Index#

acceptance probability, 29 algorithm pseudocode analytic continuation, 393 comb reconfiguration, 319 constrained-path method, 356 B propagation, 358 continuous-time cluster algorithm, 95 directed loop algorithm, 152 “on the fly,” 165 “on-the-fly” S = ¹ XY model, 166 discrete probability sampling elegant, 19 standard, 18 DMFT self-consistency loop, 222 finite T determinant method, 195 heat-bath algorithm, 34 LDA+DMFT self-consistency loop, 226 Levenberg-Marquardt algorithm, 465 long-range interactions, 430 loop algorithm directed, 109 quantum, 106 Metropolis algorithm, 30 single-spin flip, 69 modified Gram-Schmidt method, 207 column orthonormalization, 448 UDV factorization, 207 Monte Carlo program structure, 61 Newton’s method, 297 power method Monte Carlo, 309 shifted, 303 shifted inverse, 305 quantum loop and cluster algorithm, 129 rejection method, 418 sampling time-ordered sequence of imaginary times faster, 99 simple, 97 single-spin update, 400 distributed-memory computer, 402 shared-memory computer, 401 stochastic series expansion method, 143 loops and clusters, 144 Swendsen-Wang algorithm, 71 union-find algorithm serial, 406 serial (improved), 407 shared-memory computer, 410 wave function optimization methods linearized, 299 Newton, 300 weight window reconfiguration, 317 Wolff algorithm, 74 world-line method, 103 worm algorithm, 78 simplified, 79 alias method, 18, 416 Anderson impurity model, 189, 215, 220 action, 218 chain representation, 217 Anderson lattice model, 189 antiferromagnetic structure factor, 197 area law, 289 autocorrelation time, 46, 49, 52 auxiliary field method, 180 BSS algorithm (determinant algorithm), 189 continuous-time auxiliary field algorithm, 455 continuous-time determinant algorithm, 459 Hirsch transformation, 183 Hirsch-Fye algorithm, 198 Hubbard-Stratonovich transformation, 182 auxiliary fields continuous, 182 discrete, 183 average spectrum method, 382 back propagation, 328 bath Green’s function, 218 Kondo model, 251 Bayesian constrained fit, 380 Bayesian statistical inference, 367 average spectrum method, 382

Bryan’s method, 382, 464 constrained fit, 380 default model, 377 dominant subspace, 391 evidence, 374 historic maximum entropy method, 380 Jeffery’s prior, 381 likelihood function, 374 Ockham factor, 387 posterior probability, 374 prior probability, 374, 379 Bayes’s theorem, 14, 374 Binder ratio, 55 binomial distribution, 40 bipartite lattice, 102 block inversion, 200 blocking analysis, 50 averages, 50 bond dimension, 288 bootstrap analysis, 59 Bose-Hubbard model, 112 Box-Muller method, 22 Bryan’s method, 382, 464 BSS algorithm, 180

central limit theorem, 48 Chapman-Kolmogoroff equation, 15 checkerboard decomposition, 191 representation, 101 classic maximum entropy method, 380 classicization, 84, 181 closed-shell effects, 361 cluster algorithm, 70 cluster DMFT, 226 coherent states, 451 conditional mean, 63 conditional probability chain rule, 15 conditional variance, 63 confidence intervals, 48 configuration basis, 270 configuration energy, 271 constrained fit, 380 constrained LDA, 224 constrained-path method, 345 constrained-phase method, 360 constrained RPA, 224 continuous-time auxiliary field method, 455 continuous-time impurity solvers determinant ratios and matrix updates, 234 determinant structure, 255 expansion order, 258 Kondo impurity model, 249 strong-coupling approach, 237 weak-coupling approach, 230 weight of a configuration, 230 control chart, 53 correlated sampling, 462

correlation coefficient, 55 correlation function quantum Monte Carlo method, 329 covariance, 55 critical slowing down, 44 cumulative distribution function, 12 curse of dimensionality, 5

