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 [arXiv:1703.00861], this method mitigates 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 it does not require the computation of the Jacobian in generating configurations. A crucial step in this method is the RATTLE algorithm, which projects at each molecular dynamics step a transported configuration onto a submanifold (worldvolume) in the complex space. In this paper, we simplify the RATTLE algorithm by using a simplified Newton method with an improved initial guess, which can be similarly implemented to the HMC algorithm for the generalized thimble method (GT-HMC method). We perform a numerical test for the convergence of the simplified Newton equation, 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.