Exploring the Kondo Effect of an Extended Impurity with Chains of Co Adatoms in a Magnetic Field

Motivated by recent STM experiments, we explore the magnetic field induced Kondo effect that takes place at symmetry protected level crossings in finite Co adatom chains. We argue that the effective two-level system realized at a level crossing acts as an extended impurity coupled to the conduction electrons of the substrate by a distribution of Kondo couplings at the sites of the chain. Using auxiliary-field quantum Monte Carlo simulations, which quantitatively reproduce the field dependence of the zero-bias signal, we show that a proper Kondo resonance is present at the sites where the effective Kondo coupling dominates. Our modeling and numerical simulations provide a theoretical basis for the interpretation of the STM spectrum in terms of level crossings of the Co adatom chains.

Figure 1

The spectral function computed using stochastic analytical continuation algorithm [30] at given level crossings up to $L=4$. For $L=1$ we choose $J_k/t=1.5$ and for $L>1$ we choose $J_h/t=1.8$ and $J_k/t=2$. The corresponding Kondo scale is extracted in Fig. 4 of the Supplemental Material [31].

Figure 2

The $\tilde{d}$-spectral function at $\omega =0$ computed by QMC simulations for a Heisenberg chain in a field together with the zero bias conductance measured in the STM experiment (in atomic units) for an $XXZ$ chain in transverse field [11]. The magnetic field axis is normalized by the maximum values in both cases. The continuous gray vertical lines and the dashed gray lines denote the expected exact positions of the level crossings for a Heisenberg chain and for an $XXZ$ one [11], respectively.

Figure 3

Exact zero-bias conductance as a function of temperature normalized by the corresponding Kondo scale at each level crossing up to $L=4$.

Figure 4

A chain of Co adatoms on a substrate implement, at a level crossing, the Kondo model of an extended impurity.