Feynman-Vernon influence functional approach to quantum transport in interacting nanojunctions: An analytical hierarchical study

We present a nonperturbative and formally exact approach for the charge transport in interacting nanojunctions based on a real-time path-integral formulation of the reduced system dynamics. For reservoirs of noninteracting fermions, the exact trace over the leads' degrees of freedom results in the time-nonlocal Feynman-Vernon influence functional, a functional of the Grassmann-valued paths of the nanojunction, which induces correlations among the tunneling transitions in and out of the nanojunction. An expansion of the influence functional in terms of the number of tunneling transitions, and integration of the Grassmann variables between the tunneling times, allows us to obtain a still exact generalized master equation for the populations of the reduced density matrix in the occupation-number representation, as well as a formally exact expression for the current. By borrowing the nomenclature of the famous spin-boson model, we parametrize the two-state dynamics of each single-particle fermionic degree of freedom, in the occupation-number representation, in terms of blips and sojourns. We apply our formalism to the exactly solvable resonant level model (RLM) and to the single-impurity Anderson model (SIAM), the latter being a prototype system for studying strong correlations. For both systems, we demonstrate a hierarchical diagrammatic structure. While the hierarchy closes at the second tier for the RLM, this is not the case for the interacting SIAM. Upon inspection of the current kernel, known results from various perturbative and nonperturbative approximation schemes to quantum transport in the SIAM are recovered. Finally, a noncrossing approximation for the hierarchical kernel is developed, which enables us to systematically decrease temperature at each next level of the approximation. Analytical results for a simplified fourth-tier scheme are presented both in equilibrium and nonequilibrium and with an applied magnetic field.

Figure 1

General transport setting where a central interacting region, the system $\text{S}$, is tunnel coupled to several noninteracting fermionic leads with given temperature and chemical potential.

Figure 2

The elementary processes displayed in the phase of the influence functional (6): each line joining a pair of tunneling transitions represents either a creation/annihilation or annihilation/creation process, giving a total of eight elementary processes. In the lower panel, the paths in the forward (f) and backward (b) time branched are coupled by the influence phase $\mathrm{\Phi }$.

Figure 3

Forward (nonbarred) and backward (barred) Grassmann variables arranged in a single (forward) time axis with reversal of the time direction for the backward branch. This requires associating to ${\overline{\xi }}^{*}$ the annihilation and to $\overline{\xi }$ the creation of an electron in the central system.

Figure 4

Two examples of paths of an individual electron state in the central system. The transition times and the corresponding Grassmann variables ${\xi }_k\equiv \xi \left(t_k\right)$ contained in the influence functions, Eq. (26), are highlighted by red boxes. Zeros and ones indicate the occupation of the state in the forward and backward branches.

Figure 5

Blip and sojourn parametrization. Left: Forward and backward paths associated to the occupation of an individual electron state with the corresponding collapsed, single-branch path parametrized in terms of blip and sojourns (below). Right: The time evolution now occurs along the four sides of the square, whose corners define the four elements of the density matrix of a two-state system. The red full dot on the top-left corner denotes the starting (and final) state of the path.

Figure 6

Sequence of Grassmann variables associated to the transition times of an individual degree of freedom of the central system in the blip and sojourn parametrization [see Eqs. (22) and (27)].

Figure 7

Path-dependent energies in the SIAM. The energies $E_{\sigma }$ in the phase factors [Eq. (42)] associated to the overlap of blip and sojourn of the two electron states $\sigma =\uparrow ,\downarrow$ are $E_{\sigma }\left(\eta \right)={\epsilon }_{\sigma }+(1+\eta )U/2$ for blip-sojourn overlap and $E_{\sigma }={\epsilon }_{\sigma }+U/2$ for blip-blip overlap. The time axis is divided in five intervals ${\tau }_k$ separated by four tunneling transitions ($m=2$) distributed between the two paths. The phase factors descends from the action of the central system [see Eqs. (4) and (5)] after integrating out the Grassmann variables between the tunneling transitions.

