Rapid evaluation of oscillating lattice sums

We have generalized a method for the rapid evaluation of oscillating lattice sums for layered geometries, based on the Ewald's summation technique. Different types of long range interactions as well as different lattice symmetries can be considered, in particular modulated magnetic structures, periodic domain patterns, or periodic lattice distortions. As examples the demagnetizing energies for different thin films systems and magnetic excitations in a single atomic layer are calculated.     

Computer simulation study of a simple cubic dipolar lattice

Summary The present work reports Monte Carlo calculations on a classical simple cubic lattice, consisting of point dipoles with identical dipole moments. Dipolar interactions are known not to be essential for nematic ordering, but they can play a significant role in some specific cases; along the lines of the lattice model approximation already used in simulation work on liquid crystals, our system can thus be considered as an extreme case of nematogenic potential model. The system possesses an antiferroelectric ground state and, at higher temperature, it undergoes a transition to an orientationally disordered phase; comparison with mean-field and spherical-model treatments of the transition is also reported. The structural properties were investigated by calculating orientational correlation functions and found to exhibit some qualitative differences with respect to other potential models previously investigated for nematics.

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Riassunto Il presente lavoro riporta calcoli Monte Carlo per un reticolo classico cubico semplice, costituito da dipoli puntiformi con identico momento dipolare. è noto che le interazioni dipolari non sono essenziali per l’ordinamento nematico, ma esse possono avere un ruolo significantivo in alcuni casi specifici; lungo le linee dell’approssimazione a modello reticolare già applicata in lavori di simulazione per cristalli liquidi, il nostro sistema può pertanto essere considerato come un caso estremo di modello di potenziale nematogenico. Il sistema possiede stato fondamentale antiferroelectrico ed, a temperature piú elevata, transisce ad una fase orientazionalmente disordinata; sono riportati per confronto i risultati delle trattazioni a campo medio, ed a modello sferico per tale transizione. Le proprietà strutturali furono investigate calcolando funzioni di correlazione orientazionale, e si è trovato che esse mostrano qualche differenza qualitativa rispetto ad altri modelli di potenziale previamente studiati per nematici.

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Резюме В работе предлагается моделирование по методу Монте-Карло классической простой кубической решетки, образованной точечными диполями с тождественными дипольными моментами. Известно, что дипольные взаимодействия не являются существенными для нематическояо упорядочения, но они игроют существенную роль в некоторых специфических случаях. Рассматриваемая система обладает антиферроэлектрическим основным состоянием и, при высокой темлературе, претерпевает переход в ориентационно упорядоченную фазу. Проводится сравнение с приближ=yeнием среднего поля и рассмотрением в рамках сферической модели этого перехода. Псследуутся структурные свойства, посредством вычисления ориентационных корреляционных функций. Обнаружены нектторые ксчественные различия по сравнению с другими потенциалыми моделями, ранее рассмотренными для описания нематики.

     

A time-reversal lattice Boltzmann method

In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor–Green vortex decay and the vortex pair–wall collision.     

Generalized lattice gauge theory, spin foams and state sum invariants

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (≤3, ≤4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate-free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold.

In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension ≤4) and with supersymmetric gauge group (in any dimension).

Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover.

The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to topological quantum field theories with or without embedded spin networks.     

Bogoliubov transformations and fermion condensates in lattice field theories

We apply generalized Bogoliubov transformations to the transfer matrix of relativistic field theories regularized on a lattice. We derive the conditions these transformations must satisfy to factorize the transfer matrix into two terms which propagate fermions and antifermions separately, and we solve the relative equations under some conditions. We relate these equations to the saddle point approximation of a recent bosonization method and to the Foldy-Wouthuysen transformations which separate positive from negative energy states in the Dirac Hamiltonian.     

Convergence analysis of the Wolf method for Coulombic interactions

A rigorous proof for convergence of the Wolf method (Wolf et al., 1999 [9]) for calculating electrostatic energy of a periodic lattice is presented. In particular, we show that for an arbitrary lattice of unit cells, the lattice sum obtained via Wolf method converges to the one obtained via Ewald method.     

