Random and cooperative sequential adsorption on infinite ladders and strips

We analyze various processes where particles are added irreversibly and sequentially at the sites of infinite ladders or broader strips (i.e., on terraces) of adsorption sites. For sufficiently narrow strips or ladders, exact solution in closed form is possible for a variety of processes. Often this is most naturally achieved by mapping the process onto an equivalent one-dimensional process typically involvingcompetitive adsorption. We demonstrate this procedure for sequential adsorption with nearest-neighbor exclusion on a 2× square ladder. For other select processes on strips slightly too broad for exact solution, almost exact analysis is possible exploiting an empty-site shielding property. In this way, we determine a jamming coverage of 0.91556671 for random sequential adsorption of dimers on a 2× square ladder. For broader strips, we note that the complexity of these problems quickly approaches that for × lattices.     

Exactly Solvable su(N) Mixed Spin Ladders

It is shown that solvable mixed spin ladder models can be constructed from su(N) permutators. Heisenberg rung interactions appear as chemical potential terms in the Bethe Ansatz solution. Explicit examples given are a mixed spin- spin-1 ladder, a mixed spin- spin- ladder and a spin-1 ladder with biquadratic interactions.     

Critical Behavior of the Blume-Emery-Griffiths Model for a Simple Cubic Lattice on the Cellular Automaton

The spin-1 Ising model with the nearest-neighbour bilinear and biquadratic interactions and single-ion anisotropy is simulated on a cellular automaton which improved from the Creutz cellular automaton (CCA) for a simple cubic lattice. The simulations have been made for several k=K/J and d=D/J in the 0≤d<3 and −2≤k≤0 parameter regions. We confirm the existence of the re-entrant and the successive re-entrant phase transitions near the phase boundary. The phase diagrams characterizing phase transitions are presented for comparison with those obtained from other calculations. The static critical exponents are estimated within the framework of the finite-size scaling theory at d=0, 1 and 2 in the interval −2≤k≤0. The results are compatible with the universal Ising critical behavior.     

Log‐concavity of genus distributions for circular ladders

A well‐known conjecture in topological graph theory says that the genus distribution of every graph is log‐concave. In this paper, the genus distribution of the circular ladder is re‐derived, using overlap matrices and Chebyshev polynomials, which facilitates proof that this genus distribution is log‐concave.     

Electronic spectra of one-dimensional nano-quasi-periodic systems under bias

We investigate the properties of the energy spectra of quantum one-dimensional nano-quasi-crystals in the presence of external electric fields. These systems are modelled by means of finite sequences, ordered according to a Fibonacci rule which constituted of two blocks A$A$ (constant potentials of different heights defined on finite intervals) and B$B$ (delta potentials of different intensities). We use the electric field ability of producing Stark ladders in periodic systems to obtain well separated energy levels and to study the evolution of these levels when disorder is introduced. We show that this effect also allows us to predict the approximate position of the levels in the disordered system, in spite of its chaotic appearance at first view. We show, against the usual belief, that the n$n$th Stark ladder in general is not formed exclusively from the levels of the n$n$th band. The disorder is introduced in two different ways: by changing the distribution of the blocks or by changing the values of the delta potential intensities. In both cases we start from electrified periodic structures which are gradually perturbed to obtain electrified quasi-periodic structures. We show that the use of Fibonacci sequences as a particular case is not crucial and one can use the electric field to analyze any other type of quasi-periodic systems.     

Synthesis, crystal structure and magnetic properties of the spin ladder compounds La2(Cu1−xZnx)2O5 and La8(Cu1−xZnx)7O19

The influence of Zn-doping on the crystal structure and magnetic properties of the spin ladder compounds La₂Cu₂O₅ (4-leg) and La₈Cu₇O₁₉ (5-leg) have been investigated. The La₂(Cu_1−xZn_x)₂O₅ and La₈(Cu_1−xZn_x)₇O₁₉ solid solutions were obtained as single phases with x=0-0.1 via the solid-state reaction method in the temperature range between 1005-1010 °C and 1015-1030 °C in oxygen and air atmospheres, respectively. The lattice parameters a and c of the monoclinic crystal structures as well as the unit cell volume V increase with increasing x, while b and β decrease for both series. The magnetic susceptibilities χ of both series show a very similar behavior on temperature as well as on Zn-doping, which is supposed to be due to the similar Cu-O coordination in both La₂Cu₂O₅ and La₈Cu₇O₁₉. For low Zn-doping (x?0.04), a spin-chain like behavior is found. This quasi-one-dimensional behavior is strongly suppressed in both series for x?0.04. Here, the maximum (characteristic for spin chains) in χ(T) disappears and χ(T) decreases monotonically with increasing temperature.     

The ferrimagnetic phase of the Blume-Emery-Griffiths model on the cellular automaton

The Blume-Emery-Griffiths model is simulated using the cooling algorithm which is improved from the Creutz cellular automaton (CCA) under periodic boundary conditions. The simulations are carried out on a simple cubic lattice at K/J = −1.5 in the range of −3.5 < D/J < 0.5, with J and K representing the nearestneighbour bilinear and biquadratic interactions, D being the single-ion anisotropy parameter. The phase diagram characterizing phase transition of the model is obtained. We found different kinds of phase transitions between the ferromagnetic, quadrupolar, staggered quadrupolar and ferrimagnetic phases for K/J = −1.5. In particular, the region of the phase diagram containing a ferrimagnetic phase is explored and compared to those obtained by other methods. The simulations confirm that the ferrimagnetic phase occurs in the narrow interval −3.006 ≤ D/J < −3. This result is in a good agreement with Monte Carlo renormalization group and closer to the cluster variation method result than the mean field approximation result.      

Total embedding distributions of Ringel ladders

The total embedding distributions of a graph consists of the orientable embeddings and non-orientable embeddings and are known for only a few classes of graphs. The orientable genus distribution of Ringel ladders is determined in [E.H. Tesar, Genus distribution of Ringel ladders, Discrete Mathematics 216 (2000) 235–252] by E.H. Tesar. In this paper, using the overlap matrix, we obtain nonhomogeneous recurrence relation for rank distribution polynomial, which can be solved by the Chebyshev polynomials of the second kind. The explicit formula for the number of non-orientable embeddings of Ringel ladders is obtained. Also, the orientable genus distribution of Ringel ladders is re-derived.     

Transversals in Trees

The total embedding polynomial of a graph G is the bivariate polynomial where is the number of embeddings, for into the orientable surface , and is the number of embeddings, for into the nonorientable surface . The sequence is called the total embedding distribution of the graph G; it is known for relatively few classes of graphs, compared to the genus distribution . The circular ladder graph is the Cartesian product of the complete graph on two vertices and the cycle graph on n vertices. In this article, we derive a closed formula for the total embedding distribution of circular ladders.