Taming the Dynamical Sign Problem in Real-Time Evolution of Quantum Many-Body Problems

Current nonequilibrium Monte Carlo methods suffer from a dynamical sign problem that makes simulating real-time dynamics for long times exponentially hard. We propose a new “inchworm algorithm,” based on iteratively reusing information obtained in previous steps to extend the propagation to longer times. The algorithm largely overcomes the dynamical sign problem, changing the scaling from exponential to quadratic. We use the method to solve the Anderson impurity model in the Kondo and mixed valence regimes, obtaining results both for quenches and for spin dynamics in the presence of an oscillatory magnetic field.

Figure 1

Comparison of diagrams sampled in previous approaches [bare expansion (a) [18, 19] and bold expansion (b) 31, 32] to diagrams sampled in new approach (c), and diagrams leading to dynamical sign problem in previous methods (d). Thick lines: full propagators. Thin lines: bare propagators. Medium (“bold”) lines: propagators resulting from analytical resummation of subset of diagrams (here, NCA). Wiggly lines: hybridization lines. Arrows indicate $t_{\uparrow }$.

Figure 2

Top: AIM population dynamics in the Kondo regime following a coupling quench from a fully magnetized state at $U=-2ϵ=8\mathrm{\Gamma }$ and inverse temperature $\beta \mathrm{\Gamma }=50$. The bare hybridization expansion result [Fig. 1] is shown for times $\mathrm{\Gamma }t<1.5$, along with the inchworm result [Fig. 1] up to $\mathrm{\Gamma }t=10$ (the maximum order is limited to 3, at which convergence occurs). Bottom: Error estimate of data in upper panel showing exponential increase of the error as a function of time due to the dynamical sign problem in the bare method, and roughly constant error in the inchworm method.

Figure 3

Top: Population as a function of time after a coupling quench at $U=8\mathrm{\Gamma }$ and inverse temperature $\beta \mathrm{\Gamma }=50$, computed for a system in the Kondo regime (left-hand panels) and in the mixed valence regime (right-hand panels). Different traces show the convergence as a function of inchworm expansion order. Bottom: Error estimate of the populations for different inchworm expansion orders as a function of time.

Figure 4

AIM population and magnetization dynamics at interaction strengths and temperatures shown, in the presence of a time-dependent magnetic field $h(t)=2\mathrm{\Gamma }\sin (\omega t)$, with $\omega =5\mathrm{\Gamma }$. Dot is initially in the empty state $|0⟩$ (top row) or in the fully magnetized state $|\uparrow ⟩$ (bottom). Lighter curves show time evolution for $h=0$ with otherwise identical parameters. Dashed black curves show fits to $f(t)=A+B{e}^{-\gamma t}+C\sin ({\omega }_{0}t+\varphi )$. In units where $\mathrm{\Gamma }=1$, the charge relaxation rates ($\gamma$ for $|0⟩$, $|\uparrow \downarrow ⟩$) are ${\gamma }_c=2.83$, 3.8, and 4.0 for $(U,\beta )=(0,1)$, (5.0,1), and (8.0,50), respectively. The spin relaxation rates in the presence of the field are ${\gamma }_s=3.3$, 0.81, and 0.25 (dot initially empty) and 2.4, 0.81, and 0.25 (dot initially fully magnetized). The spin relaxation rates for $h=0$ are 0.68 for $U=5$, $\beta =1$, and 0.11 for $U=8$, $\beta =50$. The final amplitudes $C$ are 0.19, 0.13, and 0.1. $\varphi =-2$ in all cases.