An alternative construction of the Ewald sum

Periodicity is a popular way to handle boundaries in atomistic simulations in the bulk. However, slowly decaying Coulomb interactions lead to ill-defined energies and forces. The Ewald sum is a common way of giving meaning to these terms, but its derivation, based on Fourier series, is not transparent. Described in this article is a derivation based on the limiting behaviour as the cutoff goes to infinity of a truncated and shifted Coulomb potential. This observation strengthens the case for the use of the original Ewald sum. And such a construction may offer advantages for computation and/or modelling.     

A perfect lattice approach to nonstoichiometry

Although nonstoichiometry implies the presence of defects in a structure and hence a departure from perfect crystallinity, it is sometimes found that the defects are ordered or, at least, show a tendency to order. In these cases, it becomes possible to model the nonstoichiometry by considering a supercell and using perfect lattice techniques. We illustrate the viability of this approach with two examples. Firstly, we show how factors controlling the long-range ordering of extended defects in transition-metal oxides may be elucidated; indeed, we find that an adequate treatment of this phenomenon requires calculations on supercells. Secondly, we discuss how perfect lattice calculations may be used, in conjunction with diffraction data, to examine possible vacancy ordering schemes in the oxidation of magnetite, Fe₃O₄, to maghemite, the defect spinel structured γ-Fe₂O₃.     

Interplay between lattice dynamics and the low-pressure phase of simple cubic polonium

Low-pressure structural properties of simple cubic polonium are explored through first-principles density-functional theory based relativistic total energy calculations using pseudopotentials and plane-wave basis set, as well as linear-response theory. We have found that Po undergoes structural phase transition at low pressure near 2 GPa, where the element transforms from simple cubic to a mixture of two trigonal phases namely, hR1 (α=86°) and hR2 (α=97.9°) structures. The lattice dynamics calculations provide strong support for the observed phase transition, and show the dynamical stability (instability) of the hR2 (hR1) phase.     

A lattice Dirac operator for QCD with light dynamical quarks

In QCD chiral symmetry is explicitly broken by quark masses, the effect of which can be described reliably by chiral perturbation theory. Effects of explicit chiral symmetry breaking by the lattice regularisation of the Dirac operator, typically parametrised by the residual mass, should be negligible for almost all observables if the residual mass of the Dirac operator is much smaller than the quark mass. However, maintaining a small residual mass becomes increasingly expensive as the quark mass decreases towards the physical value and the continuum limit is approached. We investigate the feasibility of using a new approximately chiral Dirac operator with a small residual mass as an alternative to overlap and domain wall fermions for lattice simulations. Our Dirac operator is constructed from a Zolotarev rational approximation for the matrix sign function that is optimal for bulk modes of the hermitian kernel Dirac operator but not for the low-lying parts of its spectrum. We test our operator on various ³²³×64${32}^3\times 64$ lattices, comparing the residual mass and the performance of the Hybrid Monte Carlo algorithm at a similar lattice spacing and pion mass with a hyperbolic tangent operator as used by domain wall fermions. We find that our approximations have a significantly smaller residual mass than domain wall fermions at a similar computational cost, and still admit topological charge change.     

Electric field gradients in simple tetragonal lattice at interstitial site

Electric field gradient (efg) is calculated at the centre of the cell in the simple tetragonal crystal. The method uses Euler-Maclaurin summation formula and makes the planewise summation in the direct crystal space without any special regrouping of charges in point charge model. The results are in fair agreement with previous results of de Wette on the same system using Fourier transform to reciprocal space.     

