Monte Carlo simulations on the Lefschetz thimble: taming the sign problem

Abstract

We present the first practical Monte Carlo calculations of the recently proposed Lefschetz thimble formulation of quantum field theories. Our results provide strong evidence that the numerical sign problem that afflicts Monte Carlo calculations of models with complex actions can be softened significantly by changing the domain of integration to the Lefschetz thimble or approximations thereof. We study the interacting complex scalar field theory (relativistic Bose gas) in lattices of size up to 8^4 using a computationally inexpensive approximation of the Lefschetz thimble. Our results are in excellent agreement with known results. We show that - at least in the case of the relativistic Bose gas - the thimble can be systematically approached and the remaining residual phase leads to a much more tractable sign problem (if at all) than the original formulation. This is especially encouraging in view of the wide applicability - in principle - of our method to quantum field theories with a sign problem. We believe that this opens up new possibilities for accurate Monte Carlo calculations in strongly interacting systems of sizes much larger that previously possible.

Figures (4)

• Figure 1: Average density $\langle n\rangle$ in the critical region for the lattices $V=4^4, 6^4, 8^4$.
• Figure 2: Same as in Fig. \ref{['fig:1']} for the observable $\langle |\phi|^2 \rangle$.
• Figure 3: Comparison of the average density $\langle n\rangle$ obtained with the Worm Algorithm (WA) Gattringer:private with the Aurora Algorithm (AA) presented here, for the lattice $V=8^4$. We thank C.Gattringer and T.Kloiber for providing us their results.
• Figure 4: The data on the top-right show the average phase obtained with the Aurora algorithm on lattices $4^4$, $6^4$ and $8^4$. It is interesting that the average phase is large precisely in the most interesting region just above $\mu=1$. The dashed lines on the bottom-left display, for comparison, the average phase obtained with a naive phase-quenched Monte Carlo algorithm on lattices $4^4$ and $6^4$. Even on a $4^4$ lattice, the sign problem in the phase-quenched algorithm, completely hides the interesting region.