Three-dimensional Simple Cubic Lattice Research Articles

Nematic order in a simple-cubic lattice-spin model with full-ranged dipolar interactions.

In a previous paper [Phys. Rev. E 90, 022506 (2014)PLEEE81539-375510.1103/PhysRevE.90.022506], we studied the thermodynamic and structural properties of a three-dimensional simple-cubic lattice model with dipolarlike interaction, truncated at nearest-neighbor separation, for which the existence of an ordering transition at finite temperature had been proven mathematically; here we extend our investigation, addressing the full-ranged counterpart of the model, for which the critical behavior had been investigated theoretically and experimentally. In addition, the existence of an ordering transition at finite temperature had been proven mathematically as well. Both models exhibited the same continuously degenerate ground-state configuration, possessing full orientational order with respect to a suitably defined staggered magnetization (polarization), but no nematic second-rank order; in both cases, thermal fluctuations remove the degeneracy, so that nematic order does set in at low but finite temperature via a mechanism of order by disorder. On the other hand, there were recognizable quantitative differences between the two models as for ground-state energy and critical exponent estimates; the latter were found to agree with early renormalization-group calculations and with experimental results.

Calamitic and antinematic orientational order produced by the generalized Straley lattice model

We consider here a classical model, consisting of D_{2h}-symmetric particles in a three-dimensional simple-cubic lattice; the pair potential is isotropic in orientation space, and restricted to nearest neighbors. The simplest potential model is written in terms of the squares of the scalar products between unit vectors describing the three interacting arms of the molecules, as proposed in previous literature. Two predominant antinematic couplings of equal strength (+1) are perturbed by a comparatively weaker calamitic one, parameterized by a coupling constant -z ranging in [-1,0]. This choice rules out thermodynamically stable phases endowed with macroscopic biaxiality. The antinematic terms favor states with the corresponding molecular axes mutually orthogonal. Although the low-temperature phase of the special case with null calamitic term (PP0) is uniaxial and antinematically ordered, in the general case presented here both Monte Carlo and molecular-field approaches show that, for z close to zero, the models exhibit a low-temperature uniaxial nematic phase, followed by an antinematic one, and finally by the orientationally disordered one. On the other hand, for sufficiently large values of z, we only find evidence of uniaxial calamitic behavior, as expected by following the limiting cases.

Orientationally ordered phase produced by fully antinematic interactions: A simulation study

We consider here a classical model, consisting of D(2h) symmetric particles, whose centers of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is isotropic in orientation space, and restricted to nearest neighbors. Two orthonormal triads define orientations of a pair of interacting particles; the simplest potential models proposed in the literature can be written as a linear combination involving the squares of the scalar products between corresponding unit vectors only, thus depending on three parameters, and making the interaction model rather versatile. A coupling constant with negative sign tends to keep the two interacting unit vectors parallel to each other, whereas a positive sign tends to keep them mutually orthogonal (antinematic coupling). We address here a special, extreme case of the above family, involving only antinematic couplings: more precisely, three antinematic terms whose coefficients are set to a common positive value (hence the name PPP model). The model under investigation produces a doubly degenerate pair ground state; the nearest-neighbor range of the interaction and the bipartite character of the lattice can propagate the pair ground state and increase the overall degeneracy, but without producing frustration. The model was investigated by a simplified molecular field treatment as well as by Monte Carlo simulation, whose results suggested a second-order transition to a low-temperature biaxially ordered phase; ground-state configurations producing orientational order have been selected by thermal fluctuations. The molecular field treatment also predicted a continuous transition, and was found to overestimate the transition temperature by a factor 2.

