Phase diagram of the anisotropic multichannel Kondo Hamiltonian revisited

Abstract

The phase diagram of the multichannel Kondo Hamiltonian with an $XXZ$ spin-exchange anisotropy is revisited, revealing a far richer fixed-point structure than previously appreciated. For a spin-$\frac{1}{2}$ impurity and $k>2$ conduction-electron channels, a second ferromagneticlike domain is found deep inside the antiferromagnetic regime. The new domain extends above a (typically large) critical longitudinal coupling ${J}_{z}^{*}>0$ and is separated from the antiferromagnetic domain by a second Kosterlitz-Thouless line. A similar line of stable ferromagneticlike fixed points with a residual isospin-$\frac{1}{2}$ local moment is shown to exist for large ${J}_{z}⪢\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}>0$ and arbitrary $k$ and $s$ obeying $\ensuremath{\mid}k\ensuremath{-}2s\ensuremath{\mid}>1$. Here, ${J}_{z}$ is the longitudinal spin-exchange coupling, ${J}_{\ensuremath{\perp}}$ is the transverse coupling, and $s$ is the impurity spin. Near the free-impurity fixed point, spin-exchange anisotropy generates a highly relevant term for $s>1∕2$ and arbitrary $k$. Depending on the sign of ${J}_{z}^{2}\ensuremath{-}{J}_{\ensuremath{\perp}}^{2}$ and the parity of $2s$, the system flows either to a conventional Fermi liquid with no residual degeneracy or to a $k$-channel, spin-$\frac{1}{2}$ Kondo effect or to a line of ferromagneticlike fixed points with a residual isospin-$\frac{1}{2}$ local moment. These results are obtained through a combination of perturbative renormalization-group techniques, Abelian bosonization, a strong-coupling expansion in $1∕{J}_{z}$, and explicit numerical renormalization-group calculations.