the Fredholm determinants in the thermodynamic limit. In the present paper we shall consider the $XX$ chain only and denote its Hamiltonian by $H$.

To calculate the averages  one can use a representation of the canonical Fermi variables $c_j$, $c^{\dagger}_j$, $j\in{\mathcal M}$ through the spin variables [@19], [@20]. The corresponding Jordan–Wigner transformation has the form: $$\sigma^+_n=\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr)c_n,\qquad
\sigma^-_n=c_n^{\dagger}\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr),
\qquad n\in{\mathcal M},
\label{4}$$ where $\sigma^z_j=1-2c_j^{\dagger} c_j$. The periodic boundary conditions for the spin variables lead to the following boundary conditions for the Fermi variables: $$c_{M+1}=(-1)^{\mathcal{N}}c_1,\qquad
c^{\dagger}_{M+1}=c^{\dagger}_1(-1)^{\mathcal{N}},
\label{5}$$ where $\mathcal{N}=\sum_{n=1}^Mc_n^{\dagger} c_n$ is the operator of the total number of particles. The Hamiltonian $H$  takes the following form in the fermion representation [@19], [@20] $$H=H^+P^++H^-P^-,
\label{6}$$ where $P^{+}$ ($P^{-}$) are projectors on the states characterized by an even/odd number of fermions: $$P_++P_-=\mathbb I\,,\quad P_+-P_-=(-1)^{\mathcal N}\,.$$ The operators $H^{\pm}$  are formally identical, their superscripts $s=\pm$ point out an appropriate specification of the boundary conditions : $$c_{M+1}\,=\,-s\,c_1\,,\quad c^{\dagger}_{M+1}\,=\,- s\,c^{\dagger}_1\,.$$ To put it differently, the quadratic in the fermion variables operators $H^\pm$ has the following representation: $$H^{\pm}=c^{\dagger}\widehat H^{\pm} c-\frac{Mh}2,\qquad
\widehat H^{\pm}=-\hat\Delta^{(\mp)}+h{\hat I},
\label{7}$$ where the matrices ${\widehat H}^{\pm}$ are expressed through the hopping matrices  and $\hat I$ is the unit matrix: $${\widehat H}^\pm\,=\,\left(
\begin{array}{cccccc}   h   & -1/2 