Exotic spectra, wavefunctions and transport in incommensurate lattices

We present numerical results on a range of related issues for a number of incommensurate TMB’s, each of which shows a metal-insulator type transition as a binding-to-hopping ratio is made to increase through some limiting value. These supplement a series of similar results on a couple of 1D lattices in a number of recent works (see below). A brief review pertaining to spectral properties and wavefunctions in incommensurate lattices is followed by results on the above TBM’s relating to an interesting correlation between the gross features of wavefunctions and the energies arranged in a particular sequence termed thelattice-ordered sequence, and also between the lattice-ordered energies and the on-site potentials. We present a qualitative explanation of these correlations on the basis of perturbation theory. Basic results on dynamics of wavepackets in relation to spectral characteristics of incommensurate TBM’s are also reviewed. Features of lattice-ordered energies and wavefunctions for the TBM’s under study are used in the framework of the so-called Maryland construction, leading to a qualitative prediction of criteria for recurrent and non-recurrent wavepacket dynamics in these lattices, and these predictions are checked against numerical iterations of the relevant ‘quantum maps’. Closely related to the dynamics of wavepackets are the transport properties of these lattices. Results are available to indicate that the unusual spectral characteristics of pseudorandom lattices lead to novel features in transport properties of these systems. In this context, low temperature a.c conductivity in these lattices is a good probe for the spectral characteristics and wavefunctions. However, not much is known about the a.c conductivity, excepting a set of early results pertaining to the low frequency regime, principally because of the fact that the a.c conductivity depends on global characteristics of the spectrum and the entire set of wavefunctions. We present a simple model whereby the gross structure of variation of the a.c conductivity with frequency can be obtained from a knowledge of the spectrum alone for the set of TMB’s under consideration. Numerical computations show that despite its simplicity, the model leads to results in good agreement with those from the Kubo-Greenwood formula for a.c conductivity.     

Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.     

Anharmonic Excitations,Time Correlations and Electric Conductivity

We study the influence of anharmonic mechanical excitations of a classical ionic lattice on its electric properties. First, to illustrate salient features, we investigate a simple model, an one‐dimensional (1D) system consisting of ten semiclassical electrons embedded in a lattice or a ring with ten ions interacting with exponentially repulsive interactions. The lattice is embedded in a thermal bath. The behavior of the velocity autocorrelation function and the dynamic structure factor of the system are analyzed. We show that in this model the nonlinear excitations lead to long lasting time correlations and, correspondingly, to an increase of the conductivity in a narrow temperature region, where the excitations are supersonic soliton‐like. In the second part we consider the quantum statistics of general ion‐electron systems with arbitrary dimension and express ‐ following linear response transport theory ‐ the quantum‐mechanical conductivity by means of equilibrium time correlation functions. Within the relaxation time approach an expression for the effective collision frequency is derived in Born approximation, which takes into account quantum effects and dynamic effects of the ion motion through the dynamic structure factor of the lattice and the quantum dynamics of the electrons. An evaluation of the influenec of solitons predicts for 1D‐lattices a conductivity increase in the temperature region where most thermal solitons are excited, similar as shown in the classical Drude‐Lorentz‐Kubo framework. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometry, topology, and physics of non-Abelian lattices

Conclusion  After reviewing in some detail the notion of non-Euclidean lattices, whose domain of physical realization lies mostly in the novel carbon structures of the family offullerenes, we have discussed a number of physical problems denned over such lattices. We have shown that the group-theoretical definition of these lattices leads to “designing” new tubular regular structures, endowed with symmetries unheard of in the frame of customary crystallography, which combine features of extreme complexity and, at the same time, of great regularity. We have compared the role of the non-Abelian symmetries which these super-lattices are characterized by, with that of (discrete) harmonic (Fourier) lattice symmetry typical of customary crystallographic lattices. Many novel features enter into play, due to thenon-flatness of the related lattice geometry, which led us to a novel—sometimes unexpected—insight into the dynamical and/or thermodynamical properties of various physical systems which have these lattices as ambient space. We have analyzed how lattice topology bears on the complex combinatorics (related to loop-counting) of the classical Ising model. These lattices, even though finite, are, of course, much closer to being three-dimensional than regular 2D lattices simply equipped with periodic boundary conditions. We have shown, on the other hand, how the relation between the lattice symmetry (for example, in the case of fullerene, the discrete subgroup ofSU(2) that we have denotedg ₆₀ and the symmetry proper to the Hamiltonian of quantum systems of many itinerant interacting electrons (Hubbard-like models) allows us to reduce the calculation of the system spectral properties to a “size” that can be dealt with numerically with present-day numerical exact diagonalization techniques much more easily than a regular 3D cluster with a quite smaller number of sites.     

