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Preliminaries. Massive and massless free fermions. {#preliminaries}
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The Ising model at zero magnetic field is a theory of free fermions with Euclidean action (|+ ||+ im|) d\^2 x Here the coupling $m$ measures the deviation from the critical temperature of the classical model: $m=(T_{c}-T)/T_{c}$.

We are going to put this theory on an infinite cylinder of radius $R$. Let $x\sim x + R$ be the periodic coordinate and $y$ - the coordinate along the cylinder. Two sectors arise - with periodic (Neveu-Schwarz or NS-sector for brevity) and anti periodic (Ramond or R-sector). Quantizing on the spacial $x$-circle we get the single-particle frequencies \_n = where $n\in {\mathbb Z}$ in the R-sector and $n\in 1/2 + {\mathbb Z}$ in the NS-sector. Below we will use the index $n$ (or $n_1, n_2, \dots$) for integers and $k$ (or $k_1, k_2, \dots$) for half-integers.

Using the complex coordinate $z=x+iy$ the mode expansion in the NS sector is introduced as (x,y)=\_[k1/2 + [Z]{}]{} (-)\^[1/2]{} |(x,y)=\_[k1/2 + [Z]{}]{} ()\^[1/2]{} where (k) = {

[r@l]{} 1 , &k&lt;0\
-i[sign]{}(m) , &k&gt;0

. We also have similar expressions in the R-sector in terms of creation and annihilation operators $b^{\dagger}_{n}$, $b_{n}$ which we omit for 