{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c19e838a-0ed1-47ff-a397-ea0053ee6da7",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "outputs": [],
   "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 h5 import HDFArchive\n",
    "from itertools import product\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": "9738461c-04a1-443e-ae57-8e88d4c193f8",
   "metadata": {},
   "source": [
    "# Metal insulator transition\n",
    "\n",
    "The half-filled Hubbard model undergoes a transition from a metallic state to an (Mott) insulating state when increasing  the interaction strength $U$. One model where this can be solved exactly is the Hubbard model on the Bethe graph in the limit of infinite connectivity. In this limit Dynamical Mean Field Theory (DMFT) is exact and can be used to study the transition.\n",
    "\n",
    "The DMFT lattice self-consistency relation on the Bethe graph has the form\n",
    "\n",
    "$$\n",
    "\\Delta_\\sigma(\\tau) = t^2 G_\\sigma(\\tau)\n",
    "$$\n",
    "\n",
    "where $\\Delta_\\sigma(\\tau)$ is the hybridization function, $G_\\sigma(\\tau)$ is the local single particle Green's function with spin $\\sigma \\in \\{\\uparrow, \\downarrow\\}$, and we set the nearest neighbour hoppint $t$ to unity ($t=1$).\n",
    "\n",
    "At low temperature the transition is first order, but at higher temperatures it turns into a cross over. Here we will study this crossover at inverse temperature $\\beta = 5$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6aa5e06a-d60a-4184-ae2d-b820b0cbf519",
   "metadata": {},
   "source": [
    "## Solver setup\n",
    "\n",
    "Since we will be performing self-consistent DMFT calculations we modularize the solver setup using a helper function `setup_solver()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9372debf-a505-423b-a0f6-5d517fc0b3c7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Warning: could not identify MPI environment!\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Starting serial run at: 2025-08-19 17:41:44.159873\n"
     ]
    }
   ],
   "source": [
    "from triqs_soehyb import Solver\n",
    "\n",
    "from triqs.gf import make_gf_dlr_imtime, make_gf_dlr_imfreq, SemiCircular\n",
    "\n",
    "def setup_solver():\n",
    "    \n",
    "    S = Solver(\n",
    "        beta=5.0, # Inverse temperature\n",
    "        gf_struct=[('up', 1), ('do', 1)], # Green's function structure 1st index: name, 2nd index: dimension of subspace\n",
    "        eps=1e-12, # Accuracy of Discrete Lehmann Representation (DLR) used for imaginary time response functions\n",
    "        w_max=10.0, # DLR freqiency cut-off (the spectrum of the model must be in the range [-w_max, +w_max]\n",
    "        )\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)\n",
    "\n",
    "    return S"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e76af5c-55eb-48c1-a4d7-dee2b28a4793",
   "metadata": {},
   "source": [
    "## Local Hubbard interaction\n",
    "\n",
    "The local interaction in the Hubbard model has the form\n",
    "\n",
    "$$\n",
    "H_{int} = U n_\\uparrow n_\\downarrow - \\mu ( n_\\uparrow + n_\\downarrow )\n",
    "$$\n",
    "\n",
    "where $U$ is the so called Hubbard-$U$ and $\\mu$ is the chemical potential, which at half filling is given by $\\mu = U/2$. The helper function `get_h_int(U)` constructs the local Hamiltonian using Triqs second-quantization operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4359cae7-6969-4ad4-9b12-67b5eb707d70",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "h_int = -2.5*c_dag('do',0)*c('do',0) + -2.5*c_dag('up',0)*c('up',0) + 5*c_dag('do',0)*c_dag('up',0)*c('up',0)*c('do',0)\n"
     ]
    }
   ],
   "source": [
