Computational many-particle physics / H. Fehske, R. Schneider, A. Weisse (eds.).

Format:

Book

Contributor:

Fehske, H. (Holger)

Schneider, R. (Ralf)

Weisse, A. (Alexander)

Series:

Lecture notes in physics 0075-8450 ; 739.

Lecture notes in physics, 0075-8450 ; 739

Language:

English

Subjects (All):

Many-body problem--Numerical solutions--Congresses.

Many-body problem.

High performance computing--Congresses.

High performance computing.

Many-body problem--Numerical solutions.

Genre:

Conference papers and proceedings.

Physical Description:

xv, 780 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, [2008]

Summary:

Complicated many-particle problems abound in nature and in research alike. Plasma physics, statistical physics and condensed matter physics, as primary examples, are all heavily dependent on efficient methods for solving such problems. Addressing graduate students and young researchers, this book presents an overview and introduction to state-of-the-art numerical methods for studying interacting classical and quantum many-particle systems. A broad range of techniques and algorithms are covered, and emphasis is placed on their implementation on modern high-performance computers.

Contents:

Part I Molecular Dynamics

1 Introduction to Molecular Dynamics / Ralf Schneider, Amit Raj Sharma, Abha Rai 3

1.1 Basic Approach 3

1.2 Macroscopic Parameters 6

1.3 Inter-Atomic Potentials 8

1.4 Numerical Integration Techniques 14

1.5 Analysis of MD Runs 18

1.6 From Classical to Quantum-Mechanical MD 23

1.7 Ab Initio MD 24

1.8 Car-Parrinello Molecular Dynamics 25

1.9 Potential Energy Surface 28

1.10 Advanced Numerical Methods 29

References 37

2 Wigner Function Quantum Molecular Dynamics / V. S. Filinov, M. Bonitz, A. Filinov, V. O. Golubnychiy 41

2.1 Quantum Distribution Functions 41

2.2 Semiclassical Molecular Dynamics 43

2.3 Quantum Dynamics 50

2.4 Time Correlation Functions in the Canonical Ensemble 54

2.5 Discussion 58

References 59

Part II Classical Monte Carlo

3 The Monte Carlo Method, an Introduction / Detlev Reiter 63

3.1 What is a Monte Carlo Calculation? 63

3.2 Random Number Generation 67

3.3 Integration by Monte Carlo 71

3.4 Summary 77

References 78

4 Monte Carlo Methods in Classical Statistical Physics / Wolfhard Janke 79

4.1 Introduction 79

4.2 Statistical Physics Primer 80

4.3 The Monte Carlo Method 85

4.4 Cluster Algorithms 93

4.5 Statistical Analysis of Monte Carlo Data 99

4.6 Reweighting Techniques 108

4.7 Finite-Size Scaling Analysis 114

4.8 Generalized Ensemble Methods 129

4.9 Concluding Remarks 135

References 135

5 The Monte Carlo Method for Particle Transport Problems / Detlev Reiter 141

5.1 Transport Problems and Stochastic Processes 141

