SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries

We show how the density-matrix numerical renormalization group method can be used in combination with non-Abelian symmetries such as $\text{SU}\left(N\right)$. The decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the $\text{SU}\left(3\right)$ symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.

Figure 1

The occupation number as a function of ${\varepsilon }_d$ and for different broadening parameters $\Delta$. The temperature is $T=0$ and the Coulomb interaction is fixed to $U=0.2W.$

Figure 2

NRG finite size spectrum at half filling and $T$ $=$ 0. Upper (lower) panel represents the even (odd) part of the spectrum. In both plots we represent the lowest 50 energy levels. The parameters are as follows: $\Lambda =2$, $U=0.2W$, ${\varepsilon }_d=-U$, and $U/\pi \Delta =5$. The convergence was reached after approximately 27 iterations.

Figure 3

(a) The normalized spectral function for the on-site creation operator $d_{\alpha }^{†}$ for two different fillings $n=1.5$ (half filling, red dashed) and $n=1$ (1/3 filling, black solid). In the singly occupied case, $n=1$, the dot is in the Kondo regime, and the resonance is shifted away from the Fermi energy, $\omega =0$. The value of the spectral function at the Fermi energy is determined by the Friedel sum rule, $\pi \Delta A\left(0\right)\to 3/4$. (b) Evolution of the normalized spectral function as a function of $U/\pi \Delta$ in the Kondo regime, for ${\varepsilon }_d/U=-0.5$. The deviation from the Friedel sum rule is less than 1$\%$.

Figure 4

(Upper panel) Spectral function of the spin operator $d_{\alpha }^{†}{d}_{\beta }$ on a logarithmic scale for two different fillings, $\langle n\rangle =1.5$ (half filling, red dashed) and $\langle n\rangle \approx 1$ (1/3 filling, black solid) for $U=0.2W$. In the singly occupied case, $\langle n\rangle \approx 1$, the dot is in the Kondo regime. The inset indicates the linear decay in $A_S\left(\omega \right)$ in the small frequency limit. The blue, dashed-dotted line is a guideline for the eye. (Lower panel) The spectral function of the spin operator in the Kondo regime, $\langle n\rangle \approx 1$, and for three different ratios $U/\pi \Delta$: $U/\pi \Delta =15$ (solid black line), $U/\pi \Delta =10$ (dashed red line), and $U/\pi \Delta =5$ (dashed-dotted green line). The inset indicates the universal scaling collapse of the spin spectral function in the Kondo regime.

Figure 5

Impurity states for the $\text{SU}\left(3\right)$ Anderson model in the Kondo (upper panel) and in the mixed valence regime (lower panel). The green shaded area emphasizes that quantum fluctuations mix the $n$ $=$ 1 and $n$ $=$ 0 levels. The arrows indicate possible direct transitions that are responsible for the peaks observed in the spectral functions of the superconducting ($d_{\alpha }^{†}{d}_{\beta }^{†}$) and trion ($d_1^{†}{d}_2^{†}{d}_3^{†}$) operators.

Figure 6

Spectral function of the superconducting operator for $U=0.2D$ and $\langle n\rangle =1.5$ (half filling, red dashed) and $\langle n\rangle \approx 1$ (1/3 filling, black solid).

Figure 7

The spectral function for the trion operator for two different fillings $n=1.5$ (half filling, red dashed) and $n=1$ (1/3 filling, black solid). The inset represents the same data on a logarithmic scale. The trionic spectral function scales as ${\omega }^2$ in the small energy limit.

Figure 8

Possible realization of $\text{SU}\left(3\right)$ Kondo states using four edge states and three quantum dots. The upper edge state is splitted and allows transport measurements between the two upper external leads.[9]

Figure 9

The conductance as a function of ${\varepsilon }_d$ and for different broadening parameters $\Delta$. The temperature is $T=0$ and the Coulomb interaction is fixed to $U=0.2W.$

Figure 10

Finite temperature conductance as a function of ${\varepsilon }_d$. (Inset) The temperature dependence of the conductance on a logarithmic scale, for two different fillings, $n=1.5$ (half filling, red dashed) and $n=1$ (1/3 filling, black solid). In both panels the Coulomb energy and the broadening to the leads were fixed to $U=0.2W$ and $U/\pi \Delta =5$.

Figure 11

Weight diagram of the $\text{SU}\left(3\right)$ irrep (nonorthogonal axes chosen to emphasize the symmetric structure of the irrep). Each dot represents a weight; we also indicate the Young tableaux of the corresponding states. The circled dot indicates a weight with inner multiplicity 2. The blue solid and dashed arrows represent the action of $J_{-}^{\left(1\right)}$ and $J_{-}^{\left(2\right)}$, respectively.