Bosonization solution of the Kondo lattice in a Luttinger liquid

TL;DR

This work develops a bosonization-based analysis of a one-dimensional Kondo lattice embedded in a Luttinger liquid host and identifies two exactly solvable Toulouse points, $K_s=1/2$ and $K_s=2$, where refermionization yields bilinear Hamiltonians. At $K_s=1/2$, spinon-mediated interactions produce algebraic, RKKY-like inter-impurity correlations, while at $K_s=2$ the spin sector becomes gapped and impurities are locally Kondo-screened with no long-range coherence. The results demonstrate that the sign and strength of bulk spin interactions control the competition between Kondo screening and RKKY ordering, offering controlled benchmarks for 1D Kondo lattices and guiding extensions to higher dimensions and multichannel variants.

Abstract

We address the physics of a regular arrangement of independent magnetic impurities embedded in a band of interacting electrons. We focus on the one-dimensional case that can be studied using bosonization and in which the electron bulk is described by a Luttinger liquid. The impurity spins interact with the electrons via magnetic exchange that introduces the possibility of Kondo and RKKY physics. We find that for two special values of the interactions, the model can be refermionized as a non-interacting electron band hybridized with a regular array of resonant levels. These solvable limits provide access to impurity correlators that correspond to either extended algebraic order or local screening. A physical picture emerges of how the interelectron interactions can stabilize either Kondo or RKKY physics depending on the sign of the interaction.

Figures (3)

• Figure 1: Doniach's phase diagram. The crossing of the characteristic scales for Kondo screening [$\approx\sqrt{g}\exp(-1/g)$, where $g$ is the Kondo coupling $J$ multiplied by the electron density of states at the Fermi surface] and RKKY interactions [$\approx g^2/4$] indicates a competition between the tendency for the local moments to form singlets with the itinerant electrons or interact among themselves (mediated by the same itinerant electrons) to form a collective singlet. The latter corresponds to a long-range magnetically ordered state shaded under a Néel dome. Inset: Schematic depiction of the ordering diagram in a 3D Hubbard model (depicted at the mean field level, for a more accurate picture see Ref. staudt2000). The two ordering tendencies complement each other resulting in a dome region below both scales where one finds a long-range antiferromagnetic insulating phase.
• Figure 2: Regions plot in the $K_s$-$K_c$ plane indicating the term with the smallest scaling dimension. We are interested in the orange region where the incommensurate part of the spin-flip Kondo interaction, $H_\nparallel^\mathrm{inc} \to H_\nparallel^\mathrm{fs}$, is the dominant one. The different panels show how the relative importance of the different terms changes with the unitary transformation of Eq. (\ref{['eq:U']}). Notice the plot for $\lambda_\Theta={3}/{2}$ would be identical to the one in the middle panel.
• Figure 3: Plot of the impurity energy bands for the case of $M=1$ with $\tilde{d}=1$. The number of bands is $4M+3$, with $2M+2$ above zero and $2M+1$ below it.