A Appendix: Abbreviations

Abbreviation Explanation

1D,2D,3D one-, two-, three-dimensional1D3V 1D in a usual space and 3D in a velocity spaceABINIT DFT software package (open source)AIREBO Adaptive Intermolecular Reactive Bond OrderAPI Application Programming InterfaceARPES Angle-Resolved Photo-Emission SpectroscopyBE Boltzmann EquationBIT1 1D3V PIC codeBLAS Basic Linear Algebra SubsystemBO Born-OppenheimerCASTEP DFT software package (commercial)ccNUMA cache-coherent Non-Uniform Memory ArchitectureCDW Charge Density WaveCF Correlation FunctionsCI Configuration InteractionCIC Cloud in CellCISC Complex Instruction Set ComputingCO Complex ObjectCP Car-ParrinelloCPA Coherent Potential ApproximationCP-PAW Car-Parrinello software packageCPT Cluster Perturbation TheoryCPU Central Processing UnitCRS Compressed Row Storagedc direct currentDDCF Density-Density time Correlation FunctionsDDMRG Dynamical Density Matrix Renormalization GroupDFT Density Functional TheoryDMC Diagrammatic Monte CarloDMFT Dynamical Mean-Field TheoryDMRG Density-Matrix Renormalization Group

(continued)

770 A Appendix: Abbreviations

(continued)

Abbreviation Explanation

DOS Density of StatesDP Double PrecisionDRAM Dynamic Random Access MemoryDTMRG Dynamical TMRGED Exact DiagonalizationEDIP Environment-Depedent Interaction PotentialEIRENE A Monte Carlo linear transport solverEPIC Explicitly Parallel Instruction ComputingFD Feynman DiagramFFT Fast Fourier TransformFFTW Fastest Fourier Transform in the West (FFT library)FHI98md DFT software packageFMM Fast Multipole MethodFP Floating PointFPGA Field Programmable Gate ArraysFSB Front Side BusGAUSSIAN computational chemistry software programGF Green FunctionGTO Gaussian Type OrbitalsGMRES Generalized Minimum Residual MethodGPU Graphics Processing UnitHF Hartree-FockHPC High Performance ComputingHPF High Performance FortranHT HypertransportIKP Improved Kelbg PotentialILP Instruction-Level ParallelismJDS Jagged Diagonals StorageKPM Kernel Polynomial MethodLAPACK Linear Algebra PackageLD Local DistributionLDA Local Density ApproximationLDA-KS Local Density Approximation in the Kohn-Sham schemeLDOS Local Density of StatesLINPACK Linear Algebra Package (superseeded by LAPACK)LJ Lennard-JonesLR Lanczos RecursionLRU Least Recently UsedMC Monte CarloMD Molecular DynamicsMEM Maximum Entropy MethodMESI Modified/Exclusive/Shared/Invalid protocolMIPS Microprocessor without Interlocked Pipeline StagesMIT Metal-Insulator TransitionMMM Matrix Matrix Multiplication

A Appendix: Abbreviations 771

Abbreviation Explanation

MOLPRO quantum chemistry software packageMP Message PassingMPI Message Passing InterfaceMPMD Multiple Program Multiple DataMVM Matrix Vector MultiplicationNGP Nearest Grid PointNI Network InterfaceNL NUMA LinkNRG Numerical Renomalization GroupNUMA Non-Uniform Memory ArchitectureNWChem computational chemistry software packageOpenMP Open Multi-ProcessingOS Operating SystemPDP1 Programmed Data Processor 1PES Potential Energy SurfacePIC Particle-in-CellPIC-MCC Particle-in-Cell Monte Carlo CollisionPIMC Path Integral Monte CarloPJT Pseudo Jahn-TellerPLPA Pageable Link Pack AreaPOSIX Portable Operating System InterfaceQMC Quantum Monte CarloQMD Quantum Molecular DynamicsQMR Quasi Minimum Residual MethodQP Quantum ParticleQPT Quantum Phase TransitionREBO Reactive Empirical Bond OrderRFO Read For OwnershipRG Renormalization GroupRISC Reduced Instruction Set ComputingRKHS Reproducing Kernel Hilbert SpaceSIAM Single Impurity Anderson Model;

