Quantum Monte Carlo solution of the dynamical mean field equations in real time

We present real-time inchworm quantum Monte Carlo results for single-site dynamical mean field theory on an infinite coordination number Bethe lattice. Our numerically exact results are obtained on the L-shaped Keldysh contour and, being evaluated in real time, avoid the analytic continuation issues typically encountered in Monte Carlo calculations. Our results show that inchworm Monte Carlo methods have now reached a state where they can be used as dynamical mean field impurity solvers and the dynamical sign problem can be overcome. As nonequilibrium problems can be simulated at the same cost, we envisage the main use of these methods as dynamical mean field solvers for time-dependent problems far from equilibrium.

Figure 1

Real and imaginary parts of the greater dynamical mean field Green's function $G^{>}\left(t\right)$ as a function of real time $t$ up to a maximum time of $tv=2$. $U/v=4,T/v=0.05$, equilibrium. Shown is the convergence with DMFT iteration.

Figure 2

Real and imaginary parts of the greater DMFT Green's function $G^{>}\left(t\right)$, for times up to $tv=2$, with $T/v=0.5$, half filling, equilibrium, at on-site interaction strengths $U/v\in \{2.0,4.0,6.0,8.0\}$.

Figure 3

Real and imaginary parts of the greater DMFT Green's function $G^{>}\left(t\right)$, for times up to $tv=2$, half filling, equilibrium, for $U/v=4$, at temperatures $T/v\in \{0.5,0.1,0.05\}$.

Figure 4

Retarded components of the DMFT Green's function, bare Green's function, and self-energy computed for $U/v=8.0$ at half filling and temperature $T/v=0.5$. $G_{\mathrm{rec}}^{\mathrm{ret}}\left(t\right)$ (dashed orange curve lying on top of the black one) is a Green's function reconstructed by iterative substitution of ${\mathrm{\Sigma }}^{\mathrm{ret}}\left(t\right)$ into the Dyson equation. Data obtained using 2001 interpolation slices.

Figure 5

Imaginary part of the retarded component of the DMFT self-energy for interaction strengths $U/v\in \{2.0,4.0,6.0,8.0\}$ at half filling and temperature $T/v=0.5$.

Figure 6

The converged DMFT spectral function $A\left(\omega \right)$ obtained by directly performing the Fourier transform on the real time Green's function with a cutoff at $t_{\text{max}}v=2$ with $T/v=0.5$ and at on-site interaction strengths $U/v\in \{2.0;4.0;6.0;8.0\}$.

Figure 7

Comparison of raw Green's function data up to $tv=2$ (black) and $tv=4$ (red) with results from linear prediction of the $tv=2$ data (blue) and $tv=4$ data (green) for $U/v=8,T/v=0.5$. For the linear prediction we use $p=9,t_{\mathrm{fit}}v=1.0$.

Figure 8

Comparison of spectral function obtained from raw Green's function data up to $tv=2$ (black) and $tv=4$ (red) with results from linear prediction of the $tv=2$ data (blue) and $tv=4$ data (green) for $U/v=8,T/v=0.5$. For the linear prediction we use $p=9,t_{\mathrm{fit}}v=1.0$.

Figure 9

The converged DMFT spectral function $A\left(\omega \right)$ obtained by extrapolating the real-time Green's function from $tv=2.0$ to $tv=10.0$ using linear prediction with $p=9,t_{\mathrm{fit}}v=1.0$ for temperature $T/v=0.5$ at on-site interaction strengths $U/v\in \{2.0,4.0,6.0,8.0\}$.

Figure 10

The converged DMFT spectral function $A\left(\omega \right)$ obtained by extrapolating the real-time Green's function from $tv=2.0$ to $tv=10.0$ using linear prediction with $p=9,t_{\mathrm{fit}}v=1.0$ for $U/v=4$ at temperatures $T/v\in \{0.5,0.1,0.05\}$.

Figure 11

Matsubara Green's function $G\left(\tau \right)$ computed for the impurity model in equilibrium with $U/v=4.6$ at two temperatures and with different inchworm order truncations. Results from an equilibrium hybridization expansion solver (TRIQS/CTHYB [49, 50]) are given as a reference.

Figure 12

The converged DMFT spectral function $A\left(\omega \right)$ obtained by extrapolating the real-time Green's function from $t=2.0$ to $t=10.0$ using linear prediction with $p=\{5,10\},t_{\mathrm{fit}}=\{1.0,2.0\}$ for $U/v=4$ at temperatures $T/v=0.05$.

Figure 13

The converged DMFT spectral function $A\left(\omega \right)$ obtained by extrapolating the real-time Green's function from $tv=2.0$ to $tv=10.0$ using linear prediction with $p=\{5,10\},t_{\mathrm{fit}}v=\{1.0,2.0\}$ for $U/v=6$ at temperature $T/v=0.5$.

Figure 14

Retarded components of the DMFT Green's function, bare Green's function, and self-energy computed in equilibrium with $U/v=8.0$ at $T/v=0.5$. The self-energy curves are obtained by a direct solution of the Dyson equation in its discretized matrix form. $G^{\mathrm{ret}}\left(t\right)$ has been measured on 41 time slices, while a larger number of slices and cubic interpolation have been used to perform matrix inversions. The four subplots correspond to different numbers of interpolation points. $G_{rec}^{\mathrm{ret}}\left(t\right)$ (dashed orange curve) is the Green's function reconstructed by iterative substitution of ${\mathrm{\Sigma }}^{\mathrm{ret}}\left(t\right)$ into the Dyson equation. Top left: 41 slices. Top right: 101 slices. Bottom left: 1001 slices. Bottom right: 2001 slices.