brevity. The physical state space on a circle ${\cal H}_{m}$ is spanned by the states \[mbasis1\] |k\_1, …, k\_[N]{}\_[NS]{} = b\^\_[k\_1]{}…b\^\_[k\_N]{}|0\_[NS]{} , k\_[i]{}1/2 + [Z]{} with $N$ even and by the states \[mbasis2\] |n\_1, …, n\_[M]{}\_[R]{} = b\^\_[n\_1]{}…b\^\_[n\_M]{}|0\_[R]{} , n\_[i]{} where when $m>0$ the number of oscillators $M$ is even and when $m<0$ $M$ is odd.

The Hamiltonian blocks are \[HFF1\] H\^[NS]{} = \_[n=-]{}\^ \_[n+1/2]{}b\^\_[n+1/2]{}b\_[n+1/2]{} + E\_[0]{}\^[([NS]{})]{} , \[HFF2\] H\^[R]{} = \_[n=-]{}\^ \_[n]{}b\^\_[n]{}b\_[n]{} + E\_[0]{}\^[([R]{})]{} where the vacuum energies are E\_[0]{}\^[([NS]{})]{}=( ) -|m|\_[-]{}\^ () ( 1 + e\^[-|m|R()]{} ) , E\_[0]{}\^[([R]{})]{}=( ) -|m|\_[-]{}\^ () ( 1 - e\^[-|m|R()]{} ) . Here $\mu$ is a subtraction scale for the logarithmic divergence that arises from the OPE of two energy operators and is responsible for the Onsager singularity of free energy.

The theory with an arbitrary value of $m$ can be described in the $m=0$ state space using a Bogolyubov transformation. The massless fermionic fields which we denote $\chi(z)$, $\bar \chi(\bar z)$ have mode expansions && (z) = ( -i )\^[1/2]{} \_[k1/2+[Z]{}]{} a\_[k]{}e\^[i(x+iy)k]{} ,\
&& |(|z) = ( i )\^[1/2]{} \_[k1/2+[Z]{}]{} |a\_[k]{}e\^[-i(x-iy)k]{} . The operators $a_{k}, \bar a_{k}, k>0$ are the annihilation operators and $a^{\dagger}_{k}=a_{-k}$, $\bar a^{\dagger}_{k}=\bar a_{-k}$, $k>0$ are the corresponding 