Figure 8

The three topologies of second-order diagrams. Each fermion line can belong to each of the $N$ states of the central system S. The first diagram is reducible while the second and third are irreducible.

Figure 9

Compact representation for the 15 third-order diagrams. The five diagrams in the left column are reducible and can be seen as combinations of the first- and second-order ones (see Fig. 8). The 10 diagrams on the right are irreducible and their explicit forms are the same as the ones listed in Appendix pp8 for an individual degree of freedom of S. The lower-right class of diagrams vanishes when each of the three fermion lines belongs to the same state.

Figure 10

Examples of diagrams contributing to ${\mathcal{A}}_l$. The sojourn of the degree of freedom $i$ associated to the last transition is constrained to be ${{\eta }^i}^{\prime }=+1$.

Figure 11

Irreducible diagram of order 1. The vertex $\mathbf{v}$ is denoted by the full dot. The two ends of the fermion line are contracted, namely, the indices ${\zeta }^i,\alpha , k$, and $\sigma$ are summed over. The time interval $t_2-t_1$ is a blip only for the state associated to the fermion line. Note that the states $i$ are summed over in contracting the fermion line [see Eq. (90)].

Figure 12

Irreducible diagrams contributing to $J^{\left(2\right)}$. A vertex is denoted by a full dot. The two ends of the lines connecting two blocks carry path indices that are summed over in the connection. The first block is contracted with the vertex $\mathbf{v}$ and the right end of the last block is also contracted.

Figure 13

The free propagator, bubble, and crossing blocks involved in the diagrams of Figs. 11 and 12.

Figure 14

Dressed, irreducible bubble and crossing blocks that generalize the ones shown in Fig. 13. The sum $\tilde{\mathbf{B}}+\tilde{\mathbf{X}}$ is the self-energy for the exact propagator $\mathit{\varphi }$. Within the dressed bubble approximation (DBA), the irreducible bubble diagram $\tilde{\mathbf{B}}$ is the self-energy of the propagator ${\mathit{\varphi }}_{\mathrm{DBA}}$.

Figure 15

Scheme of a spinless level of energy ${\epsilon }_0$ tunnel coupled to two noninteracting leads.

Figure 16

Blocks involved in the propagator of the resonant level model. Full dots represent the vertices defined in Eqs. (91) and (92). Dashed and solid lines at the bottom denote blip and sojourn states, respectively. The sojourn index $\eta$ assumes the values $+1$ and $-1$ for occupied and empty dot, respectively. These internal sojourns are summed over, the sum being included in the definitions of the blocks.

Figure 17

Single-impurity Anderson model realized by a quantum dot tunnel coupled to two noninteracting leads.

Figure 18

The different approximation schemes for the SIAM discussed in this work, in their range of validity ($k_{\mathrm{B}}=1$). Dashed lines indicate the expected regime of validity. Sequential tunneling (ST) and cotunneling (CT) are perturbative in $\mathrm{\Gamma }$ (first and second order, respectively). The second-tier noncrossing approximation (NCA2) and the resonant tunneling approximation (RTA) are nonperturbative in $\mathrm{\Gamma }$ and neglect diagrams with overlap of more than two fermion lines (second-tier schemes). The first is obtained from the second by neglecting the crossing diagrams and yields the Meir-Wingreen-Lee result [39] for the retarded Green's function. Higher-tier approximation schemes deepen the hierarchy of fermion lines. We propose infinite- and fourth-tier schemes which neglect the crossings at the first (DBA) or at all levels (NCA, NCA4, sNCA4) and can be seen as dressed versions of the NCA2.

Figure 19

Differential conductance vs the gate voltage $V_g$ and bias voltage $V$ in the sequential tunneling (ST) approximation. At low-bias voltages, the current is strongly suppressed inside the regions enclosed by so-called Coulomb diamonds due to charging effects. The ST approximation does not account for virtual processes (of higher order in $\mathrm{\Gamma }$) which enable transport of charge also inside the Coulomb diamonds. Degenerate case ${\epsilon }_{\sigma }={\epsilon }_0=-U/2+e{V}_{\mathrm{g}}$, with temperature $k_{\mathrm{B}}T=0.1U$ and tunneling rates ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=0.005U$.

