Lattice Solution Research Articles

Symmetry classes of triplet vortex lattice solutions of the Bogoliubov de-Gennes equation in a square lattice

We give a group theoretical classification of the triplet vortex lattice states of the two-dimensional Hubbard model with a nearest neighbor ferromagnetic exchange interaction in a uniform magnetic field. We obtain 11 types of tetragonal vortex lattice states for the magnetic flux φ=φ0/p2 (φ0=ch/2e is the flux quantum, p is an integer) through a unit cell of crystal lattice. We show the configurations of the order parameters corresponding to axial phase, up spin phase, planar phase and bipolar phase. It is clarified what types of vortex lattice phase are possible in triplet superconductors such as Sr2RuO4 with basal square lattice when the symmetry of magnetic translation is considered.

Ionization of helium by antiprotons: Fully correlated, four-dimensional lattice approach

The single ionization of helium by antiproton impact has been studied utilizing a fully correlated, planar model in which the time-dependent Schr\"odinger equation was solved on a four-dimensional (4D) Cartesian lattice. The choice of a ``softcore'' Coulomb potential was shown to enable approximate reproduction of the full six-dimensional $\overline{p}\mathrm{He}$ adiabatic electronic eigenenergy curves in the dynamically important region of the 4D space, thus providing a reasonable model to treat $\overline{p}+\mathrm{He}$ collisions. In addition, the applicability of this approach was demonstrated by favorable comparison of analogous two-dimensional lattice solutions for ionization of atomic hydrogen by antiprotons with previously computed three-dimensional lattice results. The present work aids in the interpretation of existing theoretical and experimental studies considering the low-energy behavior of the single-ionization cross section for antiproton impact of few-electron targets and explores the applicability of the planar model in describing correlated two-electron systems.

Rational solutions to the Pfaff lattice and Jack polynomials

The finite Pfaff lattice is given by a commuting Lax pair involving a finite matrix L (zero above the first subdiagonal) and a projection onto sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t_1,t_2,\dotsc, after conjugation by a diagonal matrix. The sequence of polynomial \tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of \tau-functions is given inductively by the action of a fixed vertex operator.As an example, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

Dynamics of twists on antiferromagnetic spin chains: Theory

The equation of motion of twists on classical antiferromagnetic Heisenberg spin chains are derived. It is shown that twists interact via position- and velocity-dependent long-range two-body forces. A quiescent regime is identified wherein the system conserves momentum. With increasing kinetic energy the system exits this regime and momentum conservation is violated due to walls annihilation. A bitwist system is shown to be integrable and its exact solution is analysed. Many-twist systems are discussed and novel periodic twist lattice solutions are found on closed chains. The stability of these solutions is discussed.

Nonlinear Maxwell Theory and Electrons¶in Two Dimensions

We consider a system of nonlinear equations that extends the Maxwell theory. It was pointed out in a previous paper that symmetric solutions of these equations display properties characteristic of magnetic oscillations. In this paper I study a discrete model of the equations in two dimensions. This leads to the discovery of a new mechanism of vortex lattice formation. Namely, when a parameter corresponding to a magnetic field normal to the surface increases above a certain critical level, the trivial uniform-magnetic-field solution becomes, in a certain sense, unstable and a periodic vortex lattice solution emerges. The discrete vortex solutions are proven to exist, and can also be found numerically with high accuracy. Description of magnetic vortices given by the equations is optical in spirit, and may be particularly attractive in the context of high-T c superconductivity and the quantum Hall effects. Moreover, analysis of parameters involved in the discrete theory suggests existence of continuous domain solutions – a conjecture that seems unobvious on grounds of the current topological and variational methods.

A comparison of self-assembly in lattice and off-lattice model amphiphile solutions

Lattice Monte Carlo and off-lattice molecular dynamics simulations of h1t4 and h4t1 (head/tail) amphiphile solutions have been performed as a function of surfactant concentration and temperature. The lattice and off-lattice systems exhibit quite different self-assembly behavior at equivalent thermodynamic conditions. We found that in the weakly aggregating regime (no preferred-size micelles), all models yield similar micelle size distributions at the same average aggregation number, albeit at different thermodynamic conditions (temperatures). In the strongly aggregating regime, this mapping between models (through temperature adjustment) fails, and the models exhibit qualitatively different micellization behavior.

