Adaptive stepsize and instabilities in complex Langevin dynamics

TL;DR

The paper addresses the instability and convergence challenges of complex Langevin dynamics for theories with complex actions. It introduces two local, time-adaptive stepsize schemes that prevent runaway trajectories by modulating updates based on the instantaneous drift, stabilizing simulations in cases where fixed timesteps fail. Demonstrations on the three-dimensional XY model at finite chemical potential and the heavy dense limit of QCD show that the adaptive approach eliminates runaways and yields stable evolution across wide parameter ranges. The work highlights adaptive stepping as a practical, general strategy for rendering complex Langevin simulations reliable, with implications for sign-problem plagued theories and beyond.

Abstract

Stochastic quantization offers the opportunity to simulate field theories with a complex action. In some theories unstable trajectories are prevalent when a constant stepsize is employed. We construct algorithms for generating an adaptive stepsize in complex Langevin simulations and find that unstable trajectories are completely eliminated. To illustrate the generality of the approach, we apply it to the three-dimensional XY model at nonzero chemical potential and the heavy dense limit of QCD.

Figures (5)

• Figure 1: Example of a classical flow diagram in the XY model at nonzero chemical potential ($\mu=2$). The arrows denote the normalized drift terms ($K^{\rm R},K^{\rm I})$ at ($\phi^{\rm R},\phi^{\rm I}$). The dots are classical fixed points.
• Figure 2: Example of the Langevin evolution of the maximal drift term $K^{\rm max}/\beta$ and the adaptive stepsize $\epsilon_n$ in the three-dimensional XY model with $\beta=0.55$ and $\mu=0.25$ (left) and $\beta=0.1$ and $\mu=2$ (right) on lattices of size $4^3$ (top), $10^3$ (middle) and $16^3$ (bottom). The input stepsize is $\bar{\epsilon}=0.01$. Note the vertical logarithmic scale.
• Figure 3: Action density $\langle S\rangle/\Omega$ in the three-dimensional XY model as a function of $\mu^2$ at $\beta=0.55$ on lattices of size $6^3$ and $8^3$, for the full theory (circle, square, $\mu^2>0$), at imaginary $\mu$ (circle, square, $\mu^2<0$), and phase quenched (triangles, $\mu^2>0$). The vertical lines at $\mu_{\rm I}=\pi/N_\tau$ indicate the Roberge-Weiss lines at imaginary $\mu$. The dashed lines are the second-order fits (\ref{['eq:fit1']}, \ref{['eq:fit2']}), incorporating the RW reflection symmetry.
• Figure 4: Example of the Langevin evolution of the maximal drift term $\epsilon K^{\rm max}$ (left) and the adaptive stepsize $\epsilon$ (right) in the heavy dense limit of QCD with $\beta=5$, $\kappa=0.12$ and $\mu=0.7$ on a lattice of size $2^4$, using ${\cal K}=2\times 10^{-4}$.
• Figure 5: As above for $\frac{1}{3}\hbox{Tr}\, U_4U_4^{\dag}$, indicating the deviation from unitarity during the evolution.