Society for Industrial and Applied MathematicsSIMD Single Instruction Multiple DataSMP Symmetric Multi-ProcessingsMVM Sparse Matrix Vector MultiplicationSO Stochastic OptimizationSPEC Standard Performance Evaluation CorporationSP Single PrecisionSPMD Single Program Multiple DataSTL Standard Template LibrarySTM Scanning Tunneling MicroscopySTO Slater Type OrbitalsTCP/IP Transmission Control Protocol / Internet ProtocolTLB Translation Look-aside Buffer

(continued)

772 A Appendix: Abbreviations

(continued)

Abbreviation Explanation

TMRG Transfer Matrix Renormalization GroupUMA Uniform Memory ArchitectureUPC Unified Parallel CVASP ab initio molecular dynamics software packageVBS Valence Bond SolidWF Wave FunctionXOOPIC X-windows Object Oriented PICXPDP1 X-windows PDP1 plasma code

Index

ab initiomethod, 415, 490molecular dynamics, 24packages, 466transport coefficients, 227, 239

ageing, 100Alfven wave instability, 210Amdahl’s law, 704, 707, 732, 756Anderson localization, 58, 505, 516

polaron, 522Anderson model

disorder, 564single-impurity, 454, 481, 482, 520

antiferromagnetism, 81, 278, 303, 474, 477,478, 487, 496

Arnoldi method, 638, 642arrays, multi-dimensional, 738atomic pseudopotentials, 266, 425autocorrelation

density, 47, 48exponential autocorrelation time, 91integrated autocorrelation time, 103momentum, 58in Monte Carlo, 90, 103, 278, 357, 358numerical estimation, 104spin, 674

balancecode, 737detailed, 86, 229, 291machine, 737

bandwidthmemory, 684, 693network, 714

basis function, 422biorthogonal, 238valence bond, 305

benchmark

applications, 686low-level, 684time measurement, 685vector triad, 684

Berendsen controlpressure, 7temperature, 7

Bethe ansatz, 540, 570, 606, 629, 634Bethe lattice, 509, 510binary alloy model, 515, 555binary collision approximation, 146, 184Binder parameter, 114, 121binning analysis, 106, 360bisection algorithm, 401BLAS, 617, 749Boltzmann equation, 146, 227

heuristic derivation, 228integral representation, 234

Boris method, 164Born-Oppenheimer approximation, 415,

505Bose-Einstein condensation, 411, 638bosonic bath, 367, 371, 389bound state, 44, 368, 411Box-Muller method, 71, 364branch

elimination, 735miss, 735prediction, 735

branching process, 151Buffon’s needles, 64

cache, 684, 687, 692associativity, 743coherence, 718, 722direct-mapped, 696directory, 723effective size, 697, 743

774 Index

fully associative, 696hit, 693, 695instruction cache, 693levels, 693line, 695

replacement strategy, 696zero, 696, 738

miss, 693, 723read for ownership, 696, 738reuse ratio, 694set-associative, 697thrashing, 697, 742unified, 693way, 697write-back, 696write-through, 741

cache-bound, 695Cauchy distribution, 70ccNUMA, 717

congestion problem, 721locality domain, 720, 761locality problem, 721memory mapping, 762

central limit theorem, 66, 102, 380central processing unit

floating-point units, 683instruction queues, 683integer units, 683load/store units, 683multi-core, 699register, 683, 692

Chebyshev expansion, 545–575convergence, 549discrete Fourier transform, 553kernel polynomials, 549maximum entropy method, 570multi-dimensional, 552resolution, 551time evolution, 566

Chebyshev polynomial, 546CISC architecture, 687cloud-in-cell algorithm, 172cluster

embedded, 96geometrical, 94, 98simple-metal, 265stochastic, 94

cluster mean-field theory, 494

cluster Monte Carlo, see Monte Carlomethod

cluster perturbation theory, 568coherent potential approximation, 477, 506,

511collision density, 147collision integral, 145, 229column major order, 739compiler

directives, 746logs, 736, 746

compressed row storage, 751conductivity

electric, 232optical, 319, 563

confidence interval, 67, 102configuration interaction method, 431conformal field theory, 588, 591, 658constellation cluster, 723correlation function

density autocorrelation, 47, 48dynamic, 560, 621finite temperature, 557, 563momentum autocorrelation, 58pair, 20spin autocorrelation, 674static, 557time, 20, 47, 54zero momentum, 126

correlation sampling technique, 76Coulomb hole, 226Courant condition, 182CPU, see central processing unitCPU set, 766CPU time, 685Crank-Nicolson method, 567, 638critical amplitude, 82critical exponent, 82, 83, 118–125critical slowing down, 92cross section, macroscopic, 146crossbar switch, 719cumulant, 114, 242

