Narrow Results

We introduce a variant of the Bethe lattice, the fully connected Bethe lattice (FCBL); with every site in each shell of the Bethe lattice connected to every other site in the shell. A recently introduced decimation renormalization group (RG) method is utilized to obtain the quantum transmission through the Bethe lattice and FCBL. Although classically the extra intra-shell connections would not change the transmission, they have a large effect on the quantum transport and the wave functions.

The mixed spin-1 and spin- Blume–Capel Ising ferrimagnetic system for the central spin with spin-1 is studied on the Bethe lattice using the exact recursion equations. The exact expressions for the magnetization, the quadrupolar moment, the Curie temperature and the free energy are found and the phase diagrams are constructed on the Bethe lattice with the coordination numbers q = 3, 4 and 6 for the various values of the single-ion anisotropy constants d_A = D_A/J for spin-1 and d_B = D_B/J for spin-. The existence of a tricritical point is investigated for different values of q and the single-ion anisotropy constants. The phase diagrams in the (kT_c/J, d_A) plane for the central spin are obtained for two different cases; (1) d_A = d_B and (2) d_A is varied for selected values of d_B. The results are compared with those of other approximate methods.

The thermal behaviors of the order-parameters, susceptibility and specific heat of the mixed spin-½ and spin-1 Blume–Capel Ising ferrimagnetic system is studied on the Bethe lattice by using the exact recursion equations. The exact equations for the order parameters and the susceptibility are obtained for the sublattices with spin-1 and spin-½. Also, the exact expressions for the free energy and the specific heat are obtained on the Bethe lattice as a whole system. Then the thermal behaviors of the order-parameters, susceptibility, inverse susceptibility and specific heat are studied for the coordination numbers q=4 and q=6. The obtained results are illustrated and discussed in detail.

A spin-2 system consisting of two layers of Bethe lattices each with a branching ratio of q Ising spins was analyzed by the use of the exact recursion relations in a pairwise approach. The upper layer interacting with nearest-neighbor (NN) bilinear interaction J₁ is laid over the top of the lower layer interacting with bilinear NN interaction J₂, and the two layers are tied together via the bilinear interaction between the vertically aligned adjacent NN spins denoted as J₃. The study of the ground state phase diagrams on the (J₂/|J₃|, J₁/|J₃|) plane with J₃>0 and J₃<0 and on the (J₂/J₁, J₃/q J₁) plane with J₁>0 has yielded five distinct ground state configurations. The temperature dependent phase diagrams are obtained for the case with intralayer coupling constants of the two layers with ferromagnetic type J₁ and J₂>0, and the interlayer coupling constant of the layers with either ferromagnetic J₃>0 or antiferromagnetic type J₃<0 on the (kT/J₁, J₃/J₁) planes for given values of the J₂/J₁ for various values of the coordination numbers. As a result, we have found that the model presents both second- and first-order phase transitions, therefore, tricritical points.

The bilayer spin-2 Ising model on the Bethe lattice is investigated by taking into account the intralayer coupling constants of the two layers J₁ and J₂, interlayer coupling constant between the layers J₃ and crystal field interaction Δ by using the exact recursion equations in a pairwise approach. The ground state (GS) phase diagrams of the model are obtained on the (J₂/|J₁|, J₃/q|J₁|) planes for given Δ values and on the (Δ/qJ, J₃/qJ) plane when J₁ = J₂ = J, and thus 33 distinct GS configurations are found. The temperature-dependent phase diagrams are obtained for J₁ > 0, J₂ > 0, and for J₃ > 0 or J₃ < 0 on the (kT/J₁, J₃/J₁) planes for given Δ/qJ₁ and J₂/J₁ and on the (Δ/J, kT/J) plane for given J₃/J when J₁ = J₂ = J for the coordination number q = 3. It was found that the system exhibits both first- and second-order phase transitions and tricritical points. The paramagnetic phases are also classified by studying the thermal variations of the quadrupolar moments.

Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.

Using the recursion equations technique, the influences of the single-ion anisotropies or crystal-fields interactions on the magnetic properties of the mixed spin-1 and spin-7/2 Blume-Capel (BC) Ising ferrimagnetic system are studied on the Bethe lattice (BL). The ground-state phase diagram is constructed, the thermal behaviors of the order-parameters and the free-energy are thoroughly investigated in order to characterize the nature of the phase transitions and to obtain the phase transition temperature. Then, the temperature phase diagrams are obtained in the case of equal crystal-field interactions on the ($kT/\left|J\right|$ and $D/\left|J\right|$) planes when q = 3, 4 and 6 and in the case of unequal crystal-fields interactions on the ($kT/\left|J\right|$ and $D_A/\left|J\right|$) and $(kT/|J|$ and $D_B/\left|J\right|$) planes for selected values of $D_B/\left|J\right|$ and $D_A/\left|J\right|$ respectively when q = 3. The model shows first-order and second-order phase transitions, and where the lines are connected is the tricritical point. Besides the first-order and second-order phase transitions, the system also exhibits compensation temperatures depending on appropriate values of the crystal-fields interactions.

