Abstract
We review problems involving the use of Grassmann techniques in the eld of classical spin systems in two dimensions. These techniques are useful to perform exact correspondences between classical spin Hamiltonians and eld-theor y fermionic actions. This contributes to a better understanding of critical behavior of these models in term of non-quadratic effective actions which can be seen as an extension of the free fermion Ising model. Within this method, identication of bare masses allows for an accurate estimation of critical points or lines and which is supported by Monte-Carlo results and diagrammatic techniques.
Highlights
Classical and quantum spin models such as Ising model play an important role in the field of statistical physics as they allow for an accurate understanding of critical phenomena in general
An exact mathematical description of the two-dimensional (2D) Ising model relies on the JordanWigner transformation [3] which maps the product of Boltzmann weights onto a fermionic action of free fermions with a m√ass vanishing at the second order critical temperature given in dimensionless units Tc = 2/ ln(1 + 2) 2. 2691851
We have presented a method which tries to operate a correspondence between classical spin models and fermionic systems
Summary
Classical and quantum spin models such as Ising model play an important role in the field of statistical physics as they allow for an accurate understanding of critical phenomena in general. A direct introduction of Grassmann variables as an alternative tool to solve the Ising model was done long ago in the 80’s by Bugrij [5] and Plechko [6] (see a later discussion by Nojima [7]) It is based on a simple integral representation of the individual Boltzmann weights and which has the property to decouple the spins. In order to deal with this particular representation, Bugrij used two families of Grassmann variables which commute with each other, identified the resulting functional integral of the partition function with a determinant From another point of view, Plechko introduced symmetries which order the non-commuting quantities so that the sum over the spins can be performed exactly. We explain the main ideas of this method
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