Narrow Results

Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature $T_c$ of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature $T_c$ is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition $D\ge -2$. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.

We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles through finite-size subsets of vertices in a deterministic order and updates the colours assigned to the vertices of each subset. The systematic scan Markov chain that we present cycles through subsets consisting of 2×2 sub-grids and updates the colours assigned to the vertices using a procedure known as heat-bath. We give a computer-assisted proof that this systematic scan Markov chain mixes in O(log n) scans, where n is the size of the rectangular sub-grid. We make use of a heuristic to compute required couplings of colourings of 2×2 sub-grids. This is the first time the mixing time of a systematic scan Markov chain on the grid has been shown to mix for less than 8 colours. We also give partial results that underline the challenges of proving rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the grid by considering 2×3 and 3×3 sub-grids.

The Sznajd model, where two people having the same opinion can convince their neighbors on the square lattice, is modified in the sense of Deffuant et al. and Hegselmann, that only neighbors of similar opinions can be convinced. Then consensus is easy for the competition of up to three opinions but difficult for four and more opinions.

We propose a three-opinion Sznajd model with limited persuasion in which two neighbors still can influence others when the two opinions differ by a small amount. The simulation results show that the initial condition plays a crucial role in the final results and the interaction between people's opinion occurs in the common boundaries of the small districts constituted by the people holding the same opinion. We apply the model to describe the interaction on the attitude toward studying between students in a class or school which is similar with the opinion evolution and give a rough explanation about why a class or school mostly consists of ordinary students.

Tight-binding Hamiltonians with off-diagonal disorder are studied extensively in connection with localization phenomena. For instance, applying a random magnetic field to a square lattice amounts to assigning phases to the off-diagonal entries of H. A gauge transformation simplifies the resulting H to an equivalent Hamiltonian with apparently less disorder. This paper concerns the special case when non-zero entries of a Hamiltonian H are ± 1 between nearest-neighbor sites. Some thought shows that a one-dimensional chain Hamiltonian with this type of randomness can be transformed to that of an ordered chain by flipping the signs of selected basis functions, i.e. by a unitary transformation. Hence such a chain is not disordered at all. On the other hand, whether or not a similar H for a square lattice implies actual disorder is a topological question. The question can be put this way: Can one find a set of simple closed curves such that reversing the signs of basis functions inside the curves will change all non-zero H entries to +1? More generally one can ask about the extent of residual disorder and its effect on electronic properties of the model. We take up these questions on a periodic square lattice.

In this study, we investigate magnetic entropy variation and magnetocaloric properties as a function of frustration by considering a spin-1/2 antiferromagnetic Heisenberg model on a square lattice with nearest (J₁) and next-nearest neighbor exchange interaction (J₂). We show that the degree of frustration in the square lattice increases with α=J₂/J₁. While the square lattice is unfrustrated for α = 0, it becomes fully frustrated for α=0.5. Numerical results show that the entropy of the square lattice approaches zero for the unfrustrated case at T=1 K. In contrast, finite entropy can survive in the frustrated case at the same temperature. We also calculate the magnetic entropy change (ΔS_m) in the square lattice and show that the maximum value of ΔS_m increases with increasing α. These results indicate that the magnetic entropy change and consequently the magnetocaloric effect can be enhanced by increasing the degree of frustration. We conclude that the enhanced magnetocaloric effect is related to quantum fluctuations and disordered ground state present in the frustrated square lattice.

We investigate the optical absorption and the density of states of a Frenkel exciton system on a square lattice with nearest-neighbor interactions and a Gaussian distribution of transition frequencies (i.e. Gaussian diagonal disorder). Results are presented for the absorption and the density of states of direct and indirect edge systems for a range of variances. There is reasonable agreement with the corresponding finite array calculations of Schreiber and Toyozawa. The existence of an Urbach tail is also investigated.

In many diverse pattern-forming systems, the observed arrangements are square-symmetric or hexagonal. However, while many investigations have been devoted separately to the competition between these patterns and rolls, important questions still remain regarding their competition with one another.

This paper demonstrates that both square and hexagonal patterns can be accounted for by the resonant interaction of two sets of modes with wave numbers in the ratio , aligned to lie on a square lattice. The behavior of this system is investigated by expanding about the multiple bifurcation point where the stability of both sets of modes is marginal. The corresponding amplitude equations, constructed through simple symmetry arguments, are shown to encompass various forms of hexagonal and square-symmetric patterns within a single framework. This framework is then applied to the Swift–Hohenberg partial differential equation, where both hexagons and squares with up–down-asymmetry are found to occur subcritically.

We elucidate the mechanism of the self-organization of square agglomeration patterns that are described by spatial economic models on a square lattice with periodic boundary conditions. Focusing on the symmetry of the square lattice, we conduct a group-theoretic analysis and obtain bifurcating patterns from the uniform distribution. Furthermore, for the replicator dynamics, which are widely used in economics, we pay attention to the existence of invariant patterns that are solutions to the governing equation for any value of the bifurcation parameter (the trade freeness for spatial economic models). We advance invariant patterns on the square lattice as candidates of stable equilibria. Using a prototype spatial economic model proposed by Forslid and Ottaviano [2003], we numerically show a tendency that bifurcating solutions arrive at invariant patterns after bifurcation. This tendency is advanced as the underlying mechanism of the progress of economic agglomerations that is to be considered in the study of spatial economic agglomerations.

This paper develops a spatial platform for economic agglomerations that can represent a hierarchical structure of cities, towns, and so on. A global system models geographical distribution of a system of cities, while population size and local geography of each city are modeled by an individual square lattice network. The mechanism of economic agglomeration is described by economic geography models with the replicator dynamics. As a major theoretical contribution of this paper, we elucidate the bifurcation mechanism of the mono-centric distribution in a square domain. Such bifurcation expresses how satellite places appear around a large city. This bifurcation behavior, called sustaining bifurcation, is different from symmetry-breaking bifurcation studied in nonlinear mathematics, and is not given much attention in economic geography, despite its importance. An air-network of cities with the seven largest hub airports in USA is employed as a realistic example, and is modeled as a global–local system comprising a series of local square lattices with different sizes connected by an equidistant economy. This system for an economic geography model displays successive sustaining bifurcations occurring many times leading to gradual emergence of satellite places around a large city.

Using the two-dimensional Jordan–Wigner Fermionization procedure, we calculate the energy spectrum of the in-phase flux antiferromagnetically spin-1/2 coupled Heisenberg model in a square lattice, the formalism used introduces the notion of isotropic parameters. The energy spectrum is analyzed for various regimes of the exchange interactions and the isotropic parameters. The thermodynamic parameters of the lattice, notably the ground state energy, the free energy, mean energy, entropy and specific heat are calculated. It is seen that the specific heat undergoes a phase transition at a temperature below the critical temperature due to spontaneous magnetization. Its entropy for homogeneous and completely isotropic regime is compared for two regimes of the exchange interaction and it is observed that the entropy decreases with an increase in the coupling strength. All the thermodynamic parameters calculated for this spin model are seen to be in conformity with the principles and laws of Statistical Thermodynamics.

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or $k$-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a $4\times 4$ secular determinant.

In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.