Finite Two-dimensional Lattice Research Articles

Eigenvalue amplitudes of the Potts model on a torus

We consider the Q-state Potts model in the random-cluster formulation, defined on finite two-dimensional lattices of size L × N with toroidal boundary conditions. Due to the non-locality of the clusters, the partition function Z ( L , N ) cannot be written simply as a trace of the transfer matrix T L . Using a combinatorial method, we establish the decomposition Z ( L , N ) = ∑ l , D k b ( l , D k ) K l , D k , where the characters K l , D k = ∑ i ( λ i ) N are simple traces. In this decomposition, the amplitudes b ( l , D k ) of the eigenvalues λ i of T L are labelled by the number l = 0 , 1 , … , L of clusters which are non-contractible with respect to the transfer ( N) direction, and a representation D k of the cyclic group C l . We obtain rigorously a general expression for b ( l , D k ) in terms of the characters of C l , and, using number theoretic results, show that it coincides with an expression previously obtained in the continuum limit by Read and Saleur.

Character decomposition of Potts model partition functions, II: Toroidal geometry

We extend our combinatorial approach of decomposing the partition function of the Potts model on finite two-dimensional lattices of size L × N to the case of toroidal boundary conditions. The elementary quantities in this decomposition are characters K l , D labeled by a number of bridges l = 0 , 1 , … , L and an irreducible representation D of the symmetric group S l . We develop an operational method of determining the amplitudes of the eigenvalues as well as some of their degeneracies.

Character decomposition of Potts model partition functions, I: Cyclic geometry

We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L × N , with boundary conditions that are free in the L-direction and periodic in the N-direction. The decomposition of the partition function in terms of the characters K 1 + 2 l (with l = 0 , 1 , … , L ) has previously been studied using various approaches (quantum groups, combinatorics, transfer matrices). We first show that the K 1 + 2 l thus defined actually coincide, and can be written as traces of suitable transfer matrices in the cluster picture. We then proceed to similarly decompose constrained partition functions in which exactly j clusters are non-contractible with respect to the periodic lattice direction, and a partition function with fixed transverse boundary conditions.

Structure and dynamics of a clonal plant population: Classical model results in a non-classic formulation

The structure of a clonal plant population is studied in Calamagrostis canescens, a perennial grass colonizing forest areas which have become open due either to tree-felling or windthrow. A scale of ontogenetic stages is defined where the stages of individual plants ( tussocks) are described in terms of their morphology. No strict correspondence has been revealed between the stage of the tussock and its chronological age. Therefore, the formal classification of individuals considers both the age and the stage in ontogenesis, which results in a life-cycle graph defined on a finite two-dimensional lattice. The origin of individuals (seed reproduction or vegetative propagation) has also appeared to predetermine different pathways within the life cycle. The traditional way to formally relate population dynamics to the population structure leads to a matrix model, while the special pattern of the life-cycle graph, together with that of the corresponding population vector, results in a non-traditional form of the projection matrix. This form differs from the classical Leslie and Lefkovitch ones with certain nonzero entries (vital rates), which complicates the routine mathematical analysis of the model. We demonstrate that the classical model outcomes, like the asymptotic rate of population growth (or decline), the equilibrium population structure and the reproductive values of the population groups can still be obtained from a given set of the age-stage-specific vital rates. The potential-growth indicator is also shown calculable from the vital rates and it appeared useful in a procedure of model calibration on a short series of empirical data. The model can thus serve as a research tool specialised for applied problems which may be associated with the double-structured population dynamics in a clonal plant species.

Structural instability of two- and three-dimensional pyrochlore spin lattices in high magnetic fields

We consider the quantum spin-$s$ $XXZ$ Heisenberg antiferromagnet on the two- and three-dimensional pyrochlore lattices and examine a spin-Peierls mechanism of lowering the total energy by a lattice distortion in a high magnetic field. For the exact eigenstates consisting of several independent localized magnons in a ferromagnetic environment we show the existence of a spin-Peierls instability by rigorous analytical calculations. In addition we report on exact diagonalization data for finite two-dimensional pyrochlore lattices up to N=64 sites. We discuss a scenario of the field-tuned changes in the ground-state properties of the frustrated spin lattices due to the coupling between spin and lattice degrees of freedom.

