Uncovering the secrets of the 2D random-bond Blume-Capel model

The effects of bond randomness on the ground-state structure, phase diagram and critical behavior of the square lattice ferromagnetic Blume-Capel (BC) model are discussed. The calculation of ground states at strong disorder and large values of the crystal field is carried out by mapping the system onto a network and we search for a minimum cut by a maximum flow method. At finite temperatures the system is studied by an efficient two-stage Wang-Landau (WL) method for several values of the crystal field, including both the first- and second-order phase transition regimes of the pure model. We attempt to explain the enhancement of ferromagnetic order and we discuss the critical behavior of the random-bond model. Our results provide evidence for a strong violation of universality along the second-order phase transition line of the random-bond version.     

Universality classes of Ising models

We discuss a transformation of Ising spins which maps a d-dimentional Ising problem into a series of different problems in the same universality class.     

Initial perpendicular isothermal susceptibility formulas for the transverse general-spin Ising and Blume-Capel models

Using a Kubo formula and the Suzuki identities, expressions are derived for the initial perpendicular susceptibilities χ_⊥ of the transverse spin-S Ising and spin-S Blume-Capel models on regular and irregular lattices. χ_⊥ is given in terms of the thermal average of a function of the peripheral sumO _i= ε_j J _i,j S _j, where coupling to distant neighbors may be included, as well as arbitrary local parallel magnetic fieldsh _j. For the Ising model on a Bravais lattice, e.g., the susceptibility is given by $$\chi _ \bot = Nm^2 S^{ - 2} \langle B_s (\beta [O_i + h_i ])/[O_i + h_i ]\rangle $$ whereB _s is the Brillouin function. ForS=1/2, the formula of Fisher and the results of Horiguchi and Morita are regained. A connection is made with the general-spin work of Essam and Garelick.     

On the correlation equalities for the random-bond Ising model

We obtain several equalities of the configurationally averaged spin correlation functions for the random-bond Ising model by means of a gauge transformation. These equalities are shown to be useful to find the exact results for the internal energy, an upper bound of the specific heat, the equality for the zero-field susceptibility and the zero-field spin glass susceptibility, and so on.     

Universality and a 3d Ising model whose critical exponent γ varies continuously with a parameter

A high temperature series expansion for the susceptibility of a double 3-dimensional Ising model with added four spin interactions indicates a continuous dependence of λ on a parameter.     

Magnetic properties of the mixed spin-5/2 and spin-3/2 Blume-Capel Ising system on the two-fold Cayley tree

We use exact recursion relations to study the magnetic properties of the half-integer mixed spin-5/2 and spin-3/2 Blume-Capel Ising ferromagnetic system on the two-fold Cayley tree that consists of two sublattices A and B. Two positive crystal-field interactions Δ₁ and Δ₂ are considered for the sublattice with spin-5/2 and spin-3/2 respectively. For different coordination numbers q of the Cayley tree sites, the phase diagrams of the model are presented with a special emphasis on the case q = 3, since other values of q reproduce similar results. First, the T = 0 phase diagram is illustrated in the (D _A = Δ₁/J,D _B = Δ₂/J) plane of reduced crystal-field interactions. This diagram shows triple points and coexistence lines between thermodynamically stable phases. Secondly, the thermal variation of the magnetization belonging to each sublattice for some coordination numbers q are investigated as well as the Helmoltz free energy of the system. First-order and second-order phase transitions are found. The second-order phase transitions become sharper and sharper when D _A or D _B increases. The first-order transitions only exist for some appropriate non-zero values of D _A and/or D _B . The corresponding transition lines never connect to the second-order transition lines. Thus, the non-existence of tricritical points remains one of the key features of the present model. The magnetic exponent β ₀ of the model is estimated and found to be ¼ at small values of D _A = D _B = D and β ₀ = ½ at large values of D. At intermediate values of D, there is a crossover region where the magnetic exponent displays interesting behaviours.     

Efficient algorithm for random-bond ising models in 2D

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.     

Universality class of a Fibonacci Ising model

The specific heat of a certain ferromagnetic Fibonacci Ising model is shown to have a logarithmic singularity.     

Universality in the partially anisotropic three-dimensional Ising lattice

Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths \(\mathop J\limits^ \to = (J',J',J)\). The universality of both fundamental critical exponents y _t and y _h is confirmed. It is shown that the critical finite-size scaling amplitude ratios \(U = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chi\), and \(Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }\) are independent of the lattice anisotropy parameter Δ=J′/J. For the Y² invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y²≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).     

Wetting in Potts and Blume-Capel models

We discuss the wetting of the interface between two ordered phases by the disordered one in the Potts model withq large. We argue that a low-temperature expansion can be used in this situation, with logq replacing. This model is analogous to the Blume-Capel model at low temperatures, which we use as an example to review the low-temperature expansions.     

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.     

Phase transition in the 3d random field Ising model

We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,d _l for the theory isd _l 2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.