{
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   "source": [
    "%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 mpi4py import MPI\n",
    "\n",
    "from triqs.plot.mpl_interface import plt, oplot, oplotr, oploti\n",
    "plt.rcParams[\"figure.figsize\"] = (6,4) # set default size for all figures"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e51b0091-8880-4691-9ad1-7cddcff025bd",
   "metadata": {},
   "source": [
    "# Step-by-step guide"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa879df5-ca9d-45fa-ae03-988074b80939",
   "metadata": {},
   "source": [
    "## Step 1 - Spawn solver instance\n",
    "\n",
    "The first step in using `soehyb` is to instantiate an instance of the `triqs_soehyb.Solver` class."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a210740f-d7db-4017-a81b-7ccd915f8758",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Warning: could not identify MPI environment!\n",
      "  ___      ___    _  ___   _____\n",
      " / __| ___| __|__| || \\ \\ / / _ )\n",
      " \\__ \\/ _ \\ _|___| __ |\\ V /| _ \\\n",
      " |___/\\___/___|  |_||_| |_| |___/  [github.com/TRIQS/soehyb]\n",
      "\n",
      "beta = 5.0, lamb = 2.50E+02, eps = 1.00E-08, N_DLR = 32\n",
      "fundamental_operators = [1*c('up',0), 1*c('up',1), 1*c('up',2), 1*c('do',0), 1*c('do',1), 1*c('do',2)]\n",
      "H_mat.shape = (64, 64)\n",
      "H_loc = 0\n",
      "\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Starting serial run at: 2025-08-19 11:38:46.470781\n"
     ]
    }
   ],
   "source": [
    "from triqs_soehyb import Solver   \n",
    "\n",
    "norb = 3\n",
    "\n",
    "S = Solver(\n",
    "    beta=5.0, \n",
    "    gf_struct=[('up', norb), ('do', norb)], \n",
    "    eps=1e-8, w_max=50.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b4605f0-84d6-493b-a012-688506faf2c1",
   "metadata": {},
   "source": [
    "The solver constructor takes the input\n",
    "\n",
    "- `beta`: inverse temperature\n",
    "- `gf_struct`: Green's function structure 1st index: name, 2nd index: dimension of subspace\n",
    "- `eps`: Accuracy of Discrete Lehmann Representation (DLR) used for imaginary time response functions\n",
    "- `w_max`: DLR freqiency cut-off (the spectrum of the model must be in the range `[-w_max, +w_max]`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "549bb71a-f72c-47a5-9ba0-1f512be65465",
   "metadata": {},
   "source": [
    "## Step 2 - Impurity many-body Hamiltonian\n",
    "\n",
    "We also have to construct the local many-body Hamiltonian (`h_int`) of the Anderson impurity model we want to solve."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "534105f9-15b8-4631-af12-336fcfa9470e",
   "metadata": {},
   "source": [
    "### Examples\n",
    "\n",
    "- For the **single band Hubbard model** the interaction is density-density only and can be constructed using the Triqs second-quantization operators found in `triqs.operators.*`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a1eac349-5da2-4675-9659-417357d3d0f7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1*c_dag('do',0)*c('do',0) + -1*c_dag('up',0)*c('up',0) + 2*c_dag('do',0)*c_dag('up',0)*c('up',0)*c('do',0)\n"
     ]
    }
   ],
   "source": [
    "from triqs.operators import n\n",
    "\n",
    "U = 2.0\n",
    "mu = U / 2 # Chemical potential for half-filling\n",
    "h_int = U * n('up',0) * n('do', 0) - mu * ( n('up', 0) + n('do', 0) )\n",
    "\n",
    "print(h_int)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dedae90a-a720-47cf-b67c-5ae4482bf198",
   "metadata": {},
   "source": [
