Nonperturbative effects and indirect exchange interaction between quantum impurities on metallic (111) surfaces

The (111) surface of noble metals is usually treated as an isolated two-dimensional (2D) triangular lattice completely decoupled from the bulk. However, unlike in topological insulators, bulk bands also cross the Fermi level. We here introduce an effective tight-binding model that accurately reproduces results from first-principles calculations, accounting for both surface and bulk states. We numerically solve the many-body problem of two quantum impurities sitting on the surface by means of the density matrix renormalization group. By performing simulations in a star geometry, we are able to study the nonperturbative problem in the thermodynamic limit with machine precision accuracy. We find that there is a nontrivial competition between Kondo and RKKY physics and as a consequence, ferromagnetism is never developed, except at short distances. The bulk introduces a variation in the period of the RKKY interactions, and therefore the problem departs considerably from the simpler 2D case. In addition, screening and the magnitude of the effective indirect exchange are enhanced by the contributions from the bulk states.

Figure 1

Diagram of the chain with sites $A$, $B$, and $C$ and hoppings $t_1,t_2,t_3$, and $t_p$.

Figure 2

Band structure of our model copper system as obtained with the method described in the text. Red lines show bulk bands while the green line is the surface state. Blue points are theoretical values taken from Ref. [43].

Figure 3

LDOS for a site on the Shockley surface as well as a 2D triangular lattice with the same bandwidth, where zero energy corresponds to the Fermi level. The 2D results are in the thermodynamic limit, while the Shockley data correspond to 500 poles. Bulk states have very small weight below the Fermi level.

Figure 4

A representation of the star geometry. The blue circles represent sites in real space or the “seed” states in the Lanczos transformation. The new bath sites are shown in green and red. Green represents the relevant sites while red shows the discarded ones indicated by the dashed red line where the system is effectively truncated. The red sites are either doubly occupied (“2”) or empty (“0”). The orange arrows correspond to Kondo spin-1/2 impurities coupled via $J_K$.

Figure 5

(a) Orbital occupation as a function of on-site energy. Red vertical lines indicate where the system is truncated. Inset shows a zoomed-in region near the cutoff energy; only sites with occupation exactly 0 or 2 are discarded. Parameters are $J_K=1$ and $R=4$. (b) Block entanglement entropy between left and right half of the system, when orbitals are placed along a chain in increasing order by their local energy. In the DMRG simulation, the impurities are located at the position indicated by the arrow.

Figure 6

Spin-spin correlations as a function of window size or number of bath sites with the same parameters used in Fig. 5.

Figure 7

Spin-spin correlations as a function of the impurity separation for (a) triangular lattice and (b) 2D Shockley surface of a (111) metal. (c) Lindhard function for the corresponding lattices. All plots are along the nearest-neighbor direction, and the units of the coupling, $J_K$, are eV.

Figure 8

Energy gain as a function of the interimpurity distance as described in the text. The horizontal lines indicate the characteristic energy gains for forming two Kondo singlets $\mathrm{\Delta }=2E_K$ at infinite distance apart and $R=5$.

Figure 9

Spin-spin correlations for different values of $t_1/t_2$ for two spin $S=1/2$ impurities on the Shockley surface with $J_K=1.0$.