathrm{O}}_{6}$ is studied using a quantum Monte Carlo calculation (QMC) and a bond-operator mean field theory. We propose that a new type of boson, which, rather than being hard core, allows up to two occupancies at each site, is responsible for the Bose-Einstein condensation of field induced ordering. Its superfluid density is identified as the square of the in-plane staggered magnetization ${m}_{xy}$ in the ordered phase. We also compare our QMC result of the spin Heisenberg model to those predicted by the mean field theory as well as by the simple hard-core boson model for both large and small intradimer coupling $J$. The asymmetry of the phase diagram of ${m}_{xy}(h)$ of small coupling $J$ in related systems such as $\mathrm{Ni}{\mathrm{Cl}}_{2}\text{\ensuremath{-}}4\mathrm{S}\mathrm{C}(\mathrm{N}{\mathrm{H}}_{2})$ is explained with our new boson operator.

Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d-regular bipartite graph. For any λ > 0 let πλ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ|I| (πλ is the hard-core measure with activity λ on ∑). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is πλ. We show that when λ is large enough (as a function of d and the expansion of subsets of single-parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d-dimensional hypercube {0,1}d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Hardcore bosons on checkerboard lattices near half filling: Geometric frustration, vanishing charge order, and a fractional phase

We study a spinless hardcore boson model on checkerboard lattices by Green function Monte Carlo method. At half filling, the ground state energy is obtained up to a $28\ifmmode\times\else\texttimes\fi{}28$ lattice and extrapolated to infinite size; the staggered pseudospin magnetization is found to vanish in the thermodynamic limit. Thus the $(\ensuremath{\pi},\ensuremath{\pi})$ charge order is absent in this system. Away from half filling, two defects induced by each hole (particle) may carry fractional charge $(\ifmmode\pm\else\textpm\fi{}e∕2)$. For the one hole case, we study how the defect-defect correlation changes with $t∕J$, which is the ratio between the hopping integral and cyclic exchange, and is equal to $V∕2t$ when $V⪢t$. Moreover, we argue that these fractional defects may propagate independently when the concentration of holes (or defects) is large enough.

Universality of Ionic Criticality: Size- and Charge-Asymmetric Electrolytes

Grand-canonical simulations designed to resolve critical universality classes are reported for z:1 hard-core electrolyte models with diameter ratios lambda=a+/a- less than or approximately equal 6. For z=1 Ising-type behavior prevails. Unbiased estimates of Tc(lambda) are within 1% of previous (biased) estimates but the critical densities are approximately 5% lower. Ising character is also established for the 2:1 and 3:1 equisized models, along with critical amplitudes and improved Tc estimates. For z=3, however, strong finite-size effects reduce the confidence level although classical and O (n>or=3) criticality are excluded.

Why forces between proteins follow different Hofmeister series for pH above and below pI

The relative effectiveness of different anions in crystallizing proteins follows a reversed Hofmeister sequence for pH < pI and a direct Hofmeister sequence for pH > pI. The phenomenon has been known almost since Hofmeister's original work but it has not been understood. It is here given a theoretical explanation. Classical electrolyte and double layer theory deals only with electrostatic forces acting between ions and proteins. Hydration and hydration interactions are dealt with usually only in terms of assumed hard core models. But there are, at and above biological salt concentrations, other non-electrostatic (NES) ion-specific forces acting that are ignored in such modeling. Such electrodynamic fluctuation forces are also responsible for ion-specific hydration. These missing forces are variously comprehended under familiar but generally unquantified terms, typically, hydration, hydrogen bonding, π-electron–cation interactions, dipole–dipole, dipole-induced dipole and induced dipole-induced dipole forces and so on. The many important body electrodynamic fluctuation force contributions are accessible from extensions of Lifshitz theory from which, with relevant dielectric susceptibility data on solutions as a function of frequency, the forces can be extracted quantitatively, at least in principle. The classical theories of colloid science that miss such contributions do not account for a whole variety of ion-specific phenomena. Numerical results that include these non-electrostatic forces are given here for model calculations of the force between two model charge-regulated hen-egg-white protein surfaces. The surfaces are chosen to carry the same charge groups and charge density as the protein. What emerges is that for pH < pI (where the anions are counter-ions) the repulsive double layer forces increase in the order NaSCN < NaI < NaCl, while at higher pH > pI (where anions are co-ions) the forces increase in the order NaCl < NaI < NaSCN. This is in excellent agreement with both solubility experiments and experiments using SAXS. The results are also consistent with cation effects observed in protein solutions. Our results may provide some insights into a long-standing problem in solution chemistry and biology.

