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Register a new userTuning HMC using Poisson brackets
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We discuss how the integrators used for the Hybrid Monte Carlo (HMC) algorithm not only approximately conserve some Hamiltonian $H$ but exactly conserve a nearby shadow Hamiltonian (tilde H), and how the difference $Delta H equiv tilde H - H $ may be expressed as an expansion in Poisson brackets. By measuring average values of these Poisson brackets over the equilibrium distribution $propto e^{-H}$ generated by HMC we can find the optimal integrator parameters from a single simulation. We show that a good way of doing this in practice is to minimize the variance of $Delta H$ rather than its magnitude, as has been previously suggested. Some details of how to compute Poisson brackets for gauge and fermion fields, and for nested and force gradient integrators are also presented.
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We show how the integrators used for the molecular dynamics step of the Hybrid Monte Carlo algorithm can be further improved. These integrators not only approximately conserve some Hamiltonian $H$ but conserve exactly a nearby shadow Hamiltonian $tilde{H}$. This property allows for a new tuning method of the molecular dynamics integrator and also allows for a new class of integrators (force-gradient integrators) which is expected to reduce significantly the computational cost of future large-scale gauge field ensemble generation.
Numerical lattice gauge theory computations to generate gauge field configurations including the effects of dynamical fermions are usually carried out using algorithms that require the molecular dynamics evolution of gauge fields using symplectic integrators. Sophisticated integrators are in common use but are hard to optimise, and force-gradient integrators show promise especially for large lattice volumes. We explain why symplectic integrators lead to very efficient Monte Carlo algorithms because they exactly conserve a shadow Hamiltonian. The shadow Hamiltonian may be expanded in terms of Poisson brackets, and can be used to optimize the integrators. We show how this may be done for gauge theories by extending the formulation of Hamiltonian mechanics on Lie groups to include Poisson brackets and shadows, and by giving a general method for the practical computation of forces, force-gradients, and Poisson brackets for gauge theories.
We apply the Hybrid Monte Carlo method to the simulation of overlap fermions. We give the fermionic force for the molecular dynamics update. We present early results on a small dynamical chiral ensemble.
We find a new d-parameter family of ultra-local boundary Poisson brackets that satisfy the Jacobi identity. The two already known cases (hep-th/9305133, hep-th/9806249 and hep-th/9901112) of ultra-local boundary Poisson brackets are included in this new continuous family as special cases.
We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
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