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

Alternative description of the 2D Blume-Capel model using Grassmann algebra

We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He$^3$-He$^4$. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.Comment: 32 pages, 2 figures

Characteristics of 2D lattice models from fermionic realization: Ising and XYZXYZ models

We develop a field theoretical approach to the classical two-dimensional models, particularly to 2D Ising model (2DIM) and $XYZ$ model, which is simple to apply for calculation of various correlation functions. We calculate the partition function of 2DIM and $XY$ model within the developed framework. Determinant representation of spin-spin correlation functions is derived using fermionic realization for the Boltzmann weights. The approach also allows formulation of the partition function of 2DIM in the presence of an external magnetic field.Comment: 18 pages, RevTex, 2 figures, extended the appraoch to the XYZ model, the published versio

Exact partition functions of the Ising model on MxN planar lattices with periodic-aperiodic boundary conditions

The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on MxN square (sq), plane triangular (pt) and honeycomb (hc) lattices with periodic-periodic (pp), periodic-antiperiodic (pa), antiperiodic-periodic (ap) and antiperiodic-antiperiodic (aa) boundary conditions. The partition functions are used to calculate and plot the specific heat, $C/k_B$, as a function of the temperature, $\theta =k_BT/J$. We find that for the NxN sq lattice, $C/k_B$ for pa and ap boundary conditions are different from those for aa boundary conditions, but for the NxN pt and hc lattices, $C/k_B$ for ap, pa, and aa boundary conditions have the same values. Our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite-size corrections of the Ising model.Comment: 17 pages, 13 Postscript figures, uses iopams.sty, submitted to J. Phys. A: Math. Ge