Critical entropy of quantum Heisenberg magnets on simple-cubic lattices

We analyze the temperature dependence of the entropy of the spin-1/2 Heisenberg model on the three-dimensional simple-cubic lattice, for both the case of antiferromagnetic and ferromagnetic nearest neighbor exchange interactions. Using optimized extended ensemble quantum Monte Carlo simulations, we extract the entropy at the critical temperature for magnetic order from a finite-size scaling analysis. For the antiferromagnetic case, the critical entropy density equals 0.341(5)$k_B$, whereas for the ferromagnet, a larger value of 0.401(5) $k_B$ is obtained. We compare our simulation results to estimates put forward recently in studies assessing means of realizing the antiferromagnetic N\'eel state in ultra-cold fermion gases in optical lattices.

Dilution effect in correlated electron systems with orbital degeneracy

Theory of dilution effect in orbital ordered system is presented. The ${e}_{g}$ orbital model without spin degree of freedom and the spin-orbital coupled model in a three-dimensional simple-cubic lattice are analyzed by the Monte Carlo simulation and the cluster expansion method. In the ${e}_{g}$ orbital model without spin degree of freedom, reduction in the orbital ordering temperature due to dilution is steeper than that in the dilute magnet. This is attributed to a modification of the orbital wave function around vacant sites. In the spin-orbital coupled model, it is found that magnetic structure is changed from the A-type antiferromagnetic order into the ferromagnetic one. Orbital-dependent exchange interaction and a sign change in this interaction around vacant sites bring about this phenomenon. Present results explain the recent experiments in transition-metal compounds with orbital dilution.

Biaxial and uniaxial phases produced by partly repulsive mesogenic models involvingD2hmolecular symmetries

The present paper considers biaxial nematogenic lattice models, involving particles of D2h symmetry, whose centers of mass are associated with a three-dimensional simple-cubic lattice. The pair potential is isotropic in orientation space and restricted to nearest neighbors. Let two orthonormal triads define orientations of a pair of interacting particles. The investigated potential models are quadratic with respect to the nine scalar products between the two sets of unit vectors. Actually, based on available geometric identities, these expressions can be reduced to diagonal form containing only the scalar products between corresponding unit vectors and depending on three parameters. Over the years, this comparatively simple functional form has also proven to be rather versatile. By now, various sets of potential parameters capable of producing mesogenic behavior of some kind have been proposed and studied in the literature. A new and simplified form was recently proposed and investigated by Sonnet, Virga, Durand, and De Matteis [A. M. Sonnet, E. G. Virga, and G. E. Durand, Phys. Rev. E 67, 061701 (2003); G. De Matteis and E. G. Virga, Phys. Rev. E 71, 061703 (2005)] and is known to support a biaxial phase at sufficiently low temperature. Following the idea of the above authors, we have studied a more extended range of parameters, including cases where biaxiality cannot be sustained in the pair ground state. In cases where a biaxial phase survives, an appropriate mean-field analysis may predict the existence of a direct second-order transition to the isotropic phase as well as a second-order sequence isotropic-to-uniaxial-to-biaxial. A second-order phase transition is also predicted, which involves isotropic and uniaxial phases only. Monte Carlo simulations have been carried out as well, for a few points in the parameter space, and found to produce results which partly confirm mean-field predictions.

Computer simulation study of a biaxial nematogenic lattice model associated with a three-dimensional lattice and involving dispersion interactions

The present paper considers a biaxial nematogenic lattice model, involving particles of D 2 h symmetry, whose centres of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is defined by a London–De Boer–Heller dispersion interaction, restricted to nearest neighbours; to the best of our knowledge, this model has not previously been studied by simulation. On the other hand, upon isotropically averaging over the intermolecular unit vector, another interaction model is obtained (which, in turn, can be introduced in other ways and guises as well), extensively studied by Mean Field treatment and Monte Carlo simulation. We present here Monte Carlo simulation results, showing mild but recognizable differences between the two models, actually equivalent at Mean Field level.

