Cooperation of different exchange mechanisms in confined magnetic systems

The diluted Kondo lattice model is investigated at strong antiferromagnetic local exchange couplings $J$, where almost-local Kondo clouds drastically restrict the motion of conduction electrons, giving rise to the possibility of quantum localization of conduction electrons for certain geometries of impurity spins. This localization may lead to the formation of local magnetic moments in the conduction-electron system, and the inverse indirect magnetic exchange (IIME) provided by virtual excitations of the Kondo singlets couples those local moments to the remaining electrons. Exemplarily, we study the one-dimensional two-impurity Kondo model with impurity spins near the chain ends, which supports the formation of conduction-electron magnetic moments at the edges of the chain for sufficiently strong $J$. Employing degenerate perturbation theory as well as analyzing spin gaps numerically by means of the density-matrix renormalization group, it is shown that the low-energy physics of the model can be well captured within an effective antiferromagnetic Ruderman–Kittel–Kasuya–Yosida-like two-spin model (“RKKY from IIME”) or within an effective central-spin model, depending on edge-spin distance and system size.

Figure 1

(a) Two-spin Kondo model: Impurity spins ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ at a distance $d=L-3$ couple via a strong antiferromagnetic exchange $J$ to the local spins ${\mathbf{s}}_{i_1}$ and ${\mathbf{s}}_{i_2}$ at sites $i_1=2$ and $i_2=L-1$ of a one-dimensional chain of $L$ sites. $t$ is the nearest-neighbor hopping in the half filled system of conduction electrons. (b) Effective model for $J\gg t$: The impurity spins form local Kondo singlets with ${\mathbf{s}}_{i_1}$ and ${\mathbf{s}}_{i_2}$. This leads to local-moment formation at the edge sites $i=1$ and $i=L$. An “inverse” indirect magnetic exchange with coupling constant $\alpha$ emerges and mediates a ferromagnetic coupling of the edge spins ${\mathbf{s}}_1$ and ${\mathbf{s}}_L$ to the rest of the conduction-electron sea. (c) Effective RKKY and effective central-spin model [20]: For even $L$, i.e., for odd distance $d^{\prime }=L-5$ between the edge spins in the effective model, the low-energy model is a two-spin antiferromagnetic RKKY model with RKKY exchange ${\alpha }^2$. For odd $L$ ($d^{\prime }$ even), the low-energy model is a central-spin model where ${\mathbf{s}}_1$ and ${\mathbf{s}}_L$ couple ferromagnetically with strength $\alpha$ to the (delocalized) spin ${\mathbf{s}}_{\mathrm{F}}$ of the singly occupied Fermi orbital $k_{\mathrm{F}}$. (d) Low-energy spectrum: For even $L$ (odd $d^{\prime }$), the ground state is a spin singlet; ${\Delta }_{\mathrm{s}}$ is the singlet-triplet excitation energy. For odd $L$ (even $d^{\prime }$), the ground state is a spin quartet. For noninteracting conduction electrons this quartet is degenerate with a doublet.

Figure 2

Spin gap ${\Delta }_{\mathrm{s}}$ as a function of the edge spin distance $d^{\prime }$ on a double-logarithmic scale for different couplings $J$ as indicated. Dots show DMRG results for even $L$ and odd edge spin distance $d^{\prime }=L-5$, i.e., antiferromagnetic coupling. Colored dashed lines show the spin gap as given by second-order perturbation theory in $\alpha /t$; see Eq. (8).

The black dashed line shows the expected $d^{\prime }$ dependence of ${\Delta }_s$ for large $d^{\prime }$ and large $J$. The inset shows the $J$ dependence of the spin gap. For different system sizes $L$, the DMRG data follow the expected power law ${\Delta }_{\mathrm{s}}\propto 1/J^6\propto {\alpha }^2$ at strong $J$. The nearest-neighbor hopping $t=1$ fixes the energy scale.

Figure 3

Spin gap ${\Delta }_{\mathrm{s}}$ as a function of the edge spin distance $d^{\prime }$ on a double-logarithmic scale for different couplings $J$ as indicated. Dots show DMRG results for odd $L$ and even edge-spin distance $d^{\prime }=L-5$, i.e., ferromagnetic coupling. Colored dashed lines show the spin gap as given by first-order perturbation theory in $\alpha /t$; see Eq. (13).

The black dashed line shows the expected $d^{\prime }$ dependence of ${\Delta }_s$ for large $d^{\prime }$ and large $J$. The inset shows the $J$ dependence of the spin gap. The DMRG data are approximately described by the expected power law ${\Delta }_{\mathrm{s}}\propto 1/J^3\propto \alpha$ at strong $J$. The nearest-neighbor hopping $t=1$ fixes the energy scale.