Fachbereich Physik WUB-DIS 2001-7 BUGH Wuppertal D-42097 Wuppertal Germany Ph.D. Thesis Advanced Algorithms for the Simulation of Gauge Theories with Dynamical Fermionic Degrees of Freedom Wolfram Schroers November 2, 2001 arXiv:hep-lat/0304016v1 25 Apr 2003 Abstract The topic of this thesis is the numerical simulation of quantum chromodynamics including dynamical fermions. Two major problems of most simulation algorithms that deal with dynamical fermions are (i) their restriction to only two mass-degenerate quarks, and (ii) their limitation to relatively heavy masses. Realistic simulations of quantum chromodynamics, however, require the inclusion of three light dynamical fermion flavors. It is therefore highly important to develop algorithms which are efficient in this situation. This thesis is focused on the implementation and the application of a novel kind of algorithm which is expected to overcome the limitations of older schemes. This new algorithm is named Multiboson Method. It allows to simulate an arbitrary number of dynamical fermion flavors, which can in principle have different masses. It will be shown that it exhibits better scaling properties for light fermions than other methods. Therefore, it has the potential to become the method of choice. An explorative investigation of the parameter space of quantum chromodynamics with three flavors finishes this work. The results may serve as a starting point for future realistic simulations. iv Contents List of Figures ix List of Tables xi 1 Introduction 3 2 Quantum Field Theories and Hadronic Physics 7 2.1 PhenomenologyofStrongInteractions . .......... 8 2.2 Classical Field Theories . .... 12 2.3 Quantization ....................................... ... 13 2.3.1 Non-relativistic Quantum Mechanics . .... 14 2.3.2 The Axioms of Relativistic Quantum Field Theories . ... 14 2.3.3 ThePathIntegral ................................. .. 18 2.3.4 Ensembles........................................ 25 2.4 GaugeTheories..................................... .... 26 2.5 QuantumChromodynamics . .. .. .. .. .. .. .. .. .. .. .. .. ..... 28 2.5.1 Running Coupling and Energy Scales . .. 29 2.5.2 FactorizableProcesses .. .. .. .. .. .. .. .. .. .. .. .. .... 30 2.5.3 LatticeQCD ...................................... 31 2.6 Discretization...................................... .... 32 2.6.1 Scalar Fields . 32 2.6.2 GaugeFields ...................................... 33 2.6.3 D-Theory ........................................ 34 2.6.4 Fermion Fields . 36 2.6.5 Yang-Mills Theory . 41 3 Numerical Methods 43 3.1 Monte-CarloAlgorithms ............................... ..... 43 3.1.1 MarkovChains ..................................... 44 3.2 Autocorrelation.................................... ..... 46 3.2.1 Autocorrelation Function . ... 46 3.2.2 Exponential Autocorrelation Time . .... 47 3.2.3 Integrated Autocorrelation Time . ..... 48 3.2.4 Scaling Behavior . 49 3.2.5 ShortTimeSeries ................................... 50 3.3 MeasuringHadronMasses.. .. .. .. .. .. .. .. .. .. .. .. ...... 53 3.4 Bosonic Sampling algorithms . .... 54 3.4.1 Metropolis Algorithm . 55 3.4.2 HeatbathAlgorithm ................................. 56 3.4.3 Overrelaxation .................................... 61 3.5 Fermionic Sampling Algorithms . .... 61 3.5.1 Sampling with the Wilson Matrix . 62 3.5.2 Hybrid Monte-Carlo Algorithm . .. 62 v Contents 3.5.3 Multiboson Algorithms . 64 3.6 Matrix Inversion Algorithms . ...... 68 3.6.1 Static Polynomial Inversion . ... 70 3.6.2 Conjugate-GradientIteration . ...... 72 3.6.3 GMRESAlgorithm................................... 74 3.6.4 Stabilized Bi-Conjugate Gradient Algorithm . ..... 75 3.7 Eigenvalue Algorithms . .... 75 4 Tuning of Multiboson Algorithms 79 4.1 Optimizing the Polynomial Approximation . ..... 79 4.1.1 Tuning the Quadratically Optimized Polynomials . ... 80 4.1.2 Algorithm for Polynomials . 84 4.1.3 Computing the Reweighting Correction . .... 85 4.2 TuningtheDynamicalParameters . ....... 87 4.2.1 Practical Determination of Autocorrelations . ........ 87 4.2.2 Acceptance Rates vs. Polynomial Approximation Quality . ....... 92 4.2.3 Dynamical Reweighting Factor . ... 99 4.2.4 UpdatingStrategy ................................. 102 4.3 ImplementationSystems. .. .. .. .. .. .. .. .. .. .. .. .. ...... 108 4.3.1 APE Platform ..................................... 108 4.3.2 ALiCE Cluster..................................... 109 4.3.3 Accuracy Considerations and Test Suites . ...... 109 4.3.4 ArchitecturesandEfficiency. .... 112 4.4 Summary .......................................... 113 5 Comparison of Dynamical Fermion Algorithms 115 5.1 Simulation Runs at Different Parameters . ........ 115 5.2 Efficiency of Multiboson Algorithms . ..... 116 5.2.1 Tuning the MB-HB Algorithm . 