Exact solution of the impure one-dimensional n-vector model with bilinear and biquadratic exchange interactions

We demonstrate explicitly that the impure one-dimensional n-vector model with bilinear and biquadratic exchange interactions can be solved exactly. Results are provided for the free energy, equal-time spin-spin correlation functions and the initial susceptibility of the corresponding bond and site model in the annealed and quenched limit. A numerical analysis of the thermodynamic response functions in the n = 3 case is performed, and therein some interesting consequences of the simultaneous influence of the biquadratic interactions and impurities are established. The presence of the biquadratic interaction is interpreted as an effect of the lattice compressibility.     

Transitions involving conical magnetic phases in a model with bilinear and biquadratic interactions

In a previous work a model was proposed for the phase transitions of crystals with localized magnetic moments which at low temperature have a “conical” arrangement that at higher T transforms into a more symmetrical structure (depending on the compound) before becoming totally disordered. The model assumes bilinear and biquadratic interactions between magnetic moments up to the fifth neighbours, and for any given T the structure with the least free energy is obtained by a mean-field approximation (MFA). The interaction constants are derived from ab initio   energy calculations. In this work we improve upon that model modifying the MFA in such a way that a continuous (instead of discontinuous) spectrum of excited states is available to the system. In the previous work, which dealt with _LaMn2Ge2${\mathrm{LaMn}}_2{\mathrm{Ge}}_2$ and _LaMn2Si2${\mathrm{LaMn}}_2{\mathrm{Si}}_2$, we found that transitions to different structures can be obtained for increasing T, in good qualitative agreement with experiment. The critical temperatures, however, were exaggeratedly high. With the new MFA we obtain essentially the same behaviour concerning the phase transitions, and critical temperatures much closer to the experimental ones.     

An exactly soluble model of a shallow double well

Shallow one-dimensional double-well potentials appear in atomic and molecular physics and other fields. Unlike the “deep” wells of macroscopic quantum coherent systems, shallow double wells need not present low-lying two-level systems. We argue that this feature, the absence of a low-lying two-level system in certain shallow double wells, may allow the finding of new test grounds for quantum mechanics in mesoscopic systems. We illustrate the above ideas with a family of shallow double wells obtained from reflectionless potentials through the Darboux–Bäcklund transform.     

An exactly soluble model of quadratic magnetostriction

Spin-phonon coupling, quadratic in the lattice displacements, is studied assuming that the spin interactions are of infinite range. The magnetic phase transition changes from second to first order for a sufficiently strong coupling. The ferromagnetic spinodal temperature becomes infinite and the paramagnetic spinodal temperature becomes zero for couplings of appropriate signs and strength. Renormalization of the phonon frequencies, within both the Einstein and the Debye schem schemes, is found.     

The Glauber dynamics for a spin-1 metamagnetic Ising system with bilinear and biquadratic interactions

We present a study, within a mean-field approximation, of the dynamics of a spin-1 metamagnetic Ising system with bilinear and biquadratic interactions in the presence of a time-dependent oscillating external magnetic field. First, we employ the Glauber transition rates to construct the set of mean-field dynamic equations. Then, we study the time variation of the average order parameters to find the phases in the system. We also investigate the thermal behavior of dynamic order parameters to characterize the nature (first- or second-order) of the dynamic transitions. The dynamic phase transitions are obtained and the phase diagrams are constructed in two different the planes. The phase diagrams contain a disordered and ordered phases, and four different mixed phases that strongly depend on interaction parameters. Phase diagrams also display one or two dynamic tricritical points, a dynamic double critical end and dynamic quadruple points. A comparison is made with the results of the other metamagnetic Ising systems.     

Magnetotransport in simple metals. An exactly soluble model

The magnetoconductivity tensors of a metal with a weakly anisotropic Fermi surface are determined from analytic solutions to the Boltzmann equation without any restrictions on the magnitude of the cyclotron frequency compared to the collision rate. Results are given for both a two- and a three-dimensional model, the former being analytically simpler to handle. The Hall coefficient and magnetoresistance are obtained as functions of the magnetic field, and we show by explicit calculation how the thermoelectric coefficients at high magnetic fields are determined by the thermodynamic entropy.     

On the mean-field theory of magnetic multilayers with bilinear and biquadratic Heisenberg exchange

A system consisting of several layers of magnetic ions interacting by both bilinear and biquadratic Heisenberg exchange is studied within the framework of the mean-field approximation. It is shown that for S = 1 there exist two types of ordering: ferromagnetic and ferroquadrupolar. The stability of phases as the function of temperature, biquadratic exchange and surface exchange is discussed analytically and numerically and it was shown that similar to bulk samples there appear first- and second-order transitions and a tricritical point may appear depending on system parameters.     

An exactly soluble model with tricritical points for structural-phase transitions

We present and study a lattice-dynamical model whose static and dynamic properties can be described exactly for all dimensionsd≧3 (d an integer) and which, in addition, exhibits tricritical points. For certain model parameters, the tricritical behaviour is found to be identical to that of the spherical model. By changing the model parameters continuously however, the transition suddenly becomes of first order at a tricritical point (TCP). The order parameter and the susceptibility are given explicitly ford≧3. The tricritical exponents are Gaussian. The critical dynamics is also discussed.     

