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"%matplotlib inline\n",
"%config InlineBackend.figure_format = 'svg'\n",
"\n",
"import warnings \n",
"warnings.filterwarnings(\"ignore\") #ignore some matplotlib warnings\n",
"\n",
"import numpy as np\n",
"\n",
"from triqs.plot.mpl_interface import plt\n",
"plt.rcParams[\"figure.figsize\"] = (6,5) # set default size for all figures\n",
"\n",
"from triqs.utility.redirect import start_redirect\n",
"start_redirect()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Dynamical spin-spin susceptibility $\\chi_{S_z S_z}(\\tau)$\n",
"\n",
"CTHYB can measure dynamical susceptibilities $\\chi_{\\hat{O}_1 \\hat{O}_2}(\\tau)$ of operators $\\hat{O}_i$ that commute with the local Hamiltonian $[H_{loc}, \\hat{O}_i] = 0$. This is performed using \"sampling by insertion\".\n",
"\n",
"Here we reuse the Anderson model in a Wilson bath from the first tutorail and sample its spin-spin susceptibility\n",
"\n",
"$$\\chi_{S_z S_z} = \\langle S_z(\\tau) S_z(0) \\rangle$$ \n",
"\n",
"using the ``measure_O_tau`` parameter, by passing the pair of operators to sample. In this case the spin-z operator $S_z = (n_\\uparrow - n_\\downarrow)/2$.\n",
"\n",
"The one-orbital Anderson impurity embedded in a flat (Wilson) conduction bath, with non-interacting Green's function $G^{-1}_{0,\\sigma} (i\\omega_n) = i \\omega_n - \\epsilon_f - V^2 \\Gamma_\\sigma(i \\omega_n)$, and local interaction\n",
"$H_\\mathrm{int} = U n_\\uparrow n_\\downarrow$, with Hubbard $U$, local energy $\\epsilon_f$, bandwidth $D$ and hybridization $V$ at the inverse temperature of $\\beta$."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"╔╦╗╦═╗╦╔═╗ ╔═╗ ┌─┐┌┬┐┬ ┬┬ ┬┌┐ \n",
" ║ ╠╦╝║║═╬╗╚═╗ │ │ ├─┤└┬┘├┴┐\n",
" ╩ ╩╚═╩╚═╝╚╚═╝ └─┘ ┴ ┴ ┴ ┴ └─┘\n",
"\n",
"The local Hamiltonian of the problem:\n",
"(-2.5,0)*c_dag('down',0)*c('down',0) + (-2.5,0)*c_dag('up',0)*c('up',0) + (5,0)*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0)\n",
"Using autopartition algorithm to partition the local Hilbert space\n",
"Found 4 subspaces.\n",
"\n",
"Warming up ...\n",
"\n",
"Accumulating ...\n",
"17:19:09 1% ETA 00:00:07 cycle 140 of 10000\n",
"17:19:11 35% ETA 00:00:03 cycle 3587 of 10000\n",
"17:19:14 78% ETA 00:00:01 cycle 7869 of 10000\n",
"\n",
"\n",
"[Rank 0] Collect results: Waiting for all mpi-threads to finish accumulating...\n",
"[Rank 0] Timings for all measures:\n",
"Measure | seconds \n",
"Average sign | 0.00148106\n",
"G_tau measure | 0.0117463 \n",
"O_tau insertion measure | 4.88952 \n",
"Total measure time | 4.90274 \n",
"[Rank 0] Acceptance rate for all moves:\n",
"Move set Insert two operators: 0.0881532\n",
" Move Insert Delta_up: 0.0870787\n",
" Move Insert Delta_down: 0.0892352\n",
"Move set Remove two operators: 0.0876721\n",
" Move Remove Delta_up: 0.0871474\n",
" Move Remove Delta_down: 0.0881937\n",
"Move Shift one operator: 0.441899\n",
"[Rank 0] Warmup lasted: 0.0381063 seconds [00:00:00]\n",
"[Rank 0] Simulation lasted: 5.81935 seconds [00:00:05]\n",
"[Rank 0] Number of measures: 10000\n",
"Total number of measures: 10000\n",
"Average sign: (1,0)\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Starting on 1 Nodes at : 2019-06-05 17:19:09.074932\n"
]
}
],
"source": [
"from triqs.operators import n\n",
"from h5 import HDFArchive\n",
"from triqs.gf import inverse, iOmega_n, Wilson\n",
"\n",
"from triqs_cthyb import Solver\n",
"\n",
"U, e_f, D, V, beta = 5., -2.5, 1., 1., 10.\n",
"Sz = 0.5 * ( n('up', 0) - n('down', 0) )\n",
"\n",
"S = Solver(\n",
" beta=beta, gf_struct=[('up',[0]), ('down',[0])],\n",
" n_tau=400, n_iw=50,)\n",
"\n",
"S.G0_iw << inverse(iOmega_n - e_f - V**2 * Wilson(D))\n",
"\n",
"S.solve(\n",
" h_int=U*n('up',0)*n('down',0),\n",
" n_cycles=10000,\n",
" length_cycle=20,\n",
" n_warmup_cycles=100,\n",
" measure_O_tau=(Sz, Sz),\n",
" measure_O_tau_min_ins=100,\n",
" )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The measured susceptibility is stored in the member property ``S.O_tau`` and can be plotted using the TRIQS ``oplot`` command."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0, 0.25)"
]
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"output_type": "execute_result"
},
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"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"from triqs.plot.mpl_interface import oplot, oplotr, plt\n",
"\n",
"oplotr(S.O_tau)\n",
"plt.ylabel(r'$\\chi_{S_z S_z}(\\tau)$');\n",
"plt.ylim([0, 0.25])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Author: H. U.R. Strand (2019)"
]
}
],
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