Gauge mean-field theories of the underscreened Kondo lattice
Various mean-field decoupling schemes have been introduced thus far to study the important and challenging problem of the spin-$1$ underscreened Kondo lattice where magnetic order can coexist with Kondo hybridization. We use a single control parameter $N$ and an unbiased variational ansatz to unify and connect the previously proposed decouplings to standard Read-Newns theory, where fluctuations are small in $1/N$. We compute the corrections around the large-$N$ limit. In particular, we make contact with Nozi\`eres strong-coupling theory by finding the residual ferromagnetic Hund interaction that decays logarithmically in the case of a heavy-fermion metal. We map out the ground state phase diagram as a function of $N$ and the Kondo coupling. We find crucial differences between the previously proposed mean-field theories in the strength of the hybridization and total magnetization of the coexistent phase. We show that, within our unifying variational theory, the previously proposed decouplings correspond to either taking $N=2$ from the start, or performing a $1/N$ expansion and then extrapolating to $N=2$. Finally, we summarize the generalization of our theory to $S>1$.