Spin-charge separation and simultaneous spin and charge Kondo effect

We study the spin-charge separation in a Kondo-like model for an impurity with a spin and a charge (isospin) degree of freedom coupled to a single conduction channel (the “spin-charge” Kondo model). We show that the spin and charge Kondo effects can occur simultaneously at any coupling strength. In the continuum (wide-band or weak coupling) limit, the Kondo screening in each sector is independent, while at finite bandwidth and strong coupling the lattice effects lead to a renormalization of the effective Kondo exchange constants; nevertheless, universal spin and charge Kondo effects still occur. We find similar behavior in the two-impurity Anderson model with positive and negative electron-electron interaction and in the two-impurity Anderson-Holstein model with a single phonon mode. We comment on the applicability of such models to describe the conductance of deformable molecules with a local magnetic moment.

Figure 1

(Color online) Schematic representations of models discussed in the paper. (a) Spin-charge Kondo model. (b) Two-impurity Anderson model with positive and negative electron-electron interaction. (c) Single-impurity Anderson-Holstein model. (d) Two-impurity Anderson-Holstein model with electron repulsion in one orbital and electron-phonon coupling in the other.

Figure 2

(Color online) Impurity susceptibilities and entropy for the spin-charge Kondo model. Filled symbols are a fit to the exact Bethe-ansatz results for susceptibility. From this fit we deduce $T_K^S=T_K^C=2.7\times {10}^{-7}D$, in approximate agreement with $T_K=D\sqrt{\rho J}\exp (-1∕\rho J)=4.4\times {10}^{-7}D$.

Figure 3

(Color online) (a) Spin susceptibilities in the SCKM model with $J_C=J_S$ (solid lines) compared with spin susceptibilities in the corresponding SKM with the same Kondo exchange constant $J_S$ (dashed lines). (b) Effective exchange constant $J_{S\mathrm{eff}}$ as a function of $J_S$ for the case $J_C=J_S$ (red, circles), compared with the single-impurity results (black, squares). Blue straight line is a fit to the single-impurity results and serves as a guide to the eye. (c) Effective exchange constant $J_{S\mathrm{eff}}$ as a function of the charge Kondo exchange constant $J_C$ while keeping the bare spin exchange constant $J_S$ constant at $\rho J_S=0.1$ (black, circles) or $\rho J_S=0.05$ (red, squares).

Figure 4

(Color online) Schematic representation of the strong coupling fixed point Hamiltonian of the spin-charge Kondo model for (a) $J_S>J_C$ and (b) $J_C>J_S$. For $J_C=J_S$, the fixed point corresponds to the symmetric combination of (a) and (b).

Figure 5

(Color online) (a) Spin susceptibility for 2IAM (full lines). We choose $U_1=-U_2=U$ and ${\Gamma }_1={\Gamma }_2=\Gamma$. For comparison we plot results of the SIAM with the same $\Gamma$ and $U$ (dashed lines). Parameters are $U∕D=0.01$ and $\Gamma ∕U=0.02,0.03,0.04,0.05,0.1,0.2$, and $0.5$ (left to right). (b) Charge fluctuations and correlations in 2IAM for increasing hybridization strength $\Gamma$.

Figure 6

(Color online) Numerical renormalization group eigenvalue flows for the case when (a) isospin impurity only, (b) spin impurity only, (c) both impurities are present. $U_1=U=0.01$, ${\Gamma }_{1,2}∕U=0.05$ [unless turned to zero in (a) and (b)], $U_2=-3U$. The levels are labeled by the total isospin and spin quantum numbers $(I,S)$.

Figure 7

(Color online) Impurity susceptibilities and entropy for the two-impurity Anderson model. Impurity $1$ has $U_1=U=0.01$, $\Gamma ∕U=0.05$. Impurity $2$ has $U_2=-3U=-0.03$ and equal hybridization $\Gamma$. $T_K^S∕D\sim 1.1\times {10}^{-6}$, $T_K^C∕D\sim 2.9\times {10}^{-13}$. Note that the magnetic susceptibility curves nearly overlap for all values of $\delta$.

Figure 8

(Color online) Diagonal ($A_1$, $A_2$) and out-of-diagonal spectral density $\left(A_{12}\right)$ of the two impurity Anderson model for different asymmetry parameters $x$. We plot only the positive frequencies. Due to the particle-hole symmetry, we have $A_i\left(\omega \right)=A_i(-\omega )$ and $A_{12}\left(\omega \right)=-A_{21}(-\omega )$.

Figure 9

(Color online) Diagonal ($A_1$, $A_2$) and out-of-diagonal spectral densities $\left(A_{12}\right)$ of the two impurity Anderson model. Same parameters as in Fig. 7.

Figure 10

(Color online) Impurity susceptibilities and entropy for the single-impurity Anderson-Holstein model for increasing $e$-ph interaction strength $g$. $U∕D=0.01$, $\Gamma ∕U=0.04$, ${\omega }_0∕D=0.1$.

Figure 11

(Color online) Impurity susceptibilities and entropy for the two-impurity model with one Holstein impurity with ${\omega }_0∕D=0.1$ and varying $e$-ph interaction strength $g$, and one Anderson impurity with $U∕D=0.01$. Both impurities are coupled with $\Gamma ∕D=0.0004$ to the conduction band.