Soliton assisted control of source to drain electron transport along natural channels – crystallographic axes – in two-dimensional triangular crystal lattices

We present computational evidence of the possibility of fast, supersonic or subsonic,nearly loss-free ballistic-like transport of electrons bound to lattice solitons (a formof electron surfing on acoustic waves) along crystallographic axes in two-dimensionalanharmonic crystal lattices. First we study the structural changes a soliton creates inthe lattice and the time lapse of recovery of the lattice. Then we study the behavior ofone electron in the polarization field of one and two solitons with crossing pathways withsuitably monitored delay. We show how an electron surfing on a lattice soliton may switchto surf on the second soliton and hence changing accordingly the direction of its path.Finally we discuss the possibility to control the way an excess electron proceeds from asource at a border of the lattice to a selected drain at another border by followingappropriate straight pathways on crystallographic axes.     

Asymmetric vortex solitons in nonlinear periodic lattices

We reveal the existence of asymmetric vortex solitons in ideally symmetric periodic lattices and show how such nonlinear localized structures describing elementary circular flows can be analyzed systematically using the energy-balance relations. We present the examples of rhomboid, rectangular, and triangular vortex solitons on a square lattice and also describe novel coherent states where the populations of clockwise and anticlockwise vortex modes change periodically due to a nonlinearity-induced momentum exchange through the lattice. Asymmetric vortex solitons are expected to exist in different nonlinear lattice systems, including optically induced photonic lattices, nonlinear photonic crystals, and Bose-Einstein condensates in optical lattices.     

Spontaneous magnetism of quantum dot lattices

The magnetism of square lattices of quantum dots with up to 12 electrons per dot is studied using the spin-density functional formalism. At small values of the lattice constant, all lattices are nonmagnetic and gapless. When the lattice constant is increased, the shell structure of the single dots governs the magnetism of the lattice. At closed shells, the lattices are nonmagnetic and have a gap at the Fermi level. At the beginning and at the end of a shell, they become ferromagnetic and stay gapless up to large values of the lattice constant. Antiferromagnetism was observed only at midshell after a band gap was opened.     

Merrifield-Simmons Index and Its Entropy of the 4-8-8 Lattice

The Merrifield-Simmons index is a topological index used in structural chemistry for quantifying relevant properties of molecular structure. This index, in different names, has been studied on the square, triangular and hexagonal lattices by using the method of transfer matrix. To compute the Merrifield-Simmons index of more complicated lattices such as the 4-8-8 and triangular Kagomé lattices, the use of multi-step transfer matrices is needed. In this paper we introduce the concept of transfer multiplicity and use multi-step transfer matrices to derive a rigorous upper and lower bound for the entropy of Merrified-Simmons index of the 4-8-8 lattice. We show that this method can be used to deal with other complex 2-dimensional lattices.     

Nonlinear magnetotransport in a two-dimensional electron system in a square array of antidots in GaAs/AlAs heterostructures

Nonlinear magnetotransport of two-dimensional electrons in square antidot lattices prepared on the basis of selectively doped GaAs/AlAs heterostructures with the period that is much less than the electron mean free path in the initial GaAs quantum wells but is much larger than their Fermi wavelength has been studied. It has been shown that the character of the nonlinear transport of the two-dimensional electrons in the lateral lattices under study changes from classical to quantum with the decrease in the antidot radius. It has been found that the quantum lifetime increases in the magnetic field corresponding to the condition of equality of the cyclotron diameter of two-dimensional electrons and the antidot lattice period.     

Diffusion in Lorentz lattice gas cellular automata: The honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found, ranging from propagation, superdiffusion, normal, quasi-normal, and anomalous, to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far is summarized.     

Vortex solitons in the semi-infinite gap of optically induced periodic lattices

Vortex solitons with a ring vortex core residing in a single lattice site in the semi-infinite gap of square optical lattices are reported. These solitons are no longer bound states of the Bloch-wave unit (Bloch-wave distribution in one lattice site) at the band edge of the periodic lattice, and consequently they do not bifurcate from the corresponding band edge. For saturable nonlinearity, one family of such solitons is found, and its existing curve forms a closed loop, which is very surprising. For Kerr nonlinearity, two families of such vortex solitons are found.     

Designing isotropic interactions for self-assembly of complex lattices

We present a direct method for solving the inverse problem of designing isotropic potentials that cause self-assembly into target lattices. Each potential is constructed by matching its energy spectrum to the reciprocal representation of the lattice to guarantee that the desired structure is a ground state. We use the method to self-assemble complex lattices not previously achieved with isotropic potentials, such as a snub square tiling and the kagome lattice. The latter is especially interesting because it provides the crucial geometric frustration in several proposed spin liquids.     

Coulomb and liquid dimer models in three dimensions

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.     

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.     

Vortex-lattice dynamics in rotating spinor Bose-Einstein condensates

We observe interlaced square vortex lattices in rotating dilute-gas spinor Bose-Einstein condensates (BEC). After preparing a hexagonal vortex lattice in a one-component BEC in an internal atomic state |1, we coherently transfer a fraction of the superfluid to a different state |2. The subsequent evolution of this pseudo-spin-1/2 superfluid towards a state of offset square lattices involves an intriguing interplay of phase-separation and -mixing dynamics, both macroscopically and on the length scale of the vortex cores, and a stage of vortex turbulence. The stability of the square structure is proved by its response to applied shear perturbations. An interference technique shows the spatial offset between the two vortex lattices. Vortex cores in either component are filled by fluid of the other component, such that the spin-1/2 order parameter forms a Skyrmion lattice.     

