An exactly soluble model of a system containing arbitrary isotropic bilinear and biquadratic exchange interactions

Stanley's exact results for a Bethe Lattice of classical spins of arbitrary dimensionality are generalized to include an arbitrary biquadratic interaction term. Simple expressions are obtained for both the nearest neighbor dipolar and quadrupolar correlation functions. It is shown that the system never displays long range order at any finite temperature. A modest amount of biquadratic excahnge can even prevent the system from ordering at T = 0°K. However, except for the linear chain, two distinct temperatures are found for the divergence of the dipolar and quadrupolar susceptibilities.     

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.     

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.     

Classical one-dimensional Heisenberg magnet with biquadratic exchange interaction

The pair- and four-spin correlation functions and susceptibility have been calculated rigorously for a classical Heisenberg linear chain with biquadratic exchange interaction by extending Fisher's method.     

Exact solution of the one-dimensional fermionic model with correlated hopping

We extend the Bethe Ansatz solution of a onedimensional integrable fermionic model with correlated hopping to the parameter regime Δ t > 1. It is found that the model is equivalent to one with interaction 2 ? Δ t, but with twisted boundary conditions. Apart from the ground state energy we investigate the low-lying excitations and the asymptotic behaviour of the correlation functions. As in the ease of Δt < 1 we find dominating superconducting correlations for small doping. The behaviour in this regime therefore differs from that of the non-integrable model with symmetric bond-charge interaction (Hirsch model).     

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.     

Exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions

We study the exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions. The energy and Bethe ansatz equations of the Gaudin model can be obtained via the off-diagonal Bethe ansatz method. Based on the off-diagonal Bethe ansatz solutions, we construct the Bethe states of the inhomogeneous X X X Heisenberg spin chain with the generic open boundaries. By taking a quasi-classical limit, we give explicit closed-form expression of the Bethe states of the Gaudin model. From the numerical simulations for the small-size system, it is shown that some Bethe roots go to infinity when the Gaudin model recovers the U(1) symmetry. Furthermore,it is found that the contribution of those Bethe roots to the Bethe states is a nonzero constant. This fact enables us to recover the Bethe states of the Gaudin model with the U(1) symmetry. These results provide a basis for the further study of the thermodynamic limit, correlation functions, and quantum dynamics of the Gaudin model.     

Solitary excitations in one-dimensional ferromagnetic spin chains with biquadratic exchange interaction

By using the traveling wave method, the solutions of the elliptic function wave and the solitary wave are obtained in a ferromagnetic spin chain with a biquadratic exchange interaction, a single ion anisotropic interaction and an anisotropic nearest neighbour interaction. The effects of the biquadratic exchange interaction and the single ion anisotropic interaction on the properties (width, peak and stability) of the soliton are investigated. It is also found that the effects vary with the strengths of these interactions.     

Planar classical Heisenberg model with biquadratic interactions

Seven coefficients in the high temperature series expansions for the zero-field susceptibility and the specific heat are derived for the planar classical Heisenberg model with biquadratic interactions. The critical temperatures and the susceptibility exponents are determined for cubic lattices.     

Equilibrium properties of a spin-1 Ising system with bilinear,biquadratic and odd interactions

The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, α = J/K; therefore, the dependence of the nature of the phase transition on α is investigated extensively and it is found that for α ⩽ 1 and α ⩾ 2000 a second-order phase transition occurs, and for 1 < α < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of α calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.     

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.     

Exact solution of the one-dimensional deterministic fixed-energy sandpile

By reason of the strongly nonergodic dynamical behavior, universality properties of deterministic fixed-energy sandpiles are still an open and debated issue. We investigate the one-dimensional model, whose microscopical dynamics can be solved exactly, and provide a deeper understanding of the origin of the nonergodicity. By means of exact arguments, we prove the occurrence of orbits of well-defined periods and their dependence on the conserved energy density. Further statistical estimates of the size of the attraction's basins of the different periodic orbits lead to a complete characterization of the activity vs energy density phase diagram in the limit of large system's size.     

Bound electron dynamics: Exact solution for a one-dimensional oscillator-string model

The dynamical problem of a harmonically bound electron with standard dipole model coupling to the electromagnetic field in a finite one-dimensional space is solved exactly in a simple manner. It is easily shown that in this model the coupling between the electron and the field is “rigid”, in the sense of and in complete analogy with a recent treatment of a purely mechanical particle on a string. As a consequence the electron's quantum mechanical momentum fluctuations exhibit a logarithmic ultraviolet divergence. In the limit of infinite spatial extension of the field, and apart from quantal noise, the electron behaves exactly as a simple linearly damped harmonic oscillator.     

Upper bound to the spin correlation function of the n-vector model with random anisotropic interactions

We obtain an upper bound to the spin correlation function in the thermodynamic limit of the zero external field limit for the n-vector model with random anisotropic interactions. We find a sufficient condition for disappearance of the spontaneous long-range order.     

Exact solution of the 1D Hubbard model with NN and NNN interactions in the narrow-band limit

In this paper, we propose a family of weighted extended Koch networks based on a class of extended Koch networks. They originate from a r-complete graph, and each node in each r-complete graph of current generation produces mr-complete graphs whose weighted edges are scaled by factor h in subsequent evolutionary step. We study the structural properties of these networks and random walks on them. In more detail, we calculate exactly the average weighted shortest path length (AWSP), average receiving time (ART) and average sending time (AST). Besides, the technique of resistor network is employed to uncover the relationship between ART and AST on networks with unit weight. In the infinite network order limit, the average weighted shortest path lengths stay bounded with growing network order (0 < h < 1). The closed form expression of ART shows that it exhibits a sub-linear dependence (0 < h < 1) or linear dependence (h = 1) on network order. On the contrary, the AST behaves super-linearly with the network order. Collectively, all the obtained results show that the efficiency of message transportation on weighted extended Koch networks has close relation to the network parameters h, m and r. All these findings could shed light on the structure and random walks of general weighted networks.