Once again about interchain hopping

Renormalization group equations and a phase diagram are derived for a system of two chains with a single-electron hopping between chains in order to correct the results of a recent paper by F. V. Kusmartsev, A. Luther, and A. Nersesyan, Pis’ma v Zh. Exp. Teor. Fiz. 55, 692 (1992) [JETP Lett. 55, 724 (1992)]. In a recent paper [1], the effect of a single-electron hopping (SEH) between chains in a system of two chains was studied by means of the bosonization technique and the Coulomb gas description. Possible phase transitions of the Berezinskii-Kosterlitz-Thouless type, which result in generation of pair hoppings between the chains, were considered. Unfortunately, despite the spirit of the paper is correct, Eq. (4) turns out to be incomplete and leads to a wrong phase diagram shown in Fig. 1. The considered problem is the old one which recently attracted attention again [2]. The generation of the pair hoppings in a gapless case was discovered for the first time in Ref. [3] via a perturbational theory. Later, these results were confirmed by a renormalization group (RG) approach [4]. The aim of the present paper is to correct the RG equation (4) and the phase diagram of Ref. [1] and achieve in this way an agreement with the previously known results [3], [4]. Due to the generation of the pair-hopping terms, the action of the model,