    "from triqs.operators import n\n",
    "\n",
    "def get_h_int(U, mu=None):\n",
    "    if mu is None:\n",
    "        mu = U / 2\n",
    "    h_int = U * n('up',0) * n('do', 0) - mu * ( n('up', 0) + n('do', 0) )\n",
    "    return h_int\n",
    "\n",
    "h_int = get_h_int(U=5.0)\n",
    "\n",
    "print(f'h_int = {h_int}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71edb549-fc4e-40a0-ba27-c6da437a6989",
   "metadata": {},
   "source": [
    "## DMFT self-consistency\n",
    "\n",
    "Finally we design a function `solve_dmft_self_consistency(S, h_int, order, ...)` that takes a solver `S`, a local Hamiltonian `h_int`, the wanted expansion `order`, and performs the DMFT self-consistency iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "0f9aa03f-15a5-4480-a5e2-ce0d99290d9d",
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve_dmft_self_consistency(S, h_int, order, \n",
    "                                dmft_maxiter=100, dmft_tol=1e-6):\n",
    "\n",
    "    for dmft_iter in range(1, dmft_maxiter+1):\n",
    "\n",
    "        S.solve(h_int=h_int, order=order, maxiter=100, verbose=False)\n",
    "        \n",
    "        diff = np.max(np.abs(S.Delta_tau['up'].data - S.G_tau['up'].data))\n",
    "\n",
    "        print(f'DMFT: dmft_iter = {dmft_iter}, diff = {diff:2.2E}')\n",
    "        \n",
    "        if diff < dmft_tol: break\n",
    "        \n",
    "        S.Delta_tau[:] = S.G_tau # DMFT Bethe lattice self consistency\n",
    "\n",
    "    S.d = S.S.get_expectation_value(n('up', 0) * n('do', 0)).real\n",
    "    S.n_up = S.S.get_expectation_value(n('up', 0)).real\n",
    "    S.n_do = S.S.get_expectation_value(n('do', 0)).real\n",
    "\n",
    "    return S"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dc1313a-0310-492c-a9b0-02555ac952b7",
   "metadata": {},
   "source": [
    "## Solve NCA and OCA for a sweep in $U$\n",
    "\n",
    "To study the metal-insulator cross over we solve the Hubbard model on the Bethe graph for a range of $U$ values at first and second expansion order (a.k.a. the non-crossing and one-crossing approximation NCA/OCA)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "66c1de8d-a6e5-4eb1-9d95-ed8b0fd41d70",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  ___      ___    _  ___   _____\n",
      " / __| ___| __|__| || \\ \\ / / _ )\n",
      " \\__ \\/ _ \\ _|___| __ |\\ V /| _ \\\n",
      " |___/\\___/___|  |_||_| |_| |___/  [github.com/TRIQS/soehyb]\n",
      "\n",
      "beta = 5.0, lamb = 5.00E+01, eps = 1.00E-12, N_DLR = 27\n",
      "fundamental_operators = [1*c('up',0), 1*c('do',0)]\n",
      "H_mat.shape = (4, 4)\n",
      "H_loc = 0\n",
      "\n",
      "--> order = 1, U = 3.0\n",
      "DMFT: dmft_iter = 1, diff = 1.27E-01\n",
      "DMFT: dmft_iter = 2, diff = 1.24E-02\n",
      "DMFT: dmft_iter = 3, diff = 1.58E-03\n",
      "DMFT: dmft_iter = 4, diff = 3.43E-04\n",
      "DMFT: dmft_iter = 5, diff = 6.34E-05\n",
      "DMFT: dmft_iter = 6, diff = 1.24E-05\n",
      "DMFT: dmft_iter = 7, diff = 2.39E-06\n",
      "DMFT: dmft_iter = 8, diff = 4.62E-07\n",
      "--> Storing: data_order_1_U_03.00000000.h5\n",
      "--> order = 1, U = 4.0\n",
      "DMFT: dmft_iter = 1, diff = 4.31E-02\n",
      "DMFT: dmft_iter = 2, diff = 1.11E-02\n",
      "DMFT: dmft_iter = 3, diff = 3.24E-03\n",
      "DMFT: dmft_iter = 4, diff = 9.52E-04\n",
      "DMFT: dmft_iter = 5, diff = 2.83E-04\n",
      "DMFT: dmft_iter = 6, diff = 8.44E-05\n",
      "DMFT: dmft_iter = 7, diff = 2.52E-05\n",
      "DMFT: dmft_iter = 8, diff = 7.50E-06\n",
      "DMFT: dmft_iter = 9, diff = 2.24E-06\n",
      "DMFT: dmft_iter = 10, diff = 6.68E-07\n",
      "--> Storing: data_order_1_U_04.00000000.h5\n",
