Erratum: Effect of interactions on the topological expression for the chiral separation effect

In our paper, Eq. (3) as well as Eqs. (47), (83) are valid in the presented form only when (Formula presented) is the hypersurface consisting of the two hyperplanes (Formula presented). The same refers also to Eq. (147) of [1], where the noninteracting case was considered. Here, at each (Formula presented) matrix (Formula presented) should commute or anticommute with (Formula presented) and (Formula presented) and (Formula presented) in a vicinity of the space depending Fermi surface. By the space depending Fermi surface, we understand the position in momentum space of the singularities of expression standing in the integrals in Eqs. (3), (47), and (83). This vicinity is of the size that is much larger than the inverse correlation length of the considered system. In practice, this means that the space depending Fermi surface should be situated sufficiently far from the ultraviolet region of momentum space, where the chiral symmetry is violated. These expressions may be formulated also for the more general form of (Formula presented), but then certain clarifications should be added. More specifically, in Eq. (3), we should understand function (Formula presented) as depending on spatial momenta (Formula presented) and spatial coordinates (Formula presented), i.e. it should be substituted by (Formula presented) (A) Here the three-dimensional vector (Formula presented) parametrizes surface (Formula presented). Function (Formula presented) represents the dependence of Matsubara frequency on (Formula presented), the upper sign is to be chosen for the upper piece of (Formula presented), the lower sign-for the lower piece. Both pieces are closed through the boundary of the Brillouin zone. The piece of (Formula presented) with nonzero (Formula presented) should also be situated sufficiently far from the region of momentum space, where there is no chiral symmetry. We took into account that in the considered equilibrium systems (Formula presented) as well as (Formula presented) does not depend on (Formula presented). In turn, (Formula presented) in Eq. (3) should be substituted by the (Formula presented)-inverse with respect to (Formula presented), i.e. (Formula presented) that obeys Here the three-dimensional vector (Formula presented) parametrizes surface (Formula presented). Function (Formula presented) represents the dependence of Matsubara frequency on (Formula presented), the upper sign is to be chosen for the upper piece of (Formula presented), the lower sign-for the lower piece. Both pieces are closed through the boundary of the Brillouin zone. The piece of (Formula presented) with nonzero (Formula presented) should also be situated sufficiently far from the region of momentum space, where there is no chiral symmetry. We took into account that in the considered equilibrium systems (Formula presented) as well as (Formula presented) does not depend on (Formula presented). In turn, (Formula presented) in Eq. (3) should be substituted by the (Formula presented)-inverse with respect to (Formula presented), i.e. (Formula presented) that obeys (Formula presented) (B) With these clarifications Eq. (3) reads (for a general form of (Formula presented) ), (Formula presented) (C) with the three-dimensional volume (Formula presented), which is assumed to be large. The above clarifications refer also to the interacting version of Eq. (3) derived in our paper, i.e., to Eqs. (47), (80). These equations are valid in their original form only for (Formula presented) given by (Formula presented). For the case of the hypersurface of more general form, they are to be understood as Eq. (C), when (Formula presented) (D) while (Formula presented) is the (Formula presented)-inverse to (Formula presented). With this clarification, Eqs. (47), (80) represent topological invariants robust to both smooth modification of the system and smooth modification of the form of hypersurface (Formula presented). In both cases, the singularities should be avoided; i.e., modifying the system, one should not pass over a phase transition, while modifying (Formula presented) one should not cross the position of singularities that extends the notion of Fermi surface to the interacting nonhomogeneous systems.