Imaginary-time quantum many-body theory out of equilibrium. II. Analytic continuation of dynamic observables and transport properties

Within the imaginary-time theory for nonequilibrium in quantum dot systems the calculation of dynamical quantities like Green's functions is possible via a suitable quantum Monte Carlo algorithm. The challenging task is to analytically continue the imaginary-time data for both complex voltage and complex frequency onto the real variables. To this end a function-theoretical description of dynamical observables is introduced and discussed within the framework of the mathematical theory of several complex variables. We construct a feasible maximum-entropy algorithm for the analytical continuation by imposing a continuity assumption on the analytic structure and provide results for spectral functions in stationary nonequilibrium and current-voltage characteristics for different values of the dot charging energy.

Figure 1

Contents of the paper as a flowchart. Solid arrows represent numerical procedures within our approach. Open arrows represent the formal steps necessary for a derivation of the MaxEnt kernel used for numerical results. Shaded areas are discussed in the corresponding sections. The rather unconventional two-dimensional function theory with respect to the simultaneously variables $(z_{\phi },z_{\omega })$ is explained by analogies ($\widehat{=}$) with the conventional theory in the light-shaded area. It may be skipped at first reading. Concepts such as the biholomorphic maps are used to obtain explicit equations. Dash-dotted lines denoting either Vladimirov's formula or Cauchy's integral equation put an emphasis on the fact that the content of a domain (of holomorphy) is parametrized by its (Bergman-Shilov) boundary. In the numerical implementation it is thus an inverse problem to reconstruct function values on the (Bergman-Shilov) boundary, which requires a MaxEnt approach.

Figure 2

Effective-equilibrium data as obtained from CT-QMC simulations[4] for $U=8\Gamma$, $e\Phi =0.1\Gamma$, $\beta =5{\Gamma }^{-1}$. The integer number $m$ specifies the respective index of the Matsubara voltage ${\phi }_m=4\pi m/\beta$. The discontinuity at ${\omega }_n={\phi }_m/2$ is the principal branch cut which will, in particular, be discussed in Sec. 3d.

Figure 3

Spectral function of the dot electrons as inferred for the nonequilibrium weak-coupling case $U=2\Gamma$, $e\Phi =\Gamma$, $\beta =5{\Gamma }^{-1}$, compared to second-order perturbation theory.

Figure 4

Nonequilibrium spectral functions for $U=4\Gamma$ and inverse temperature $\beta =5{\Gamma }^{-1}$ of the leads, as compared to second-order perturbation theory. Line legends are the same as in Fig. 3.

Figure 5

Nonequilibrium spectral functions for $U=6\Gamma$ and inverse temperature $\beta =5{\Gamma }^{-1}$, as compared to second-order perturbation theory. Line legends are the same as in Fig. 3.

Figure 6

Nonequilibrium spectral functions for $U=8\Gamma$ at inverse temperature $\beta =5{\Gamma }^{-1}$, as compared to second-order perturbation theory. Line legends are the same as in Fig. 3.

Figure 7

Nonequilibrium spectral functions for $U=10\Gamma$ at inverse temperature $\beta =5{\Gamma }^{-1}$, as compared to second-order perturbation theory. Line legends are the same as in Fig. 3.

Figure 8

Lower-temperature spectral functions as inferred for $U=4\Gamma$, $\beta =10{\Gamma }^{-1}$ with or without an annealing step. The default model for the single-step procedure is taken from temperature $\beta =5{\Gamma }^{-1}$.

Figure 9

Transport characteristics as compared to real-time QMC data from Ref. 15.

Figure 10

Current as a function of the voltage for different interaction strengths.

Figure 11

Comparison of the single-wedge approach from Ref. 4 and the multiwedge approach at $U=6\Gamma$, $\beta \Gamma =10$, $e\Phi =0.5\Gamma$.

Figure 12

Comparison of Cauchy and Bergman-Weil integral representation theories. Integrations run over (a) the full topological boundary $\partial D$ and (b) the Bergman-Shilov boundary $S\subset \partial D$ of an analytic polyhedron, respectively.

Figure 13

Branch cut structure of the Green's function. The fully interacting structure is obtained from the perturbative expansion in $U/\Gamma$.

Figure 14

An artist's impression of asymptotically filling the wedge $T^C$ with the sequence ${\mathcal{D}}_n$. ${\mathcal{D}}_n$ is a domain with piecewise smooth boundaries created by intersecting with a growing bicylinder ${\mathcal{B}}_n$, creating the analytic polyhedra ${\mathcal{D}}_n$. The corresponding sequence of Bergman-Shilov boundaries improperly converges to the set $\mathrm{Edge}\cup \left\{\infty \right\}$. Therefore, the sheaf of holomorphic functions on $T^C$ is solely characterized by its edge values and its asymptotic behavior $\underline{z}\to \infty$.

Figure 15

On the distinguished boundary surface of the bicylinder, the picture of ${\mathrm{Edge}}_{T^C}$ under the componentwise Möbius transformation ${\mathfrak{m}}_C$ is delimited by the dash-dotted lines ${\mathfrak{p}}_{\infty }$. A discontinuity of ${G|}_{T^C}\circ {\mathfrak{m}}_C^{-1}$ occurs at the intersection point $(1,1)$ of the two circles and prevents a Cauchy representation from being applicable. The violation of the Cauchy-Bochner condition (24) appears to be related to the occurrence of the discontinuity.

Figure 16

An arbitrary cone $C$ with 0 as vertex can be parametrized by an opening ratio $r$ and an orientation angle $\vartheta$. The rotation $R_{\vartheta }$ which maps $C_{r,0}$ to $C_{r,\vartheta }$ induces a biholomorphism between $T^{C_{r,0}}$ and $T^{C_{r,\vartheta }}$.

