Numerical Methods for Evading the Sign Problem: A Primer on Complex Langevin and Lefschetz Thimble Methods
Most simulations in Lattice QCD involve the use of algorithms based on Monte Carlo importance sampling. However, more often than not, the probability weights that the method utilizes turn out to be negative, or worse, complex valued. This renders Monte Carlo methods unusable for such problems. In such cases, one says that the theory on the lattice is plagued by the sign problem. The most standard example of such a problem is QCD at finite chemical potential. In this report, we will discuss and review two main candidates which might help in evading the sign problem - the complex Langevin method, and the Lefschetz thimble method. The former can be viewed as a complexified generalization of stochastic quantization using the Langevin equation, whereas the latter is a deformation of the original integration contour using the holomorphic gradient flow equation. We will apply these methods to two well-known toy models - the quartic model with a linear term, and the