An exactly soluble case of the triangular ising model in a magnetic field

An anisotropic triangular Ising model in which the first- and second-order parameters and the field parameters are functionally related is solved exactly by representing the distribution of the atom patterns in terms of a suitably constructed Markov process. The probabilities of patterns, defined as the probabilities generated by this process, are a mathematically tractable alternative to the classical representation of these probabilities in terms of the partition function. The interaction and field parameters of this Ising model, its magnetization, free energy, and its nearest neighbor correlation functions, are expressed in terms of the parameters of this Markov process. Special cases are worked out in detail and numerical examples are given.     

Dimerized phase and entanglement in the one-dimensional spin-1 bilinear biquadratic model

Dimerized phase and quantum entanglement are investigated in the one-dimensional spin-1 bilinear biquadratic model. Employing the infinite matrix product state representation, groundstate wavefunctions are numerically obtained by using the infinite time evolving block decimation method in the infinite lattice system. From a bipartite entanglement measure of the groundstates, i.e., von Neumann entropy, the phase transition points can be clearly extracted. Moreover, the even-bond and odd-bond von Neumann entropies show two different values in the spontaneous dimerized phase. It implies that the quantum entanglement can distinguish the two degenerate groundstates. Then, we define a dimer entropy in the spontaneous dimerized phase. Comparing to the dimer order parameter, the dimer entropy can play a role of a local order parameter to characterize the spontaneous dimerized phase.     

Nonlinear chemical dynamics in low dimensions: An exactly soluble model

Restricting space to low dimensions can cause deviations from the mean-field behavior in certain statistical systems. We investigate, both numerically and analytically, the behavior of the chemical reaction A+2X3X in one and two dimensions. In one dimension, we produce exact results showing that the trimolecular reaction system stabilizes in a nonequilibrium, locally frozen, asymptotic state in which the ratior of A to X particles is a constant number,r=0.38, quite different from the mean-field ratio,r _MF=1. The same trimolecular model, however, reaches the mean-field limit in two dimensions. In contrast, the bimolecular chemical reaction A+X2X is shown to agree with the mean-field predictions in all dimensions. For both models, we show that the adoption of certain types of transition rules in the laws of evolution can lead to oscillatory steady states.     

A simple and exactly soluble model of a deep two-electron trap

A simple and exactly soluble model is presented for a two-electron capture process accompanied by a large lattice relaxation. The model is a system of two interacting electrons in a deformable lattice with a single donor-type impurity. It is demonstrated that the model has the following two advantages. First, it exhibits a larger lattice relaxation for the binding of the second electron than for the first. Second, it enables us to discuss possible magnetic behaviors of the impurity in the ground state.     

On a class of exactly soluble statistical mechanical models with nonpolynomial interactions

The approximating Hamiltonian method of N. N. Bogolubov, Jr. is generalized to models with nonpolynomial intensive-observable interactions. The original Hamiltonian is proved to be thermodynamically equivalent to one linear in the intensive-observable trial Hamiltonian. We show that the exact expression for the free energy density in the thermodynamic limit can be obtained from a min-max principle for the system with trial Hamiltonian.On leave of absence from Institute for Nuclear Research and Nuclear Energetics, Bulgarian Academy of Sciences, Sofia, Bulgaria.On leave of absence from Institute for Solid State Physics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Theory of quasimagnetic impurity in a metal: An exactly soluble model of interacting electrons

We study an impurity atom, on which two-body forces are important, dissolved in a metal, where they are negligible. With the aid of the well-known boson excitation spectrum of the electronic Fermi sea, we predict the low-energy effects of one- and two-body potentials on the impurity, in the nonmagnetic regime. We obtain for the first time exact expressions for the cutoff independent contributions to the specific heat and paramagnetic susceptibility, the spectral amplitudes or one-electron density of states on the impurity, and the scattering cross-section. The entire spectrum of manybody eigenstates is explicitly obtained. The onset of a local magnetic moment appears as a sudden breakdown of the model Hamiltonian, and occurs when the two-body potential exceeds a critical value U_c which is O(E_F) in magnitude. A study of the renormalization of the interaction parameters terminates the paper.     

Renormalization group treatment of a spin-one Ising model with biquadratic exchange interaction

The critical behavior of a spin-one Ising model with biquadratic exchange interactions is studied using the mean field renormalization group method. The phase diagram for various lattices is obtained. The nature of the phase transition with biquadratic interaction is analysed and comparison with other methods is made.     

Pairing effects at finite temperature and finite rotational frequencies: An exactly soluble model

We propose a new type of experiment to look for parity violation due to neutral current interactions in deuterium. This involves measuring a certain angular momentum transfer from an atomic beam of polarized 2S _1/2 deuterium to a capacitor producing a constant electric field. The torque exerted on the capacitor which is to be measured turns out to be of order 10⁷ ?/s in a realistic situation.     

An exactly solved model with a wetting transition

A model of a binary mixture, showing a wetting transition, is examined. No prewetting phenomena are found. The scaling functions are obtained for the film thickness and for the correlation lengths.