Numerical Methods for the QCD Overlap Operator IV: Hybrid Monte Carlo

Abstract

The extreme computational costs of calculating the sign of the Wilson matrix within the overlap operator have so far prevented four dimensional dynamical overlap simulations on realistic lattice sizes, because the computational power required to invert the overlap operator, the time consuming part of the Hybrid Monte Carlo algorithm, is too high. In this series of papers we introduced the optimal approximation of the sign function and have been developing preconditioning and relaxation techniques which reduce the time needed for the inversion of the overlap operator by over a factor of four, bringing the simulation of dynamical overlap fermions on medium-size lattices within the range of Teraflop-computers. In this paper we adapt the HMC algorithm to overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. We then derive the fermionic force for the overlap operator, elaborating on the problem of Dirac delta-function terms from zero crossings of eigenvalues of the Wilson operator. The crossing scheme proposed shows energy violations which are better than O($\Delta {\tau }^2$) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. Finally, we test our algorithm on small $4^4$, $6^4$, and $8^4$ lattices at large masses.

v2 60 pages; substantial changes to all parts of the article; v3 minor revsions

References in corpus (51)

• Exactly massless quarks on the lattice
• A Method for Simulating Chiral Fermions on the Lattice
• Locality properties of Neuberger's lattice Dirac operator
• Chiral Fermions on the Lattice
• Abelian chiral gauge theories on the lattice with exact gauge invariance
• Prospects for perfect actions
• A practical implementation of the Overlap-Dirac operator
• Testing and tuning symplectic integrators for Hybrid Monte Carlo algorithm in lattice QCD
• Numerical Methods for the QCD Overlap Operator: I. Sign-Function and Error Bounds
• An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD
• Numerical techniques for lattice QCD in the $\epsilon$--regime
• Dynamical overlap fermions, results with hybrid Monte-Carlo algorithm
• A Study of Practical Implementations of the Overlap-Dirac Operator in Four Dimensions
• Scaling tests with dynamical overlap and rooted staggered fermions
• Finite-size scaling of the quark condensate in quenched lattice QCD
• Staggered versus overlap fermions: a study in the Schwinger model with $N_f=0,1,2$
• Light Quark Masses with Overlap Fermions in Quenched QCD
• Chiral Condensate in the Deconfined Phase of Quenched Gauge Theories
• Physics issues in simulations with dynamical overlap fermions
• Optimal lattice domain-wall fermions
• A note on Zolotarev optimal rational approximation for the overlap Dirac operator
• Chiral Logs in Quenched QCD
• Topology with Dynamical Overlap Fermions
• On the Neuberger overlap operator
• Numerical Methods for the QCD Overlap Operator:III. Nested Iterations
• Chiral properties of two-flavor QCD in small volume and at large lattice spacing
• Topological susceptibility with the improved Asqtad action
• Quark mass effects on the topological susceptibility in QCD
• Topological susceptibility in full QCD: lattice results versus the prediction from the QCD partition function with granularity
• Scaling test of fermion actions in the Schwinger model
• Locality of optimal lattice domain-wall fermions
• Lanczos Approach to the Inverse Square Root of a Large and Sparse Matrix
• The overlap lattice Dirac operator and dynamical fermions
• B_K from quenched QCD with exact chiral symmetry
• Topology conserving gauge action and the overlap-Dirac operator
• Schwinger Model with the Overlap-Dirac Operator: exact results versus a physics motivated approximation
• Low-lying Wilson Dirac operator eigenvector mixing in dynamical overlap Hybrid Monte-Carlo
• Topological tunneling with Dynamical overlap fermions
• Simulating an arbitrary number of flavors of dynamical overlap fermions
• Current status of Dynamical Overlap project
• Hybrid Monte Carlo algorithm for lattice QCD with two flavors of dynamical Ginsparg-Wilson quarks
• A note on the practical feasibility of domain-wall fermions
• The overlap operator as a continued fraction
• Efficient computation of low-lying eigenmodes of non-Hermitian Wilson-Dirac type matrices
• Speed and Adaptability of Overlap Fermion Algorithms
• Overlap and domain wall fermions: what is the price of chirality?
• Dynamical overlap fermions with increased topological tunnelling
• Chiral symmetry breaking from Ginsparg-Wilson fermions
• The saga of rooted staggered quarks
• Optimal lattice domain-wall fermions with finite Ns
• The Overlap-Dirac Operator: Topology and Chiral Symmetry Breaking

Cited by in corpus (29)

• Light Hadron Masses from Lattice QCD
• Deconfinement transition in two-flavour lattice QCD with dynamical overlap fermions in an external magnetic field
• Revisiting the Hybrid Quantum Monte Carlo Method for Hubbard and Electron-Phonon Models
• Two-flavor QCD simulation with exact chiral symmetry
• Hadron Spectroscopy with Dynamical Chirally Improved Fermions
• Lattice QCD for Cosmology
• The Semimetal-Mott Insulator Quantum Phase Transition of the Hubbard Model on the Honeycomb Lattice
• QCD thermodynamics with dynamical overlap fermions
• Topology with Dynamical Overlap Fermions
• Numerical Methods for the QCD Overlap Operator:III. Nested Iterations
• Topology of dynamical lattice configurations including results from dynamical overlap fermions
• Quark Hadron Duality at Finite Temperature
• Avoiding Ergodicity Problems in Lattice Discretizations of the Hubbard Model
• Low-lying Wilson Dirac operator eigenvector mixing in dynamical overlap Hybrid Monte-Carlo
• Upper and lower Higgs boson mass bounds from a chirally invariant lattice Higgs-Yukawa model
• Fast Matrix Square Roots with Applications to Gaussian Processes and Bayesian Optimization
• Monte-Carlo simulations of overlap Majorana fermions
• Brief Review of Saturation Physics
• Two Lectures on QCD at Short Distances: Jets and Factorization
• Lattice simulations of adjoint QCD with one Dirac overlap fermion
• Physics results from dynamical overlap fermion simulations
• A lattice Dirac operator for QCD with light dynamical quarks
• A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential
• Dynamical overlap fermions with increased topological tunnelling
• The eigSUMR inverter for overlap fermion
• Block renormalization group transformations and overlap fermions
• Modifying the molecular dynamics action to increase topological tunnelling rate for dynamical overlap fermions
• Deconfinement transition in two-flavour lattice QCD with dynamical overlap fermions
• The Hubbard Model on the Honeycomb Lattice with Hybrid Monte Carlo