A theorem concerning twisted and untwisted partition functions in and lattice gauge theories

In order to get a clue to understanding the volume-dependence of vortex free energy (which is defined as the ratio of the twisted against the untwisted partition function), we investigate the relation between vortex free energies defined on lattices of different sizes. An equality is derived through a simple calculation which equates a general linear combination of vortex free energies defined on a lattice to that on a smaller lattice. The couplings in the denominator and in the numerator however shows a discrepancy, and we argue that it vanishes in the thermodynamic limit. Comparison between our result and the work of Tomboulis is also presented. In the appendix we carefully examine the proof of quark confinement by Tomboulis and summarize its loopholes.     

光学傅里叶变换系统频谱位置的一种简单推导方法

以不同光照条件下各种光学傅里叶变换系统为研究对象,利用光学变换矩阵和柯林斯公式推导了其频谱位置所满足的条件.推导过程表明,该方法具有简单、便捷和容易在有限的课堂教学时间内讲解清楚的特点.     

A simple phase unwrapping approach based on filtering by windowed Fourier transform

Phase unwrapping is an interesting yet challenging problem in optical interferometry. In this paper, we limit our interest to unwrapping noisy phase maps as it is a common but difficult task. Our aim is to propose a simple solution to phase unwrapping. We first remove the noise by a novel windowed Fourier transform approach and then use sequential line scanning method for phase unwrapping. This simple approach is verified to be very effective.     

A method for measuring the Witten index using lattice simulation

We propose a method to measure the Witten index using lattice simulation. A requirement for the lattice model is that it has at least one exact supersymmetry at finite lattice spacing. We prove the validity of the method in case of the supersymmetric quantum mechanics, where the index is well known.     

A simple approach to nonlinear oscillators

A very simple and effective approach to nonlinear oscillators is suggested. Anyone with basic knowledge of advanced calculus can apply the method to finding approximately the amplitude-frequency relationship of a nonlinear oscillator. Some examples are given to illustrate its extremely simple solution procedure and an acceptable accuracy of the obtained solutions.     

WChPT analysis of twisted mass lattice data

We perform a Wilson Chiral Perturbation Theory (WChPT) analysis of quenched twisted mass lattice data. The data were generated by two independent groups with three different choices for the critical mass. For one choice, the so-called pion mass definition, one observes a strong curvature for small quark masses in various mesonic observables (“bending phenomenon”). Performing a combined fit to the next-to-leading (NLO) expressions, we find that WChPT describes the data very well and the fits provide very reasonable values for the low-energy parameters.     

Thermodynamic and transport properties of simple fluids using lattice sums: bulk phases and liquid-vapour interface

Molecular dynamics simulations in the canonical ensemble have been performed to obtain the thermodynamic and transport properties of the Lennard-Jones fluid. The dispersion interactions were calculated using lattice sums. This method makes it possible to simulate the full potential avoiding the inclusion of the long range corrections (LRC) during or at the end of simulations. In the calculation of dynamic properties in bulk phases and thermodynamic quantities of inhomogeneous systems where the interface is physically present, in general the LRC cannot easily be included. By using the lattice sums method, the results are independent of the truncation of the potential. In the liquid-vapour interface simulations it is not necessary to make any pre-judgments about the form of the LRC formula to calculate coexisting properties such as the surface tension. The lattice sums method has been applied to evaluate how well the full interaction can be calculated in the liquid phase and in the liquid-vapour interface. In the liquid phase the pressure, configurational energy, diffusion coefficient and shear viscosity were obtained. The results of the thermodynamic properties are compared with those obtained using the spherically truncated and shifted (STS) potential with the LRC added at the end of simulations, and excellent agreement is found. The transport properties are calculated on different system sizes for a state near the triple point. The diffusion coefficient using the lattice sums method increases with the number of molecules, and the results are higher than those of the STS model truncated at 2.5σ (STS2.5). The shear viscosity does not show any system size dependence for systems with more than 256 molecules, and the lattice sums results are essentially the same as those for the STS2.5. In the liquid-vapour equilibria the coexisting densities and vapour pressures for the full potential agree well with those obtained using the Gibbs ensemble and the NPT + test particle methods. The surface tension using lattice sums and truncation of forces at 2.5σ agrees well with STS results using large system sizes and cutoff distances.