Entropic rigidity of randomly diluted two- and three-dimensional networks

In recent work, we presented evidence that site-diluted triangular central-force networks, at finite temperatures, have a nonzero shear modulus for all concentrations of particles above the geometric percolation concentration p(c). This is in contrast to the zero-temperature case where the (energetic) shear modulus vanishes at a concentration of particles p(r)>p(c). In the present paper we report on analogous simulations of bond-diluted triangular lattices, site-diluted square lattices, and site-diluted simple-cubic lattices. We again find that these systems are rigid for all p>p(c) and that near p(c) the shear modulus mu approximately (p-p(c))(f), where the exponent f approximately 1.3 for two-dimensional lattices and f approximately 2 for the simple-cubic case. These results support the conjecture of de Gennes that the diluted central-force network is in the same universality class as the random resistor network. We present approximate renormalization group calculations that also lead to this conclusion.     

Preparation,structural, and calorimetric characterization of bicomponent metallic photonic crystals

We report preparation and characterization of novel bicomponent metal-based photonic crystals having submicron three-dimensional (3D) periodicity. Fabricated photonic crystals include SiO₂ sphere lattices infiltrated interstitially with metals, carbon inverse lattices filled with metal or metal alloy spheres, Sb inverse lattices, and Sb inverse lattices filled with Bi spheres. Starting from a face centered SiO₂ lattice template, these materials were obtained by sequences of either templating and template extraction or templating, template extraction, and retemplating. Surprising high fidelity was obtained for all templating and template extraction steps. Scanning electron microscopy (SEM), small angle X-ray scattering (SAXS) and differential scanning calorimetry (DSC) were used to characterize the structure and the effects of the structure on calorimetric properties. To the best of our knowledge, SAXS data on metallic photonic crystals were collected for first time. PACS 42.70.Qs; 68.37.Hk; 61.10.Eq; 81.70.Pg     

Calculation of mechanical strength of crystalline lattices of iron,molybdenum, and tungsten

Semiempirical tensor equations of state were obtained for iron, molybdenum, and tungsten and used to examine the mechanical properties of their crystalline lattices under conditions of a complex system of stresses. The most hazardous systems of stresses were distinguished, and stresses and deformations were calculated. The obtained values of strength are in good agreement with the existing empirical data.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 74–78, 1972.     

Use of the pseudopotential model method to study mechanical properties of alkali metal lattices in case of complex stress systems

Based on the Krasko-Gurskii pseudopotential model, the mechanical properties of alkali metal lattices are considered for deformations conserving the symmetry elements D_3d and D_2h of bcc lattices. An estimate is given of the parameters of lattice distortion, and the most dangerous stress systems are derived.     

Critical temperature and condensed fraction of Bose-Einstein condensation in optical lattices

Critical temperature and condensate fraction of Bose-Einstein condensation in the optical lattice are studied. The results show that the critical temperature in optical lattices can be characterized with an equivalent critical temperature in a single lattice, which provide a fast evaluation of critical temperature and condensate fraction of Bose-Einstein condensation confined with pure optical trap. Critical temperature can be estimated with an equivalent critical temperature. It is predicted that critical temperature is proportional to q in q number lattices for superfluid state and should be equal to that in a single lattic for Mott insulate state. Required potential depth or Rabi frequency and maximum atom number in the lattices both for superfluid state and Mott state are presented based on views of thermal mechanical statistics.     

Piron's and Bell's Geometrical Lemmas

The famous Gleason's Theorem gives a characterization of measures on lattices of subspaces of Hilbert spaces. The attempts to simplify its proof lead to geometrical lemmas that possess also easy proofs of some consequences of Gleason's Theorem. We contribute to these results by solving two open problems formulated by Chevalier, Dvure?enskij and Svozil. Besides, our use of orthoideals provides a unified approach to finite and infinite measures.     

Solitons in complex optical lattices

We present an overview of our recent results in the area of soliton excitation and control in optical lattices induced by different types of nondiffracting beams featuring unique symmetries. Optical lattices offer the possibility to engineer and to control the diffraction of light beams in media with transversally modulated optical properties, to manage the corresponding reflection and transmission bands, and to form specially designed defects. Consequently, they afford the existence of a rich variety of new families of nonlinear stationary waves and solitons, lead to new rich dynamical phenomena, and offer novel conceptual opportunities for all-optical shaping, switching and routing of optical signals encoded in soliton formats. In this overview, we consider different types of solitons, including fundamental, multipole, and vortex solitons in reconfigurable lattices optically induced by nondiffracting radially symmetric and azimuthally modulated single Bessel beams, soliton control in networks, couplers, and switches induced by several mutually coherent or incoherent Bessel beams, we address soliton properties in three-dimensional Bessel lattices, as well as in lattices produced by Mathieu and parabolic optical beams.     

Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice

Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schr?dinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < π) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes. Received 6 October 2001 / Received in final form 1st March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: mjn@ifm.liu.se     

Multifractal analysis of electronic states on random Voronoi-Delaunay lattices

We consider the transport of non-interacting electrons on two- and three-dimensional random Voronoi-Delaunay lattices. It was recently shown that these topologically disordered lattices feature strong disorder anticorrelations between the coordination numbers that qualitatively change the properties of continuous and first-order phase transitions. To determine whether or not these unusual features also influence Anderson localization, we study the electronic wave functions by multifractal analysis and finite-size scaling. We observe only localized states for all energies in the two-dimensional system. In three dimensions, we find two Anderson transitions between localized and extended states very close to the band edges. The critical exponent of the localization length is about 1.6. All these results agree with the usual orthogonal universality class. Additional generic energetic randomness introduced via random potentials does not lead to qualitative changes but allows us to obtain a phase diagram by varying the strength of these potentials.     

Elasticity and melting of skyrmion flux lattices in p-wave superconductors

We analytically calculate the energy, magnetization curves [B(H)], and elasticity of Skyrmions flux lattices in p-wave superconductors near the lower critical field H(c1), and we use these results with the Lindemann criterion to predict their melting curve. In striking contrast to vortex flux lattices, which always melt at an external field H>H(c1), Skyrmion flux lattices never melt near H(c1). This provides a simple and unambiguous test for the presence of Skyrmions.     

Structural, electronic, mechanical, and magnetic properties and relative stability of polymorphic modifications of ReN2 from Ab initio calculation data

A comparative analysis of the structural, electronic, mechanical, and magnetic properties and relative stability has been carried out in terms of ab initio calculations for four possible polymorphic modifications of rhenium dinitride, whose nonmetallic lattices contain both individual nitrogen atoms and dimers N₂. It has been found that the recently synthesized hexagonal polymorph ReN₂ (structural type 2H-MoS₂) is a weak d ⁰ magnet in which the magnetic state is formed due to spin splitting of N 2p states.     

New metallic allotropes of planar and tubular carbon

We propose a new family of layered sp(2)-like carbon crystals, incorporating five-, six-, and seven-membered rings in 2D Bravais lattices. These periodic sheets can be rolled so as to generate nanotubes of different diameter and chirality. We demonstrate that these sheets and tubes are metastable and more favorable than C60, and it is also shown that their mechanical properties are similar to those of graphene. Density of states calculations of all structures revealed an intrinsic metallic behavior, independent of orientation, tube diameter, and chirality.     

Height Representation,Critical Exponents,and Ergodicity in the Four-State Triangular Potts Antiferromagnet

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a (2+2)-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagomé lattice and to bond-coloring models on the triangular and honeycomb lattices. We also calculate critical exponents for the ground-state ensemble of the model. We find that the exponents governing the spin–spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang–Swendsen–Kotecký cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices whose topology has no noncontractible loops of infinite order, such as the projective plane. To guard against biases introduced by lack of ergodicity, we perform our simulations on both the torus and the projective plane.     

Analytical results for the electronic shell structure in mesoscopic systems: Balian-Bloch type theory for spheres,discs, rings and anti-dot lattices

The electronic shell structure resulting from the interference of closed orbital paths is determined for mesoscopic systems like spherical clusters, discs and rings by extending the semiclassical theory of Balian and Bloch. Analytical results for the shell structure in the density of states are obtained. Thus, the dependence of the shell structure on dimension, size and geometry and potential of the mesoscopic system and on an external magnetic field can be studied systematically. Comparison of the semiclassical results and those of quantum mechanical calculations permits analysis of typical quantum mechanical effects and shows the validity of the semiclassical theory. Our results should stimulate new experiments, can be used to calculate oscillations in the binding-energy, ionization-potential, and can be applied to analyze oscillations in the electronic density of states of quantum dot systems like anti-dot lattices.     

Interval order and semiorder lattices

When a set of closed intervals of the reals is partially ordered by decreeing that A<B when A lies strictly to the left of B, the resulting structure is called an interval order. Semiorders may be viewed as interval orders that arise from closed intervals having a fixed length. The paper initiates a careful study of interval orders and semiorders that happen also to be lattices. A structure theory is obtained for a class of interval order lattices that includes all such lattices of finite length. Characterizations are given of when these lattices are modular or distributive, as well as when they are semiorders. The theory is of some interest because the completion by cuts of an interval order is necessarily an interval order lattice. Though it is shown that the completion by cuts of a semiorder need not be a semiorder, necessary and sufficient conditions are given for a lattice of finite length to be isomorphic to the completion by cuts of a semiorder.The author wishes to dedicate this paper to the memory of his late colleague Professor Charles H. Randall.