default model, 377 detailed balance condition, 25, 29, 125 determinant method, 180 finite T, 180 zero T, 445 diffusion Monte Carlo, 320 direct sampling method, 17 directed-loop algorithm, 106, 147 distributed-memory architecture, 398 dominant subspace, 391 double counting, 224 dynamical cluster approximation, 227 dynamical critical exponent, 170, 173 dynamical mean field theory (DMFT), 219 approximation, 221 self-consistency condition, 221 efficiency computational, 32 statistical, 32 entangled state, 290 entropy content distribution, 38 Markov process, 38 equal-time expectation values, 115 equilibration, 27 time, 46, 52 equilibrium distribution, 43 ergodicity, 25, 26, 30, 110 error bars, 48 error propagation, 55 evidence, 374 excited states correlation function Monte Carlo, 329 modified power method, 331 expectation value, 46 exponential distribution, 22 extended-ensemble Monte Carlo method, 420 extended Hubbard model, 189 extrapolated estimate, 328 extrapolated estimator, 352 Falicov-Kimball model, 189 Fermi, E., 5 Fermi sea, 276 Fermion sign problem, 114, 338 Feynman, R., 6 Feynman path integral, 85, 88 finite-size scaling, 170, 172 corrections, 173 free energy scaling form, 172 fixed node method, 341 forward walking method, 353 freezing of clusters, 146 generalized Hubbard model, 189 generalized inverse, 20 generalized Metropolis algorithms, 31 Goldstone’s theorem, 167 graph, 71 closed, 77 edge, 71 vertex, 71 Grassmann algebra, 451 Green’s function equal time, 188 exact Green’s function Monte Carlo, 320 imaginary-time dependent, 321 impurity, 218 linear, 320 single particle, 188 time independent, 321 unequal time, 208 Weiss (DMFT), 218 Green’s function Monte Carlo, 320 diffusion, 320 exact, 320 growth estimator, 328 guiding function, 327 Gutzwiller trial state, 275 H-theorem for Monte Carlo simulations, 36 Haldane gap, 139 hard-core Bosons, 111 Hartree-Fock approximation, 275 restricted, 275 unrestricted, 276 heat-bath algorithm, 33 Hessian matrix, 297 high-temperature series expansion. 85. 140 Hirsch-Fye algorithm, 199, 201 historic maximum entropy method, 380 Hoshen-Kopelman clustering algorithm, 73, 404 Householder transformation, 449 Hubbard model, 189, 274 Hubbard-Stratonovich transformation continuous, 182 discrete. 183 general, 189, 435 hybrid Monte Carlo method, 211 hybrid parallelization, 399 hybridization expansion impurity solver, 237 kinetic energy, 260 Krylov formalism, 247 matrix formalism, 244 segment representation, 239 hybridization function, 218, 238 importance sampling, 326 interaction representation, 229 inverse power method, 304 Ising model, 6, 30, 44, 168 cluster algorithm, 70 cluster updating cycle, 72 continuous-time limit, 92 finite-size scaling, 172 Fortuin-Kasteleyn decomposition, 73 high-temperature representation, 76 kink representation, 93 longitudinal field, 86 mean field solution, 219 quantum-to-classical mapping, 90 single spin-flip algorithm, 68 transverse field, 87, 92 jackknife analysis, 57 Jastrow function, 273 Jeffery’s prior, 381 Jensen’s inequality, 37 Jordan-Wigner transformation, 112 kink operator, 112 Kondo impurity model, 215, 249 strong-coupling approach, 252 weak-coupling approach, 249 Krylov space, 248 Kullback-Leibler information, 37 kurtosis, 52 Lagrange multiplier, 368 lattice fixed-node method, 342 LDA+DMFT, 223 Legendre polynomials, 243 Levenberg-Marquardt method, 393, 464 likelihood function, 374 loop algorithm, 105 Lorentz invariance, 173 marginal probability density, 14 distribution, 15 Markov chain definition. 24 Monte Carlo, 10 Markov property, 24 Marshall’s transformation, 102, 130, 339 matrix acceptance probability, 29 aperiodic, 25 block inversion, 200 condition number, 202 ergodic, 26 irreducible, 25 nonnegative, 91 positive, 26, 91 positive-definite, 91 primitive, 26 proposal probability, 28 reducible, 25

spectral radius, 310 stochastic, 25 transition probability, 24, 28 matrix product stabilization, 196, 202 matrix product states, 287, 288 Matsubara frequency, 218 Matsubara-Matsuda transformation, 11 Matsubara transformation, 218 maximum entropy method Bryan’s method, 382, 464 classic, 380 historic, 380 Metropolis algorithm, 9, 25, 28 Metropolis, N., 3, 4, 24 Metropolis-Barker algorithm, 31 Metropolis-Hastings algorithm, 31 mixed estimator, 327 models Anderson impurity, 189, 215, 220 Anderson lattice, 189 Bose-Hubbard, 112 extended Hubbard, 189 Falicov-Kimball, 189 generalized Hubbard, 189 Hubbard, 189, 274 Ising, 6, 68, 70, 72, 73, 76, 92, 93 Kondo impurity, 249 longitudinal field Ising, 86 nonlinear sigma, 139 Potts, 75 quantum, 90 spin- ¹ XY, 104 spinless Fermion, 112 tJ, 282 transverse field Ising, 87, 92 XXZ, 111 zero field XY, 98 modified Gram-Schmidt method, 206 modified power method, 331 Monte Carlo Markov chain, 10 Monte Carlo method, 3 multicanonical, 421 Monte Carlo, name, 5 multielectron propagator, 441 Newton’s method, 296 nonlinear sigma model, 139 nuclear magnetic relaxation time, 373 Ockham factor, 387 operator irrelevant, 171 positive-definite, 91 relevant, 171 optical lattice, 174 partially screened interaction, 224 particle-hole transformation, 212 Perron-Frobenius theorem, 26, 313

