We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.
We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.
We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region $(\ensuremath{\beta}\ensuremath{\ll}1).$ Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter $\ensuremath{\alpha}=(〈\overline{s}{}^{4}〉\ensuremath{-}3〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1})/(〈\overline{s}{}^{4}〉\ensuremath{-}〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1}),$ where $\overline{s}=s\ensuremath{-}〈s〉,$ and 〈sk〉 denotes the kth moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass $m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.$ Assuming $〈{s}^{3}〉\ensuremath{\ne}0$ we show that for $\ensuremath{\alpha}>0,$ \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold 2m. For $\ensuremath{\alpha}<0$ bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to 2m.
We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d -dimensional lattice ferromagnetic classical N -component vector spin systems at high temperature (0 < β << 1). Each system is characterized by a single site a priori spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe–Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe–Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.
We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.
An antiferromagnetic triangular Ising model with ferromagnetic next nearest neighbor interactions is studied by means of a transfer matrix method and a finite size scaling. Using the Roomany-Wyld approximation, it cannot be concluded definitely whether the system has only one critical point or not. By studying decays of the correlation functions of each sublattice, it is found that there is a finite temperature region where the correlation functions of each sublattice show a power law decay, namely, in this region every temperature is a critical temperature. The results by means of the present transfer matrix method are in good agreement with that by Monte Carlo methods, namely, the discrepancy between both results, reported recently, can be got rid of.
We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, in general anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a ≥ 0, b ≥ 0, and the symbols uj, α denote cartesian components of the spins. For some specific values of the two parameters a and b, and on the basis of available theoretical results, one can prove the existence of an ordering transition taking place at finite temperature, and obtain rigorous upper and lower bounds on the transition temperatures. This holds, for example, when n = 2, 3, a > b = 0 (the models studied in our previous paper), as well as for n = 2, a = b > 0 and n = 3, b > a = 0, where a continuous O(2) symmetry of the interaction is involved. We have studied these two latter cases by computer simulation, and made comparison with mean-field treatment; simulation results show a broad qualitative similarity between the four models, and a closer, quantitative one, between pairs of models with the same number of spin components, especially for n = 2.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.