deadlock, 725density functional theory, 432–435

constrained, 463, 490LDA+DMFT, 490

density matrix, 256, 397canonical, 54

Index 775

group property, 398high-temperature approximation, 398one-particle, 408, 409reduced, 582, 583

density matrix renormalization group, 562,581, 583, 589, 592, 593

additive quantum numbers, 611–613computational cost, 616correction vector, 626discarded weight, 614dynamical, 626–629finite system algorithm, 607–611infinite system algorithm, 602–607long-ranged interactions, 656–657optimization, 616quantum data compression, 654sweeping, 607time evolution, 639–643truncation error, 613–616, 654two-dimensional lattice, 617

density of states, 480, 485–488, 491, 497,507, 555, 622

density operator, see density matrixdetailed balance, 86, 229, 291detector function, 150directed loop, 307disordered system, 477, 493, 506, 555, 556,

564distribution

bimodal, 555Boltzmann, 85Cauchy, 70, 628Fermi-Dirac, 261Gauss, 66, 69, 185, 364Gaussian flux, 70local Green function, 509Lorentz ansatz, 236Maxwell-Boltzmann, 6, 169momentum, 408, 628, 632, 644–650multicanonical, 131for particle injection, 168Poisson, 300, 303, 304quasiparticle, 228Student’s t-distribution, 67uniform, 68Wigner, 41, 257

DMFT, see dynamical mean-field theory

DMRG, see density matrix renormalizationgroup

domain decomposition, 706, 708downfolding approach, 456DRAM gap, 693dynamical cluster approximation, 494–499dynamical mean-field theory, 477–484, 505,

520in density functional theory, 490extension to clusters, 492LDA+DMFT, 490

eigenvalue problemgeneralized, 423implicit, 226LAPACK, 424sparse, 539–543

energy hypersurface, 437ensemble

canonical, 5, 54, 80expanded, 129extended, 130generalized, 129generalized Gibbs, 651Gibbs, 5grand-canonical, 5, 313isothermal-isobaric, 5micro-canonical, 5, 573multi-canonical, 131

ensemble average, 18, 241, 477entanglement, 581, 593, 653entropy

entanglement, 589, 653von Neumann, 589, 653

EPIC architecture, 687, 755ergodic hypothesis, 6, 18, 240estimator, 66, 150

biased, 104collison, 151conditional expectation, 72, 153improved, 97, 306improved cluster estimator, 97path integral, 402track-length, 153

Ewald summation, 30exchange energy, 263exciton, 368, 371, 385–389exciton-polaron, 371, 372extinction coefficient, 147

776 Index

false sharing, 723, 758fast Fourier transform, 175, 181, 209, 303,

309, 423, 554Fermi gas, 261Fermi liquid, 223, 485Fermi surface, 227, 497, 592

harmonics, 237ferromagnetism, 81, 99, 119, 474, 477, 489,

529, 658, 660Feynman expansion, 375, 383, 479field weighting, 173finite-size scaling, 84, 114–128, 307, 475,

591, 630first touch policy, 763flop, 683Fortuin-Kasteleyn representation, 93, 289,

303Fredholm integral equation, 63, 141, 374front-side bus, 718

Gauss distribution, 66, 69, 185, 364Gaussian flux distribution, 70Gibbs oscillation, 549Glauber algorithm, 89global optimization, 443goodness-of-fit parameter, 117Green function, 148, 478, 485, 552, 554,

562, 568local, 480, 481, 509

Gustafson’s law, 705gyrofluid model

three-field, 204two-fluid equations, 193vorticity equation, 207

gyrokineticsdispersion relation and fluctuation

spectrum, 209guiding center drift velocity, 197gyro-averaged potential, 201gyro-center eq. of motion, 195gyrophase-averaged eq. of motion, 197history, 192one-form, 198particle simulation, 207polarization drift, 202

Hartree approximation, 427, 477Hartree-Fock approximation, 427–432heat-bath algorithm, 88

Heisenberg model, 278, 303, 474–477, 529,537, 671

hidden free energy barriers, 134High Performance Fortran, 709Hilbert transform, 480, 561Hirsch-Fye algorithm, 337–343, 482Holstein model, 358, 521, 523, 562, 567Holstein-Hubbard model, 368Hubbard model, 455, 473, 480, 484–490,

