Monte Carlo Study of Real Time Dynamics on the Lattice

Monte Carlo studies involving real time dynamics are severely restricted by the sign problem that emerges from a highly oscillatory phase of the path integral. In this Letter, we present a new method to compute real time quantities on the lattice using the Schwinger-Keldysh formalism via Monte Carlo simulations. The key idea is to deform the path integration domain to a complex manifold where the phase oscillations are mild and the sign problem is manageable. We use the previously introduced “contraction algorithm” to create a Markov chain on this alternative manifold. We substantiate our approach by analyzing the quantum mechanical anharmonic oscillator. Our results are in agreement with the exact ones obtained by diagonalization of the Hamiltonian. The method we introduce is generic and, in principle, applicable to quantum field theory albeit very slow. We discuss some possible improvements that should speed up the algorithm.

Figure 1

The Schwinger-Keldysh contour (left) and its discretized form (right) in the complex time plane. $\mathrm{\Delta }{t}_n$ refers to either $\pm a$ or $-ia$ depending on the location of $n$ on the contour.

Figure 2

Retarded correlators $⟨\dot{x}(t)\dot{x}(t^{\prime })⟩$ and $⟨x(t)x(t^{\prime })⟩$. The dotted and solid lines represent the exact results obtained by diagonalizing the Hamiltonian.

Figure 3

Histogram of ${\mathbb{S}}_I$ (mod $2\pi$) for the $T_{\mathrm{flow}}=0$ calculation corresponding to an integration over ${\mathbb{R}}^N$ (in red) and the $T_{\mathrm{flow}}=0.2$ calculation corresponding to an integration over $\mathrm{\Gamma }$ (in blue). It is clear that the modest flow $T_{\mathrm{flow}}=0.2$ reduces the sign problem significantly.