Simplified algorithm for the Worldvolume HMC and the Generalized-thimble HMC

The Worldvolume Hybrid Monte Carlo method (WV-HMC method) [arXiv:2012.08468] is a reliable and versatile algorithm towards solving the sign problem. Similarly to the tempered Lefschetz thimble method, this method removes the ergodicity problem inherent in algorithms based on Lefschetz thimbles. In addition to this advantage, the WV-HMC method significantly reduces the computational cost because one needs not compute the Jacobian of deformation in generating configurations. A crucial step in this method is the RATTLE algorithm, where the Newton method is used at each molecular dynamics step to project a transported configuration onto a submanifold (worldvolume) in the complex space. In this paper, we simplify the RATTLE algorithm by employing a simplified Newton method (the fixed-point method) along with iterative solvers for orthogonal decompositions of vectors, and show that this algorithm further reduces the computational cost. We also apply this algorithm to the HMC algorithm for the generalized thimble method (GT-HMC method). We perform a numerical test for the convergence of the simplified RATTLE algorithm, and show that the convergence depends on the system size only weakly. The application of this simplified algorithm to various models will be reported in subsequent papers.