Density-operator approaches to transport through interacting quantum dots: Simplifications in fourth-order perturbation theory

Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling while accounting for the strong interactions in such systems. In this paper we report a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM). Central to the BR and RT is the generalized master equation for the reduced density matrix. We demonstrate the exact equivalence of these two techniques. By accounting for coherences (nondiagonal elements of the density matrix) between nonsecular states, we show how contributions to the transport kernels can be grouped in a physically meaningful way. This not only significantly reduces the numerical cost of evaluating the kernels but also yields expressions similar to those obtained in the TM approach, allowing for a detailed comparison. However, in the TM approach an ad hoc regularization procedure is required to cure spurious divergences in the expressions for the transition rates in the stationary (zero-frequency) limit. We show that these problems derive from incomplete cancellation of reducible contributions and do not occur in the BR and RT techniques, resulting in well-behaved expressions in the latter two cases. Additionally, we show that a standard regularization procedure of the TM rates employed in the literature does not correctly reproduce the BR and RT expressions. All the results apply to general quantum dot models and we present explicit rules for the simplified calculation of the zero-frequency kernels. Although we focus on fourth-order perturbation theory only, the results and implications generalize to higher orders. We illustrate our findings for the single impurity Anderson model with finite Coulomb interaction in a magnetic field.

Figure 1

(Color online) (a) Typical transport measurement circuit setup. The source-drain bias voltage $V_{\mathsf{b}}$ drives a current $I$ through the quantum dot. Applying a voltage $V_{\mathsf{g}}$ to the gate electrode (in this case a backgate beneath an insulating substrate) shifts the energy required to add an additional electron to the dot by $-e\alpha V_{\mathsf{g}}$, where $-e$ is the electron charge and $\alpha$ the gate coupling. (b) The measured differential conductance is usually presented in the form of a stability diagram, i.e., in a gate voltage–bias voltage plane. Here we show the characteristics for a single interacting, spin-split level, i.e., an Anderson model in magnetic field, which we return to in Secs. 5, 6 (see Fig. 8 for details).

Figure 2

Time ordering in a diagram associated with a fourth-order process. Every term arising from the BR approach, Eq. (31), can uniquely be translated into a specific diagram. This gives a one-to-one mapping between the BR and RT approaches. While the time order is crucial for this mapping, the resulting diagram can also directly be used to represent the Laplace transformed expression (see Fig. 3).

Figure 3

On the left we give the contributions to the time-evolution kernel ${\mathcal{K}}^{\left(2\right)}$ and ${\mathcal{K}}^{\left(4\right)}$ as they arise from the BR in the interaction picture. Here, $p=\pm ,\overline{p}=-p$. The corresponding diagrammatic RT representations, standing directly for the Laplace transformed contributions to $K^{\left(2\right)}$ and $K^{\left(4\right)}$, are shown on the right. All minus signs of the BR contributions are incorporated into the diagrams.

Figure 4

(Color online) Example of one possible fourth-order contribution to the kernel element $K_{b{b}^{\prime }}^{a{a}^{\prime }}$. Here, $\omega$ and ${\omega }^{\prime }$ are the energies which are assigned to the fermion lines. Unless the intermediate states of a diagram are labeled (like, for example, in Sec. 4 concerning reducible diagrams), it is implicitly meant to contain the sum over any intermediate state. As described in the main text, the quantities ${\delta }_i$ can by read off from cuts through the diagram. For our example: ${\delta }_0=E_{a^{\prime }}-E_a$, ${\delta }_1=i0+E_d-E_a+{\omega }^{\prime }$, ${\delta }_2=i0+E_{b^{\prime }}-E_a-\omega +{\omega }^{\prime }$, and ${\delta }_2=i0+E_{b^{\prime }}-E_c-\omega$.

Figure 5

(Color online) The 16 irreducible fourth-order diagrams, together with the eight reducible correction diagrams, can be sorted by topology into three diagram classes $\text{G}=\text{A}$, B, and C. Within each class, there are three groups G.(0), G.(1), and G.(2), labeled by the number of vertices on the upper part of the contour. In G.(1), the latest (leftmost) vertex distinguishes between stand-alone diagrams (s) and triple diagram subgroups (t).

Figure 6

(Color online) Example of the construction of gain and loss partners in second order. From the diagram rules it follows that the contribution of both diagrams are identical except for their sign.

Figure 7

(Color online) Example of the gain-loss chain among class C diagrams in fourth order.