496, 529–537, 540, 543, 570, 574,632, 655

multi-orbital, 490hypertransport, 720

importance sampling, 73, 85, 151, 375instruction throughput, 686instruction-level parallelism, 686interaction representation, 302, 311, 375Ising model, 81, 586

jackknife method, 107, 360Jacobi-Davidson algorithm, 541jagged diagonals storage, 751

Kelbg potential, improved, 44kernel

collision, 146Dirichlet, 550Fejer, 551Jackson, 551Lorentz, 552subcritical, 152transport, 148

kernel polynomial method, see Chebyshevexpansion

Kholevo bound, 654Kohn-Sham method, 433Kondo problem, 341, 482, 600Krylov space, 540, 625, 638Kubo formalism, 253, 560

Lanczos algorithm, 638, 642correlation functions, 572, 625DMRG, 625eigenvalues, 539eigenvectors, 540

latency, 693, 698of network, 715

leap-frog algorithm, 16, 164

Index 777

least-recently-used strategy, 696, 749Lebesgue-Stieltjes integral, 72Lehmann function, 370, 391Lenard-Balescu equation, 224Lennard-Jones potential, 8Lewis diagram, 420Li-Sokal bound, 96Lie transform, 199limit D → ∞, 477–480linear response, 253, 267, 560LINPACK, 701Liouville equation, 42, 51load imbalance, 703, 708, 725, 756local density approximation, 258, 262, 435local density of states, 507, 556local distribution approach, 506, 509local scale invariance, 101locality

of reference, 694spatial, 695, 740temporal, 694

loopblocking, 747fusion, 744interchange, 742nest, 744peeling, 753unroll and jam, 746unrolling, 745

loop algorithm, 277, 288, 300, 303directed, 307

loop operators, 303

Markov chain, 85, 143, 287, 375master equation, 249matrix-product state, 593, 598–600, 639maximum entropy method, 391, 497, 570Maxwell-Boltzmann distribution, 6, 169mean-field theory, 475–491memory

bandwidth, 693, 717bus, 719distributed, 707latency, 693, 717shared, 717

memory-bound, 695Mermin-Wagner theorem, 492MESI protocol, 722message passing interface

barrier, 714benchmarks, 715blocking communication, 714collective communication, 712communicator, 711derived types, 712non-blocking communication, 714point-to-point communication, 712rank, 709, 711wildcards, 712wrapper scripts, 710

metal-insulator transition, 344, 486–487,516

Metropolis algorithm, 86, 378mobility edge, 518, 519, 522molecular dynamics, 3–37

quantum, 41, 50semiclassical, 43, 50, 58

momentum distribution, 408, 628, 632,644–650

Monte Carlo method, 52, 63, 511, 520δf method, 76cluster, 93–98, 277, 303

multiple-cluster update, 94Swendsen-Wang algorithm, 93Wolff algorithm, 94

continuous imaginary time, 299, 302diffusion, 141directed loop, 307importance sampling, 73loop algorithm, 277, 288, 300, 303multibondic simulations, 133quantum, 357

auxiliary field, 277, 312–325determinant, 359diagrammatic, 374–390Hirsch-Fye algorithm, 337–343, 482path integral, 397–405projector, 305, 483world-line method, 277, 358

sampling of permutations, 404sign problem, 292, 365, 404stochastic series expansion, 301, 302Wigner-Liouville equation, 43worm algorithm, 307

Moore’s law, 417, 686Morse potential, 11Mott-Hubbard insulator, 486–487

778 Index

MPI, see message passing interface

Neel state, 487, 496Nagaoka theorem, 474, 489network, 707

bandwidth, 714non-blocking, 708

neutral gas transport, 156Newton-Lorentz force, 162Newton-Raphson method, 441non-temporal stores, 696, 738nudged elastic band method, 442null-collision approximation, 186NUMALink, 720numerical renormalization group, 483–484,

