Anticommuting Integrals and Fermionic Field Theories for Two-Dimensional Ising Models

摘要

We review the applications of the integral over anticommuting Grassmannvariables (nonquantum fermionic fields) to the analytic solutions and thefield-theoretical formulations for the 2D Ising models. The 2D Ising modelpartition function $Q$ is presentable as the fermionic Gaussian integral. Theuse of the spin-polynomial interpretation of the 2D Ising problem is stressed,in particular. Starting with the spin-polynomial interpretation of the localBoltzmann weights, the Gaussian integral for $Q$ appears in the universal formfor a variety of lattices, including the standard rectangular, triangular, andhexagonal lattices, and with the minimal number of fermionic variables (two persite). The analytic solutions for the correspondent 2D Ising models then followby passing to the momentum space on a lattice. The symmetries and the questionon the location of critical point have an interesting interpretation withinthis spin-polynomial formulation of the problem. From the exact lattice theorywe then pass to the continuum-limit field-theoretical interpretation of the 2DIsing models. The continuum theory captures all relevant features of theoriginal models near $T_c$. The continuum limit corresponds to the low-momentumsector of the exact theory responsible for the critical-point singularities andthe large-distance behaviour of correlations. The resulting field theory is themassive two-component Majorana theory, with mass vanishing at $T_c$. Bydoubling of fermions in the Majorana representation, we obtain as well the 2DDirac field theory of charged fermions for 2D Ising models. The differencesbetween particular 2D Ising lattices are merely adsorbed, in thefield-theoretical formulation, in the definition of the effective mass.