Real-time renormalization group and cutoff scales in nonequilibrium applied to an arbitrary quantum dot in the Coulomb blockade regime

We apply the real-time renormalization group (RG) in nonequilibrium to an arbitrary quantum dot in the Coulomb blockade regime. Within one-loop RG equations, we include self-consistently the kernel governing the dynamics of the reduced density matrix of the dot. As a result, we find that relaxation and dephasing rates generically cut off the RG flow. In addition, we include all other cutoff scales defined by temperature, energy excitations, frequency, and voltage. We apply the formalism to transport through single molecular magnets, realized by the fully anisotropic Kondo model (with three different exchange couplings $J_x$, $J_y$, and $J_z$) in a magnetic field $h_z$. We calculate the differential conductance as function of bias voltage $V$ and discuss a quantum phase transition which can be tuned by changing the sign of $J_x{J}_y{J}_z$ via the anisotropy parameters. Finally, we calculate the noise $S\left(\Omega \right)$ at finite frequency $\Omega$ for the isotropic Kondo model and find that the dephasing rate determines the height of the shoulders in $dS\left(\Omega \right)∕d\Omega$ near $\Omega =V$.

Figure 1

Example of a sequence of two irreducible blocks. The double line represents time propagation in Liouville space of the dot (time increases to the right). The black dots represent the vertices $G_{{11}^{\prime }}^{p{p}^{\prime }}$. The lines connecting the vertices are the reservoir contractions. Whereas the auxiliary vertical lines $a$ and $c$ hit reservoir contractions, line $b$ does not and separates the two irreducible blocks.

Figure 2

The RG diagrams determining the renormalization of (a) $G$ and (b) $L_{\mathrm{D}}$. The cross indicates differentiation of the lead contractions with respect to $\Lambda$. Time ordering is defined with respect to the middle time.

Figure 3

Generation of a transition rate from state $s$ to $s^{\prime }$ by the RG (for simplicity, we consider two leads L, R with $V={\mu }_{\mathrm{L}}-{\mu }_{\mathrm{R}}$ and the case $E_s=E_{s^{\prime }}$ here). Energy-conserving transitions from one of the intervals $d\Lambda$ which are integrated out to the other lead or vice versa can only occur in situation (b).

Figure 4

The differential conductance as function of bias voltage at $h_z={10}^{-4}$ for different values of $B_2$ and $B_4$. The easy-axis anisotropy of the molecule is $D=0.05$ and the coupling to the leads $J=0.01$. We have set ${\Lambda }_0=1$.

Figure 5

Energy landscape and coupling of the states of a molecular magnet with $S=7∕2$. The longitudinal anisotropy parameter $D$ determines the height of the parabola $-D{S}_z^2$. The transverse anisotropy constants $B_2$ and $B_4$ couple every second or fourth state. For a pure $B_4$ term, this leads to $⟨+\mid S_{+}\mid -⟩$ being zero or finite depending on the size of $B_4$ since different states form the ground states ∣±⟩.

Figure 6

Derivative of the noise for the isotropic Kondo model without magnetic field at $V=50T_{K,a=0}$. The asymmetry of the couplings is described by the asymmetry parameter $a$, where $J_{\mathrm{L},\mathrm{R}}=J_0(1\pm a)$. Inset: Noise for symmetric couplings and $V=50T_K$ (solid line), $V=100T_K$ (dotted line), and $V=150T_K$ (dash-dotted line), showing a dip at $\Omega =\pm V$.

Figure 7

Left: Interpretation of the finite frequency noise. It corresponds to the sum of two currents with different voltages $\mid V\pm \Omega \mid$ and different renormalized couplings. If both couplings were bare, i.e., not renormalized and therefore equal, the sum would be independent of $\Omega$ for $0<\Omega <V$. Right: For $V<\Omega$, the sum increases with $\Omega$ even for bare couplings because the absolute value of the currents is important.