Dual Lattice Research Articles

Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative index behavior

Arrays of metallic nanoparticles support individual and collective plasmonic excitations that contribute to unusual phenomena like surface enhanced Raman scattering, anomalous transparency, negative index, and subwavelength resolution in various metamaterials. We have examined the electromagnetic response of dual Kron's lattice and films containing up to three monolayers of metallic nanospheres. It appears that open cubic Kron's lattice exhibits `soft' electromagnetic response but no negative index behavior. The close-packed arrays behave similarly: there are plasmon resonances and very high transmission at certain wavelengths that are much larger than the separation between the particles, and a `soft' magnetic response, with small but positive effective index of refraction. It would be interesting to check those predictions experimentally.

Investigation of phase separation within the generalized Lin–Taylor model for a binary liquid mixture of large hexagonal and small triangular particles

The generalized Lin–Taylor model defined on the hexagonal lattice is used to investigate the phase separation in an asymmetric binary liquid mixture consisting of large A (hexagons) and small B (triangles) particles. By considering interaction energies between A–A and A–B pairs of particles that occupy nearest-neighbour cells of the hexagonal lattice, we have derived an exact solution for the considered model system having established a mapping correspondence with the two-dimensional Ising model on its dual triangular lattice. Altogether, six different types of coexistence curves including those with re-entrant miscibility regions (i.e. closed-loop coexistence curves) were found with dependence on the relative strength between both coupling constants.

Free and nonfree Voronoi polyhedra

The Voronoi polyhedron of some point v of a translation lattice is the closure of the set of points in space that are closer to v than to any other lattice points. Voronoi polyhedra are a special case of parallelohedra, i.e., polyhedra whose parallel translates can fill the entire space without gaps and common interior points. The Minkowski sum of a parallelohedron with a segment is not always a parallelohedron. A parallelohedron P is said to be free along a vector e if the sum of P with a segment of the line spanned by e is a parallelohedron. We prove a theorem stating that if the Voronoi polyhedron Pv(f) of a quadratic form f is free along some vector, then the Voronoi polyhedron Pv(g) of each form g lying in the closure of the L-domain of f is also free along some vector. For the dual root lattice E6*, and the infinite series of lattices D2m+, m ≥ 4, we prove that their Voronoi polyhedra are nonfree in all directions.

The mobility of dual vortices in honeycomb, square, triangular, Kagome and dicelattices

It was known that by a duality transformation, interacting bosons at filling factorf = p/q hopping on a lattice can be mapped to interacting vortices hopping onthe dual lattice subject to a fluctuating dual ‘magnetic field’ whoseaverage strength through a dual plaquette is equal to the boson densityf = p/q. So the kinetic term of the vortices is the same as the Hofstadterproblem of electrons moving in a lattice in the presence off = p/q flux per plaquette. Motivated by this mapping, we study theHofstadter bands of vortices hopping in the presence of magnetic fluxf = p/q per plaquette on five most common bipartite and frustrated lattices namelysquare, honeycomb, triangular, dice and Kagome lattices. We count the totalnumber of bands, and determine the number of minima and their locations inthe lowest band. We also numerically calculate the bandwidths of the lowestHofstadter bands in these lattices that directly measure the mobility of the dualvortices. The less mobile the dual vortices are, the more likely are the bosonsto be in a superfluid state. We find that apart from the Kagome lattice at oddq, they all satisfy theexponential decay law W = Ae−cq even at the smallest q. At given q, thebandwidth W decreases in the order of triangle, square and honeycomb lattice. This indicates thatthe domain of the superfluid state of the original bosons increases in the orderof the corresponding direct lattices: honeycomb, square and triangular. Whenq = 2, we find that the lowest Hofstadter band is completely flat for both Kagome and dicelattices. There is a gap on the Kagome lattice, but no gap on the dice lattice. Thisindicates that the boson ground state at half filling with nearest neighbour hopping onKagome lattice is always a superfluid state. The superfluid state remains stable slightlyaway from the half filling. Our results show that the behaviours of bosons at or near halffilling on Kagome lattices are quite distinct from those in square, honeycomb andtriangular lattices studied previously.

Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices

A conjecture is given for the exact location of the multicritical point in the phase diagram of the +/- J Ising model on the triangular lattice. The result p_c=0.8358058 agrees well with a recent numerical estimate. From this value, it is possible to derive a comparable conjecture for the exact location of the multicritical point for the hexagonal lattice, p_c=0.9327041, again in excellent agreement with a numerical study. The method is a variant of duality transformation to relate the triangular lattice directly with its dual triangular lattice without recourse to the hexagonal lattice, in conjunction with the replica method.

Determination of the effects of the pore size distribution and pore connectivity distribution on the pore tortuosity and diffusive transport in model porous networks

An efficient computation method to study flow and transport process of small molecules in porous media using a dual site-bond lattice model, DBSM, is described. The microscopic properties of the porous network take into account the influence of local heterogeneities during the simulations. The numerical experiments demonstrated the combined effect of pore size distribution and connectivity distribution on the mass transport properties and the structural tortuosity. The results indicate that the pore size distribution and percolation phenomena related with pore shielding effects, influence significantly the tortuosity and the effective diffusivity of the porous network. Also, the simulations raise the important role of the connectivity distribution among the various pores in the gas diffusive properties of the poorly connected networks.

A Columnar Phase of Dendritic Lipid−Based Cationic Liposome−DNA Complexes for Gene Delivery: Hexagonally Ordered Cylindrical Micelles Embedded in a DNA Honeycomb Lattice

Gene therapy holds great promise as a future approach to fighting disease and is explored in worldwide clinical trials. Cationic liposome (CL)-DNA complexes are a prevalent nonviral delivery vector, but their efficiency requires improvement and the understanding of their mechanism of action is incomplete. As part of our effort to investigate the structure-transfection efficiency relationships of self-assembled CL-DNA vectors, we have synthesized a new, highly charged (16+) multivalent cationic lipid, MVLBG2, with a dendritic headgroup. Our synthetic scheme allows facile variation of the headgroup charge and the spacer connecting hydrophobic and headgroup moieties as well as gram-scale synthesis. Complexes of DNA with mixtures of MVLBG2 and neutral 1,2-dioleoyl-sn-glycerophosphatidylcholine (DOPC) exhibit the well-known lamellar phase at 90 mol % DOPC. Starting at 20 mol % dendritic lipid, however, two novel nonlamellar phases are observed by synchrotron X-ray diffraction. The structure of one of these phases, present in a narrow range of composition around 25 mol % MVLBG2, has been solved. In this novel dual lattice structure, termed H(I)C, hexagonally arranged tubular lipid micelles are surrounded by DNA rods forming a three-dimensionally continuous substructure with honeycomb symmetry. Complexes in the H(I)C phase efficiently transfect mouse and human cells in culture. Their transfection efficiency, as well as that of the lamellar complexes containing only 10 mol% dendritic lipid, reaches and surpasses that of commercially available, optimized DOTAP-based complexes. In particular, complexes containing MVLBG2 are significantly more transfectant over the entire composition range in mouse embryonic fibroblasts, a cell line empirically known to be hard to transfect.

Dual formulations of non-Abelian spin models: Local representation and low-temperature asymptotics

Non-Abelian lattice SU ( N ) or U ( N ) spin models can be formulated in terms of link variables which are subject to the Bianchi constraints. Using this representation we derive exact and local dual formulation for the partition function and some observables of such models on a cubic lattice in arbitrary dimension D. An interaction between dual variables ranges over one given hypercube of the dual lattice. We use our construction to study the dual of SU ( 2 ) model in two dimensions. Leading terms of the asymptotic expansion of the dual weight are computed and it is proven that at low temperatures it converges to a certain Gaussian distribution uniformly in fluctuations of dual variables. This result enables us to define the semiclassical limit of the dual formulation and to extract leading perturbative contribution to the correlation function which shows power-like decay. We present some analytical evidences that the low-temperature limit of the dual formulation is described by ISO ( 2 ) -like approximation of SU ( 2 ) matrices.

