Green's functions from real-time bold-line Monte Carlo

We present two methods for computing two-time correlation functions or Green's functions from real-time bold-line continuous-time quantum Monte Carlo. One method is a formally exact generalized auxiliary lead formalism by which spectral properties may be obtained from single-time observables. The other involves the evaluation of diagrams contributing to two-time observables directly on the Keldysh contour. Additionally, we provide a detailed description of the bold-line Monte Carlo method. Our methods are general and numerically exact, and able to reliably resolve high-energy features such as band edges. We compare the spectral functions obtained from real-time methods to analytically continued spectral functions obtained from imaginary-time Monte Carlo, thus probing the limits of analytic continuation.

Figure 1

Left: an illustration of the single probe, wide-band auxiliary current setup. Right: the double probe, narrow-band variation of the auxiliary current formalism. The dot is depicted as the central circle, and is coupled by thick (thin) lines to the physical (auxiliary) reservoirs. The curved region within each reservoir sketches the shape and filling of its coupling density. Whereas in the single probe apparatus a single high-bandwidth auxiliary reservoir is coupled to the dot and the integral of the spectral function is obtained by measuring the auxiliary current while varying its chemical potential, in the double probe scheme (suggested as a theoretical tool rather than an experiment) the spectral function is obtained without the need for a derivative by attaching two low-bandwidth leads, of which one is empty and the other is full.

Figure 2

The elements of atomic state propagator diagrams. Upper left: full and bare propagator lines. Upper right: interaction (“hybridization”) vertices. Bottom: examples of low-order diagrams. The upper line represents spin up, the lower line spin down. Solid lines denote occupied orbitals, dashed lines empty orbitals. Thick lines denote dressed propagators, thin lines bare propagators. Wiggly lines denote the ejection of an electron to the bath or its propagation back to the dot.

Figure 3

The matrix elements of the NCA self-energy (two terms on the left with a single hybridization line) and of the OCA self-energy (all four terms) in diagrammatic form.

Figure 4

Top left panel: a diagram on the two-branch Keldysh contour. Note the branch crossing hybridization line, which is not contained in single-branch propagators. Top right panel: a diagram contributing to the current or a two-time correlation function, with a special hybridization line (grey) coupling to the final time. Lower panel: two of the NCA vertex equations which take into account diagrams near the beginning of the contour. The gray parts of the lines are not part of the diagrams, but instead illustrate their location on the Keldysh contour.

Figure 5

The dependence of the probed spectral function $A\left(\omega \right)\equiv -\frac{1}{\pi }\Im \left\{G^R\left(\omega \right)\right\}$ on time and frequency, as obtained from the auxiliary current method for an initially decoupled quantum dot at ${\Omega }_c=10\Gamma$. At $U=0$ and $\beta \Gamma =1$ (left), a simple Lorentzian shape develops. At $U=6\Gamma$ and $\beta \Gamma =1$ (center), the interaction distorts and widens the spectrum. At $U=6\Gamma$ and $\beta \Gamma =3$ (right), the spectral profile typical to the Kondo problem develops, exhibiting a central Kondo peak between two lower Hubbard peaks.

Figure 6

Several cuts through the data of Fig. 5 are shown at fixed $\omega$, making it easier to observe both the convergence and the behavior at short times.

Figure 7

The real part of the retarded two-time correlation function $G^r(t,t^{\prime })$ as a function of the time difference $t-t^{\prime }$, shown for several values of the final time $t$. The parameters chosen are $U=6\Gamma$, $\beta \Gamma =3$ and ${\Omega }_C=10\Gamma$.

Figure 8

The real part of the retarded two-time correlation function $G^r(t,t^{\prime })$ as a function of the time difference $t^{\prime }-t$, for a final time $t$ of $10/\Gamma$. The dependency on different parameters is illustrated by taking different values of $U$ (top panel), $\beta$ (middle panel) and ${\Omega }_C$ (lower panel) with the other parameters fixed.

Figure 9

The spectral function $A\left(\omega \right)$ in the noninteracting limit $U=0$ is shown for several combinations of the inverse temperature $\beta \Gamma$ and band width ${\Omega }_c/\Gamma$. The exact result is shaded in gray for comparison to the imaginary-time data analytically continued with MaxEnt using a flat model (solid red), the result obtained from directly Fourier transforming the real-time bold-line CTQMC correlation function (green circles), and the real-time bold-line CTQMC auxiliary current data (thick blue line with error bars). Both real-time results are obtained by propagating a decoupled initial state to $\Gamma t=10$. The inset in the top left panel zooms in on the region near the band edge.

Figure 10

The spectral function $A\left(\omega \right)$ in the weakly interacting limit $U=2\Gamma$ is shown for several combinations of the inverse temperature $\beta \Gamma$ and band width ${\Omega }_c/\Gamma$. The imaginary-time data analytically continued with MaxEnt using a flat model (solid red) is shown along with the result obtained from directly Fourier transforming the real-time bold-line CTQMC correlation function (green circles), and the real-time bold-line CTQMC auxiliary current data (thick blue line with error bars). Both real-time results are obtained by propagating a decoupled initial state to $\Gamma t=10$.

Figure 11

The spectral function $A\left(\omega \right)$ in the strongly interacting limit $U=6\Gamma$ is shown for several combinations of the inverse temperature $\beta \Gamma$ and band width ${\Omega }_c/\Gamma$. The imaginary-time data analytically continued with MaxEnt using a flat model (solid red) is shown along with the result obtained from directly Fourier transforming the real-time bold-line CTQMC correlation function (green circles), and the real-time bold-line CTQMC auxiliary current data (thick blue line with error bars). Both real-time results are obtained by propagating a decoupled initial state to $\Gamma t=10$. Real-time NCA data is also shown for comparison (dashed purple line).