      "--> order = 1, U = 5.0\n",
      "DMFT: dmft_iter = 1, diff = 3.63E-02\n",
      "DMFT: dmft_iter = 2, diff = 5.84E-03\n",
      "DMFT: dmft_iter = 3, diff = 1.63E-03\n",
      "DMFT: dmft_iter = 4, diff = 4.16E-04\n",
      "DMFT: dmft_iter = 5, diff = 1.06E-04\n",
      "DMFT: dmft_iter = 6, diff = 2.69E-05\n",
      "DMFT: dmft_iter = 7, diff = 6.85E-06\n",
      "DMFT: dmft_iter = 8, diff = 1.74E-06\n",
      "DMFT: dmft_iter = 9, diff = 4.42E-07\n",
      "--> Storing: data_order_1_U_05.00000000.h5\n",
      "--> order = 1, U = 6.0\n",
      "DMFT: dmft_iter = 1, diff = 3.04E-02\n",
      "DMFT: dmft_iter = 2, diff = 2.63E-03\n",
      "DMFT: dmft_iter = 3, diff = 5.08E-04\n",
      "DMFT: dmft_iter = 4, diff = 1.05E-04\n",
      "DMFT: dmft_iter = 5, diff = 2.09E-05\n",
      "DMFT: dmft_iter = 6, diff = 4.09E-06\n",
      "DMFT: dmft_iter = 7, diff = 8.00E-07\n",
      "--> Storing: data_order_1_U_06.00000000.h5\n",
      "  ___      ___    _  ___   _____\n",
      " / __| ___| __|__| || \\ \\ / / _ )\n",
      " \\__ \\/ _ \\ _|___| __ |\\ V /| _ \\\n",
      " |___/\\___/___|  |_||_| |_| |___/  [github.com/TRIQS/soehyb]\n",
      "\n",
      "beta = 5.0, lamb = 5.00E+01, eps = 1.00E-12, N_DLR = 27\n",
      "fundamental_operators = [1*c('up',0), 1*c('do',0)]\n",
      "H_mat.shape = (4, 4)\n",
      "H_loc = 0\n",
      "\n",
      "--> order = 2, U = 3.0\n",
      "DMFT: dmft_iter = 1, diff = 8.02E-02\n",
      "DMFT: dmft_iter = 2, diff = 3.40E-03\n",
      "DMFT: dmft_iter = 3, diff = 3.65E-04\n",
      "DMFT: dmft_iter = 4, diff = 5.09E-05\n",
      "DMFT: dmft_iter = 5, diff = 7.03E-06\n",
      "DMFT: dmft_iter = 6, diff = 9.69E-07\n",
      "--> Storing: data_order_2_U_03.00000000.h5\n",
      "--> order = 2, U = 4.0\n",
      "DMFT: dmft_iter = 1, diff = 4.58E-02\n",
      "DMFT: dmft_iter = 2, diff = 1.46E-02\n",
      "DMFT: dmft_iter = 3, diff = 6.23E-03\n",
      "DMFT: dmft_iter = 4, diff = 2.71E-03\n",
      "DMFT: dmft_iter = 5, diff = 1.20E-03\n",
      "DMFT: dmft_iter = 6, diff = 5.39E-04\n",
      "DMFT: dmft_iter = 7, diff = 2.42E-04\n",
      "DMFT: dmft_iter = 8, diff = 1.09E-04\n",
      "DMFT: dmft_iter = 9, diff = 4.91E-05\n",
      "DMFT: dmft_iter = 10, diff = 2.22E-05\n",
      "DMFT: dmft_iter = 11, diff = 9.99E-06\n",
      "DMFT: dmft_iter = 12, diff = 4.50E-06\n",
      "DMFT: dmft_iter = 13, diff = 2.03E-06\n",
      "DMFT: dmft_iter = 14, diff = 9.16E-07\n",
      "--> Storing: data_order_2_U_04.00000000.h5\n",
      "--> order = 2, U = 5.0\n",
      "DMFT: dmft_iter = 1, diff = 4.13E-02\n",
      "DMFT: dmft_iter = 2, diff = 1.34E-02\n",
      "DMFT: dmft_iter = 3, diff = 6.27E-03\n",
      "DMFT: dmft_iter = 4, diff = 2.80E-03\n",
      "DMFT: dmft_iter = 5, diff = 1.23E-03\n",
      "DMFT: dmft_iter = 6, diff = 5.41E-04\n",
      "DMFT: dmft_iter = 7, diff = 2.37E-04\n",
      "DMFT: dmft_iter = 8, diff = 1.04E-04\n",
      "DMFT: dmft_iter = 9, diff = 4.56E-05\n",
      "DMFT: dmft_iter = 10, diff = 2.00E-05\n",
      "DMFT: dmft_iter = 11, diff = 8.75E-06\n",
      "DMFT: dmft_iter = 12, diff = 3.84E-06\n",
      "DMFT: dmft_iter = 13, diff = 1.68E-06\n",
      "DMFT: dmft_iter = 14, diff = 7.37E-07\n",
      "--> Storing: data_order_2_U_05.00000000.h5\n",
      "--> order = 2, U = 6.0\n",
      "DMFT: dmft_iter = 1, diff = 3.30E-02\n",
      "DMFT: dmft_iter = 2, diff = 5.61E-03\n",
      "DMFT: dmft_iter = 3, diff = 1.83E-03\n",
      "DMFT: dmft_iter = 4, diff = 6.47E-04\n",
      "DMFT: dmft_iter = 5, diff = 2.18E-04\n",
      "DMFT: dmft_iter = 6, diff = 7.20E-05\n",
      "DMFT: dmft_iter = 7, diff = 2.37E-05\n",
      "DMFT: dmft_iter = 8, diff = 7.77E-06\n",
      "DMFT: dmft_iter = 9, diff = 2.55E-06\n",