Figure 20

Linear conductance within the NCA2, in units of the conductance quantum $G_0=2e^2/h$, vs the gate voltage ${\epsilon }_0-\mu$ in the degenerate case ${\epsilon }_{\sigma }={\epsilon }_0$. The peaks shrink and move towards the center as temperature is decreased. The curves at the lowest temperatures display an unphysical pinning at the particle-hole symmetry point. The tunnel coupling is $\mathrm{\Gamma }=0.1U$ with ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }/2$.

Figure 21

Linear conductance within the NCA2 and the DSO schemes compared with the curves from the numerically exact DMNRG. We consider the degenerate case ${\epsilon }_{\sigma }={\epsilon }_0$ for three different temperatures. The tunnel coupling is $\mathrm{\Gamma }=0.2U$ with ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }/2$. The parameters are chosen to allow a direct comparison with the results of [35].

Figure 22

Linear conductance within the $\mathrm{\Gamma }$-broadened ST [Eq. (206)] vs the gate voltage ${\epsilon }_0-\mu$ in the degenerate case ${\epsilon }_{\sigma }={\epsilon }_0$. Lowering the temperature, the peaks shrink until $k_{\mathrm{B}}T\simeq \mathrm{\Gamma }$ but they do not move towards the center. Their height grows as $T^{-1}$. The tunnel coupling is $\mathrm{\Gamma }=0.01U$ with ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }/2$.

Figure 23

Linear conductance of the simplified NCA4 vs the gate voltage for three different temperatures. Degenerate case ${\epsilon }_{\sigma }={\epsilon }_0$ with $\mathrm{\Gamma }=0.2U$. The dashed lines depict the expectation value $\langle {\widehat{n}}_{\uparrow }\rangle =\langle {\widehat{n}}_{\downarrow }\rangle$. The parameters are chosen to allow a direct comparison with the results of [35]. The Kondo temperature [Eq. (199)] for this choice of parameters is $k_{\mathrm{B}}{T}_{\mathrm{K}}\simeq 0.004U$. The sNCA4 conductance has a very weak dependence on temperature in the lowest two panels, a hint that the Fermi-liquid regime is approached (see also Fig. 25). However, in contrast to the DMNRG curve, the unitary limit $G=G_0$ is not approached.

Figure 24

Linear conductance of the simplified NCA4 vs the gate voltage for three different temperatures. Degenerate case ${\epsilon }_{\sigma }={\epsilon }_0$ with $\mathrm{\Gamma }=0.4U$. The dashed lines depict the expectation value $\langle {\widehat{n}}_{\uparrow }\rangle =\langle {\widehat{n}}_{\downarrow }\rangle$. The Kondo temperature [Eq. (199)] for this choice of parameters is $k_{\mathrm{B}}{T}_{\mathrm{K}}\simeq 0.04U$. Note that the sNCA4 performs better at low temperature with respect to the case $\mathrm{\Gamma }=0.2U$ (cf. Fig. 23).

Figure 25

Temperature dependence of the linear conductance in the simplified NCA4 and NCA2 at the particle-hole symmetry point ${\epsilon }_0-\mu =-U/2$. Degenerate case ${\epsilon }_{\sigma }={\epsilon }_0$ with $\mathrm{\Gamma }=0.2U$ (upper panel) and $\mathrm{\Gamma }=0.4U$ (lower panel). The red dashed lines are at the values of the Kondo temperature $T_{\mathrm{K}}$ given by Eq. (199) while the vertical solid lines correspond to the three temperatures considered in in Figs. 23 and 24. Consistently with the DMNRG results, well below $T_{\mathrm{K}}$ the sNCA4 curves depend only weakly on the temperature. However, the unitary value $G_0$ is not attained.

