Quantum Criticality Perspective on the Charging of Narrow Quantum-Dot Levels

Understanding the charging of exceptionally narrow levels in quantum dots in the presence of interactions remains a challenge within mesoscopic physics. We address this fundamental question in the generic model of a narrow level capacitively coupled to a broad one. Using bosonization we show that for arbitrary capacitive coupling charging can be described by an analogy to the magnetization in the anisotropic Kondo model, featuring a low-energy crossover scale that depends in a power-law fashion on the tunneling amplitude to the level. Explicit analytical expressions for the exponent are derived and confirmed by detailed numerical and functional renormalization-group calculations.

Figure 1

The exponent $\alpha$ computed using the NRG, FRG, and Eqs. (8). NRG parameters: ${\Gamma }_{+}/D=0.04$, $\Lambda =1.7$, and 2800 states are retained. Inset: The model system. Two localized QD levels are coupled by tunneling to separate baths. Spinless electrons residing on the two levels experience a Coulomb repulsion $U$.

Figure 2

The exponent $\beta$ computed using the NRG, FRG, and Eqs. (8). NRG parameters: ${\Gamma }_{+}/D=0.02$, $\Lambda =1.6$, and 2000 states retained. Inset: Representative NRG data for $\Omega$ vs $b$, along with the log-log fits used to extract $\beta$.

Figure 3

A scaling plot of $\Omega$ for two (fixed) ratios $U/{\Gamma }_{+}$ and different combinations of ${\Gamma }_{-}$ and $b$, obtained using the FRG. The coefficients $A$ and $B$ were extracted from the limiting cases where $b=0$ and ${\Gamma }_{-}=0$, respectively [25]. For clarity, the data for $U/{\Gamma }_{+}=5$ were multiplied by a constant as indicated by the arrow length. Dashed lines show the asymptotes $\mathcal{F}=1$ and $\mathcal{F}=x$.