      "DMFT: dmft_iter = 10, diff = 8.34E-07\n",
      "--> Storing: data_order_2_U_06.00000000.h5\n"
     ]
    }
   ],
   "source": [
    "orders = [1, 2]\n",
    "Us = np.arange(3.0, 7.0, 1.0)\n",
    "\n",
    "for order in orders:\n",
    "\n",
    "    S = setup_solver()\n",
    "    \n",
    "    for U in Us:\n",
    "        \n",
    "        print(f'--> order = {order}, U = {U}')\n",
    "        \n",
    "        S.U = U\n",
    "        h_int = get_h_int(U)\n",
    "\n",
    "        S = solve_dmft_self_consistency(S, h_int, order)\n",
    "\n",
    "        filename = f'data_order_{order}_U_{U:011.8f}.h5'\n",
    "        print(f'--> Storing: {filename}')\n",
    "        with HDFArchive(filename, 'w') as A: \n",
    "            A['S'] = S"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12329eef-af0d-46b1-a48e-98a10798ed70",
   "metadata": {},
   "source": [
    "## Load result\n",
    "\n",
    "To visualize the crossover we load the results from file using a helper function and class."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "4b274fc5-df67-4570-8a2f-a2dad74c7fc3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--> Loading: data_order_1_U_03.00000000.h5\n",
      "--> Loading: data_order_1_U_04.00000000.h5\n",
      "--> Loading: data_order_1_U_05.00000000.h5\n",
      "--> Loading: data_order_1_U_06.00000000.h5\n",
      "--> Loading: data_order_2_U_03.00000000.h5\n",
      "--> Loading: data_order_2_U_04.00000000.h5\n",
      "--> Loading: data_order_2_U_05.00000000.h5\n",
      "--> Loading: data_order_2_U_06.00000000.h5\n"
     ]
    }
   ],
   "source": [
    "import glob\n",
    "\n",
    "class AttrList:\n",
    "    def __init__(self, l): self.l = l\n",
    "    def __getattr__(self, attr): return [ getattr(x, attr) for x in self.l ]\n",
    "\n",
    "\n",
    "def load_order(order):\n",
    "\n",
    "    filenames = np.sort(glob.glob(f'data_order_{order}_U_*.h5'))\n",
    "\n",
    "    results = []\n",
    "    for filename in filenames:\n",
    "        print(f'--> Loading: {filename}')\n",
    "        with HDFArchive(filename, 'r') as A:\n",
    "            results.append(A['S'])\n",
    "\n",
    "    return AttrList(results)\n",
    "\n",
    "nca = load_order(1)\n",
    "oca = load_order(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "95e8bf34-798a-47de-8186-b41125588626",
   "metadata": {},
   "source": [
    "## Visualization\n",
    "\n",
    "To visualize the metal insulator cross over we plot the double occupancy $\\langle n_\\uparrow n_\\downarrow \\rangle$ as a function of $U$ for NCA and OCA."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a8d689c5-71b3-4fed-be16-937572394579",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"392.14375pt\" height=\"270.315469pt\" viewBox=\"0 0 392.14375 270.315469\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2025-08-19T17:42:11.588108</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.10.3, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 270.315469 \n",
       "L 392.14375 270.315469 \n",
       "L 392.14375 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 50.14375 232.759219 \n",
       "L 384.94375 232.759219 \n",
       "L 384.94375 10.999219 \n",
       "L 50.14375 10.999219 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m802c9edb3d\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"65.361932\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 3.0 -->\n",
       "      <g transform=\"translate(57.410369 247.357656) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-33\" d=\"M 2597 2516 \n",
       "Q 3050 2419 3304 2112 \n",
       "Q 3559 1806 3559 1356 \n",
       "Q 3559 666 3084 287 \n",
       "Q 2609 -91 1734 -91 \n",
