In the landscape of approaches toward the simulation of Lattice Models with complex action the Complex Langevin Equation (CLE) appears as a straightforward method with a simple, well defined set up. Its applicability, however, is controlled by certain specific conditions which are not always satisfied, such as holomorphy and limited diffusion in the non-compact directions. We here provide some details about what this method can (and did) achieve, which are its present limitations and which are its prospects. This includes a discussion of the difficulties related with the mentioned conditions and of the procedures designed to meet them, such as various kinds of “cooling” and the estimation of systematic errors. We involve tests and insights from simple, solvable models and present results from lattice models, including QCD, especially concerning the attempt to explore the phase diagram of the latter at non-zero chemical potential.

Nonzero Temperature and Density