Multicritical point relations in three dual pairs of hierarchical-lattice Ising spin glasses

The Ising spin-glasses are investigated on three dual pairs of hierarchical lattices, using exact renormalization-group transformation of the quenched bond probability distribution. The goal is to investigate a recent conjecture which relates, on such pairs of dual lattices, the locations of the multicritical points, which occur on the Nishimori symmetry line. Towards this end we precisely determine the global phase diagrams for these six hierarchical spin-glasses, using up to 2.5 x 10^9 probability bins to represent the quenched distribution subjected to an exact renormalization-group transformation. We find in all three cases that the conjecture is realized to a very good approximation, even when the mutually dual models belong to different spatial dimensionalities d and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with d=2 or 1.5, the spin-glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality. PACS numbers: 75.10.Nr, 64.60.Kw, 05.45.Df, 05.10.Cc

THE NORMAL DUAL CONGRUENCES AND THE DUAL BIANCHI LATTICE

The main goal of the paper is to find the discrete analogue of the Bianchi system in spaces of arbitrary dimension together with its geometric interpretation. We show that the proper geometric framework for such generalization is the language of dual quadrilateral lattices and of dual congruences. After introducing the notion of the dual Koenigs lattice in a projective space of arbitrary dimension, we define the discrete dual congruences and we present, as an important example, the normal dual discrete congruences. Finally, we introduce the dual Bianchi lattice as a dual Koenigs lattice allowing for a conjugate normal dual congruence, and we find its characterization in terms of a system of integrable difference equations.

Lattice duality: The origin of probability and entropy

Bayesian probability theory is an inference calculus, which originates from a generalization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lattice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry—a generalization of information theory. In this paper we introduce this novel perspective on these spaces in which machine learning is performed and discuss the relationship between these results and several proposed generalizations of information theory in the literature.

Approximate extension of partial ε-characters of abelian groups to characters with application to integral point lattices

Let G be an abelian group, S ⊆ G be a finite set, and T denote the multiplicative group of complex unitswith the invariant arc metric | arg( a/b)|. We will show that for a mapping ƒ : S → T to be ε-close on S to a character φ : G → T it is enough that ƒ be extendable to a mapping ¯f : ( S U {1} U S −1) n → T, where n is big enough and ¯f violates the homomorphy condition at most up to an arbitrary σ < min(ε, π/2). Moreover, n can be chosen uniformly, independently of G and both ƒ and ¯f, depending just on σ, ε and the number of elements of S. The proof is non-constructive, using the ultraproduct construction and Pontryagin duality, hence yielding no estimate on the actual size of n. As one of the applications we show that, for a vector u ∈ R q to be ε-close to some vector from the dual lattice H ★ of a full rank integral point lattice H ⊆ ℤ q , it is enough for the scalar product ux to be δ-close (with δ < 1/3) to an integer for all vectors xH satisfying Σ i| x i | < n, where n depends on δ, ε and q only.

An Exact Percolation Point of Surface Filling in a Four-Dimensional Hyper-Cubic Lattice

An exact percolation threshold in a four-dimensional hyper-cubic lattice is obtained as p c = 0.5 from geometrical consideration. In a similar way, a macroscopically percolated structure consisting of (2n - n)-dimensional structure is formed at the critical point p c = 0.5 in a 2n-dimensional hyper-cubic lattice, because of the existence of a self-dual lattice in 2n-dimensional space and the orthogonal relation of the (2n - n)-dimensional structures in the original and dual hyper-cubic lattices.

Dual processes to solve single server systems

A new approach is used to determine the transient probability functions of the classical queueing systems: M/M/1, M/M/1/H, and M/M/1/H with catastrophes. This new solution method uses dual processes, randomization and lattice path combinatorics. The method reveals that the transient probability functions for M/M/1/H and M/M/1/H with catastrophes have the same mathematical form.

Exact location of the multicritical point for finite-dimensional spin glasses: a conjecture

We present a conjecture on the exact location of the multicritical point in the phase diagram of spin glass models in finite dimensions. By generalizing our previous work, we combine duality and gauge symmetry for replicated random systems to derive formulas which make it possible to understand all the relevant available numerical results in a unified way. The method applies to non-self-dual lattices as well as to self dual cases, in the former case of which we derive a relation for a pair of values of multicritical points for mutually dual lattices. The examples include the +-J and Gaussian Ising spin glasses on the square, hexagonal and triangular lattices, the Potts and Z_q models with chiral randomness on these lattices, and the three-dimensional +-J Ising spin glass and the random plaquette gauge model.