The quaternary alloy (QA) is simulated on the Bethe lattice (BL) in the form of AB_pC_qD_r and its phase diagrams are calculated by using the exact recursion relations (ERR) for the coordination number z = 3. The QA is designed on the BL by placing A atoms (spin-1/2) on the odd shells and randomly placing B (spin-3/2), C (spin-5/2) or D (spin-1) atoms with probabilities p, q and r, respectively, on the even shells. A compact form of formulation for the QA is obtained in the standard-random approach which can easily be reduced to ternary alloy (TA) and mixed-spin models by the appropriate values of the random variables p, q and r. The phase diagrams are calculated on the temperature and ratio of bilinear interaction parameter planes for given values of probabilities.

The mixed spin-1/2 and spin-1 Blume–Capel model is studied with randomly alternated coordination numbers (CN) on the Bethe lattice (BL) by utilizing the exact recursion relations. Two different CNs are randomly distributed on the BL by using the standard–random (SR) approach. It is observed that this model presents first-order phase transitions and tricritical points for variations of CNs 3 and 4, even if these behaviors are not displayed for the regular mixed-spin on the BL. The phase diagrams are mapped by obtaining the phase transition temperatures of the first- and second-order on several planes.

The A, B and C atoms with spin-1/2, spin-3/2 and spin-5/2 are joined together sequentially on the Bethe lattice in the form of ABCABC$\dots$ to simulate a molecule as a triple mixed-spin system. The spins are assumed to be interacting with only their nearest-neighbors via bilinear exchange interaction parameter in addition to crystal and external magnetic fields. The order-parameters are obtained in terms of exact recursion relations, then from the study of their thermal variations, the phase diagrams are calculated on the possible planes of our system. It is found that the model gives only second-order phase transitions in addition to the compensation temperatures.

The triangular-type Ising nanowire is constructed on the Bethe lattice (BL) by using the core-shell structure consisting of spin-3/2 atoms as the core and spin-1/2 atoms as the triangular shell. Each triangular plaquette of spins forms a nanoparticle which is connected to upper and lower plaquettes symmetrically. The additions of the plaquettes continue indefinitely until the thermodynamic limit to construct the nanowire. The inter- and intra-bilinear interaction parameters (J) are assumed to be positive or negative to simulate the ferromagnetic (FM) or antiferromagnetic (AFM) interactions, respectively. The crystal field for spin-3/2 and external magnetic field for all sites are also included into the model. After obtaining the formulation of the model in terms of exact recursion relations (ERRs), the thermal variations of magnetizations are studied in detail to obtain the phase diagrams. It is found that the model leads to different types of FM and AFM regions with various forms of phase transitions. It is also interesting that the model presents random or oscillatory magnetization behavior regions for the appropriate values of our system parameters.

When gas molecules bind to a surface they may do so in such a way that the adsorption of one molecule inhibits the arrival of others. Two models which have frequently been studied are the "dimer model" and the "blocking model", and rather complete solutions for these are known on the Bethe lattice. In this letter, the probability that a connected set of sites remains unoccupied is calculated.

In this work, the ternary alloy (TA) of the form $A{B}_p{C}_{1-p}$ with spin-$\frac{3}{2}$, spin-2 and spin-$\frac{5}{2}$, respectively, is studied on the Bethe lattice in terms of exact recursion relations in the standard random approach. The bilinear interaction parameter $J_{AB}$ is assumed to be ferromagnetic between the nearest-neighbor spins with spin-$\frac{3}{2}$ and spin-2, while $J_{AC}$ is taken to be antiferromagnetic between spin-$\frac{3}{2}$ and spin-$\frac{5}{2}$. The possible phase diagrams are obtained from the thermal analysis of the order parameters for the given coordination numbers z = 3,4,5 and 6. This analysis has also revealed that the model gives both second- and first-order phase transitions in addition to the compensation temperatures.