Steady-state mode I cracks in a viscoelastic triangular lattice

We construct exact solutions for Mode I steady-state cracks in an ideally brittle viscoelastic triangular lattice model. Our analytic solutions for the infinite lattice are compared to numerical results for finite width systems. The issues we address include the crack velocity versus driving curve as well as the onset of additional bond breaking, signaling the emergence of complex spatio-temporal behavior. Somewhat surprisingly, the critical velocity for this transition becomes a decreasing function of the dissipation for sufficiently large values thereof. Lastly, we briefly discuss the possible relevance of our findings for experiments on mode I crack instabilities.

Explicit solutions of integrable lattices

Explicit examples of quadrilateral lattices and their integrable reductions of pseudo-circular, symmetric and pseudo-Egorov types are presented.

Phase diagram of dilute lattice surfactant solution

We have studied the temperature–composition phase diagram for a dilute surfactant model solution on the square lattice through Monte Carlo simulations. A micellar region and a solubility line were identified from the measurement of different properties, among which the volume fraction of aggregate, specific heat, and convergence of relaxation from different initial conditions. We have compared properties of the above system with those of a symmetric lattice solution in the dilute regime. Comparison of data allow an interpretation of the micellar phase as a transition region between a microscopically homogeneous phase and a solid phase.

Interaction between sine-Gordon breathers.

Using the exact breather lattice solution of the sine-Gordon equation, we obtain the asymptotic interaction between two breathers. We identify the exponential dependence of the interaction on the breather separation as well as its power-law dependence on the breather frequency. Numerical simulation of the breather lattice demonstrates its instability. However, stabilization of such structures is found to be feasible through ac driving and damping. Finally, other limits of the original periodic solution are traced, obtaining the leading-order terms in the interaction between "pseudosphere" solutions and kink-antikink pairs.

Atomic slide puzzle: self-diffusion of an impure atom.

In a series of recent papers [Phys. Rev. Lett. 86, 1562 (2001); Nature (London) 408, 665 (2000)] van Gastel and co-workers have presented what may be the first experimental evidence, based on a series of scanning tunnel microscope images, that impure, Indium atoms, embedded into the first, close-packed layer of a Cu(001) surface, are not localized but make concerted, long excursions. Such excursions occur due to continuous reshuffling of the surface following the position exchanges of both impure and host Cu atoms with the naturally occurring surface vacancies. van Gastel and co-workers have also formulated an original lattice-gas type model with asymmetric exchange probabilities, whose numerical solution is in a good agreement with the experimental data. In this paper we propose an exact lattice solution of several versions of this model.

Parametric simultons in one-dimensional nonlinear lattices

Parametric simultaneous solitary wave (simulton) excitations are shown to be possible in nonlinear lattices. Taking a one-dimensional diatomic lattice with a cubic potential as an example, we consider the nonlinear coupling between the upper cut-off mode of acoustic branch (as a fundamental wave) and the upper cut-off mode of optical branch (as a second harmonic wave). Based on a quasi-discreteness approach the Karamzin-Sukhorukov equations for two slowly varying amplitudes of the fundamental and the second harmonic waves in the lattice are derived when the condition of second harmonic generation is satisfied. The lattice simulton solutions are given explicitly and the results show that these lattice simultons can be nonpropagating when the wave vectors of the fundamental wave and the second harmonic waves are exactly at π/a (where a is the lattice constant) and zero, respectively.

Soliton lattice structure of a spin density wave in the one-dimensional t- J model

We determine the soliton lattice solution of the self-consistent equation for antiferromagnetic order in the one-dimensional t- J model. The region of parameters and limits of applicability of this solution are examined. We also discuss the effect of the charge density modulation accompanying the spin density wave antiferromagnetic order.

