Stochastic quantization at finite chemical potential

TL;DR

This work investigates stochastic quantization with complex Langevin dynamics as a nonperturbative approach to QCD at finite chemical potential, where the sign problem impedes traditional importance sampling. It tests the method on a hierarchy of models: a one-link U(1) model, a one-link SU(3) model, and QCD in the hopping expansion, all with the correct complex determinant property $\det M(\mu)=[\det M(-\mu)]^*$. Across these settings, observables such as Polyakov loops and density agree with exact or reweighting results, even in regimes where the average phase factor is small; the study connects convergence to the positivity of complex FP eigenvalues and to the stability of classical flow, providing a partial understanding of why the sign problem does not obstruct Langevin dynamics here. The findings suggest that complex Langevin dynamics can extend nonperturbative explorations of QCD at finite density beyond small chemical potentials and motivate further work toward real QCD in more challenging density regimes, including cross-checks with imaginary chemical potential and extensions to broader parameter ranges.

Abstract

A nonperturbative lattice study of QCD at finite chemical potential is complicated due to the complex fermion determinant and the sign problem. Here we apply the method of stochastic quantization and complex Langevin dynamics to this problem. We present results for U(1) and SU(3) one link models and QCD at finite chemical potential using the hopping expansion. The phase of the determinant is studied in detail. Even in the region where the sign problem is severe, we find excellent agreement between the Langevin results and exact expressions, if available. We give a partial understanding of this in terms of classical flow diagrams and eigenvalues of the Fokker-Planck equation.

Figures (15)

• Figure 1: Real part of the Polyakov loop $\langle e^{ix}\rangle$ (left) and the conjugate Polyakov loop $\langle e^{-ix}\rangle$ (right) as a function of $\mu$ for three values of $\beta$ at fixed $\kappa=1/2$. The lines are the analytical results, the symbols are obtained with complex Langevin dynamics.
• Figure 2: Left: Real part of the density $\langle n\rangle$. Right: Real part of the plaquette $\langle \cos x\rangle$ versus $\mu^2$. Results at positive (negative) $\mu^2$ have been obtained with complex (real) Langevin evolution.
• Figure 3: Left: Real part of $\langle e^{2i\phi}\rangle = \langle \det M(\mu)/\det M(-\mu)\rangle$. Right: Real part of $\langle e^{-2i\phi}\rangle = \langle \det M(-\mu)/\det M(\mu)\rangle = Z(-\mu)/Z(\mu)$.
• Figure 4: Scatter plot of $e^{2i\phi} = \det M(\mu)/\det M(-\mu)$ during the Langevin evolution for various values of $\mu$ at $\beta=1$, $\kappa=1/2$. Note the different scale in the middle box.
• Figure 5: Classical flow diagram in the $x-y$ plane for $\beta=1$, $\kappa=1/2$, $\mu=0.1$ (top) and $\mu=1$ (bottom). The big dots indicate the fixed points at $x=0$ and $\pi$. The small circles indicate a trajectory during the Langevin evolution, each dot separated from the previous one by 500 steps. Note the periodicity $x\to x+2\pi$.