Currents and Green's functions of impurities out of equilibrium: Results from inchworm quantum Monte Carlo

We generalize the recently developed inchworm quantum Monte Carlo method to the full Keldysh contour with forward, backward, and equilibrium branches to describe the dynamics of strongly correlated impurity problems with time-dependent parameters. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We then illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state.

Figure 1

Illustration of hybridization expansion diagrams on the Keldysh contour with equilibrium branch. (Top) Full propagator on the $+$ branch from $t_2$ to $t_1$ (left diagram) is given by the bare propagator (middle diagram) plus all possible combinations of hybridization events between $t_2$ and $t_1$, one of which is drawn as the right diagram. (Bottom) Full propagator spanning the $-$, equilibrium, and $+$ contour containing diagrams that span the contour.

Figure 2

Illustration of the diagrams generating a propagator in the inchworm formalism. The propagator from $t_2$ to $t_1$ (left diagram) is, at lowest inchworm order, given by the propagator from $t_2$ to $t_{\text{split}}$ joined with a bare propagator from $t_{\text{split}}$ to $t_1$ $①$. In higher-order diagrams, hybridization lines are either contained in the region $t_{\text{split}}$ and $t_1$ as in $②$; or, connected by crossing to an endpoint in that region, as in $③$. Diagrams with inclusions not obeying this rule, such as $④$, are already included in the inchworm propagator and should not be summed over.

Figure 3

(Left) Illustration of the causal structure of the inchworm propagators. A propagator $p_{\varphi }(t_1,t_2)$ (red) depends on all propagators with start point on or after $t_2$ and end point up to $t_{\text{split}}$ (blue). (Middle) Illustration of the inchworm algorithm showing the values known at some step (blue) and values that can be computed using known values in blue (red). Evaluation of the full propagator proceeds diagonally towards the upper right corner from the initial values as dictated by the causal structure of the propagators. (Right) Step-by-step illustration of the algorithm. Cells with circles are evaluated with the bare algorithm and solid cells with inchworm. Red cells represent propagators computed at a given step, while blue cells represent already computed propagators. Green cells in both panels are trivial and can be evaluated analytically.

Figure 4

(Top) Time evolution of the impurity occupation $N$ after a voltage quench using the noncrossing and one-crossing approximations (NCA and OCA, respectively). Black lines: semianalytically computed NCA and OCA solutions. Blue line: NCA solution generated from an inchworm expansion truncated to order one. Red line: OCA solution from an inchworm expansion truncated to order 2. (Bottom) Statistical error estimate of the quantities shown in the upper panel.

Figure 5

Order distribution (red) and sampled “flat” histogram (blue) for a Wang-Landau simulation of the inchworm propagators. The large overlap of the reweighted distribution with order zero allows normalization to low-order diagrams.

Figure 6

Illustration of the diagrams generating a Green's function. The Green's function $G(t,t^{\prime })$ is given at lowest order by the middle diagram, where the dashed line is a virtual hybridization line from the Green's function creation/annihilation operators. Higher-order diagrams contain at least one hybridization line that crosses the virtual line. A sample term is shown in the right diagram.

Figure 7

(Top) Time evolution of the density on the impurity after a voltage quench with $\mathrm{\Gamma }=1,U=10,{\epsilon }_d=0,D=5,T=1$, and $V=6$. Results obtained from a bare QMC calculation are shown for $t\le 0.6$. The inchworm results with different orders agree with the bare result for $t\le 0.6$ and coincide with each other for longer times. (Bottom) Error estimates. Data obtained using the bare method shows an exponential increase of the errors as a function of time, whereas inchworm errors grow slower as a function of time.

Figure 8

(Top) The current dynamics after a voltage quench with $\mathrm{\Gamma }=1,U=4,{\epsilon }_d=-2,D=5,T=1$, and $V=4$. The inchworm results with different orders converge as maximum-order increases. (Bottom) Error estimates of inchworm data obtained by averaging eight independent calculations. Errors increase as a function of time but avoid the exponential amplification seen in bare calculations.

Figure 9

(Top) The imaginary time Green's function in equilibrium (half-filling) with $\mathrm{\Gamma }=1,U=4,{\epsilon }_d=-2,D=5,T=1$ and $V=0$. Inchworm results with different orders all coincide and agree with the bare calculation. (Bottom) The error estimate for the inchworms data is approximately constant in imaginary time.

Figure 10

(Top) A contour plot of the dynamics of auxilary current spectrum $A_{\text{aux}}(\omega ,t)$ after a voltage quench with $\mathrm{\Gamma }=1,U=4,{\epsilon }_d=-2,D=5,T=1$, and $V=4$. The maximum-order cutoff for the inchworm calculation is 6. A formation and a splitting of the Kondo peak are observed as a function of time. (Middle) Slices of auxiliary current spectrum at different times from the top contour plot. A clear splitting of the spectrum is shown. (Bottom) Error estimate on the spectral function obtained from eight independent simulations.

Figure 11

(Top) The (half-filling) spectrum at $t=2.0$ after a voltage quench with $\mathrm{\Gamma }=1,U=4,{\epsilon }_d=-2,D=5,T=1$, and $V=4$. The spectral function shows the establishment of a split Kondo peak as the diagram order is increased. The data for order 6 are identical to the data shown in Fig. 10. (Bottom) Error estimate for data shown in main panel. The error remains constant as a function of frequency and increases as the maximum order is increased.

Figure 12

(Top) The (half-filling) spectrum at $t=2.0$ with no applied voltage with $\mathrm{\Gamma }=1,U=4,{\epsilon }_d=-2,D=5,T=1$, and $V=0$. (Bottom) Error estimate for data shown in the main panel.

Figure 13

(Top) Spectral function away from half-filling at $t=2.0$ after a voltage quench with $\mathrm{\Gamma }=1,U=10,{\epsilon }_d=-2,D=5,T=1$, and $V=4$. (Bottom) Error estimate for data shown in the main panel.