Computer simulation of a biaxial nematogenic model on a three-dimensional lattice and based on a recently proposed interaction potential

The present paper considers a biaxial nematogenic lattice model, involving particles of D 2h symmetry, whose centres of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is isotropic in orientation space, restricted to nearest neighbours, and has the simplified form recently proposed by Sonnet, et al. (Phys. Rev. E 67 (2003) 061701); the mean field analysis carried out by the named authors predicts a range of parameters where a direct first-order transition occurs between biaxial and isotropic phases; we have performed Monte Carlo simulations, and found results which qualitatively confirm their prediction.

COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE-GAS MODEL WITH FOURTH-RANK INTERACTIONS

We have considered a classical lattice-gas model, consisting of a three-dimensional simple-cubic lattice, whose sites host three-component unit vectors; pairs of nearestneighboring sites interact via the nematogenic potential [Formula: see text] here P4(τ) denotes the fourth Legendre polynomial, nuj=0,1 are occupation numbers, uj are unit vectors (classical spins), and ∊ is a positive quantity setting the energy and temperature scales (i.e. T* =k B T / ∊). The total Hamiltonian is given by [Formula: see text] where ∑{j < k} denotes sum over all distinct nearest-neighboring pairs of lattice sites. The saturated-lattice version of this model defines a nematogenic lattice model, already studied in the literature, and found to possess a transition to an orientationally ordered phase at low temperature; moreover, according to available rigorous results, there exists a μ0<0, such that, for all μ>μ0, the system supports an ordering transition at a finite, μ-dependent, temperature. We present here a detailed study of the case μ=0, and characterize it by means of Monte Carlo simulation, Mean Field and Two Site Cluster treatments; the latter significantly improves the agreement with simulation results.

MEAN FIELD, TWO-SITE CLUSTER, AND COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE-GAS MODEL

We have considered a classical lattice-gas model, consisting of a three-dimensional simple-cubic lattice, whose sites host three-component unit vectors; pairs of nearest-neighbouring sites interact via the nematogenic potential [Formula: see text] here P2(τ) denotes the second Legendre polynomial, νj = 0, 1 are occupation numbers, uj are unit vectors (classical spins), and ∊ is a positive quantity setting energy and temperature scales (i.e. T* = k B T/∊); the total Hamiltonian is given by [Formula: see text] where ∑{j<k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model defines the extensively studied Lebwohl–Lasher model, possessing a transition to an orientationally ordered phase at low temperature; according to available rigorous results, there exists a μ0 < 0, such that, for all μ > μ0, the system supports an ordering transition at a finite, μ-dependent, temperature. Continuing along the lines of our previous communication [S. Romano, Int. J. Mod. Phys.B14, 1195 (2000)], we present here a detailed study of the case μ = 0, using Monte Carlo simulation, Mean Field and Two Site Cluster treatments; the latter significantly improves the agreement with simulation results.

COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE-GAS MODEL

We have considered a classical lattice-gas model, consisting of a three-dimensional simple-cubic lattice, whose sites host three-component unit vectors; pairs of nearest-neighbouring sites interact via the nematogenic potential [Formula: see text] here P2(τ) denotes the second Legendre polynomial, νj=0,1 are occupation numbers, uj are the unit vectors (classical spins), and ∊ is a positive quantity setting energy and temperature scales (i.e. T*=k B T/∊); the total Hamiltonian is given by [Formula: see text] where ∑{j<k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model defines the extensively studied Lebwohl–Lasher model, possessing a transition to an orientationally ordered phase at low temperature; according to available rigorous results, there exists a μ0<0, such that, for all μ>μ0, the system supports an ordering transition at a finite, μ-dependent, temperature. We have studied here the case μ=0, and found evidence of a transition, taking place at a lower temperature, and possessing a more pronounced first-order character than its Lebwohl–Lasher counterpart; a Mean Field treatment has also been worked out, and found to yield results in qualitative agreement with simulation.