       "Q 1441 -91 1130 -33 \n",
       "Q 819 25 488 141 \n",
       "L 488 750 \n",
       "Q 750 597 1062 519 \n",
       "Q 1375 441 1716 441 \n",
       "Q 2309 441 2620 675 \n",
       "Q 2931 909 2931 1356 \n",
       "Q 2931 1769 2642 2001 \n",
       "Q 2353 2234 1838 2234 \n",
       "L 1294 2234 \n",
       "L 1294 2753 \n",
       "L 1863 2753 \n",
       "Q 2328 2753 2575 2939 \n",
       "Q 2822 3125 2822 3475 \n",
       "Q 2822 3834 2567 4026 \n",
       "Q 2313 4219 1838 4219 \n",
       "Q 1578 4219 1281 4162 \n",
       "Q 984 4106 628 3988 \n",
       "L 628 4550 \n",
       "Q 988 4650 1302 4700 \n",
       "Q 1616 4750 1894 4750 \n",
       "Q 2613 4750 3031 4423 \n",
       "Q 3450 4097 3450 3541 \n",
       "Q 3450 3153 3228 2886 \n",
       "Q 3006 2619 2597 2516 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-33\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"116.089205\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 3.5 -->\n",
       "      <g transform=\"translate(108.137642 247.357656) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-35\" d=\"M 691 4666 \n",
       "L 3169 4666 \n",
       "L 3169 4134 \n",
       "L 1269 4134 \n",
       "L 1269 2991 \n",
       "Q 1406 3038 1543 3061 \n",
       "Q 1681 3084 1819 3084 \n",
       "Q 2600 3084 3056 2656 \n",
       "Q 3513 2228 3513 1497 \n",
       "Q 3513 744 3044 326 \n",
       "Q 2575 -91 1722 -91 \n",
       "Q 1428 -91 1123 -41 \n",
       "Q 819 9 494 109 \n",
       "L 494 744 \n",
       "Q 775 591 1075 516 \n",
       "Q 1375 441 1709 441 \n",
       "Q 2250 441 2565 725 \n",
       "Q 2881 1009 2881 1497 \n",
       "Q 2881 1984 2565 2268 \n",
       "Q 2250 2553 1709 2553 \n",
       "Q 1456 2553 1204 2497 \n",
       "Q 953 2441 691 2322 \n",
       "L 691 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-33\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"166.816477\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 4.0 -->\n",
       "      <g transform=\"translate(158.864915 247.357656) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-34\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"217.54375\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 4.5 -->\n",
       "      <g transform=\"translate(209.592187 247.357656) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-34\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"268.271023\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 5.0 -->\n",
       "      <g transform=\"translate(260.31946 247.357656) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-35\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"318.998295\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 5.5 -->\n",
       "      <g transform=\"translate(311.046733 247.357656) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-35\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_7\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m802c9edb3d\" x=\"369.725568\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- 6.0 -->\n",
       "      <g transform=\"translate(361.774006 247.357656) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-36\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_8\">\n",
       "     <!-- U -->\n",
       "     <g transform=\"translate(213.884375 261.035781) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-55\" d=\"M 556 4666 \n",
       "L 1191 4666 \n",
       "L 1191 1831 \n",
       "Q 1191 1081 1462 751 \n",
       "Q 1734 422 2344 422 \n",
       "Q 2950 422 3222 751 \n",
       "Q 3494 1081 3494 1831 \n",
       "L 3494 4666 \n",
       "L 4128 4666 \n",
       "L 4128 1753 \n",
       "Q 4128 841 3676 375 \n",
       "Q 3225 -91 2344 -91 \n",