The mixed spin-1/2 and spin-3/2 Blume–Capel (BC) model is considered on the Bethe lattice (BL) with randomly changing coordination numbers (CN) and examined in terms of exact recursion relations. A couple of two different CNs are changed randomly on the shells of the BL in terms of a standard–random approach to obtain the phase diagrams on possible planes of the system parameters. It is found from the thermal analysis of the order-parameters that the model only gives the second-order phase transitions as in the regular mixed case. As the probability of having larger CN increases, the temperatures of the critical lines also increase as expected.

The Ising model with nearest- and next-nearest-neighbor (NNN) bilinear interactions is examined on the Bethe lattice (BL) in terms of exact recursion relations (ERR) when the external magnetic $H$ is turned on. The thermal variation of the magnetization belonging to the central spin is investigated to calculate the possible phase diagrams of the model for given coordination numbers. Different phase regions, ferromagnetic (FM), antiferromagnetic (AFM) and paramagnetic (PM), are discovered and the phase lines in terms of first-order or second-order phase transitions are calculated. These lines are found to be order–disorder or order–order phase transition lines. It is also found that they combine at some special points or terminate at some end points for appropriate values of the model parameters.

A triple mixed-spin Ising system defined on the Bethe lattice is numerically investigated by means of exact recursion relations (ERRs) calculations. The lattice is constituted by three types of magnetic atoms A, B, C with spins $1/2$, $1$, $3/2$ respectively arranged in the form ABCABC. The effects of bilinear exchange and crystal-field interactions as well as those of thermal fluctuations on the order parameters and phase diagrams are thoroughly studied and specified. First-order transitions and tricritical points are present for the coordination number $q=4$ whereas at $q=3$ they are absent. Global compensation phenomena are absent for the magnetic system. Instead, it is shown that it can only occur between the sublattice magnetizations B and C of the system. Several novel kinds of reentrance of the phase boundaries while varying the values of model parameters have been reported.

The external random magnetic field $(H_i)$ with three nodes, i.e. acting up and down along the $z$-axis and zero, effective on the spins in the Blume-Capel model is analyzed on the Bethe lattice in terms of the exact recursion relations. All the nodes are assumed to have the same probability, $p=1/3$, so that the model could give various kinds of phase transitions. As a mapping of the phase transitions, the phase diagrams are constructed on two different planes which present very rich and interesting phase diagrams. In addition to the second- and first-order phase transitions, a few critical points, reentrant and double reentrant behaviors are also observed.

The outcome of the random crystal field effects on the antiferromagnetic spin-1 Blume–Capel model and external magnetic field are examined on the Bethe Lattice in terms of exact recursion relations. It is assumed that the crystal field is either turned on or off randomly with probability $p$ and $1-p$, respectively. The phase diagrams are constructed from the thermal analysis of the order parameters with the coordination number $z=3$ which corresponds to honeycomb lattice. It is explored that the system goes both second- and first-order phase transitions, along with the reentrant behavior and a few critical points. The reentrant behavior is stronger for lower values of $p$ and disappears as $p$ gets closer to 1.0. The first-order lines are observed to be either linked to the tricritical points or decomposed. The critical end points and double critical points are also observed.

The molecular approach of a spin model is constructed on the Bethe lattice (BL), and then it is examined in terms of exact recursion relations. Rather than assuming that each BL site is inhabited by a single spin, each site is occupied by two spin-1/2 atoms A and B, forming a molecule. Each molecule is considered to contain two spin-1/2 atoms, as well as $q=3,4,$ or 6 nearest-neighbor molecules. In addition to the internal interactions between the atoms of each molecule, the molecules interact via their atoms in terms of bilinear interaction parameters J. Atoms of a molecule interact with $J_{A^i{B}^i}$, while the molecules interact via their atoms in terms of $J_{A^i{B}^{i+1}}=J_{B^i{A}^{i+1}}$ and $J_{A^i{A}^{i+1}}=J_{B^i{B}^{i+1}}$. After obtaining the magnetizations of each atom in the central molecule of the BL, the average magnetization of the molecule is determined. It is found that the model presents first-and second-order and random phase transitions. The model also displays tricritical, bicritical and end points, in addition to reentrant behavior for appropriate J values.

The Anderson model on the Bethe lattice is historically among the first for which an energy regime of extended states and a separate regime of localized states could be established. In this paper, we review recently discovered surprises in the phase diagram. Among them is that even at weak disorder, the regime of diffusive transport extends well beyond energies of the unperturbed model into the Lifshitz tails. As will be explained, the mechanism for the appearance of extended states in this non-perturbative regime are disorder-induced resonances. We also present remaining questions concerning the structure of the eigenfunctions and the associated spectral statistics problem on the Bethe lattice.