Momentum Lattice Simulation on a Small Lattice Using Stochastic Quantization

We have studied the scalar ${\varphi }^4$-model in the symmetric phase and the non–compact $U(1)$ gauge theory on a momentum lattice using the Langevin equation for generating configurations. In the ${\varphi }^4$-model we have analyzed the renormalized mass and in the $U(1)$-model we have analyzed the Wilson loop operator. We used a second order algorithm for solving the Langevin equation, and we looked for the convergence rate of the method. We studied the stochastic time needed to generate equilibrium configurations and compared first and second order schemes for both models.