Interaction-induced local moments in parallel quantum dots within the functional renormalization group approach

We propose a version of the functional renormalization-group (fRG) approach, which is, due to including Litim-type cutoff and switching off (or reducing) the magnetic field during fRG flow, capable of describing a singular Fermi-liquid (SFL) phase, formed due to the presence of local moments in quantum dot structures. The proposed scheme allows one to describe the first-order quantum phase transition from the “singular” to the “regular” paramagnetic phase with applied gate voltage to parallel quantum dots, symmetrically coupled to leads, and shows sizable spin splitting of electronic states in the SFL phase in the limit of vanishing magnetic field $H\to 0$; the calculated conductance shows good agreement with the results of the numerical renormalization group. Using the proposed fRG approach with the counterterm, we also show that for asymmetric coupling of the leads to the dots the SFL behavior similar to that for the symmetric case persists, but with occupation numbers, effective energy levels, and conductance changing continuously through the quantum phase transition into the SFL phase.

Figure 1

Left: Schematic representation of parallel double quantum dots (DQ1 and DQ2) connected to two left (L) and right (R) leads by equal tunneling matrix elements $t_i^{\alpha }$. Right: Schematic representation of a configuration of parallel quantum dots, symmetrically coupled to the leads, obtained as a result of the transformation (5). The vertical up-down arrow indicates the presence of two-particle interactions between quantum dots [see Eq. (17)].

Figure 2

Linear conductance $2G/G_0$ $(G_0=2e^2/h)$ at $T=0$ as a function of magnetic field $H$ for the symmetric case ${\mathrm{\Gamma }}_i^{\alpha }=U/4$ and $V_g=0$. Dashed green and dash-dotted blue lines correspond to fRG approximation in the sharp cutoff and Litim-type cutoff scheme, respectively. Solid black upper line: NRG calculation. Solid red line: fRG approach with the counterterm ${\chi }_1^{\mathrm{\Lambda }}$, Eq. (16) with $\tilde{H}=0.1U$ and ${\mathrm{\Lambda }}_c=0.05U$.

Figure 3

Renormalization of the energy levels ${\epsilon }_{\sigma }^{\mathrm{\Lambda }}={\epsilon }_{1\left(2\right),\sigma }^{\mathrm{\Lambda }}$ with spin up $\sigma =\uparrow$ (lower curves) and down $\sigma =\downarrow$ (upper curves) in the fRG approximation with different counterterms for ${\mathrm{\Gamma }}_i^{\alpha }=U/4$ (note that in this case ${\epsilon }_{1,\sigma }^{\mathrm{\Lambda }}={\epsilon }_{2,\sigma }^{\mathrm{\Lambda }})$, and $H=V_g=0$. Solid (black) line and dash-dot-dotted (red) lines: The linear counterterm $({\mathrm{\Lambda }}_c=\tilde{H}/2)$ with $\tilde{H}=0.2U$ and $0.02U$, respectively. Dashed (green) and dash-dotted (blue) lines: Exponential counterterm $({\mathrm{\Lambda }}_c=10{\mathrm{\Lambda }}_0=0.05U)$ with $\tilde{H}=0.2U$ and $0.1U$, respectively.

Figure 4

Upper panel: Effective energy levels ${\epsilon }_{1\left(2\right),\sigma }^{\mathrm{\Lambda }\to 0}$ (thick solid (red) and dashed (black) line for $\sigma =\uparrow ,\downarrow$, respectively) and off-diagonal self-energies (effective hopping between the levels) ${\mathrm{\Sigma }}_{12,\sigma }^{\mathrm{\Lambda }\to 0}$ [thin solid (green) and dashed (blue) line for $\sigma =\uparrow ,\downarrow ]$ in units of $U$ as a function of the gate voltage for ${\mathrm{\Gamma }}_i^{\alpha }=U/4$ and $H\to 0$. Lower panel: The average occupation numbers $\langle n_{1\left(2\right),\sigma }\rangle$ [solid (red) line and dashed (black) line for $\sigma =\uparrow ,\downarrow ]$ as a function of the gate voltage for the same parameters. The calculations were performed within the fRG approach with the linear counterterm $(\tilde{H}/U=0.1,{\mathrm{\Lambda }}_c/U=0.05)$.

Figure 5

(a, b) The occupation numbers $\langle n_{e\left(0\right)}\rangle$ and the average square of magnetic moment $\langle {\mathbf{S}}_{e\left(0\right)}^2\rangle$ in the even [dashed (black) lines] and odd [solid (red) lines] orbitals, as well as the average of the square of the total spin $\langle {\mathbf{S}}_t^2\rangle =\langle {\left({\mathbf{S}}_{\mathbf{1}}+{\mathbf{S}}_{\mathbf{2}}\right)}^2\rangle$ [dash-dotted (blue) line] as a function of the gate voltage for ${\mathrm{\Gamma }}_i^{\alpha }=U/4$. (c) The interaction energy $E_{\mathrm{int}}$ of the double quantum dot system as a function of $V_g/U$.

Figure 6

The dependence of linear conductance $G$ on gate voltage $V_g$ for ${\mathrm{\Gamma }}_i^{\alpha }=U/4$ (upper plot), ${\mathrm{\Gamma }}_i^{\alpha }=U/2$ (middle plot), and ${\mathrm{\Gamma }}_i^{\alpha }=U/6$ (lower plot) at $H\to 0$ (and temperature $T=0)$ within the fRG approach with the linear counterterm $(\tilde{H}/U=0.1,{\mathrm{\Lambda }}_c/U=0.05$, solid black line) and NRG calculation (dashed red line). Dash-dotted green and dash-dash-dotted blue lines on upper plot are fRG approaches in the sharp cutoff and Litim-type cutoff scheme without counterterm.

Figure 7

The conductance (a), effective energy levels (b), ${\epsilon }_{1,\sigma }^{\mathrm{\Lambda }\to 0}$ [thick dashed (red) line for $\sigma =\uparrow$ and thick solid (black) line for $\sigma =\downarrow ]$ and ${\epsilon }_{2,\sigma }^{\mathrm{\Lambda }\to 0}$ [thin dashed (blue) line for $\sigma =\uparrow$ and thin solid (green) line for $\sigma =\downarrow ]$, as well as the spin-resolved average occupation numbers (c) of each quantum dot $\langle n_{1\left(2\right),\sigma }\rangle$ [with the same notation of the curves as on (b)] in the asymmetric case with ${\mathrm{\Gamma }}_1^L={\mathrm{\Gamma }}_1^R=U/3$ and ${\mathrm{\Gamma }}_2^L={\mathrm{\Gamma }}_2^R=U/6$ obtained by the fRG approach with the linear counterterm $(\tilde{H}/U=0.1,{\mathrm{\Lambda }}_c/U=0.05)$.