Figure 26

Effect of the temperature on the sNCA4 differential conductance for $\mathrm{\Gamma }=0.1U$, with ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }/2$, in the degenerate case. (a) Scheme of the stability diagram for the SIAM (cf. Fig. 19). The curves are the differential conductance for two values of the gate voltage at $k_{\mathrm{B}}T=0.04U$. (b) Differential conductance vs voltage bias $eV={\mu }_L-{\mu }_R$, for different temperatures. The gate voltage is set to the values shown in (a), namely, ${\epsilon }_0-\mu =-U/2$ (left) and ${\epsilon }_0-\mu =-U/4$ (right). At the particle-hole symmetry point, ${\epsilon }_0-\mu =-U/2$, the Kondo temperature, Eq. (199), is $k_{\mathrm{B}}{T}_{\mathrm{K}}\simeq 0.00006U$.

Figure 27

Effect of the temperature on the sNCA4 differential conductance vs voltage bias $eV={\mu }_L-{\mu }_R$ in the degenerate case and for (a) $\mathrm{\Gamma }=0.2U$ and (b) $\mathrm{\Gamma }=0.4U$. The gate voltage is set to the same values as in Fig. 26, namely, ${\epsilon }_0-\mu =-U/2$ (particle-hole symmetry point, left) and ${\epsilon }_0-\mu =-U/4$ (right). At the particle-hole symmetry point, the Kondo temperature, Eq. (199), is $k_{\mathrm{B}}{T}_{\mathrm{K}}\simeq 0.004U$ in (a) and $k_{\mathrm{B}}{T}_{\mathrm{K}}\simeq 0.04U$ in (b).

Figure 28

Effect of the magnetic field on the sNCA4 conductance. Symmetric coupling to the leads with $\mathrm{\Gamma }=0.1U$, and temperature $k_{\mathrm{B}}T=0.01U$. (a) Scheme of the stability diagram for the SIAM. The solid black curves are the differential and linear conductances at zero magnetic field and for two values of the bias voltage $eV={\mu }_L-{\mu }_R$, namely, $eV=0$ and $eV=U$. (b) Linear and differential conductances vs the gate voltage, calculated for the two values of the bias voltage shown in (a), for different values of the Zeeman splitting ${\mathrm{\Delta }}_B={\epsilon }_{\uparrow }-{\epsilon }_{\downarrow }$.

Figure 29

Two examples of paths with two transitions, either in the forward (a) or in the backward (b) time branch. The Grassmann variables boxed in red are the ones appearing in the influence functional for the examples of paths considered. Note that we use the same time direction for the two branches with the consequence that $\overline{\xi }$ and ${\overline{\xi }}^{*}$ creates and annihilates an electron in the dot, respectively.

Figure 30

Paths with two transitions, one in the forward and the other in the backward branch. The boxed Grassmann variables in red are the ones appearing in the influence functional for the examples of paths considered.

Figure 31

Path with four transitions, two in the forward and the others in the backward branch. Here, only the collective path is represented, with diagonal and off-diagonal states depicted with continuous and dashed lines, respectively. The boxed Grassmann variables in red are the ones appearing in the influence functional in this example.

Figure 32

Path with four transitions distributed in the two subpaths of the spin variables $\uparrow$ and $\downarrow$. Note that the path for $\sigma =\uparrow$ is of the type (c) in Fig. 30, while $\sigma =\downarrow$ undergoes a sequence of the type (a) in Fig. 29.

Figure 33

Path with four transitions distributed in the two subpaths of the spin variables $\uparrow$ and $\downarrow$. Note that the subpaths are both of the type (c) in Fig. 30.

Figure 34

Path with four transitions distributed in the two subpaths of the spin variables $\uparrow$ and $\downarrow$. Note that the subpath $\sigma =\uparrow$ is of the type (b) in Fig. 29, while $\sigma =\downarrow$ undergoes a sequence of the type (c) (see Fig. 30).

Figure 35

Six-transition path. General case (a) and two specific examples: In the first ${\eta }_2$ is left unspecified while ${\eta }_0={\eta }_1={\eta }_3=-1$ and ${\zeta }_1={\zeta }_2={\zeta }_3=+1$ (b). In the second also ${\zeta }_3$ is left unspecified (c).