       "Q 1459 -91 1007 375 \n",
       "Q 556 841 556 1753 \n",
       "L 556 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-55\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <defs>\n",
       "       <path id=\"mbb4bd66744\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"232.759219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.00 -->\n",
       "      <g transform=\"translate(20.878125 236.558437) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"195.799219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.02 -->\n",
       "      <g transform=\"translate(20.878125 199.598437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"158.839219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.04 -->\n",
       "      <g transform=\"translate(20.878125 162.638437) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"121.879219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.06 -->\n",
       "      <g transform=\"translate(20.878125 125.678437) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"84.919219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 0.08 -->\n",
       "      <g transform=\"translate(20.878125 88.718437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"47.959219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_14\">\n",
       "      <!-- 0.10 -->\n",
       "      <g transform=\"translate(20.878125 51.758437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbb4bd66744\" x=\"50.14375\" y=\"10.999219\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_15\">\n",
       "      <!-- 0.12 -->\n",
       "      <g transform=\"translate(20.878125 14.798437) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" transform=\"translate(63.623047 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" transform=\"translate(95.410156 0)\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" transform=\"translate(159.033203 0)\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_16\">\n",
       "     <!-- $\\langle n_{\\uparrow} n_{\\downarrow} \\rangle$ -->\n",
       "     <g transform=\"translate(14.798438 141.029219) rotate(-90) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-27e8\" d=\"M 572 2006 \n",
       "L 1453 4856 \n",
       "L 1984 4856 \n",
       "L 1103 2006 \n",
       "L 1984 -844 \n",
       "L 1453 -844 \n",
       "L 572 2006 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-Oblique-6e\" d=\"M 3566 2113 \n",
       "L 3156 0 \n",
       "L 2578 0 \n",
       "L 2988 2091 \n",
       "Q 3016 2238 3031 2350 \n",
       "Q 3047 2463 3047 2528 \n",
       "Q 3047 2791 2881 2937 \n",
       "Q 2716 3084 2419 3084 \n",
       "Q 1956 3084 1622 2776 \n",
       "Q 1288 2469 1184 1941 \n",
       "L 800 0 \n",
       "L 225 0 \n",
       "L 903 3500 \n",
       "L 1478 3500 \n",
       "L 1363 2950 \n",
       "Q 1603 3253 1940 3418 \n",
       "Q 2278 3584 2650 3584 \n",
       "Q 3113 3584 3367 3334 \n",
       "Q 3622 3084 3622 2631 \n",
       "Q 3622 2519 3608 2391 \n",
       "Q 3594 2263 3566 2113 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2191\" d=\"M 2541 4688 \n",
       "L 2822 4688 \n",
       "L 4047 3456 \n",
       "L 3672 3081 \n",
       "L 2947 3813 \n",
       "L 2947 0 \n",
       "L 2416 0 \n",
       "L 2416 3813 \n",
       "L 1684 3081 \n",
       "L 1309 3456 \n",
       "L 2541 4688 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2193\" d=\"M 2822 -22 \n",
       "L 2541 -22 \n",
       "L 1309 1209 \n",
       "L 1684 1584 \n",
       "L 2416 853 \n",
       "L 2416 4666 \n",
       "L 2947 4666 \n",
       "L 2947 853 \n",
       "L 3672 1584 \n",
       "L 4047 1209 \n",
       "L 2822 -22 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-27e9\" d=\"M 1925 2006 \n",