Metallic surface of a Mott insulator–Mott insulating surface of a metal

The dynamical mean-field theory (DMFT) is employed to study the Mott transition in the semi-infinite Hubbard model at half-filling and zero temperature. We consider the low-index surfaces of the three-dimensional simple-cubic lattice and systematically vary the model parameters at the very surface. Within the DMFT the problem is self-consistently mapped onto a set of coupled effective impurity models corresponding to the inequivalent layers parallel to the surface. Assuming that the influence of the Hubbard bands on the low-energy quasi-particle resonance can be neglected at the critical point, a simplified ``linearized DMFT'' becomes possible which is formally equivalent to the Weiss molecular-field theory for the semi-infinite Ising model. This implies that qualitatively the rich phenomenology of the Landau description of second-order phase transitions at surfaces has a direct analogue for the surface Mott transition. Motivated by this formal analogy, we work out the predictions of the linearized DMFT in detail. It is found that under certain circumstances the surface of a Mott insulator can be metallic while a Mott-insulating surface of a normal metal is not possible. The corresponding phase diagrams, the (mean-field) critical exponents and the critical profiles of the quasi-particle weight are derived. The results are confirmed by a fully numerical evaluation of the DMFT equations using the exact-diagonalization (ED) method.

Degenerate Rosenbluth Monte Carlo scheme for cluster counting and lattice animal enumeration

We describe an algorithm for the Monte Carlo enumeration of clusters and lattice animals. The method may also be used to calculate associated properties such as moments or perimeter multiplicities of the clusters. The scheme is an adaptation of the Rosenbluth method for growing polymer chains and may be used to estimate the number of distinct lattice animals on any lattice topology. The method is validated against exact and Monte Carlo enumerations for clusters of size 30, on two- and three-dimensional simple-cubic lattices. The method may be readily adapted to yield Boltzmann weighted averages over clusters.

Influence of chain architecture on the thermodynamic properties of lattice polymer solutions

Monte Carlo computer simulations have been performed on a three-dimensional simple-cubic lattice for mixtures of linear and branched chains with monomeric solvent molecules. Mixtures with chains of length 10 and of length 40 have been considered for both linear and branched architectures at a variety of polymer volume fractions. For the internal energy of mixing of athermal chains with solvent, the lattice cluster theory (LCT) is found to give predictions in reasonably good agreement with the results of the simulations, especially for short chains. At lower temperatures, the theory is not as satisfactory. However, it does prove to be more accurate for branched polymer with solvent than for linear polymer with solvent

Sequence-structure relationships in proteins and copolymers.

We model proteins as copolymer chains of H (hydrophobic) and P (polar) monomers configured as self-avoiding flights on three-dimensional simple-cubic lattices. The HH interaction is favorable. The folding problem is to find the ``native'' conformation(s) (lowest free energy) for an HP sequence. Using geometric proofs for self-avoiding lattice chains, we develop equations relating a monomer sequence to its native structures. These constraint relations can be used for two purposes: (1) to compute a tight lower bound on the free energy of the native state for HP sequences of any length, which is useful for testing conformational search strategies, and (2) to develop a search strategy. In its present implementation, the search strategy finds native states for HP lattice chains up to 36 monomers in length, which is a speedup of 5\char21{}15 orders of magnitude over existing brute-force exhaustive-search methods.

Spin-wave theory for anisotropic Heisenberg antiferromagnets.

The anisotropic Heisenberg antiferromagnet (AF), which is defined as a three-dimensional simple-cubic lattice with in-plane antiferromagnetic interaction {ital J}{sub } and interplane coupling {ital J}{sub {perpendicular}}={gamma}{ital J}{sub } and is believed to describe the magnetic properties of the cupric oxide materials, is studied using low-temperature spin-wave theory. The dependence of the {ital T}=0 staggered magnetization, ground-state energy, transverse susceptibility, spin-wave velocity, and the Neel temperature {ital T}{sub {ital N}} on the anisotropy parameter {gamma} (0{le}{gamma}{le}1) are obtained. These results are found to be in satisfactory agreement with existing experiments on cupric oxide materials. The apparent difference between the muon-spin resonance and neutron-scattering results for the ordered moment in the AF state is well explained.