projective_algorithm.ipynb 21.3 KB
 Jefferson Stafusa E. Portela committed Aug 07, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# ALF Tutorial 2.0 \n", "## Introductory examples and exercises" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook is part of the [Tutorial 2.0](https://git.physik.uni-wuerzburg.de/ALF/ALF_Tutorial) for the quantum Monte Carlo simulation package [*Algorithms for Lattice Fermions* - ALF](https://git.physik.uni-wuerzburg.de/ALF/ALF_code), and can be found, together with its required files, in the [pyALF repository](https://git.physik.uni-wuerzburg.de/ALF/pyALF).\n", "\n", "ALF is compiled from source, which is downloaded from the [ALF repository](https://git.physik.uni-wuerzburg.de:ALF) when not found locally.\n", "\n", "[**REMEMBER TO UPDATE GIT ADDRESSES**]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Projective algorithm " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The projective approach is the method of choice if one is interested in ground-state properties. The starting point is a pair of trial wave functions, $|\\Psi_{T,L/R} \\rangle$, that are not orthogonal to the ground state $|\\Psi_0 \\rangle$:\n", "$$\n", " \\langle \\Psi_{T,L/R} | \\Psi_0 \\rangle \\neq 0. \n", "$$\n", "The ground-state expectation value of any observable $\\hat{O}$ can then be computed by propagation along the imaginary time axis:\n", "$$\n", "\t \\frac{ \\langle \\Psi_0 | \\hat{O} | \\Psi_0 \\rangle }{ \\langle \\Psi_0 | \\Psi_0 \\rangle} = \\lim_{\\theta \\rightarrow \\infty} \n", "\t \\frac{ \\langle \\Psi_{T,L} | e^{-\\theta \\hat{H}} e^{-(\\beta - \\tau)\\hat{H} }\\hat{O} e^{- \\tau \\hat{H} } e^{-\\theta \\hat{H}} | \\Psi_{T,R} \\rangle } \n", "\t { \\langle \\Psi_{T,L} | e^{-(2 \\theta + \\beta) \\hat{H} } | \\Psi_{T,R} \\rangle } ,\n", "$$\n", "where $\\beta$ defines the imaginary time range where observables (time displaced and equal time) are measured and $\\tau$ varies from $0$ to $\\beta$ in the calculation of time-displace observables.\n", "\n", "---\n", "\n", "1. Import Simulation class from the py_alf python module, which provides the interface with ALF, as well as mathematics and plotting packages:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from py_alf import Simulation # Interface with ALF\n", "# \n", "import numpy as np # Numerical library\n", "from scipy.optimize import curve_fit # Numerical library\n", "import matplotlib.pyplot as plt # Plotting library" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "2. Create instances of Simulation, specifying the necessary parameters, in particular the Projector to True:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Theta values used:\n", "1\n", "5\n", "10\n", "20\n" ] } ], "source": [ "sims = [] # Vector of Simulation instances\n", "print('Theta values used:')\n", "for theta in [1, 5, 10, 20]: # Values of Theta\n", " print(theta)\n", " sim = Simulation(\n", " 'Hubbard', # Hamiltonian\n", " { # Model and simulation parameters for each Simulation instance\n", " 'Model': 'Hubbard', # Base model\n", " 'Lattice_type': 'N_leg_ladder', # Lattice type\n", " 'L1': 4, # Lattice length in the first unit vector direction\n", " 'L2': 1, # Lattice length in the second unit vector direction\n", " 'Checkerboard': False, # Whether checkerboard decomposition is used or not\n", " 'Symm': True, # Whether symmetrization takes place\n", " 'Projector': True, # Whether to use the projective algorithm\n", " 'Theta': theta, # Projector parameter\n", " 'ham_T': 1.0, # Hopping parameter\n", " 'ham_U': 4.0, # Hubbard interaction\n", " 'ham_Tperp': 0.0, # For bilayer systems\n", " 'beta': 1.0, # Inverse temperature\n", " 'Ltau': 0, # '1' for time-displaced Green functions; '0' otherwise \n", " 'NSweep': 400, # Number of sweeps\n", " 'NBin': 10, # Number of bins\n", " 'Dtau': 0.05, # Only dtau varies between simulations, Ltrot=beta/Dtau\n", " 'Mz': True, # If true, sets the M_z-Hubbard model: Nf=2, N_sum=1,\n", " }, # HS field couples to z-component of magnetization\n", " alf_dir='~/Programs/ALF', # Local ALF copy, if present\n", " )\n", " sims.append(sim)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "3. Compile ALF, downloading it first if not found locally. This may take a few minutes:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Compiling ALF... Done.