    "- For multi-orbital models one often use the **Kanamori interaction** which can be built using tools in `triqs.operators.util.*` as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f55b959b-4469-46d7-9d95-336acdc14e05",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.2*c_dag('do',0)*c_dag('do',1)*c('do',1)*c('do',0) + 2.2*c_dag('do',0)*c_dag('do',2)*c('do',2)*c('do',0) + 0.8*c_dag('do',0)*c_dag('up',0)*c('up',2)*c('do',2) + 0.8*c_dag('do',0)*c_dag('up',0)*c('up',1)*c('do',1) + 4.6*c_dag('do',0)*c_dag('up',0)*c('up',0)*c('do',0) + 3*c_dag('do',0)*c_dag('up',1)*c('up',1)*c('do',0) + 0.8*c_dag('do',0)*c_dag('up',1)*c('up',0)*c('do',1) + 3*c_dag('do',0)*c_dag('up',2)*c('up',2)*c('do',0) + 0.8*c_dag('do',0)*c_dag('up',2)*c('up',0)*c('do',2) + 2.2*c_dag('do',1)*c_dag('do',2)*c('do',2)*c('do',1) + 0.8*c_dag('do',1)*c_dag('up',0)*c('up',1)*c('do',0) + 3*c_dag('do',1)*c_dag('up',0)*c('up',0)*c('do',1) + 0.8*c_dag('do',1)*c_dag('up',1)*c('up',2)*c('do',2) + 4.6*c_dag('do',1)*c_dag('up',1)*c('up',1)*c('do',1) + 0.8*c_dag('do',1)*c_dag('up',1)*c('up',0)*c('do',0) + 3*c_dag('do',1)*c_dag('up',2)*c('up',2)*c('do',1) + 0.8*c_dag('do',1)*c_dag('up',2)*c('up',1)*c('do',2) + 0.8*c_dag('do',2)*c_dag('up',0)*c('up',2)*c('do',0) + 3*c_dag('do',2)*c_dag('up',0)*c('up',0)*c('do',2) + 0.8*c_dag('do',2)*c_dag('up',1)*c('up',2)*c('do',1) + 3*c_dag('do',2)*c_dag('up',1)*c('up',1)*c('do',2) + 4.6*c_dag('do',2)*c_dag('up',2)*c('up',2)*c('do',2) + 0.8*c_dag('do',2)*c_dag('up',2)*c('up',1)*c('do',1) + 0.8*c_dag('do',2)*c_dag('up',2)*c('up',0)*c('do',0) + 2.2*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0) + 2.2*c_dag('up',0)*c_dag('up',2)*c('up',2)*c('up',0) + 2.2*c_dag('up',1)*c_dag('up',2)*c('up',2)*c('up',1)\n"
     ]
    }
   ],
   "source": [
    "spin_names = ['up', 'do']\n",
    "\n",
    "U = 4.6\n",
    "J = 0.8\n",
    "\n",
    "from triqs.operators.util.U_matrix import U_matrix_kanamori\n",
    "\n",
    "KanMat1, KanMat2 = U_matrix_kanamori(norb, U, J)\n",
    "\n",
    "from triqs.operators.util.hamiltonians import h_int_kanamori\n",
    "\n",
    "h_int = h_int_kanamori(spin_names, norb, KanMat1, KanMat2, J, off_diag=True)\n",
    "\n",
    "print(h_int)\n",
    "\n",
    "from itertools import product\n",
    "N = sum([ n(spin, idx) for spin, idx in product(spin_names, range(norb)) ])\n",
    "\n",
    "mu = 0.5*(5*U - 10*J)\n",
    "\n",
    "h_int -= mu * N"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "378098ff-7233-4434-9bec-9b75a31c6342",
   "metadata": {},
   "source": [
    "## Step 3 - Hybridization function\n",
    "\n",
    "To fully specify the Anderson impurity model we also have to supply the hybridization function $\\Delta(\\tau)$ that describes the retarded fluctuation of electrons to the environment. The solver instance `S` sets up the hybridization function container available as `S.Delta_tau` and we need to explicitly set its value."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff8a13fd-5eeb-46d1-9d5d-bf30825cf4af",
   "metadata": {},
   "source": [
    "### Examples\n",
    "\n",
    "Here is a simple example with a Hybridization function $\\Delta(\\tau)$ comprised of a **single discrete pole/state**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "0ada854e-a4b4-491c-afbd-5540c6d91a25",
   "metadata": {},
   "outputs": [],
   "source": [
    "from triqs.gf import make_gf_dlr_imtime, make_gf_dlr_imfreq\n",
    "from triqs.gf import inverse, iOmega_n\n",
    "\n",
    "for bidx, delta_tau in S.Delta_tau:\n",
    "    delta_w = make_gf_dlr_imfreq(delta_tau)\n",
    "    delta_w << inverse(iOmega_n - 1.0)\n",
    "    delta_tau[:] = make_gf_dlr_imtime(delta_w)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ecac644-9fb2-43c0-b53d-8b26f055db75",
   "metadata": {},
   "source": [
    "Another common test case is a spin- and orbital-diagonal hybridzation function with a **semi-circular density of states**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d2e4b357-677c-4238-a60a-dc3497f16648",
   "metadata": {},
   "outputs": [],
   "source": [
    "from triqs.gf import make_gf_dlr_imtime, make_gf_dlr_imfreq, SemiCircular\n",
    "\n",
    "for bidx, delta_tau in S.Delta_tau:\n",
    "    delta_w = make_gf_dlr_imfreq(delta_tau)\n",
    "    delta_w << SemiCircular(2.0)\n",
    "    delta_tau[:] = make_gf_dlr_imtime(delta_w)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27ec122a-487d-490f-9762-16236168f559",
   "metadata": {},
   "source": [
    "## Step 4 - Self-consistent solution\n",
    "\n",
    "Combining the hybridization fucntion `S.Delta_tau` and the local many-body interaction `h_int` the Anderson impurity model is fully specified and we can call the `S.solve(...)` method to obtain the pseudo-particle self-consistent solution at a given expansion `order`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e7c57247-8d19-4244-a16e-d9096bc3ea82",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "AdaPol: Hybridization fit accuracy 1.31E-07, using 5 poles.\n",