116 5.2.2 Tuning the MB-OR Algorithm . 117 5.2.3 Direct Algorithmic Comparison . 118 5.2.4 Scaling Behavior of Algorithms . 120 5.3 Summary .......................................... 122 6 Exploring the Parameter Space with Three Degenerate DynamicalFlavors 123 6.1 TheNon-ZeroTemperatureCrossover . .......... 123 6.2 TheChiralLimit...................................... 124 6.3 ExplorativeStudies.................................. ..... 126 6.3.1 The Case β =5.3.................................... 126 6.3.2 The Case β =5.2.................................... 127 6.4 SummaryandOutlook................................. .... 138 7 Summary and Outlook 141 A Notations and Conventions 143 A.1 DiracMatrices ...................................... 144 vi Contents B Groups and Algebras 147 B.1 GroupsandRepresentations. ........ 147 B.2 TheU(1)Group ..................................... 149 B.3 The SU(N)Groups....................................... 149 B.3.1 TheSU(2)Group................................... 149 B.3.2 TheSU(3)Group................................... 150 B.4 ThePoincar´eGroup ................................. ..... 150 B.5 Spin-StatisticsTheorem .............................. ...... 152 B.6 GrassmannAlgebras ................................. ..... 152 B.6.1 Definitions ....................................... 153 B.6.2 Derivatives ....................................... 154 B.6.3 Integration ...................................... 154 C Local Forms of Actions Used 157 C.1 GeneralExpressions ................................ ...... 157 C.2 LocalFormsofVariousActions . ....... 158 C.2.1 PureGaugeFields ................................... 158 C.2.2 Lattice Fermion Fields . 159 D Logistics for Running Large Numerical Productions 167 D.1 DesignoftheDatabase............................... ...... 168 Bibliography 173 vii Contents viii List of Figures 2.1 Fundamental representations of the SU(3)F flavor group. The left graph shows the quark triplet (u,d,s) and the right graph shows the anti-quark triplet (¯u,d¯,¯s). 10 ′ 2.2 Pseudoscalar meson octet together with the singlet (the η state) as classified by the parameters of the SU(3)F group. 10 2.3 Vector meson octet together with the singlet (the φ state) as classified by the parameters of the SU(3)F group. 11 2.4 Baryon octet as classified by the parameters of the SU(3)F group. ............ 11 2.5 Relations of different axiomatic frameworks for quantum field theory............ 17 2.6 Relations between the different kinds of n-point functions consisting of vacuum expectation values of products of field operators. The figure has been taken from [28]. 18 2.7 Different methods for obtaining prediction in QCD. ......... 32 2.8 Spectrum of the hopping matrix, D(y, x) in Eq. (2.82), in the limit β 0......... 39 → 2.9 Spectrum of the hopping matrix, D(y, x) in Eq. (2.82), in the limit β ........ 40 → ∞ 4.1 Histogram of the 512 smallest eigenvalues of Q˜2(y, x). ................... 80 4.2 Histogram of the 512 largest eigenvalues of Q˜2(y, x). .................... 80 α=1 4.3 Test function λ Pn1=20(λ) for a quadratically optimized polynomial. 81 4.4 Norms R and R vs. the lower interval limit ǫ..................... 82 | 20| k 20k 4.5 Norms R180 and R180 as defined in Eq. (4.2) vs. the lower interval limit ǫ....... 82 | | k ˆ160k ˜2 4.6 Residual norms for R200(Q ) vs. the lower interval limit ǫ.................. 84 α=1 4.7 Polynomial λ P20(λ) which has been obtained by applying the GMRES algorithm to a thermalized gauge field configuration together with the corresponding quadratically optimized polynomial. 84 4.8 Residual vector norm for both the GMRES and the quadratically optimized polynomials. 85 2 2 4.9 Individual correction factors as computed using the 512 lowest eigenvalues of Q˜ (y, x) for the quadratically optimized polynomial P180(Q˜ ). 86 4.10 Similar to Fig. 4.9, but with the cumulative correction factor from the 512 lowest eigenvalues. 86 4.11 PlaquettehistoryofHMCrun. ....... 88 4.12 Normalized autocorrelation function and integrated autocorrelation time vs. the cutoff for the HMC plaquette history. 88 4.13 Variance σB (Plaquette) vs. the bin size B using the Jackknife method for the HMC plaquette history. 89 4.14 Order-1-lag-30 differenced series of the HMC plaquette history. .............. 89 4.15 Correlation function Γ (1) (t)togetherwithitsintegralvs.thecutoff. 90 AD30 A) 4.16 Integrated autocorrelation time as obtained from the lag-differencing method for varying lags l. 90 4.17 Similar to Fig. 4.15, but with a differencing lag l =23.................... 90 4.18 Time series of the average plaquette from the TSMB algorithm. ............. 91 4.19 Autocorrelation function and the corresponding integral as a function of the cutoff for the plaquette history from the TSMB run. 91 4.20 Jackknife