phase problem, 360 phase space, 66 symmetry constraint, 101 phase transition first order, 44 second order, 44 phaseless method, 360 plaquette, 100, 123 Poisson distribution, 96 Poisson probability, 19 Poisson process, 96 positive projection method, 347 posterior probability, 374 Potts model, 75 power methods, 302 constrained path, 345 constrained phase, 360 Green’s function diffusion, 320 exact, 320 linear, 342 linear method, 320 inverse Monte Carlo, 313 lattice fixed node, 342 modified, 331 Monte Carlo, 309 shifted, 303 shifted inverse, 304 principle of maximum entropy, 377 prior probability, 374, 379 probability change of variables, 13 conditional, 13 density, 12 event, 11 joint, 13 mutually exclusive, 12 outcome, 11 projected entangled pair states, 287 propagator, 321 proposal probability, 28

race condition, 400 random number generator, 16 random sampling, 15 random variable, 12 Rayleigh-Ritz variational bound, 267 Rayleigh’s quotient, 268 regularization, 368 rejection method, 23, 418 renormalization group transformation, 171 Renyi information, 37 repulsive Hubbard model, 189 resonant valence bond state, 279 Rosenbluth, M., 4, 24, 35 Rosenbluth’s theorem, 35, 36

sample mean, 47 sample space, 11 sample variance, 47, 48 sampling, 15 Schmidt number, 290 Schmidt’s theorem, 289 self-energy impurity, 221 lattice, 221 shared-memory architecture, 398 Sherman-Morrison formula, 359, 439 sign problem, 91, 114, 338 strong-coupling impurity solver, 255, 257 weak-coupling impurity solver, 256 sign-weighted average, 340 single-particle Green’s function, 188 skewness, 52 Slater determinant, 274 spectral density, 115, 368 spectral radius, 26 spin singlet bond, 283 spinless Fermion model, 112 split-spin representation, 137 staggered magnetization, 198 standard deviation, 48 static susceptibility, 116 stationary condition, 24 statistical correlation function, 49 statistical error, 9 statistical independence, 13 statistical mean, 46 statistical variance, 46 Stirling’s formula, 375 stochastic process, 95 stochastic reconfiguration, 314 branching, 316 comb, 318 sampling with replacement, 317 stratified sampling, 317 weight window, 316 stochastic series expansion, 140 strong-coupling continuous-time impurity solver, 237 Green’s function measurement, 242 kinetic energy (DMFT), 259 Krylov formalism, 247 Legendre measurement, 243 matrix formalism, 244 sampling, 240 segment representation, 239 state histogram, 246 structure factor, anti ferromagnetic, 197 Suzuki-Trotter decomposition, 89 sweep, 44, 61 Swendsen-Wang cluster algorithm, 71 Tchebycheff’s inequality, 46 tensor network states, 287 theory of most probable distributions, 375 Thouless’s theorem, 345, 432, 443 tJ model, 282 transverse field Ising model, 87 continuous-time cluster algorithm, 93 partition function, 90 quantum-to-classical mapping, 90 trial state optimization, 291 trial states, 272 Gutzwiller, 275 Hartree-Fock, 275 Jastrow, 273 matrix product, 287 resonant valence bond, 279 tensor product, 287 Trotter decomposition, 89 twisted-phase boundary condition, 361 two-step sampling algorithm, 69, 70 UDV factorization, 204, 206 Ulam, S., 3–5 unbiased estimator, 56 unequal time Green’s function, 208 union-find algorithm, 404 valence bond states, 282 variance reduction methods, 318 variational principle, 267 von Neuman entropy, 37 Wannier orbital basis, 223 weak-coupling continuous-time impurity solver, 230 Green’s function measurement, 234 potential energy, 259 sampling, 233 Weiss Green’s function, 218 Weiss mean field theory, 219 winding number, 110 Wolff algorithm, 73 world line, 101 configuration, 123, 141 worm algorithm forward-scattering, 162 on-the-fly vertex generation, 162 worms, 106 head, 147 tail, 147 XXZ model, 111, 129 XY model spin- ¹ , 104 zero field, 98 zero variance principle, 271