45 research outputs found
Second-order critical lines of spin-S Ising models in a splitting field with Grassmann techniques
We propose a method to study the second-order critical lines of classical
spin- Ising models on two-dimensional lattices in a crystal or splitting
field, using an exact expression for the bare mass of the underlying field
theory. Introducing a set of anticommuting variables to represent the partition
function, we derive an exact and compact expression for the bare mass of the
model including all local multi-fermions interactions. By extension of the
Ising and Blume-Capel models, we extract the free energy singularities in the
low momentum limit corresponding to a vanishing bare mass. The loci of these
singularities define the critical lines depending on the spin S, in good
agreement with previous numerical estimations. This scheme appears to be
general enough to be applied in a variety of classical Hamiltonians
Ferromagnetic phase transitions of inhomogeneous systems modelled by square Ising models with diamond-type bond-decorations
The two-dimensional Ising model defined on square lattices with diamond-type
bond-decorations is employed to study the nature of the ferromagnetic phase
transitions of inhomogeneous systems. The model is studied analytically under
the bond-renormalization scheme. For an -level decorated lattice, the
long-range ordering occurs at the critical temperature given by the fitting
function as , and the
local ordering inside -level decorated bonds occurs at the temperature given
by the fitting function as . The critical amplitude of the logrithmic singularity
in specific heat characterizes the width of the critical region, and it varies
with the decoration level as , obtained by fitting the numerical results. The cross over from a
finite-decorated system to an infinite-decorated system is not a smooth
continuation. For the case of infinite decorations, the critical specific heat
becomes a cusp with the height . The results are compared
with those obtained in the cell-decorated Ising model.Comment: 18 pages, 7 figure
Fermions and Disorder in Ising and Related Models in Two Dimensions
The aspects of phase transitions in the two-dimensional Ising models modified
by quenched and annealed site disorder are discussed in the framework of
fermionic approach based on the reformulation of the problem in terms of
integrals with anticommuting Grassmann variables.Comment: 11 pages, 1 table, no figures. The discussion is merely based on a
talk given at the International Bogoliubov Conference on Problems of
Theoretical and Mathematical Physics, MIRAS--JINR, Moscow--Dubna, Russia,
August 21--27, 200
New perspectives on the Ising model
The Ising model, in presence of an external magnetic field, is isomorphic to
a model of localized interacting particles satisfying the Fermi statistics. By
using this isomorphism, we construct a general solution of the Ising model
which holds for any dimensionality of the system. The Hamiltonian of the model
is solved in terms of a complete finite set of eigenoperators and eigenvalues.
The Green's function and the correlation functions of the fermionic model are
exactly known and are expressed in terms of a finite small number of parameters
that have to be self-consistently determined. By using the equation of the
motion method, we derive a set of equations which connect different spin
correlation functions. The scheme that emerges is that it is possible to
describe the Ising model from a unified point of view where all the properties
are connected to a small number of local parameters, and where the critical
behavior is controlled by the energy scales fixed by the eigenvalues of the
Hamiltonian. By using algebra and symmetry considerations, we calculate the
self-consistent parameters for the one-dimensional case. All the properties of
the system are calculated and obviously agree with the exact results reported
in the literature.Comment: 19 RevTeX pages, 9 panels, to be published in Eur. Phys. J.