Figure 8

(Color online) Stability diagram ($dI/d{V}_{\mathsf{b}}$ as a function of $V_{\mathsf{g}}$ and $V_{\mathsf{b}}$) for the Anderson impurity model in a magnetic field, see setup in Fig. 1a. The dot single-particle energy for spin projection $\sigma =\downarrow /\uparrow$ is given by ${ϵ}_{\sigma }={ϵ}_0-e\alpha V_{\mathsf{g}}\pm E_Z/2$. Here we set the Zeeman splitting to $E_Z=0.2U$, the level offset to ${ϵ}_0=0.25U$. We used symmetric tunnel rates associated with the leads, ${\Gamma }_s={\Gamma }_d=\Gamma =0.0004k_{B}T/\hslash$ and set the thermal energy to $k_{B}T=0.01U$. We assume equal capacitances associated with the source and drain tunnel junctions and apply the bias symmetrically. The Coulomb blockade regime, where the level is singly occupied, is the central triangular region within the shown gate range. The positions of the second-order transport resonances are marked by dashed gray lines, the fourth-order transport resonances (emphasized by white dotted lines) are (i) inelastic cotunneling, (ii) pair tunneling, and (iii) cotunneling assisted sequential tunneling (weakly seen continuation of the Zeeman-lifted $|\downarrow ⟩\to |0/2⟩$ excitation lines inside the Coulomb blockade region).

Figure 9

(Color online) Examples of diagrams with initial states $a=a^{\prime }=|\uparrow ⟩$. We determine the charge numbers of the allowed intermediate and the final states, using that at a vertex where a contraction line ends/starts the charge number changes by $\pm 1$. Similarly, the spin projection changes by $\pm \sigma /2$ at this vertex, where $\sigma /2$ is the electrode spin index of the contraction. Using this restriction, we find that the group A.(2) can only contribute to inelastic cotunneling $\left(b=b^{\prime }=|\downarrow ⟩\right)$ while C.(2) also allows the elastic process $\left(b=b^{\prime }=|\uparrow ⟩\right)$. Groups B.(1) and C.(1) yield broadening and level renormalization of sequential tunneling processes, like $|\uparrow ⟩\to |0⟩$: B.(1) accounts for the possibility of a charge fluctuation in the initial state $\left(|\uparrow ⟩\to |0⟩\to |\uparrow ⟩\right)$, C.(1) for a charge fluctuation in the final state $\left(|0⟩\to |\sigma ⟩\to |0⟩\right)$.

Figure 10

(Color online) Absolute errors occurring when neglecting contributions of a certain diagram class. Because for classes B and C, the corrections along the resonance lines (marked by gray dotted lines) exceed the ones in between by orders of magnitude, the color scales are chosen logarithmic both in the positive and in the negative regimes, i.e., ${}^{10}\text{l}\text{og}\left|\left(dI/d{V}_{\mathsf{b}}\right)-{\left(dI/d{V}_{\mathsf{b}}\right)}_{\text{approx}}\right|$. Hereby, white color indicates that the contribution from the specific diagram class lies below the threshold for the logarithmic red/blue color scale applied for positive/negative error.

Figure 11

(Color online) Top: the stability diagram for a single-level quantum dot as in Fig. 8 calculated using the TM approach. Middle: absolute deviation between the nonlinear conductance calculated with the GME and TM approaches, in logarithmic scale, i.e., ${}^{10}\text{l}\text{og}\left|\left(d{I}_{\text{GME}}/d{V}_{\mathsf{b}}\right)-\left(d{I}_{\text{TM}}/d{V}_{\mathsf{b}}\right)\right|$, with color coding just as in Fig. 10. Bottom: relative deviation between the nonlinear conductance calculated with the GME and TM approaches, $\left[\left(d{I}_{\text{GME}}/d{V}_{\mathsf{b}}\right)-(d{I}_{\text{TM}}/d{V}_{\mathsf{b}}\right)]/\left(d{I}_{\text{GME}}/d{V}_{\mathsf{b}}\right)$. Although the upper panel seems similar to the GME result in Fig. 8, indeed large relative errors exist in the vicinity of resonances and throughout the regime where transport is not suppressed by Coulomb blockade, in particular, when electron pair tunneling in the single-electron tunneling regime (Ref. 31) is energetically allowed.

Figure 12

(Color online) From the representative diagram, all members of the triple group can be constructed, exemplified here by subgroup C.(2)(t).