Comparison of diluted antiferromagnetic Ising models on frustrated lattices in a magnetic field

We study diluted antiferromagnetic Ising models on triangular and kagome lattices in a magnetic field, using the replica-exchange Monte Carlo method. We observe seven and five plateaus in the magnetization curve of the diluted antiferromagnetic Ising model on the triangular and kagome lattices, respectively, when a magnetic field is applied. These observations contrast with the two plateaus observed in the pure model. The origin of multiple plateaus is investigated by considering the spin configuration of triangles in the diluted models. We compare these results with those of a diluted antiferromagnetic Ising model on the three-dimensional pyrochlore lattice in a magnetic field pointing in the [111] direction, sometimes referred to as the “kagome-ice” problem. We discuss the similarity and dissimilarity of the magnetization curves of the “kagome-ice” state and the two-dimensional kagome lattice.     

Layered Ising models on triangular and honeycomb lattices

We study inhomogeneous Ising models on triangular and honeycomb lattices. The nearest neighbour couplings can have arbitrary strength and sign such that the coupling distribution is translationally invariant in the direction of one lattice axis, i.e. the models have a layered structure. By using a transfer matrix method we derive closed form expressions for the partition functions and free energies. The critical temperatures are calculated. Phase transitions at a finite critical temperature are universally of Ising type. Models with no phase transition may show different behaviour atT=0, which is explicitly shown for fully frustrated models on square, triangular and honeycomb lattices. Finally, generalizations to layered Ising models on more general lattices are discussed.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.     

Exactly soluble planar Ising models on compressible lattices

Two planar Ising models on compressible lattices are considered. The elastic forces act in the horizontal direction only and between nearest-neighbors, but are otherwise arbitrary. The nearest-neighbor exchange interaction is taken as constant for two spins with the same column index and depending on separation for spins on the same row. In the first model (A) the transition remains continuous, and Fisher's theory of renormalized exponents applies; in the second model (B) the additional constraint that spins on a column move as a unit changes the transition to first order.     

The potts model on bethe lattices

We analyze the properties of theq-state ferromagnetic Potts model for realq. The nature of the phase transition at the critical point is first-order forq2, and second-order forq=2. The random-bond percolation limitq1, and its second-order-like transition, are not related to the previous behaviour since they arise from non-stable phases of the system. It is suggested that this property characterizes the model on high-dimensional lattices, too.Supported by MPI and CNR     

Critical properties of high-spin Ising models on anisotropic lattices

The coordinates of the critical points of spin-S Ising models with coupling constants J and J′ are calculated for 1/2 ≤ S ≤ 13/2. The calculations are performed for several values of S and Δ ≡ J′/J independently by using the phenomenological renormalization-group method or (approximate) self-duality. Numerical results combined with a mean-field analysis show that the critical coupling strength for Δ ～ 1 (weakly anisotropic lattice) is K _c ^(S) (Δ) = K _c ^(S) (1)[1 + a(1 ? Δ)], where a = (d ? 1)/d is independent of S (d is the space dimension). Both free energy and internal energy are determined at the critical points. An extremum of the critical internal energy is found at Δ* ∈ (0, 1). The parameter Δ* can be used as a criterion that separates quasi-isotropic and quasi-one-dimensional regimes (Δ* < Δ ≤ 1 and Δ < Δ*, respectively). The finite-size scaling amplitudes A _s and A _e of the inverse spin-spin and energy-energy correlation lengths are estimated. Calculations show that the amplitudes A _s and A _e are independent of S within the accuracy of the adopted approximations. Moreover, their ratio A _e/A _s is independent of the anisotropy parameter Δ. These results support the Ising universality hypothesis.     

Some AB-percolation problems in the antiferromagnetic potts model

The AB-correlated-site/random-bond percolation problem in a q-state antiferromagnetic Potts model on Bethe lattices is solved. We find the analytic expression of the AB-percolation characteristic functions in terms of the temperature, the external field and the active bond concentration p_B. The AB-threshold and the phase boundary of the system coincide at zero temperature and at most in two other points for every constant p_B > 1?σ. The properties of the Bethe lattice allow us to find the temperature dependent p_B which defines the AB-droplets, i.e. those special AB-clusters which diverge with thermal exponents along the phase boundary.     

Ising models on the 2 × 2 × ∞ lattices

Exact analytic solutions are presented for two 2 × 2 × ∞ Ising étagères. The first model has a simple cubic lattice with fully anisotropic interactions. The second model consists of two different types of linear chains and includes noncrossing diagonal bonds on the side faces of the 2 × 2 × ∞ parallelepiped. In both cases, the solutions are expressed through square radicals and obtained by using the obvious symmetry of the Hamiltonians, Z ₂ × C _2v , and the hidden algebraic λλ symmetry of the transfer matrix secular equations. The solution found for the second model is used to analyze the behavior of specific heat in a frustrated many-chain system. The text was submitted by author in English.     

On the critical temperature of non-periodic Ising models on hexagonal lattices

The critical temperature of layered Ising models on triangular and honeycomb lattices are calculated in simple, explicit form for arbitrary distribution of the couplings.     

