projective_algorithm.ipynb 21.3 KB
Newer Older
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": [
       "<ErrorbarContainer object of 3 artists>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.errorbar(1/thetas, ener[:, 0], ener[:, 1])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## Exercises\n",
    "1. Try .......................... reference plot below (found in Sec. 3.3 - Comparison of finite and projective codes - of the [ALF documentation](https://git.physik.uni-wuerzburg.de/ALF/ALF_code/-/blob/master/Documentation/ALF_v1.0.pdf))."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}