Quantum phase transition in a gapped Anderson model: A numerical renormalization group study

We use the numerical renormalization group method to investigate the spectral properties of a single-impurity Anderson model with a gap $\delta$ across the Fermi level in the conduction-electron spectrum. For any finite $\delta >0$, at half filling the ground state of the system is always a doublet. Away from half filling a quantum phase transition (QPT) occurs as function of the gap value $\delta$, and the system evolves from the strong-coupling (SC) Kondo-type state, corresponding to $\delta <{\delta }_C$ toward a localized moment (LM) regime for $\delta >{\delta }_C$. The opening of the gap leads to the formation of one (two) bound states when the system is in the SC (LM) regime. The evolution across the QPT of their positions and the corresponding weights together with the dynamic properties of the model are investigated.

Figure 1

Renormalization group flow diagram at $T=0$ for the gapped Anderson model, when the system is away from half filling. When $\delta <{\delta }_C$ the SC fixed point is stable while for $\delta >{\delta }_C$ the LM becomes the stable fixed point. At half filling ${\delta }_C=0$ so LM is stable irrespective of the value $\delta >0$.

Figure 2

(Color online) Spectral function for the symmetrical model for different gap values. The parameters used are $-{ϵ}_d=\frac{U}{2}=0.2$, $\Gamma =0.04$. $T_K^0$ is the Kondo temperature in the absence of the gap. The up arrows indicate the positions of the bound states in each case, their magnitude being proportional to the corresponding weight rescaled with the gap value, i.e., $\pi \Gamma W_b/\delta$. Only the positive frequency is plotted, the spectrum for negative energies being symmetric.

Figure 3

(Color online) The positions of the bound states and the corresponding spectral weights rescaled with the Kondo temperature $T_K^0$ (top layer) and with the gap value $\delta$ (bottom layer).

Figure 4

(Color online) Phase diagram indicating the separation between SC and LM regimes. The parameter $\eta =1+2{ϵ}_d/U$ describes the asymmetry of the system. At half filling, the critical gap is ${\delta }_C=0$, and the system is always in the LM regime. The solid lines correspond to the analytical results obtained within the local-moment approach (see Ref. 16) with similar rescaled coupling $\tilde{U}=U/\pi \Gamma$. The inset displays the shift of the phase boundary line as function of $\Lambda$ for $0.5<\eta <0.75$.

Figure 5

Spectral functions for the asymmetrical case, close to the Fermi energy. In the upper panel the system is in the strong-coupling regime, characterized by $\delta ⪡T_K^0$. In the lower panel the system is in the local-moment regime with the gap $\delta$ of the order of $T_K^0$ or larger. The arrows indicate the positions of the bound states and their amplitude is rescaled to $\pi \Gamma W_b/\delta$. In the SC regime a single resonance develops while in the LM regime there are two. The dotted line in the left upper panel is the spectral function for the normal Anderson model.

Figure 6

(Color online) Typical evolution of the bound-state energies (left panel) and the corresponding weights (right panel) across the SC-LM quantum phase transition. In the left panel, the circles (squares) represent the positions of the bound states in the SC(LM) regime. The crosses in the right panel indicate the sum of the weights of two bound states formed in the LM regime. The arrows point to the critical gap ${\delta }_C$ where transition occurs.