The Kondo effect in spin chains

It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, 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 gapless 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 gapless 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(2)(C). We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange J(2)(C) similar or equal to 0.2412 where a bulk marginal coupling present in the spin chain model for J(2) < J(2)(C) vanishes. There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J(2)(C) the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. 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 slightly alters the beta function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T(K), with the bare Kondo coupling. In their 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, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.