The densification of Cu/Ti system by spark plasma sintering

In order to unclose the dynamics of SPS densification, a special sintering sample (Cu/Ti wires compact) was designed. Characters of the shrinkage rates during sintering process and microstructures of products fabricated by the spark plasma sintering (SPS) and hot-press sintering were investigated. The experimental results reveal that a higher temperature field is formed at the connected area and conductive net of the compact. These high-temperature parts deformed more easily than other parts, which is believed to be the main cause of SPS fast densification, according to a hard-core and soft-hell material model.

Fully Coupled Channel Approach to Doubly Stranges-Shell Hypernuclei

We describe ab initio calculations of doubly strange, S = -2, s-shell hypernuclei (4(LambdaLambda)H, 5LambdaLambda)H, 5(LambdaLambda)He, and 6(LambdaLambda)He) as a first attempt to explore the few-body problem of the full-coupled channel scheme for these systems. The wave function includes LambdaLambda, LambdaSigma, NXi, and SigmaSigma channels. Minnesota NN, D2' YN, and simulated YY potentials based on the Nijmegen hard-core model are used. Bound-state solutions of these systems are obtained. We find that a set of phenomenological B8B8 interactions among the octet baryons in S = 0,-1, and -2 sectors, which is consistent with all of the available experimental binding energies of S = 0,-1, and -2 s-shell (hyper)nuclei, can predict a particle stable bound state of 4(LambdaLambda)H. For 5(LambdaLambda)H and 5(LambdaLambda)He, LambdaN-SigmaN and XiN-LambdaSigma potentials significantly affect the net LambdaLambda-NXi coupling, and a large Xi probability is obtained even for a weaker LambdaLambda-NXi potential.

Sedimentation Equilibrium of Colloidal Suspensions in a Planar Pore Based on Density Functional Theory and the Hard-Core Attractive Yukawa Model

The sedimentation equilibrium of colloidal suspensions modeled by hard-core attractive Yukawa (HCAY) fluids in a planar pore is studied. The density profile of the HCAY fluid in a gravitational field and its distribution between the pore and uniform phases are investigated by a density functional theory (DFT) approach, which results from employing a recently proposed parameter-free version of the Lagrangian theorem-based density functional approximation (Zhou, S. Phys. Lett. A 2003, 319, 279) for hard-sphere fluids to the hard-core part of the HCAY fluid, and the second-order functional perturbation expansion approximation to the tail part as was done in a recent partitioned density functional approximation (Zhou, S. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 061201). The resultant DFT approach is, thus, the first adjustable parameter-free DFT for HCAY fluids. The validity of the present DFT for HCAY fluids of reduced range parameter z(red) = 1.8 under various external potentials is established in the first of the papers cited previously. The present DFT for HCAY fluids can predict the radial distribution function for the bulk HCAY fluid accurately in the colloidal limit (large value of z(red)), and in the hard-sphere limit, its prediction for the density profile of the hard-sphere fluid in a gravitational field is in very good agreement with the existing simulation data. The dependence of the density profile and distribution coefficient on the magnitude of the interparticle attraction, gravitational field, and degree of confinement is investigated in detail by the present DFT approach. Intuitive and qualitative analyses are also compared with the quantitative DFT calculational results.

Ferromagnetism in a hard-core boson model

The problem of ferromagnetism -- associated with a ground state with maximal total spin -- is discussed in the framework of a hard-core model, which forbids the occupancy at each site with more than one particle. It is shown that the emergence of ferromagnetism on finite square lattices crucially depends on the statistics of the particles. Fermions (electrons) lead to the well-known instabilities for finite hole densities, whereas for bosons (with spin) ferromagnetism appears to be stable for all hole densities.