5.2 The Transport Equation: Fredholm Integral Equation of Second Kind 143

5.3 The Boltzmann Equation 144

5.4 The Linear Integral Equation for the Collision Density 147

5.5 Monte Carlo Solution 150

5.6 Some Special Sampling Techniques 154

5.7 An Illustrative Example 156

References 158

Part III Kinetic Modelling

6 The Particle-in-Cell Method / David Tskhakaya 161

6.1 General Remarks 161

6.2 Integration of Equations of Particle Motion 163

6.3 Plasma Source and Boundary Effects 166

6.4 Calculation of Plasma Parameters and Fields Acting on Particles 170

6.5 Solution of Maxwell's Equations 175

6.6 Particle Collisions 183

6.7 Final Remarks 188

References 188

7 Gyrokinetic and Gyrofluid Theory and Simulation of Magnetized Plasmas / Richard D. Sydora 191

7.1 Introduction 191

7.2 Single Particle Dynamics 193

7.3 Continuum Gyrokinetics 200

7.4 Gyrofluid Model 204

7.5 Gyrokinetic Particle Simulation Model 207

7.6 Gyrokinetic Particle Simulation Model Applications 210

7.7 Summary 217

References 218

Part IV Semiclassical Approaches

8 Boltzmann Transport in Condensed Matter / Franz Xaver Bronold 223

8.1 Boltzmann Equation for Quasiparticles 223

8.2 Techniques for the Solution of the Boltzmann Equation 230

8.3 Conclusions 252

References 253

9 Semiclassical Description of Quantum Many-Particle Dynamics in Strong Laser Fields / Thomas Fennel, Jorg Kohn 255

9.1 Semiclassical Many-Particle Dynamics in Mean-Field Approximation 255

9.2 Semiclassical Ground State 261

9.3 Application to Simple-Metal Clusters 265

References 272

Part V Quantum Monte Carlo

10 World-line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons / F.F. Assaad, H.G. Evertz 277

10.1 Introduction 277

10.2 Discrete Imaginary Time World Lines for the XXZ Spin Chain 278

10.3 World-Line Representations without Discretization Error 299

10.4 Loop Operator Representation of the Heisenberg Model 303

10.5 Spin-Phonon Simulations 308

10.6 Auxiliary Field Quantum Monte Carlo Methods 312

10.7 Numerical Stabilization Schemes for Lattice Models 325

10.8 The Hirsch-Fye Impurity Algorithm 337

10.9 Selected Applications of the Auxiliary Field Method 344

10.10 Conclusion 345

10.A The Trotter Decomposition 345

10.B The Hubbard-Stratonovich Decomposition 347

10.C Slater Determinants and their Properties 349

References 353

11 Autocorrelations in Quantum Monte Carlo Simulations of Electron-Phonon Models / Martin Hohenadler, Thomas C. Lang 357

11.1 Introduction 357

11.2 Holstein Model 358

11.3 Numerical Methods 358

11.4 Problem of Autocorrelations 360

11.5 Origin of Autocorrelations and Principal Components 363

11.6 Conclusions 365

References 366

12 Diagrammatic Monte Carlo and Stochastic Optimization Methods for Complex Composite Objects in Macroscopic Baths / A. S. Mishchenko 367

