Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices

We study the face-centered cubic lattice (fcc) in up to six dimensions. In particular, we are concerned with lattice Green's functions (LGF) and return probabilities. Computer algebra techniques, such as the method of creative telescoping, are used for deriving an ODE for a given LGF. For the four- and five-dimensional fcc lattices, we give rigorous proofs of the ODEs that were conjectured by Guttmann and Broadhurst. Additionally, we find the ODE of the LGF of the six-dimensional fcc lattice, a result that was not believed to be achievable with current computer hardware.Comment: 16 pages, final versio

The Heisenberg paramagnet

Ph.D.Harold A. Gersc

Strong short-range magnetic order in a frustrated FCC lattice and its possible role in the iron structural transformation

We investigate magnetic properties of a frustrated Heisenberg antiferromagnet with a face-centered cubic (FCC) lattice and exchange interactions between the nearest- and next-nearest neighbours, J1 and J2. In a collinear phase with the wave vector Q = (pi,pi,pi) the equations of the self-consistent spin-wave theory for the sublattice magnetization and the average short range order parameter are obtained and numerically solved. The dependence of the Neel temperature T_N on the ratio J2/J1 is obtained. It is shown, that at strong enough frustration there is a wide temperature region above T_N with strong short range magnetic order. Application of this result to description of structural phase transition between alpha and gamma-phase of Fe is considered

The Korringa-Kohn-Rostoker nonlocal coherent-potential approximation : a new method for calculating the electronic structure of disordered metallic systems

The limitations of the current 'first-principles' effective medium approach to calculating the electronic structure of disordered systems are described. These limitations can be addressed by a cluster theory, and only very recently the first satisfactory cluster theory, the nonlocal coherent potential approximation, has been developed within a tight-binding framework. However an approach based on KKR multiple scattering is needed in order to treat the problem from first principles for ab-initio calculations. In this thesis, these ideas are reformulated in terms of multiple scattering theory and the Korringa-Kohn-Rostoker non-local coherent potential approximation (KKR-NLCPA) is introduced for describing the electronic structure of disordered systems. The KKR-NLCPA systematically provides a hierarchy of improvements upon the widely used local mean-field KKR-CPA approach and includes nonlocal correlations in the disorder configurations by means of a self-consistently embedded cluster. The KKR-NLCPA method satisfies all of the requirements for a successful cluster generalisation of the KKR-CPA; it determines a site-to-site translationally-invariant effective medium, it is herglotz analytic, becomes exact in the limit of large cluster sizes, reduces to the KKR-CPA for a single-site cluster, is straightforward to implement numerically, and enables the effects of short-range order upon the electronic structure to be investigated. In particular, it is suitable for combination with electronic density functional theory to give an ab-initio description of disordered systems. Future applications to charge correlation and lattice displacement effects in alloys and spin fluctuations in magnets amongst others are very promising. The method is illustrated by application to a simple one-dimensional model

Dynamics of disordered harmonic lattices

Extensive numerical and analytic studies of vibrational spectra, normal modes, thermodynamic properties, and dynamical properties of harmonic systems with varying degrees of substitutional disorder have been made. The effects on observable properties of random mixtures of two or more species of atoms with differing masses and differing couplings to nearest-and next-nearest-neighbors have been investigated. Using the IBM-7030 digital computer, spectra for linear chains of 100,000 atoms have been obtained. Calculations in two and three dimensions have been limited to arrays of approximately 1000 atoms. Varying composition, mass ratio, and order affect the spectra in two and three dimensions in ways analogous to those effected in the linear chain. A physical interpretation of the complex nature of the disordered spectrum is given. The effects of disorder on the dynamics of binary disordered harmonic chains have been studied and have been found to be quite pronounced --Abstract, page ii

The Korringa-Kohn-Rostoker nonlocal coherent-potential approximation : a new method for calculating the electronic structure of disordered metallic systems

The limitations of the current 'first-principles' effective medium approach to calculating the electronic structure of disordered systems are described. These limitations can be addressed by a cluster theory, and only very recently the first satisfactory cluster theory, the nonlocal coherent potential approximation, has been developed within a tight-binding framework. However an approach based on KKR multiple scattering is needed in order to treat the problem from first principles for ab-initio calculations. In this thesis, these ideas are reformulated in terms of multiple scattering theory and the Korringa-Kohn-Rostoker non-local coherent potential approximation (KKR-NLCPA) is introduced for describing the electronic structure of disordered systems. The KKR-NLCPA systematically provides a hierarchy of improvements upon the widely used local mean-field KKR-CPA approach and includes nonlocal correlations in the disorder configurations by means of a self-consistently embedded cluster. The KKR-NLCPA method satisfies all of the requirements for a successful cluster generalisation of the KKR-CPA; it determines a site-to-site translationally-invariant effective medium, it is herglotz analytic, becomes exact in the limit of large cluster sizes, reduces to the KKR-CPA for a single-site cluster, is straightforward to implement numerically, and enables the effects of short-range order upon the electronic structure to be investigated. In particular, it is suitable for combination with electronic density functional theory to give an ab-initio description of disordered systems. Future applications to charge correlation and lattice displacement effects in alloys and spin fluctuations in magnets amongst others are very promising. The method is illustrated by application to a simple one-dimensional model.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research CouncilGBUnited Kingdo

Solitons in nonlinear lattices

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.Comment: 169 pages, 35 figures, a comprehensive survey of results on solitons in purely nonlinear and mixed lattices, to appear in Reviews of Modern Physic