Nonlinear localization in thermalized lattices: application to DNA

In the last few years numerous studies have been devoted to intrinsic localized modes in nonlinear lattices because they provide examples of localization without disorder. The properties of “discrete breathers” as exact solutions of these nonlinear lattices are now well understood, but this is not the case for the properties of nonlinear localization and energy relaxation in thermalized lattices. In biological molecules such as DNA, where large amplitude nonlinear motions are essential for function, temporary deviations from energy equipartition could play an important role. After a brief introduction to intrinsic localized modes, we address the following questions: (a) does nonlinear localization survive in the presence of a thermal bath and how can we characterize it? (b) what is the origin of localization and how do discrete breathers contribute to it? (c) can we observe nonlinear localization in an experiment? The last point discusses recent results of an experiment performed on DNA, which suggest that the effects of nonlinear localization in a thermalized system may have already been observed.

Meron pairs and fermion zero modes

Merons, conjectured as a semiclassical mechanism for color confinement in QCD, have been described analytically by either infinite action configurations or an Ansatz with discontinuous action. We construct a smooth, finite action, stationary lattice solution corresponding to a meron pair. We also derive an analytical solution for the zero mode of the meron pair Ansatz, show that it has the qualitative behavior of the exact zero mode of the lattice solution, and propose the use of zero modes to identify meron gauge field configurations in stochastic evaluations of the lattice QCD path integral.

A method for locating symmetric homoclinic orbits using symbolic dynamics

In this paper we present a method which can identify and locate symmetric homoclinic orbits in a homoclinic tangle formed by the intersecting stable and unstable manifolds of a symmetric 2D map. The method consists of a systematic search in parameter space and determination of the order in which these orbits arise using symbolic dynamics. Each orbit corresponds to a unique sequence and it is computed by iterating the map along the unstable manifold to match a specific symmetry at the middle of the orbit. An application of the method to the determination of multibreather solutions of 1D lattices is discussed.

Exciton polaritons in quantum-dot photonic crystals

We present a theory of the photonic band structure of three-dimensional arrays of quantum dots (QDs). A system of Maxwell’s and material equations is solved and the dispersion equation for exciton-polaritons is derived making allowance for a nonlocal dielectric response of quasi-zero-dimensional excitons confined in QDs. The reflection and transmission coefficients are calculated for a single plane, a pair of planes and a stack of equidistant planes of QDs. Two different approaches are proposed to perform a calculation. One of them is based on recurrent equations relating the reflection coefficients for N + 1 and N planes, while in the other approach the Bloch solutions for an infinite QD lattice are used.

TWO-BRANCH ENERGY OF MAGNETIC SURFACES AND CONNECTION WITH ANHOLONOMY

For general boundary conditions, we show that the energy H of a classical Heisenberg ferromagnetic spin system on a curved surface satisfies the inequality H≥|Γ|. Here Γ is a certain geometric phase or anholonomy associated with the spin vector field. This is a generalization of the well known Bogomol'nyi inequality H≥4πQ (Q=integer). For a wide variety of curved surfaces, seeking solutions with certain symmetries, we find that the variational equation δH=0 can be reduced to a sine-Gordon equation with a geometry-dependent parameter. Soliton lattice solutions of this equation are analyzed. It is shown that both H and Γ develop two branches each, all of which merge at the one-soliton limit. This branching is interpreted as a topological transition of the spin textures that occurs at the one-soliton point.

Symmetries of vortex lattice solutions of the Bogoliubov–de Gennes equation in a square lattice

We give a group theoretical classification of the vortex lattice states of the attractive Hubbard model in a uniform magnetic field. We obtain four types of tetragonal vortex lattice states ( G 0, G 2, G 1, G −1) for the magnetic flux φ=φ 0/p 2 (φ 0=ch/2e is the flux quantum, p is an integer) through a unit cell of crystal lattice. We performed numerical studies for these vortex states in the presence of only on-site attraction U. We found that the Abrikosov lattice ( G 1) is the most stable. The quasi-particle spectrum for G 1 state is gapless. The local density of states of this state is obtained.

On global existence of localized solutions of some nonlinear lattices

We prove global existence and uniqueness of solutions of some important nonlinea lattices which include the Fermi-Pasta-Ularn (FPU) lattice. Our result shows (on a particular example) that the FPU lattice with high nonlinearity and its continuum limit display drastically different behaviour with respect to blow up phenomenon