Imaginary-time quantum many-body theory out of equilibrium: Formal equivalence to Keldysh real-time theory and calculation of static properties

We discuss the formal relationship between the real-time Keldysh and imaginary-time theory for nonequilibrium in quantum dot systems. The latter can be reformulated using the recently proposed Matsubara-voltage approach. We establish general conditions for correct analytic continuation procedure on physical observables, and apply the technique to the calculation of static quantities in steady-state nonequilibrium for a quantum dot subject to a finite bias voltage and external magnetic field. Limitations of the Matsubara voltage approach are also pointed out.

Figure 1

(a) Keldysh contour for real-time diagrammatics. If the time evolution along the dashed line does not contribute an extra factor, the whole contour can be deformed to one along the imaginary time from $t=-i\beta$ to $t=0$ as shown in (b).

Figure 2

(a) Keldysh contour in forward direction. Crosses mark interaction points $\widehat{V}$ and the dot an observable $\widehat{A}$. (b) Reversed series of scattering points. (c) Backward Keldysh contour with scattering events equivalent to (a) if $\widehat{A}$ is written in terms of QD operators.

Figure 3

Real part of the effective-equilibrium double occupancy as a function of the Matsubara voltage ${\varphi }_m$ at several values of interaction strength $U$ and bias voltage $\Phi$.

Figure 4

Real part of the effective-equilibrium magnetization as a function of the Matsubara voltage.

Figure 5

Kondo scaling analysis of effective-equilibrium magnetization data at ${\mu }_{B}B=T_K/2$, $e\Phi =T_K/4$. The analysis makes use of the equilibrium Kondo temperatures $k_B{T}_K(U=5\Gamma )\approx \frac{1}{10}\Gamma$, $k_B{T}_K(U=8\Gamma )\approx \frac{1}{20}\Gamma$, $k_B{T}_K(U=10\Gamma )\approx \frac{1}{40}\Gamma$. The latter ratios are chosen to be approximately identical to the results of Haldane's scaling formula.[55]

Figure 6

MaxEnt inference process for the double occupancy. Parameters are $U=5$, $e\Phi =0.25\Gamma$, $\beta =20{\Gamma }^{-1}$. Due to lack of prior knowledge, we use a flat default model, that is, the shift function ${\varrho }_{\text{shift}}\left(\varphi \right)$; see Eq. (85). Remember that the actual spectral function ${\varrho }_D\left(\varphi \right)$ was shifted to a positive one, ${\varrho }_D^{\prime }\left(\varphi \right)$, via Eq. (83). One finds that the different equivalent ways of imposing a flat default model for ${\varrho }_D\left(\varphi \right)$ yield practically the same spectral function. Nevertheless, computing the physical value (73) yields values which are distributed over a certain range. This range is displayed as error bars in the results, plots Figs. 7 and 8.

Figure 7

Double occupancy as a function of the bias voltage at different values of $U$, as compared to second-order perturbation theory. In addition, the dashed lines show the temperature dependence of $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ in equlibrium as obtained by NRG, assuming $e\Phi =k_{\mathrm{B}}T$ (see text).

Figure 8

Magnetic susceptibility as a function of bias voltage in the Kondo regime $U=8\Gamma$ at ${\mu }_{B}B=k_B{T}_K/2$, $T=T_K/2$. The dot-dashed line represents an equilibrium NRG calculation for $T\ge T_{\mathrm{K}}/2$, rescaled in both magnitude and temperature to match the low-bias behavior of historic MaxEnt (see inset). The double-dot-dash curve finally is a fit of historic MaxEnt to some scaling function (see text).

Figure 9

(a)–(d) Real-time Keldysh contour for self-energy ${\Sigma }_{\left(4\right)}^{>}(t,0)$ in the fourth-order perturbation when one intermediate time is in the finite interval $[0,t]$ and the other time in along the contour stretching to $-\infty$. The Wick's contraction is taken as shown in (g) and (h). The dummy label of (g) is used for time orderings (a), (c), (e) and the label (h) is used for (b) (d), (f). The cross represents the intermediate times $s_1$ and $s_2$ for interaction, in addition to the creation/annihilation points 0 and $t$.

Figure 10

Different time ordering with two intermediate interaction events extend to infinity. (a), (d), (e), (g) use the label in Fig. 1(g) and (b), (c), (f), (h) use the label in Fig. 1(h).