Fermi Surface and Antiferromagnetism in the Kondo Lattice: An Asymptotically Exact Solution in $d>1$ Dimensions

Interest in the heavy fermion metals has motivated us to examine the quantum phases and their Fermi surfaces within the Kondo lattice model. We demonstrate that the model is soluble asymptotically exactly in any dimension $d>1$, when the Kondo coupling is small compared with the RKKY interaction and in the presence of antiferromagnetic ordering. We show that the Kondo coupling is exactly marginal in the renormalization group sense, establishing the stability of an ordered phase with a small Fermi surface ${\mathrm{AF}}_S$. Our results have implications for the global phase diagram of the heavy fermion metals, suggesting a Lifshitz transition inside the antiferromagnetic region and providing a new perspective for a Kondo-destroying antiferromagnetic quantum critical point.

Figure 1

With the Fermi surface (FS) of the conduction-electron component not touching the antiferromagnetic zone boundary, only the uniform component ($\overrightarrow{q}\approx 0$) of the local moments can interact with two states near the FS, as shown in (a). The linear coupling involving the staggered component, $\overrightarrow{n}·{\overrightarrow{s}}_c$, is not kinematically favorable, as shown in (b).

Figure 2

The Feynman rules associate wavy lines with magnons ($\tilde{\pi }$ fields), and solid straight lines with itinerant electrons ($\psi$ fields). A slash through a boson line indicates a time derivative (i.e., $\dot{\tilde{\pi }}$). (a) Represents the four diagrams in ${\Gamma }_z$. (b) Describes the infinite number of spin-flip vertices ${\Gamma }_{\perp }$ involving an odd number of magnons.

Figure 3

(a) Shows the lowest order corrections to the vertices ${\Gamma }_z$ and ${\Gamma }_{\perp }$. (b) Is an example of a class of diagrams that do not contribute to the beta function.

Figure 4

Kinematics for the momentum-shell RG. Only the shaded region is integrated over.