Schwinger-Keldysh formalism on the lattice: A faster algorithm and its application to field theory

Abstract

A new algorithm is developed allowing the Monte Carlo study of a $1+1$-dimensional theory in real time. The main algorithmic development is to avoid the explicit calculation of the Jacobian matrix and its determinant in the update process. This improvement has a wide applicability and reduces the cost of the update in thimble-inspired calculations from $\mathcal{O}({N}^{3})$ to less than $\mathcal{O}({N}^{2})$. As an additional feature, the algorithm leads to improved Monte Carlo proposals. We exemplify the use of the algorithm to the real-time dynamics of a scalar ${\ensuremath{\phi}}^{4}$ theory with weak and strong couplings.