\n" ] } ], "source": [ "sims[0].compile() # Compilation needs to be performed only once" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "4. Perform the simulations, as specified in each element of sim:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Prepare directory \"/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=1_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\" for Monte Carlo run.\n", "Create new directory.\n", "Run /home/stafusa/Programs/ALF/Prog/Hubbard.out\n", "Prepare directory \"/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=5_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\" for Monte Carlo run.\n", "Resuming previous run.\n", "Run /home/stafusa/Programs/ALF/Prog/Hubbard.out\n", "Prepare directory \"/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=10_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\" for Monte Carlo run.\n", "Resuming previous run.\n", "Run /home/stafusa/Programs/ALF/Prog/Hubbard.out\n", "Prepare directory \"/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=20_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\" for Monte Carlo run.\n", "Resuming previous run.\n", "Run /home/stafusa/Programs/ALF/Prog/Hubbard.out\n" ] } ], "source": [ "for i, sim in enumerate(sims):\n", " sim.run() # Perform the actual simulation in ALF" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "5. Calculate the internal energies:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=1_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\n", "Analysing Ener_scal\n", "Analysing Part_scal\n", "Analysing Pot_scal\n", "Analysing Kin_scal\n", "Analysing Den_eq\n", "Analysing SpinZ_eq\n", "Analysing Green_eq\n", "Analysing SpinXY_eq\n", "Analysing SpinT_eq\n", "/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=5_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\n", "Analysing Ener_scal\n", "Analysing Part_scal\n", "Analysing Pot_scal\n", "Analysing Kin_scal\n", "Analysing Den_eq\n", "Analysing SpinZ_eq\n", "Analysing Green_eq\n", "Analysing SpinXY_eq\n", "Analysing SpinT_eq\n", "/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=10_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\n", "Analysing Ener_scal\n", "Analysing Part_scal\n", "Analysing Pot_scal\n", "Analysing Kin_scal\n", "Analysing Den_eq\n", "Analysing SpinZ_eq\n", "Analysing Green_eq\n", "Analysing SpinXY_eq\n", "Analysing SpinT_eq\n", "/home/stafusa/ALF/pyALF/Hubbard_N_leg_ladder_L1=4_L2=1_Checkerboard=False_Symm=True_Projector=True_Theta=20_T=1.0_U=4.0_Tperp=0.0_beta=1.0_Dtau=0.05_Mz=True\n", "Analysing Ener_scal\n", "Analysing Part_scal\n", "Analysing Pot_scal\n", "Analysing Kin_scal\n", "Analysing Den_eq\n", "Analysing SpinZ_eq\n", "Analysing Green_eq\n", "Analysing SpinXY_eq\n", "Analysing SpinT_eq\n" ] } ], "source": [ "ener = np.empty((len(sims), 2)) # Matrix for storing energy values\n", "thetas = np.empty((len(sims),)) # Matrix for Thetas values, for plotting\n", "for i, sim in enumerate(sims):\n", " print(sim.sim_dir) # Directory containing the simulation output\n", " sim.analysis() # Perform default analysis\n", " thetas[i] = sim.sim_dict['Theta'] # Store Theta value\n", " ener[i] = sim.get_obs(['Ener_scalJ'])['Ener_scalJ']['obs'] # Store internal energy" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "For Theta values [ 1. 5. 10. 20.] the measured energies are:\n", " [[-2.127919 0.015365]\n", " [-2.123798 0.016821]\n", " [-2.115344 0.016098]\n", " [-2.087136 0.013937]]\n" ] } ], "source": [ "print('For Theta values', thetas, 'the measured energies are:\\n', ener)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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