12.1 Introduction 367

12.2 Physical Properties of Interest 372

12.3 The Diagrammatic Monte Carlo Method 374

12.4 Stochastic Optimization Method 391

12.5 Conclusions and Perspectives 393

References 394

13 Path Integral Monte Carlo Simulation of Charged Particles in Traps / Alexei Filinov, Jens Boning, Michael Bonitz 397

13.1 Introduction 397

13.2 Idea of Path Integral Monte Carlo 397

13.3 Basic Numerical Issues of PIMC 401

13.4 PIMC for Degenerate Bose Systems 406

13.5 Discussion 410

References 411

Part VI Ab-Initio Methods in Physics and Chemistry

14 Ab-Initio Approach to the Many-Electron Problem / Alexander Quandt 415

14.1 Introduction 415

14.2 An Orbital Approach to Chemistry 419

14.3 Hartree-Fock Theory 427

14.4 Density Functional Theory 432

References 435

15 Ab-Initio Methods Applied to Structure Optimization and Microscopic Modelling / Alexander Quandt 437

15.1 Exploring Energy Hypersurfaces 437

15.2 Applied Theoretical Chemistry 444

15.3 Model Hamiltonians 451

15.4 Summary and Outlook 465

15.A Links to Popular Ab Initio Packages 466

References 467

Part VII Effective Field Approaches

16 Dynamical Mean-Field Approximation and Cluster Methods for Correlated Electron Systems / Thomas Pruschke 473

16.1 Introduction 473

16.2 Mean-Field Theory for Correlated Electron Systems 475

16.3 Extending the DMFT: Effective Cluster Theories 492

16.4 Conclusions 499

References 501

17 Local Distribution Approach / Andreas Alvermann, Holger Fehske 505

17.1 Introduction 505

17.2 Applications of the LD Approach 514

17.3 Summary 525

References 526

Part VIII Iterative Methods for Sparse Eigenvalue Problems

18 Exact Diagonalization Techniques / Alexander Weiße, Holger Fehske 529

18.1 Basis Construction 529

18.2 Eigenstates of Sparse Matrices 539

References 543

19 Chebyshev Expansion Techniques / Alexander Weiße, Holger Fehske 545

19.1 Chebyshev Expansion and Kernel Polynomial Approximation 545

19.2 Applications of the Kernel Polynomial Method 554

19.3 KPM in Relation to other Numerical Approaches 568

References 575

Part IX The Density Matrix Renormalisation Group: Concepts and Applications

20 The Conceptual Background of Density-Matrix Renormalization / Ingo Peschel, Viktor Eisler 581

20.1 Introduction 581

20.2 Entangled States 581

20.3 Reduced Density Matrices 582

20.4 Solvable Models 583

20.5 Spectra 586

20.6 Entanglement Entropy 589

20.7 Matrix-Product States 593

20.8 Summary 594

References 594

21 Density-Matrix Renormalization Group Algorithms / Eric Jeckelmann 597

21.1 Introduction 597

21.2 Matrix-Product States and (Super-)Blocks 598

21.3 Numerical Renormalization Group 600

21.4 Infinite-System DMRG Algorithm 602

21.5 Finite-System DMRG Algorithm 607

21.6 Additive Quantum Numbers 611

21.7 Truncation Errors 613

21.8 Computational Cost and Optimization 616

21.9 Basic Extensions 617

References 618

22 Dynamical Density-Matrix Renormalization Group / Eric Jeckelmann, Holger Benthien 621

22.1 Introduction 621

22.2 Methods for Simple Discrete Spectra 623

22.3 Dynamical DMRG 626

22.4 Finite-Size Scaling 630

22.5 Momentum-Dependent Quantities 631

22.6 Application: Spectral Function of the Hubbard

Model 632

References 634

23 Studying Time-Dependent Quantum Phenomena with the Density-Matrix Renormalization Group / Reinhard M. Noack, Salvatore R. Manmana, Stefan Wessel, Alejandro Muramatsu 637

23.1 Time Dependence in Interacting Quantum Systems 637

23.2 Sudden Quench of Interacting Fermions 643

23.3 Discussion 650

References 651

24 Applications of Quantum Information in the Density-Matrix Renormalization Group / O. Legeza, R.M. Noack, J. Solyom, L. Tincani 653

24.1 Basic Concepts of Quantum Information Theory 653

24.2 Entropic Analysis of Quantum Phase Transitions 657

24.3 Discussion and Outlook 662

References 663

25 Density-Matrix Renormalization Group for Transfer Matrices: Static and Dynamical Properties of 1D Quantum Systems at Finite Temperature / Stefan Glocke, Andreas Klumper, Jesko Sirker 665

25.1 Introduction 665

25.2 Quantum Transfer Matrix Theory 666

25.3 The Method - DMRG Algorithm for the QTM 669

25.4 An Example: The Spin-1/2 Heisenberg Chain with Staggered and Uniform Magnetic Fields 671

25.5 Impurity and Boundary Contributions 672

25.6 Real-Time Dynamics 673

References 676

Part X Concepts of High Performance Computing

26 Architecture and Performance Characteristics of Modern High Performance Computers / Georg Hager, Gerhard Wellein 681

26.1 Microprocessors 682

26.2 Parallel Computing 701

26.3 Conclusion and Outlook 729

References 729

27 Optimization Techniques for Modern High Performance Computers / Georg Hager, Gerhard Wellein 731

27.1 Optimizing Serial Code 732

27.2 Shared-Memory Parallelization 755

27.3 Conclusion and Outlook 766

References 767.

Notes:

"A summer school on 'computational many-body physics' [was organized] in September 2006, during the 550th anniversary of the University Greifswald"--Pref.

Includes bibliographical references and index.

ISBN:

3540746854

9783540746850

OCLC:

187294876