Metallic photonic crystals with strong broadband absorption at optical frequencies over wide angular range

We show theoretically that a finite two-dimensional square lattice of metallic cylinders in air can be designed to have almost 100% absorptance over a wide optical wavelength range and for a wide range of incidence angles. The broadband and wide-angle strong absorption is attributed to the presence of a large number of flat bands interacting with air bands and the greatly improved impedance matching between metallic photonic crystals and air. The frequency band of intense absorption is in the visible, ultraviolet, or near infrared, depending on the metallic material.

Structure of finite two-dimensional Yukawa lattices: dust crystals.

Dust particles in plasmas are often confined near the boundary between the plasma bulk and the sheath where the gravitation is balanced by electrostatic force. To keep dust particles from running away horizontally, an electrostatic potential is usually applied to the electrode surrounding these dusty plasmas and, under appropriate conditions, we have finite two-dimensional lattices of dust particles. Modeling the interaction between dust particles as the isotropic Yukawa interaction, structures of finite two-dimensional Yukawa systems at low temperatures have been analyzed both by numerical simulations and variational methods. The effect of the correlation energy between dust particles is shown to play an important role in the formation of the one-body distribution in these systems.

A Diffraction Method of Study of Thermal Quasiorder in a Finite Two-Dimensional Harmonic Lattice

Due to the non-existence of long-range order, the diffraction peaks of 2D-solids are considered to have a power-law shape q p η-2 . Taking into account the finite size effects and calculating the powder average, we show that this power-law behaviour appears only for high q p and then for very small intensities. It is therefore quite difficult and hazardous to characterise the quasiorder by using this asymptotic behaviour. Although the shape of the central part of the peak cannot be used to characterise the quasiorder, we show that, for a fairly good resolution, it is possible to determine η using this central part. This determination can be done irrespectively with the other details of the system by comparing the peak width to its value at low temperature, i.e., at low value of η. By using two diffraction peaks, we propose the simple relation : η(Q B1 )/Q 2 B 1 = η(Q B2 )/Q 2 B 2 as a check of the two-dimensional quasiorder.

Conductance fluctuations in periodic antidot arrays.

We present an exact quantum-mechanical calculation of the conductance for a large but finite two-dimensional antidot lattice. Rich structure is predicted for the mesoscopic system and is characterized by two kinds of conductance fluctuations: slow fluctuations as seen in the thermally averaged conductance and rapid fluctuations appearing at low temperature on the top of the slow ones. The former are due to electron wave interferences in a form of Bragg reflections, while the latter can be attributed to resonant tunneling via electron states of the system.

Mean-squared displacement of a hard-core tracer in a periodic lattice.

We show analytically and numerically that logarithmic contributions to the mean-squared displacement of a hard-core tracer particle in a finite two-dimensional periodic lattice can be considered as a transient, and only survive in the infinite system

Infinite-U Hubbard model in the large-spin regime: exact diagonalization study

The infinite-U Hubbard model on finite two-dimensional square lattices is investigated by the use of exact numerical diagonalization. Configurations with one spin flip away from the ferromagnetic state and up to three holes are studied for lattice sizes ranging from 4*4 to 12*12. For all these situations, the state with one spin flip is found to be more stable than the Nagaoka ferromagnetic state. The crystal momentum of the ground state in this spin sector depends on the system size. Systematic comparison with the Gutzwiller wavefunction shows that it reproduces qualitatively the crystal momentum dependence of the energy, but not the exact correlation between the spin flip and the holes.

Exact time-dependent propagation of vacancy motion in the t-J limit of the two-dimensional Hubbard Hamiltonian.

A technique to perform real-time quantum propagation of many-particle systems on finite lattices is presented and applied to the motion of a single hole in the Heisenberg and Ising magnets (t-J and t-${\mathit{J}}_{\mathit{z}}$ models) for finite two-dimensional lattices of sizes 3\ifmmode\times\else\texttimes\fi{}3 and 4\ifmmode\times\else\texttimes\fi{}4. Use of Trotter's formula together with a checkerboard decomposition allows a direct real-time calculation of the many-fermion dynamics to be made for asymptotically long times. Starting from localized initial states, the hole-density distribution in the single-hole system is evaluated as a function of time. We find statistical behavior already appears in the 4\ifmmode\times\else\texttimes\fi{}4 lattice due to the large Hilbert space of the fermion background (\ensuremath{\approxeq}${10}^{5}$ states). For J\ensuremath{\ll}t, the hole appears to move almost incoherently in both models, with small quantum fluctuations. For larger values of the spin coupling J, the t-${\mathit{J}}_{\mathit{z}}$ model shows a strong tendency to localize, while incorporation of the Heisenberg spin-flip term removes this feature and introduces an increasing component of low-frequency coherent motion for larger J.

