Generalized-gradient approximations with non-vanishing exchange-correlation magnetic torque

The description of systems of interacting electrons in the presence of magnetic fields, within spin-density functional theory (SDFT), requires the non-collinear magnetization density vector $\mathbf{m}(\mathbf{r})$ to be used as basic variable, along with the particle density $n(\mathbf{r})$. Futhermore, for a meaningful description of spin-dynamics, the magnetization density and its conjugate exchange-correlation (xc) field $\mathbf{B_{xc}}(\mathbf{r})$ must not be constrained to be locally parallel at every point in space. It is well known that the local density approximation (LDA) cannot, by construction, provide such a non-collinear configuration of the two vector fields. Here we show how popular generalized gradient approximations (GGAs), developed assuming collinear spin-density, can be used to describe non-collinear magnetization states, including the presence of non-vanishing local torque between $\mathbf{m}(\mathbf{r})$ and $\mathbf{B_{xc}}(\mathbf{r})$. Unlike previous attempts to extend the use of collinear GGAs to the domain of non-collinear magnetization densities, the approach we introduce is invariant with respect of spin-rotations, globally satisfies the \emph{zero-torque theorem}, reduces to the proper collinear limit and is numerically stable.