Generalized Wilson chain for solving multichannel quantum impurity problems

The numerical renormalization group is used to solve quantum impurity problems, which describe magnetic impurities in metals, nanodevices, and correlated materials within dynamical mean field theory. Here we present a simple generalization of the Wilson chain, which improves the scaling of computational cost with the number of conduction bands, bringing more complex problems within reach. The method is applied to calculate the $t$ matrix of the three-channel Kondo model at $T=0$, which shows universal crossovers near non-Fermi-liquid critical points. A nonintegrable three-impurity problem with three bands is also studied, revealing a rich phase diagram and novel screening and overscreening mechanisms.

Figure 1

NRG for a two-channel model. The impurity subsystem and Wilson chain orbitals are denoted by the circle and squares, respectively. Solid lines represent chain couplings; dashed boxes represent diagonalization and truncation. The energy scale decreases in the direction of the arrows. (a) Standard formulation: Channel orbitals are added symmetrically. (b) Interleaved formulation: Channel orbitals are added sequentially.

Figure 2

Scaling spectra $t_{\alpha }\left(\omega \right)$ for channel $\alpha =1,2,3$ at $T=0$ for the 3CK model, obtained using the interleaved NRG method. For parameters, see text. The $t$ matrices for equivalent channels have been averaged [24]. (a) vs $\omega /T_K^{3\mathrm{CK}}$, tuned to the critical point with ${\rho }_1{J}_1^c=0.05$, ${\rho }_2{J}_2^c\simeq 0.05088...,{\rho }_3{J}_3^c\simeq 0.05180...$ such that $T_K^{3\mathrm{CK}}\approx {10}^{-10}{D}_1$. The dotted-dashed line is for the 2CK model, with points obtained using standard NRG for equal bandwidths ($g=1$). (b) Crossover to FL fixed points due to perturbation $J_3=J_3^c+0.01\times T_K^{3\mathrm{CK}}$. The impurity flows to strong coupling [FL (SC)] with channel 3, while channels 1 and 2 decouple [FL (DC)]. (c) Crossover to the 2CK fixed point due to perturbation $J_3=J_3^c-0.01\times T_K^{3\mathrm{CK}}$. Channel 3 decouples, while the impurity is overscreened by channels 1 and 2. Asymptotes for (a)–(c) are given by Eqs. (9)–(11), and discussed further in the text.

Figure 3

Impurity contribution to entropy $S_{\mathrm{imp}}\left(T\right)$ vs $T/D$ for the 3IK model. Upper panel: $J/D=0.2$, $J^{\prime }/D=0.1$, tuning $K=K_c\pm \lambda T_K^c$ to approach the 3CK QCP (dotted line) from the 2CK phase ($K>K_c$, dashed lines) and from the Kondo-screened FL phase ($K<K_c$, solid lines). At the QCP, $K_c\simeq 7\times {10}^{-6}D$ and $T_K^c\approx {10}^{-10}D$. Plotted for $\lambda ={10}^4,{10}^3,{10}^2,10,1$ in the direction of the arrow. Circle points for $K=0$. Lower panel: Analogous plot for $J/D=0.1$, $J^{\prime }/D=0.2$. The 2IK QCP with $K_c\simeq {10}^{-9}D$, $T_K^c\approx {10}^{-9}D$ separates local-singlet and Kondo-screened phases. Plotted for $\lambda =1,{10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4}$ in the direction of the arrow. (a)–(d) illustrate the ground states on each side of the transition (see text), using solid ellipses for Kondo singlets, dashed ellipse for 2CK overscreening, and a dotted ellipse for the local singlet.