On the critical temperature of non-periodic Ising models on hexagonal lattices

The critical temperature of layered Ising models on triangular and honeycomb lattices are calculated in simple, explicit form for arbitrary distribution of the couplings.     

Collective modes in Ising lattices

The effect of collective modes on the otherwise local structure of Ising lattices is investigated by studying a number of exactly solvable models. First, the open one-dimensional Ising model serves to define sharp locality. This feature then remains upon extension to a Bethe lattice, despite the existence of a phase transition. But insertion of periodic boundary conditions creates a collective mode which breaks locality in a very specific fashion. A model interface is analyzed to show that even when locality is not broken, local uniformity can become untenable.     

Ising model on mixed two-dimensional lattices

We inform results on physical and topological magnitudes related to the ground level of Ising model on mixed two-dimensional lattices of coordination numbers 4 (Kagomé lattices) and 5 (five-point star lattices). We consider little clusters of size N, where N represents the total number of spins, subject to periodic boundary conditions. On these systems we randomly distribute ±J nearest-neighbor interactions (+J: antiferromagnetic, −J: ferromagnetic (F)). Concentration x of F interactions is varied in the interval (0,1). Two different methods are used to obtain results reported here. First, a numerical method related to multiple replicas. Second, an analytical method based on probabilistic analysis of flat and curved plaquettes. Both methods are complementary to each other. Initially, this study is restricted to calculate frustration of plaquettes and bonds, energy and bond order parameter at T=0. The results of magnitudes informed here are compared with the similar ones obtained for honeycomb, square and triangular lattices.     

Ground-state properties of the antiferromagnetic potts model in an external field

The ground-state properties of the antiferromagnetic q-state Potts model in an external field are studied. At the upper critical field this model may be mapped onto a hard-core lattice gas with activity z = q ?1. This allows us to get some exact results for the triangular lattice on which the corresponding hard hexagon problem has been recently solved by Baxter.     

Interfacial adsorption in planar potts models

Zeitschrift für Physik B Condensed Matter - The net adsorption,W, of non-boundary states at interfaces in two-dimensional Potts models is studied using Monte Carlo techniques. For the...     

The stochastic models method applied to the critical behavior of Ising lattices

The stochastic models (SM) computer simulation method for treating manybody systems in thermodynamic equilibrium is investigated. The SM method, unlike the commonly used Metropolis Monte Carlo method, is not of a relaxation type. Thus an equilibrium configuration is constructed at once by adding particles to an initiallyempty volume with the help of a model stochastic process. The probability of the equilibrium configurations is known and this permits one to estimate the entropy directly. In the present work we greatly improve the accuracy of the SM method for the two and three-dimensional Ising lattices and extend its scope to calculate fluctuations, and hence specific heat and magnetic susceptibility, in addition to average thermodynamic quantities like energy, entropy, and magnetization. The method is found to be advantageous near the critical temperature. Of special interest are the results at the critical temperature itself, where the Metropolis method seems to be impractical. At this temperature, the average thermodynamic quantities agree well with theoretical values, for both the two and three-dimensional lattices. For the two-dimensional lattice the specific heat exhibits the expected logarithmic dependence on lattice size; the dependence of the susceptibility on lattice size is also satisfactory, leading to a ratio of critical exponents/=1.85 ±0.08. For the three-dimensional lattice the dependence of the specific heat, long-range order, and susceptibility on lattice size leads to similarly satisfactory exponents:=0.12 ±0.03,=0.30 ±0.03, and=1.32 ±0.05 (assuming =2/3).     

Clusters and Ising droplets in the antiferromagnetic lattice gas

A definition of clusters of particles and holes with antiferromagnetic order is given for a lattice gas with coupling constant K < 0. In two dimensions it is shown that the Ising antiferromagnetic critical line is also a percolation line if P_b = 1 - exp(-|K|/2). Along this line these clusters called “droplets” diverge with Ising exponents.     

Real-space renormalization group investigation of pure and disordered mixed spin Ising models ond-dimensional lattices

The mixed spin Ising model (spins =1/2 andS=1) ond-dimensional hypercubic lattices with nearest-neighbour exchange interactions is studied via a renormalization group transformation in position space. The phase diagrams in (L, K) space, i.e. in dependence of the bilinear (K) and the biquadratic (L) interaction coefficients, are qualitatively different ford=2 andd>2. For any dimensiond however it is found that all transitions are of second order. At zero-temperature (K=,L=), the ferromagnetic order disappears at (L/K)₀=2, which does not depend ond. Using an extension of this real-space renormalization group analysis we study the two-dimensional random disordered version of the above model.L is kept homogeneous and the bilinear interactionsK _ij are assumed to be independent random variables with distributionP(K _ij )=p(K _ij –K)+(1–p)(K _ij –K); whereK>0. The phase diagrams for different values ofp are obtained. At zero temperature, it is found that in the bond diluted model (=0) the value (L/K)₀ depends continuously onp, whereas in the random ±K interactions (=–1) (L/K)₀ is unique and does not depend onp.Supported by the agreement of cooperation between the DFGW. Germany and the CNR-Maroc