  The free electron Kondo problem can be described by a one-dimensional (1D)
model because only the s-wave part of the electronic wave-function is affected
by the Kondo coupling. Since only the spin degrees of freedom are involved in
the Kondo interaction, and due to spin-charge separation in 1D, the universal
low energy long distance physics of the Kondo model also arises when a magnetic
impurity is coupled to the end of a gap-less antiferromagnetic J_1-J_2 spin-1/2
chain, where J_1(J_2) is the (next-)nearest neighbor coupling. Experimental
realizations of such spin chain models are possible and using various
analytical and numerical techniques, we present a detailed and quantitative
comparison between the usual free electron Kondo model and such spin chain
versions of the Kondo problem. For the gap-less J_1-J_2 spin chain two cases
are studied, with zero next nearest neighbor coupling, J_2=0, and with a
critical second neighbor coupling, J_2=J_2c. We first focus on the spin chain
impurity model at J_2c~0.2412 where a bulk marginal coupling present in the
spin chain model for J_2<J_2c vanishes. At J_2c, the usual Kondo physics is
recovered in the spin chain model in the low energy regime. We then analyze the
nearest-neighbor model (J_2=0) where a new kind of Kondo effect occurs due to
the presence of the bulk marginal coupling. This marginal coupling leads to a
slower variation of the Kondo temperature T_K with the bare Kondo coupling. In
the exact Bethe ansatz solution of this spin chain impurity model (J_2=0) Frahm
and Zvyagin noted this relation as well as the connection to the Kondo problem.
Here, we provide further evidence for the connection to Kondo physics and
present low temperature QMC results for the impurity susceptibility that
further support this connection.
