A Quasi-exact Formula for Ising critical temperatures on Hypercubic Lattices

Real Space Renormalization Group (RSRG) treatment of Ising model for square and simple cubic lattice is investigated and critical coupling strengths of these lattices are obtained. The mathematical complications, which appear inevitable in the decimated partition function due to Block-spin transformation, is treated with a relevant approximation. The approximation is based on the approximate equivalence of $\ln(1+f(K,\{\sigma_{n.n}\})) \simeq f(K,\{\sigma_{n.n}\})$ for small ...

We obtained analytically eigenvalues of a multidimensional Ising Hamiltonian on a hypercube lattice and expressed them in terms of spin-spin interaction constants and the eigenvalues of the one-dimensional Ising Hamiltonian (the latter are well known). To do this we wrote down the multidimensional Hamiltonian eigenvectors as the Kronecker products of the eigenvectors of the one-dimensional Ising Hamiltonian. For periodic boundary conditions, it is possible to obtain exact res...

We study the $\pm J$ transverse-field Ising spin glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong field limit. In the SK model and in high-dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension $d = 6$ which is below the upper critical dimension of $d=8$. In contrast, in lower dimensi...

The cluster variation - Pade` approximant method is a recently proposed tool, based on the extrapolation of low/high temperature results obtained with the cluster variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting non-classical critical exponents are also applied a...

The truncated two-point function of the nearest-neighbor ferromagnetic Ising model on $\mathbb Z^d$ ($d\ge3$) in its pure phases is proven to decays exponentially fast throughout the ordered regime ($T<T_c$). Together with known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: $(T,h) = (T_c,0)$.

We have found a simple criterion which allows for the straightforward determination of the order-disorder critical temperatures. The method reproduces exactly results known for the two dimensional Ising, Potts and $Z(N<5)$ models. It also works for the Ising model on the triangular lattice. For systems which are not selfdual our proposition remains an unproven conjecture. It predicts $\beta_c=0.2656...$ for the two coupled layers of Ising spins. Critical temperature of the th...

We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,.. and for the lattice euclidean scalar field t...

We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied Tensor Renormalization Group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically yielding a value 0.964 for the critical exponent nu much closer to the exact value 1 th...

The critical behavior of the 1/5-depleted square-lattice Ising model with nearest neighbor ferromagnetic interaction has been investigated by means of both an exact solution and a high-temperature series expansion study of the zero-field susceptibility. For the exact solution we employ a decoration transformation followed by a mapping to a staggered 8-vertex model. This yields a quartic equation for the critical coupling giving $K_{c} (\equiv\beta J_{c}) =0.695$. The series e...

We investigate the critical properties of the Ising S=1/2 and S=1 model on (3,4,6,4) and (3,3,3,3,6) Archimedean lattices. The system is studied through the extensive Monte Carlo simulations. We calculate the critical temperature as well as the critical point exponents gamma/nu, beta/nu and nu basing on finite size scaling analysis. The calculated values of the critical temperature for S=1 are k_BT_C/J=1.590(3) and k_BT_C/J=2.100(4) for (3,4,6,4) and (3,3,3,3,6) Archimedean l...