      "(Order, N_Diags) = [(1, 1), (2, 60)]\n",
      "max_order = 2\n",
      "\n",
      " iter |   conv   |    Z-1    \n",
      "------+----------+-----------\n",
      "    1 | 1.95E-01 | +2.22E-16\n",
      "    2 | 3.68E-02 | +1.11E-15\n",
      "    3 | 2.64E-02 | +2.22E-16\n",
      "    4 | 3.25E-03 | -3.33E-16\n",
      "    5 | 1.83E-03 | -2.22E-16\n",
      "    6 | 6.63E-04 | +2.22E-16\n",
      "    7 | 6.22E-05 | +2.22E-16\n",
      "    8 | 6.54E-05 | -9.99E-16\n",
      "    9 | 1.34E-05 | +2.22E-16\n",
      "   10 | 3.23E-06 | +0.00E+00\n",
      "\n",
      "Timing:                               incl.     excl.\n",
      "------------------------------------------------------------\n",
      "Dyson equation:                      17.747    17.747   8.5% |--|\n",
      "Hybridization compression (AAA):      0.245     0.245   0.1% |\n",
      "Pseudo-particle self-energy:        171.026   171.026  82.3% |--------------------------------|\n",
      "Single particle Green's function:    15.046    15.046   7.2% |--|\n",
      "Other:                                3.631     3.631   1.7% ||\n",
      "------------------------------------------------------------\n",
      "Total:                                        207.696 100.0%\n",
      "\n"
     ]
    }
   ],
   "source": [
    "S.solve(h_int=h_int, order=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bba188a4-d19d-477f-bdf0-b135f5c5d01a",
   "metadata": {},
   "source": [
    "## Step 5 - Single particle response function\n",
    "\n",
    "After convergence the solver computes the diagrammatic series for the single particle Green's function that is available as the member `S.G_tau` of the solver class instance `S`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "eeb420d4-fc66-436b-8f4a-483f35715ded",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Green Function G composed of 2 blocks: \n",
      " Greens Function G_up with mesh DLR imaginary time mesh of size 32 with beta = 5, statistics = Fermion, w_max = 50, eps = 1e-08 and target_shape (3, 3): \n",
      " \n",
      " Greens Function G_do with mesh DLR imaginary time mesh of size 32 with beta = 5, statistics = Fermion, w_max = 50, eps = 1e-08 and target_shape (3, 3): \n",
      " \n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(S.G_tau)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83a9a807-d760-424b-a80f-1cb2c66ff1d0",
   "metadata": {},
   "source": [
    "## Step 6 - Store solver to disk\n",
    "\n",
    "The solver is hdf5 serializable and can be written to disk using the `h5` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "76b80b13-05a5-49c1-ae86-ed868d5c693a",
   "metadata": {},
   "outputs": [],
   "source": [
    "from h5 import HDFArchive\n",
    "\n",
    "filename = f'data_soehyb_result.h5'\n",
    "with HDFArchive(filename, 'w') as A: \n",
    "    A['S'] = S"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b7c6ef11-3bcc-43d5-892e-f08389329da0",
   "metadata": {},
   "source": [
    "## Step 7 - Read result from disk\n",
    "\n",
    "For later visualization we can now read the solver and its data from disk."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "477ab542-bd45-4aa8-b3d2-1851e7d5ce3d",
   "metadata": {},
   "outputs": [],
   "source": [
    "with HDFArchive(filename, 'r') as A:\n",
    "        S = A['S']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c493ba7c-f201-467c-aa05-87cbe9b23db4",
   "metadata": {},
   "source": [
    "## Step 7 - Postprocessing and visualization \n",
    "\n",
    "The single particle Green's function can now be visualized using the standard Triqs plotting tools."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f202ea70-e583-4e8c-bb8d-6cf3564ad5fe",
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      ],
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from triqs.plot.mpl_interface import plt, oplot, oplotr, oploti\n",
    "\n",
    "oplotr(S.G_tau['up'][0, 0], label=None)\n",
    "plt.ylabel(r'$G(\\tau)$');"
   ]
  }
 ],
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