Lineshape calculations for two-dimensional powder samples

Diffraction studies on two-dimensional adsorbed systems are of great current interest. To obtain large surface areas, which are in particular needed for neutron scattering, one resorts to powder-like substrates. As the expressions derived for the structure factor of a finite two-dimensional (2D) lattice are usually fairly complicated, one runs into difficulties when proper powder averaging is attempted and when final formulae are to be compiled which have to be handy enough to permit fitting the Bragg reflections of the 2D system as a function of all parameters involved. Due to the fact that the 2D structure factor can be expressed with high precision as a linear combination of three elementary functions such as a Gaussian, a Lorentzian and a squared-Lorentzian, we are able to present an algorithm performing the correct powder averaging for both the isotropic case and the one involving preferred orientation. To fit the reflections an easily programmable set of formulae for the three types of cross sections of the Bragg rods is supplied, being valid for unit cells with 1 or 2 atoms. More atoms per cell require an approximation. The impact of a distribution of the size of the 2D crystallites is discussed.

On the computational complexity of Ising spin glass models

In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied. In a finite two-dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice. In contrast to this fact, the same problems are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional case. NP-hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it.

Capillary displacement and percolation in porous media

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

Ground state properties of a spin glass model

A new algorithm has been developed to find ground state properties of finite two-dimensional lattices of Ising spins with +J, −J, or O interaction strengths. The algorithm is based on a list of all possible string assignments on a boundary which divides the sample in two. Averaged ground state properties are presented for random lattices in which +J occurs with probability p, −J with probability q, and O with probability r (p+q+r = 1). Evidence for zero temperature phase transitions between paramagnetic, ferromagnetic, antiferromagnetic, and possible spin glass phases is discussed.

Analytic Form for the Static Structure Factor for a Finite Two-Dimensional Harmonic Lattice

The static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}})$ for a two-dimensional harmonic lattice of finite size $L$ is expressed analytically. Although one consequence of finite size is the absence of very-long-wavelength phonons, we find that the explicit introduction of a phonon cutoff has very little effect. The structure factor shows an universal behavior for all $L$, differing only by scale factors: $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}})$ always has the infinite-size form far from a Bragg point, but is always rounded off close to the Bragg point. Implications for the interpretation of experimental results are discussed.

Two-Dimensional Classical Anisotropic Heisenberg Ferromagnets

Modified Monte Carlo calculations are presented for a finite two-dimensional lattice of classical spins coupled by anisotropic Heisenberg exchange interactions. Most of the arrays considered had $N=100$ spins. The most extensive calculations were done for the zero-field magnetization and energy as a function of temperature for an anisotropy halfway between the isotropic Heisenberg and the Ising limits. The effects of adding a field and of varying the anisotropy were considered. Limited calculations of the constant-temperature magnetic susceptibility and the constant-field specific heat were also attempted. The available high-temperature power series for the isotropic zero-field Heisenberg case for infinite $N$ was calculated for use as a check on the Monte Carlo calculations. Lowest-order spin-wave theory, as a function of anisotropy and field, was developed explicitly for the infinite $N$ low-temperature case. As might be expected, the agreement of spin-wave theory with the Monte Carlo results improves with increasing anisotropy and field. The fairly good agreement at high temperatures that we get between the series and the Monte Carlo calculations shows that a large lattice is not needed to represent an infinite lattice in this temperature region. The fairly good agreement (especially for the mean energy) that we obtain between the Monte Carlo technique and the spin-wave theory indicates that a large lattice is not needed to approximate an infinite lattice at low temperatures as long as the anisotropy and/or field is fairly large. Owing to limited computer time, we were not able to make our method converge near the critical temperature, or below the critical temperature when the anisotropy and field were very small. Finally, we only saw evidence for one critical temperature, and its value appears to be somewhat larger than but of the same order of magnitude as the Stanley-Kaplan transition temperature.

Stability of the finite two-dimensional harmonic lattice

It is shown that, contrary to accepted fact, the two-dimensional finite, harmonic lattice is stable below temperatures of the order of magnitude of 1°K/ln N, where N is the number of atoms in the lattice. We note that for N = 10 16, ln N ≈ 30, making the region of stability easily accessible.