Analysis of the validity of perturbation density functional theory: Based on extensive simulation for simple fluid at supercritical and subcritical temperature under various external potentials

Because of the scarcity of available simulation data for confined hard-core attractive Yukawa model fluid, extensive Monte Carlo (MC) simulation research for this fluid under the influence of various external potentials were carried out. The present MC simulation results were employed to test a performance of the third-order perturbation density functional theory (DFT) based on a high order direct correlation function (DCF) [S. Zhou and E. Ruckenstein, Phys. Rev. E. 61, 2704 (2000)]. It was found that the present perturbation DFT formalism is soundly structured only if the imported second-order DCF is reliable. In this case, the accuracy of the results can be satisfactory or even very high for various types of external potentials. Further, the associated adjustable parameter can be universal, i.e., independent of the particular external field responsible for the generation of a nonuniform density profile. Dependence of both the maintenance of the reliability of the formalism and holding of the universality of the adjustable parameter on the accuracy of the imported bulk second-order DCF can be strengthened by the large difference between the external field investigated and that caused by a single hard wall used for specification of the adjustable parameter. In case the gaseous density in the subcritical region is below the coexistence density, an excellent performance of the present formalism is observed even for the mean spherical approximation's second-order DCF as an input. This advantageous property, combined with the fact that the present formalism needs only the second-order DCF of fluid at the coexistence state as an input, enables the present formalism to be a very good theoretical tool for the investigations of wetting and prewetting transitions.

Prediction of collective diffusion coefficient of bovine serum albumin in aqueous electrolyte solution with hard-core two-Yukawa potential

A new method to predict concentration dependence of collective diffusion coefficient of bovine serum albumin (BSA) in aqueous electrolyte solution is developed based on the generalized Stokes-Einstein equation which relates the diffusion coefficient to the osmotic pressure. The concentration dependence of osmotic pressure is evaluated using the solution of the mean spherical approximation for the two-Yukawa model fluid. The two empirical correlations of sedimentation coefficient are tested in this work. One is for a disordered suspension of hard spheres, and another is for an ordered suspension of hard spheres. The concentration dependence of the collective diffusion coefficient of BSA under different solution conditions, such as pH and ionic strength is predicted. From the comparison between the predicted and experimental values we found that the sedimentation coefficient for the disordered suspension of hard spheres is more suitable for the prediction of the collective diffusion coefficients of charged BSA in aqueous electrolyte solution. The theoretical predictions from the hard-core two-Yukawa model coupled with the sedimentation coefficient for a suspension of hard spheres are in good agreement with available experimental data, while the hard sphere model is unable to describe the behavior of diffusion due to its neglect of the double-layer repulsive charge-charge interaction between BSA molecules.

A Three State Hard-Core Model on a Cayley Tree

We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ∀ λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure *. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.

Exact Monte Carlo time dynamics in many-body lattice quantum systems

On the base of a Feynman-Kac--type formula involving Poisson stochastic processes, recently a Monte Carlo algorithm has been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.

The Field-Induced Magnetic Ordering Transition in TlCuCl3

In the first part we investigate in detail if the Bose-Einstein condensation scenario for magnons can quantitatively explain the observed field-induced magnetic ordering in TlCuCl$_3$. We use a bond-operator approach to map the spin system onto hard-core bosons and exactly account for the hard-core constraint in the dilute limit. We solve the hard-core model within the Hartree-Fock-Popov approximation and discuss its validity and the consequences of this approximation for the critical properties. In the second part the effects of spin-phonon and spin-orbit coupling are discussed within this framework. We show that the experimental magnetization and specific heat data are well described if a certain type of anisotropy is included. We also present predictions for the quasiparticle gap which might be tested in the future.

Bose-Einstein condensation of particle-hole pairs in ultracold fermionic atoms trapped within optical lattices.

We investigate the Bose-Einstein condensation (BEC, superfluidity) of particle-hole pairs in ultracold fermionic atoms with repulsive interactions and arbitrary polarization, which are trapped within optical lattices. In the strongly repulsive limit, the dynamics of particle-hole pairs can be described by a hard-core Bose-Hubbard model. The insulator-superfluid and charge-density-wave- (CDW) superfluid phase transitions can be induced by decreasing and increasing the potential depths with controlling the trapping laser intensity, respectively. The parameter and polarization dependence of the critical temperatures for the ordered states (BEC and/or CDW) are discussed simultaneously.