600–602

O(n) spin models, 96OpenMP, 484, 723

barrierimplicit, 727, 757

critical section, 725, 756flush, 727lock, 726overhead, 756parallel region, 724reduction clause, 728, 759sentinel, 724thread, 723thread ID, 726work sharing directives, 724

optimizationcommon sense, 732–736by compiler, 691, 734

orbital picture, 419orthogonality catastrophe, 482out-of-order execution, 686, 692

padding, 743, 759parallel efficiency, 705parallelization, 484, 702

incremental, 755particle mesh technique, 260particle mover, 163particle weighting, 170Pauli-blocking, 229peak performance, 683phase separation, 489phase transition

second-order, 82phase-ordering kinetics, 99phonons, 308, 358

acoustical, 310optical, 309

PingPong, 715pipeline

bubbles, 686, 689depth, 688, 689flush, 735latency, 688stall, 690throughput, 688wind-down, 688, 695wind-up, 688, 695

pipelining, 686, 687software, 690, 735, 737

plasmon, 48surface, 270

Poisson distribution, 300, 303, 304Poisson equation, 162

gyrokinetic, 201Poisson solver, 177polaron, 367, 369, 373, 522potential energy surface, 28Potts models, 93power-law singularity, 82predictor-corrector method, 15prefetch, 698, 737

in hardware, 699outstanding, 699in software, 698

principal component representation, 364probability

conditional, 243marginal, 243

profiling, 767pseudo-gap, 493, 497pseudopotential approximation, 266, 425

quantum impurity problem, 482–484quantum Monte Carlo, see Monte Carlo

methodquantum pair potential, 43, 58, 402quantum percolation, 555quantum phase transition, 500

entropic analysis, 657quantum transfer matrix, 667quasiparticle, 367, 485, 486, 496

Index 779

concept, 224

race condition, 725, 758radiation transfer, 145random numbers

congruential generator, 68Gauss distributed, 69hit or miss, 72inversion method, 69pseudo-random number generator, 87rejection method, 70uniform, 68

REBO potential, 13redistribution function, 146register

pressure, 746, 747spill, 746

rejection method, 70reorder buffer, 687reweighting

multi-histogram, 112range, 109single-histogram, 108

RISC architecture, 687row major order, 739Runge-Kutta method, 638

scalability, 702scaling

strong, 703weak, 703, 704

scaling relations, 83scattering probability, 229Schmidt decomposition, 582, 583, 594Schrodinger equation, 50, 255, 566, 637

Bloch electrons, 225screened exchange, 226second quantization, 452self-energy, 225, 479, 520self-force, 174semiclassical approximation, 255serialization, 702shape function, 170SIMD extension, 692, 734simulated annealing, 87single-cluster algorithm, 94single-impurity Anderson model, 454, 481,

482, 520six vertex model, 288

snoop, 723sparse matrix, 533, 539, 547, 566, 575, 750spectral function, 319, 498, 562, 571, 574,

622, 632spin Peierls transition, 308statistical error, 103steepest descent, 440stochastic fix-point equation, 511stochastic optimization, 391–393stochastic series expansion, 301, 302streaming, 694strength reduction, 733structure factor

dynamical, 48, 319, 570static, 661

structure optimization, 440supercritical slowing down, 92, 132superscalar

architecture, 686processors, 692

Swendsen-Wang cluster algorithm, 93symmetric multiprocessing, 484, 717symmetry

inversion, 531particle number, 484, 530, 531particle-hole, 531SU(2), 484, 530translation, 492, 531, 533

t-distribution, 67t-J model, 294, 474tempering

parallel, 130simulated, 129

test particle method, 259TEXTOR tokamak, 157thermoremanent magnetization, 101Thomas-Fermi model, 262thread

pinning, 719, 728, 762placement, 727POSIX, 724safety, 727

tight-binding approximation, 425time average, 18, 241time evolution, 566, 637, 673Top500 list, 701transfer matrix renormalization group,

669–671

780 Index

translation look-aside buffer, 742transport flux, 147Trotter-Suzuki decomposition, 278, 345,

399, 642, 666two-level system, 368, 371, 389–390

umbrella sampling, 131Unified Parallel C, 709uniform memory access, 717universality hypothesis, 83

variancereduction, 151statistical, 66

vector computer, 682Verlet algorithm, 14

velocity, 17Vlasov equation, 258

gyrokinetic, 200Vlasov-Poisson-Ampere equations, 203

wallclock time, 685Wang-Landau recursion, 134Weiss field, 476Wigner function, 41, 50, 52, 59Wigner representation, 41, 257Wigner-Liouville equation, quantum, 42,

50, 55, 59Wolff cluster algorithm, 94world-line method, 277, 358

continuous imaginary time, 299, 302discrete imaginary time, 278

worm algorithm, 307, 409write combine buffer, 696

XXZ model, 278, 559, 643, 672–676