The Kondo effect in spin chains

Abstract

It is well known that the free electron Kondo problem can be described by aone-dimensional (1D) model because only the s-wave part of the electronic wavefunction isaffected by the Kondo coupling. Moreover, since only the spin degrees of freedomare 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 ariseswhen a magnetic impurity is coupled to the end of a gapless antiferromagneticJ1–J2 spin- chain, where J1(J2) is the (next) nearest neighbor coupling. Experimental realizations of such spin chainmodels are possible and, using various analytical and numerical techniques, wepresent a detailed and quantitative comparison between the usual free electronKondo model and such spin chain versions of the Kondo problem. For the gaplessJ1–J2 spin chain two cases are studied, with zero next nearest neighbor coupling,J2 = 0, and with a critical second-neighbor coupling,J2 = J2c. We first focus on the spin chain impurity model with a critical second-neighborantiferromagnetic exchange where a bulk marginal coupling present in the spin chain model forJ2<J2c vanishes. There, the usual Kondo physics is recovered in the spin chain model in the lowenergy regime (up to negligible corrections, dropping as powers of inverse length or energy). AtJ2c the spin chain model is not exactly solvable and we demonstrate the equivalenceto the Kondo problem by comparing density matrix renormalization groupcalculations on the frustrated spin chain model with exact Bethe ansatz calculationsof the electronic Kondo problem. We then analyze the nearest neighbor model(J2 = 0) where a new kind of Kondo effect occurs due to the presence of thebulk marginal coupling. This marginal coupling slightly alters theβ function for the Kondo coupling, leading to a slower variation of the Kondo temperature,TK,with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model(J2 = 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 theconnection to Kondo physics and in addition we present low temperature quantum MonteCarlo results for the impurity susceptibility that further support this connection.