Thermodynamics of the Hubbard model on stacked honeycomb and square lattices

We present a numerical study of the Hubbard model on simply stacked honeycomb and square lattices, motivated by a recent experimental realization of such models with ultracold atoms in optical lattices. We perform simulations with different interlayer coupling and interaction strengths and obtain Néel transition temperatures and entropies. We provide data for the equation of state to enable comparisons of experiments and theory. We find an enhancement of the short-range correlations in the anisotropic lattices compared to the isotropic cubic lattice, in parameter regimes suitable for the interaction driven adiabatic cooling.     

Charge frustration and quantum criticality for strongly correlated fermions

We study a model of strongly correlated electrons on the square lattice which exhibits charge frustration and quantum critical behavior. The potential is tuned to make the interactions supersymmetric. We establish a rigorous mathematical result which relates quantum ground states to certain tiling configurations on the square lattice. For periodic boundary conditions this relation implies that the number of ground states grows exponentially with the linear dimensions of the system. We present substantial analytic and numerical evidence that for open boundary conditions the system has gapless edge modes.     

Vortex lattice structural transitions: a Ginzburg-Landau model approach

We analyze the rhombic to square vortex lattice phase transition in anisotropic superconductors using a variant of Ginzburg-Landau theory. The mean-field phase diagram is determined to second order in the anisotropy parameter, and shows a reorientation transition of the square vortex lattice with respect to the crystal lattice. We then derive the long-wavelength elastic moduli of the lattices, and use them to show that thermal fluctuations produce a reentrant rhombic to square lattice transition line, similar to recent studies which used a nonlocal London model.     

Quantum dimer model on the kagome lattice: solvable dimer-liquid and ising gauge theory

We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices include the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer liquid. These models offer a very natural-and maybe the simplest possible-framework to illustrate general concepts such as fractionalization, topological order, and relation to Z2 gauge theories.     

Statistical description of the system electrons on the liquid helium surface

System of electrons on the liquid helium surface is considered. General methods for obtaining free energy functional for the systems in mean field approximation are developed. These methods applied for treating systems with particles arranged in a lattice. Thus obtained functional of free energy is analyzed. The localization distance for electron and conditions for existing square or triangular lattices as well as phase transition between them are obtained.     

Arbitrary structuring of two-dimensional photonic crystals by use of phase-only Fourier gratings

The computer-generated holography technique is applied to the structuring of two-dimensional (2D) photonic crystals with inherently embedded arbitrary defects. The technique uses phase-only Fourier gratings as a generator of spot arrays in the focal plane, such that a single exposure produces a 2D array of focused spots with desired defects or modifications in the lattice structure. We demonstrate several types of large-area 2D lattice structures with square, hexagonal, or hybrid lattices embedded with point and (or) line defects. Scanning the Fourier plane in the depth direction throughout multiphoton polymerization media allows 3D lattices with stacked defect layers to be formed.     

Exact parent Hamiltonian for the quantum Hall states in a lattice

We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.     

A simple method for fabricating two- and three-dimensional photorefractive photonic lattices microstructures

We demonstrate an approach for easy fabrication of two- and three-dimensional optically induced nonlinear photonic lattices microstructures in photorefractive crystal by applying spatial filter and amplitude mask. The experimental setup of this method is very simple and flexible without complicated optical adjustment system. It can be applied in almost any optical laboratories. Two-dimensional hexagonal, square and three-dimensional hexagonal nonlinear photonic lattices microstructures have been produced in an iron-doped lithium niobate photorefractive crystal. The period of the induced photonic lattices microstructures can be dominated easily. This method is easily extended to generate more complex photonic lattices microstructures in photorefractive crystals, such as quasicrystal lattice.     

Excitonic metal–insulator phase transition of the Mott type in compressed calcium

It has been experimentally found that, under the static compression of a calcium crystal at room temperature, it undergoes a series of structural phase transitions: face-centered cubic lattice → body-centered cubic lattice → simple cubic lattice. It has been decided to investigate precisely the simple cubic lattice (because it is an alternative lattice) with the aim of elucidating the possibility of the existence of other (nonstructural) phase transitions in it by using for this purpose the Hubbard model for electrons with half-filled ns-bands and preliminarily transforming the initial electronic system into an electron–hole system by means of the known Shiba operators (applicable only to alternative lattices). This transformation leads to the fact that, in the new system of fermions, instead of the former repulsion, there is an attraction between electrons and holes. Elementary excitations of this new system are bound boson pairs—excitons. This system of fermions has been quantitatively analyzed by jointly using the equation-of-motion method and the direct algebraic method. The numerical integration of the analytically exact transcendental equations derived from the first principles for alternative (one-, two-, and three-dimensional) lattices has demonstrated that, in systems of two-species (electrons + hole) fermions, temperature-induced metal–insulator phase transitions of the Mott type are actually possible. Moreover, all these crystals are in fact excitonic insulators. This conclusion is in complete agreement with the analytically exact calculations of the ground state of a one-dimensional crystal (with half-filled bands), which were performed by Lieb and Wu with the aim to find out the Mott insulator–metal transition of another type.