On the convergence of complex Langevin dynamics: the three-dimensional XY model at finite chemical potential

Abstract

The three-dimensional XY model is studied at finite chemical potential using complex Langevin dynamics. The validity of the approach is probed at small chemical potential using imaginary chemical potential and continuity arguments, and at larger chemical potential by comparison with the world line method. While complex Langevin works for larger beta, we find that it fails for smaller beta, in the region of the phase diagram corresponding to the disordered phase. Diagnostic tests are developed to identify symptoms correlated with incorrect convergence. We argue that the erroneous behaviour at smaller beta is not due to the sign problem, but rather resembles dynamics observed in complex Langevin simulations of simple models with complex noise.

Figures (6)

• Figure 1: Real part of action density $\langle S\rangle/\Omega$ as a function of $\mu^2$ on a lattice of size $8^3$, using complex Langevin dynamics and the world line formulation at real $\mu$ ($\mu^2>0$) and real Langevin dynamics at imaginary $\mu$ ($\mu^2<0$). The vertical lines on the left indicate the Roberge-Weiss transitions at $\mu_{\rm I}=\pi/8$.
• Figure 2: Colour plot indicating the relative difference $\Delta S$ between the expectation value of the action density obtained with complex Langevin dynamics and in the world line formulation, see Eq. (\ref{['eq:relS']}). Also shown is the phase boundary $\beta_c(\mu)$ between the ordered (large $\beta$) and disordered (small $\beta$) phase Banerjee:2010kc.
• Figure 3: Width of the distribution $P[\phi^{\rm R},\phi^{\rm I}]$ in the imaginary direction for various values of $\beta$ as a function of $\mu^2$ on a $10^3$ lattice (left) and, for larger $\mu$, as a function of $\mu$ on a $8^3$ lattice (right).
• Figure 4: Distribution of action density $S/\Omega$ for various values of $\beta$ at $\mu=0$ on a $8^3$ lattice, using a hot and a cold start.
• Figure 5: Distribution of $K^{\rm max}/(6\beta)$ at $\mu=0$ on a $8^3$ lattice using a hot and a cold start.