RHMC

Rational Hybrid Monte Carlo (RHMC) is a stochastic simulation algorithm used primarily in lattice quantum chromodynamics (QCD) to study systems of quarks and gluons with dynamical fermions. The method extends the Hybrid Monte Carlo (HMC) framework by evaluating fractional powers of the fermion determinant through a rational approximation, enabling simulations with non-integer numbers of fermion flavors or with non-degenerate flavor content.

In RHMC, the inverse (or fractional power) of the Dirac operator D is approximated by a rational function R(D) that closely matches D^{-a}, where a corresponds to the desired flavor factor (for example a = 1/2 for a single degenerate pair). The rational function is written as a partial fraction expansion, which allows multiple shifted linear systems to be solved simultaneously via multi-shift solvers during the molecular dynamics evolution. The algorithm then performs a deterministic trajectory under an approximate Hamiltonian, followed by a Metropolis acceptance step to correct for the approximation error.

History and usage: The technique was introduced in the early 2000s to address limitations of polynomial approximations in HMC for non-integer flavors and has since become standard in lattice QCD codes for simulations with fractional or non-degenerate quark content, including staggered and Wilson-type fermions.

Advantages and limitations: RHMC provides stable and controllable approximation of fractional determinants and is compatible with existing HMC infrastructure, enabling efficient use of multi-time-scale integrators. However, it adds overhead for the construction and evaluation of the rational approximation and can demand substantial memory for the partial fraction representation and the associated solvers.

See also: Hybrid Monte Carlo; lattice QCD; fermion determinant.