Dynamics of forward filtration combustion at the pore‐network level

AbstractA pore‐network model is developed, based on a dual lattice, to study the dynamics of forward filtration combustion (FC) in porous media at the pore‐network scale. A novel moving coordinate method is also implemented that allows for the long‐term profiles to be obtained in a straightforward manner. The model is used to reproduce recent analytical work, including the multiplicity of steady states for the nonadiabatic case, and to study a number of other effects. Particular emphasis is placed on the effect of the heterogeneity in the microstructure, introduced by the randomness in the pore radius. It is found that the local disorder leads to less oxidant consumption, as compared to the homogeneous case. This underprediction increases with an increase in the disorder, a decrease in the thermal conductivity, and a decrease in the spatial correlation. Temporary local extinction can result for sufficiently large injection rates. As a result of the heterogeneity, the spatially averaged reaction rate is generally different from the rate calculated based on the microscale Arrhenius expression, using the averaged variables, with discrepancies of a factor of 2 or higher not uncommon. Although this effect depends on the parameter values, it can be a cause of concern when the fronts at the small scale are sufficiently distorted. Coupling of fronts occurs in a layered system, when there is sufficient oxidant supply. These results show that the microscale geometry can have a strong influence on the overall macroscopic behavior of the process. © 2005 American Institute of Chemical Engineers AIChE J, 2005

On the degrees of freedom of lattice electrodynamics

Using Euler's formula for a network of polygons for 2D case (or polyhedra for 3D case), we show that the number of dynamic degrees of freedom of the electric field equals the number of dynamic degrees of freedom of the magnetic field for electrodynamics formulated on a lattice. Instrumental to this identity is the use (at least implicitly) of a dual lattice and of a (spatial) geometric discretization scheme based on discrete differential forms. As a by-product, this analysis also unveils a physical interpretation for Euler's formula and a geometric interpretation for the Hodge decomposition.

Exact Plaquette-Ordered Ground States in the Generalized Hubbard Model in Arbitrary Dimensions

We show the existance of the exact plaquette-ordered ground states of the Hubbard model including site-off-diagonal interactions in arbitrary dimensions, by decomposing the Hamiltonian as sum of products of projection operators for each spin sector. The obtatined exact ground states are interpreted as N\'eel ordered states on the dual lattices. We demonstrate this idea in the one-dimensional chain and higher-dimensional lattices such as the Kagom\'e lattice, and determine parameter regions of the exact ground states.

Compact U(1) on a spatial lattice: duality, photons, and confinement

I present a Hamiltonian approach to compact U(1) gauge theory on a regular spatial lattice in (2 + 1) and (3 + 1) dimensions. The diagonalization of the Hamiltonian in a ’’charge network’’ basis (eigenstates of the electric field) leads in a natural way to an electromagnetic duality transformation where the dual variables can be associated with the dual lattice. I present the solutions, in particular the vacuum state, in the weak-coupling limit and compare with the (shadow) states obtained recently by Varadarajan in a UV-regularized version of loopquantized (non-compact) continuum U(1) gauge theory. Finally, I comment on the existence of an instanton condensate which destroys this simple picture in the (2 + 1)-dimensional compact case and leads to confinement.

Stochastic perturbations in vortex-tube dynamics

A dual lattice vortex formulation of homogeneous turbulence is developed, within the Martin-Siggia-Rose field theoretical approach. It consists of a generalization of the usual dipole version of the Navier-Stokes equations, known to hold in the limit of vanishing external forcing. We investigate, as a straightforward application of our formalism, the dynamics of closed vortex tubes, randomly stirred at large length scales by Gaussian stochastic forces. We find that besides the usual self-induced propagation, the vortex tube evolution may be effectively modeled through the introduction of an additional white-noise correlated velocity field background. The resulting phenomenological picture is closely related to observations previously reported from a wavelet decomposition analysis of turbulent flow configurations.