SU(4) Kondo Effect in Carbon Nanotubes

We investigate theoretically the nonequilibrium transport properties of carbon nanotube quantum dots. Owing to the two-dimensional band structure of graphene, a double orbital degeneracy plays the role of a pseudospin, which is entangled with the spin. Quantum fluctuations between these 4 degrees of freedom result in an SU(4) Kondo effect at low temperatures. This exotic Kondo effect manifests as a four-peak splitting in the nonlinear conductance when an axial magnetic field is applied.

Figure 1

NRG results for the total spectral density $A_{\mathrm{tot}}(\omega )$. The solid line is for the case of ${\Delta }_{\mathrm{orb}}={\Delta }_Z=0$, the dashed line for ${\Delta }_{\mathrm{orb}}=8T_K^{\mathrm{SU}(4)}$ ($T_K^{\mathrm{SU}(4)}=0.0133{\Gamma }_0$) with ${\Delta }_Z\approx 0$, and the dash-dotted line for ${\Delta }_{\mathrm{orb}}=4{\Delta }_Z=8T_K^{\mathrm{SU}(4)}$. (Parameters: ${ϵ}_d=-10{\Gamma }_0$, $U=200{\Gamma }_0$, ${\Gamma }_0=0.01D$, and $\delta \Gamma =0$.) Left inset: Zoom of the four-peak splitting in $A_{\mathrm{tot}}(\omega )$. Right inset: Comparison of the SU(4) (solid line) and the single-level SU(2) (dotted line) Kondo models. (Parameters: ${ϵ}_d=-5{\Gamma }_0$, $U=500{\Gamma }_0$, ${\Gamma }_0=0.02D$, and $\delta \Gamma =0$.)

Figure 2

$\mathrm{\text{NCA}}+\mathrm{\text{EOM}}$ results for the SU(4) Kondo model. (a) $A_{\mathrm{tot}}(\omega )$ and (b) $dI/dV$ versus $eV$ for different magnetic fields ranging from ${\Delta }_{\mathrm{orb}}=0.5T_K^{\mathrm{SU}(4)}$ (bottom curve) to ${\Delta }_{\mathrm{orb}}=1.5T_K^{\mathrm{SU}(4)}$ (top curve). (The curves are shifted vertically for clarity.) When $B_{\parallel }\ne 0$, the Kondo resonance splits due to the removal of both spin and orbital degeneracies. The rest of parameters are ${ϵ}_d=-4{\Gamma }_0$ and $T=0.003{\Gamma }_0$ with $\delta \Gamma =0$. The quantum dot is symmetrically coupled to two leads (${\Gamma }_L={\Gamma }_R={\Gamma }_0$) consisting of Lorentzian bands of width $2D=20{\Gamma }_0$. Inset: Schematic representation of the allowed transitions. (1) Intraorbital with spin flip. (2) Interorbital without spin flip. (3) Interorbital with spin flip.

Figure 3

NRG results for the total spectral density comparing (a) the SU(4) Kondo model ($\delta \Gamma =0$) and (b) the two-level SU(2) Kondo model ($\delta \Gamma ={\Gamma }_0$). Parameters are ${\Delta }_{\mathrm{orb}}=5{\Delta }_Z=15T_K^{\mathrm{SU}(4)}$ ($\approx 0.2{\Gamma }_0$), ${ϵ}_d=-10{\Gamma }_0$, $U=1000{\Gamma }_0$, and ${\Gamma }_0=0.01D$.