Nonequilibrium theory of Coulomb blockade in open quantum dots

We develop a nonequilibrium theory to describe weak Coulomb blockade effects in open quantum dots. Working within the Bosonized description of electrons in the point contacts, we expose deficiencies in earlier applications of this method, and address them using a $1∕N$ expansion in the inverse number of channels. At leading order this yields the self-consistent potential for the charging interaction. Coulomb blockade effects arise as quantum corrections to transport at the next order. Our approach unifies the phase functional and Bosonization approaches to the problem, as well as providing a simple picture for the conductance corrections in terms of renormalization of the dot’s elastic-scattering matrix, which is obtained also by elementary perturbation theory. For the case of ideal contacts, a symmetry argument immediately allows us to conclude that interactions give no signature in the averaged conductance. Nonequilibrium applications to the pumped current in a quantum pump are worked out in detail.

Figure 1

Schematic drawing of the system under consideration: a quantum dot (center) coupled to source and drain reservoirs (left and right) via point contacts. The two point contacts have $N_1$ and $N_2$ propagating channels at the Fermi level. Direct reflection at the point contact is described by energy-independent reflection matrices $r_{\mathrm{c}1}$ and $r_{\mathrm{c}2}$, so that their dimensionless conductances are $G_1=(e^2∕2\pi ħ)(N_1-\mathrm{tr}{r}_{\mathrm{c}1}{r}_{\mathrm{c}1}^{†})$ and $G_2=(e^2∕h)(N_2-\mathrm{tr}{r}_{\mathrm{c}2}{r}_{\mathrm{c}2}^{†})$, respectively. Interactions in the dot are described by the charging energy $E_{\mathrm{c}}$.

Figure 2

Representation of the source and drain reservoirs as one-dimensional ideal leads. The lead-dot interface is at $x=0$. The one-dimensional description (3.1) is valid from $x=-\infty$ up to slightly beyond $x=0$.

Figure 3

Inclusion of a fictitious lead at the point contact. Top panel: With the Hamiltonian $H_0$, the fictitious lead couples ideally to outgoing electrons (solid lines), so that all electrons that exit the dot enter into the fictitious lead. The Hamiltonian $H_1$ describes scattering of electrons exiting the dot into the physical lead and backscattering of electrons coming from the fictitious lead (dotted line). The amplitude for this scattering process in channel $j$ is $i{r}_j,j=1,\dots ,N$. Bottom panel: If $r_j=1$, the fictitious lead is decoupled again and the original problem is restored.

Figure 4

Integration contour for Eq. 3.16.

Figure 5

(a) Diagrammatic representation of leading-in-$1∕N$ contribution to the current. The current vertex $i$ is denoted with a filled circle, the bubble of single-fermion Green function with a solid loop, and the tree structure with a hatched triangle. The defining equation for the tree structure is shown in (b), where the dotted arrow represents the interaction line.

Figure 6

Diagrammatic representation of the effective interaction $\kappa$ (thick arrow) in terms of the bare interaction line $\ln f$ (thin arrow) and a bubble of single-fermion Green functions (solid loop).

Figure 7

Diagrammatic representation of subleading-in-$1∕N$ corrections to the current.

Figure 8

(Color online) The interaction correction to the variance of the conductance, made dimensionless by writing $G=({\nu }_s{e}^2∕2\pi \hslash )(N_1{N}_2∕N^2)g$. The solid curve shows the noninteracting contribution to the conductance fluctuations; the dashed curve shows the interaction correction for $\pi ∕E_{\mathrm{c}}N{\tau }_{\mathrm{d}}=0.1$, multiplied by $N$ to make it fit on the same scale as the noninteracting contribution.

Figure 9

(Color online) The variance of the mean-field contribution to the capacitance, made dimensionless by writing $C_{\mathrm{c}}=2\pi e^{2}c∕\left(E_{\mathrm{c}}^2{N}^2{\tau }_{\mathrm{d}}\right)$ (solid curve) and the interaction correction to the variance of the electrochemical capacitance (dashed), as a function of temperature. The interaction correction to the variance has been multiplied by $N$ in order to fit on the same vertical scale.

Figure 10

Top panel: schematic drawing of a quantum dot with a voltage probe. Bottom panel: layout of labeling of fermion fields $R,L,V$, and $W$.

Figure 11

Left diagram: Schematic drawing of the fictitious lead $\left(E\right)$ coupling of which to the left-moving fermions through the Hamiltonian (C8) gives the second term of the effective action (C7). Right diagram: Definition of the fermion fields $R,E$, and $L$ after the change of variables of Eq. (C9).