Figure 17

Action of the operator $-{\mathcal{Q}}_{\vartheta }^{\left(\mathrm{edge}\right)}$. It translates between functions living on edges of wedges with two different angular orientations. The orientation $\vartheta$ is the one of the considered data wedge, and 0 is the orientation of the physical limiting procedure (10). Initially acting on the physical edge function, the consecutive formal operations which comprise $-{\mathcal{Q}}_{\vartheta }^{\left(\mathrm{edge}\right)}$ either change the angular rotation or leave it invariant, as indicated by the respective arrows.

Figure 18

Transformation behavior of the test function $f\left(\underline{x}\right)$ as a function of the edge-to-edge map $-{\mathcal{Q}}_{\vartheta }^{\left(\mathrm{edge}\right)}$ for different values of $\vartheta$. Function values are shown within the range $[-5,5]\times [-5,5]$. Due to translational and scale invariance, it represents the edge-to-edge transformation behavior of a Dirac $\delta$ function under the continuity assumption (65).

Figure 19

Comparison of ${\chi }^2$ as a function of the MaxEnt regularization parameter for single-wedge kernel ${\mathcal{P}}_{r,\vartheta }$ and multiwedge kernel $\mathcal{Q}$ at weak interaction $U=2\Gamma ,\beta =5{\Gamma }^{-1},$ and $e\Phi =\Gamma$ as a function of the regularization parameter $\alpha$. For the same input set, the single-wedge approach clearly fails to converge due to the presence of higher-order branch cuts.

Figure 20

Application of the MaxEnt procedure for the $\mathcal{Q}$ mapping to the nonequilibrium weak-coupling case $U=2\Gamma$, $\beta =5{\Gamma }^{-1},e\Phi =\Gamma$, with CT-QMC data as input. The default model has been identified via its maximal posterior probability.

Figure 21

The wedge to be considered. The dash-dotted lines denote the data yielded by imaginary-time theory.

Figure 22

A single line contained by the cone $C$ may after biholomorphic rotation $R_{-\vartheta }$ and subsequent complexification be interpreted as the upper half plane $\mathbb{H}$ of $\mathbb{C}$.

Figure 23

Uniquely reconstructed range ${\mathrm{Edge}}_{T^C}^{(\mathrm{recon},0)}$ of $G\left(\underline{z}\right){|}_{T^C}$ on the edge of $T^C$ following from the partial argument of Appendix A1. The wiggly lines in the boundary mean that the area extends to infinity.

Figure 24

Enhancing the formal holomorphic reconstruction within the wedge.

Figure 25

Reconstructing the Green's function on the complete edge. The reconstructed area ${\mathrm{Edge}}_{T^C}^{(\mathrm{recon},0)}$ [see Fig. 23 and Eq. (A2)] may be extended by affine transformations along the $R_{{\vartheta }_0}{(0,1)}^T$ direction using the information from the set $E_1$.

Figure 26

Example for $I_3$ as a function of $y^{\prime }$, using ${\phi }_m=2.0$, ${\omega }_n=2.0$, with test function location $X=0.0$, $Y=4.0$, and test function width $\delta ={10}^{-3}$. The interacting branch cut geometry is used for the determination of $r$ and $\vartheta$.

Figure 27

Cut through ${\mathcal{Q}}_{r,\vartheta }$ in $\tilde{A}$ space for different pairs of $i{\phi }_m$ and $i{\omega }_n$, at $\beta \Gamma =5.0$. The wedge opening ratios $r$ and wedge orientations $\vartheta$ (Fig. 16) are chosen according to the interacting branch cut geometry. The “nonequilibrium” line represents the location of the dot-electron spectral function for a system with source-drain voltage $e\Phi =0.5\Gamma$. Wiggly structures at higher energies result from the increasingly coarse-grained $\tilde{A}$ grid.

Figure 28

Successive improvement of kernel quality by averaging out the local kernel structure within the local $\tilde{A}$ grid resolution. Data are shown for $U=0$, $\Phi =0$, $\beta \Gamma =5$, $n=0,\cdots ,9$, $m=-3,\cdots ,3$ and a realistic mock diagonal covariance matrix $C=\mathrm{diag}\left(\frac{{10}^{-13}}{{\Gamma }^2}\right)$. The abscissa denotes $x_{\phi }/\Gamma$, the ordinate denotes $x_{\omega }/\Gamma$, grayscale denotes $\tilde{A}(x_{\phi },x_{\omega })$ in units of ${\Gamma }^{-1}$.

Figure 29

Sensitivity to the default model, using the same parameters and scales as in Fig. 28d.

Figure 30

MaxEnt results for flat low-energy default models (E4), $R=5$. As compared to Fig. 29b, the quality of low-energy data is increased significantly, due to the correct high-energy behavior of the default model. Scales are as in Fig. 28d.

Figure 31

At high energies, one might (a) expect the lateral structure of $\tilde{A}(x_{\phi },x_{\omega })$ to be composed of two Hubbard peaks and possibly a quasiparticle resonance which combine to the physical spectrum $A\left(\omega \right)$ at the intersection point. In the complementary scenario (b), the function $\tilde{A}$ would not differ from the noninteracting one at high energies.

Figure 32

Posterior probability of the default model (E3) at $\beta \Gamma =5$, $e\Phi =\Gamma$ for several interactions strengths. The result is found to be essentially independent of the bias voltage. The kernel validated in Fig. 28d has been used.