       "L 1044 -844 \n",
       "L 513 -844 \n",
       "L 1394 2006 \n",
       "L 513 4856 \n",
       "L 1044 4856 \n",
       "L 1925 2006 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-27e8\" transform=\"translate(0 0.125)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-Oblique-6e\" transform=\"translate(39.013672 0.125)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2191\" transform=\"translate(116.030273 -16.28125) scale(0.7)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-Oblique-6e\" transform=\"translate(191.054688 0.125)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2193\" transform=\"translate(268.071289 -16.28125) scale(0.7)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-27e9\" transform=\"translate(343.095703 0.125)\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_15\">\n",
       "    <path d=\"M 65.361932 108.600873 \n",
       "L 166.816477 169.417931 \n",
       "L 268.271023 194.907459 \n",
       "L 369.725568 206.924589 \n",
       "\" clip-path=\"url(#p1e73bd6e21)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "    <defs>\n",
       "     <path id=\"m5ec4cb5270\" d=\"M 0 3 \n",
       "C 0.795609 3 1.55874 2.683901 2.12132 2.12132 \n",
       "C 2.683901 1.55874 3 0.795609 3 0 \n",
       "C 3 -0.795609 2.683901 -1.55874 2.12132 -2.12132 \n",
       "C 1.55874 -2.683901 0.795609 -3 0 -3 \n",
       "C -0.795609 -3 -1.55874 -2.683901 -2.12132 -2.12132 \n",
       "C -2.683901 -1.55874 -3 -0.795609 -3 0 \n",
       "C -3 0.795609 -2.683901 1.55874 -2.12132 2.12132 \n",
       "C -1.55874 2.683901 -0.795609 3 0 3 \n",
       "z\n",
       "\" style=\"stroke: #1f77b4\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#p1e73bd6e21)\">\n",
       "     <use xlink:href=\"#m5ec4cb5270\" x=\"65.361932\" y=\"108.600873\" style=\"fill: #1f77b4; stroke: #1f77b4\"/>\n",
       "     <use xlink:href=\"#m5ec4cb5270\" x=\"166.816477\" y=\"169.417931\" style=\"fill: #1f77b4; stroke: #1f77b4\"/>\n",
       "     <use xlink:href=\"#m5ec4cb5270\" x=\"268.271023\" y=\"194.907459\" style=\"fill: #1f77b4; stroke: #1f77b4\"/>\n",
       "     <use xlink:href=\"#m5ec4cb5270\" x=\"369.725568\" y=\"206.924589\" style=\"fill: #1f77b4; stroke: #1f77b4\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_16\">\n",
       "    <path d=\"M 65.361932 54.122435 \n",
       "L 166.816477 142.49295 \n",
       "L 268.271023 187.750597 \n",
       "L 369.725568 204.255123 \n",
       "\" clip-path=\"url(#p1e73bd6e21)\" style=\"fill: none; stroke: #ff7f0e; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "    <defs>\n",
       "     <path id=\"mfad4838c3f\" d=\"M 0 3 \n",
       "C 0.795609 3 1.55874 2.683901 2.12132 2.12132 \n",
       "C 2.683901 1.55874 3 0.795609 3 0 \n",
       "C 3 -0.795609 2.683901 -1.55874 2.12132 -2.12132 \n",
       "C 1.55874 -2.683901 0.795609 -3 0 -3 \n",
       "C -0.795609 -3 -1.55874 -2.683901 -2.12132 -2.12132 \n",
       "C -2.683901 -1.55874 -3 -0.795609 -3 0 \n",
       "C -3 0.795609 -2.683901 1.55874 -2.12132 2.12132 \n",
       "C -1.55874 2.683901 -0.795609 3 0 3 \n",
       "z\n",
       "\" style=\"stroke: #ff7f0e\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#p1e73bd6e21)\">\n",
       "     <use xlink:href=\"#mfad4838c3f\" x=\"65.361932\" y=\"54.122435\" style=\"fill: #ff7f0e; stroke: #ff7f0e\"/>\n",
       "     <use xlink:href=\"#mfad4838c3f\" x=\"166.816477\" y=\"142.49295\" style=\"fill: #ff7f0e; stroke: #ff7f0e\"/>\n",
       "     <use xlink:href=\"#mfad4838c3f\" x=\"268.271023\" y=\"187.750597\" style=\"fill: #ff7f0e; stroke: #ff7f0e\"/>\n",
       "     <use xlink:href=\"#mfad4838c3f\" x=\"369.725568\" y=\"204.255123\" style=\"fill: #ff7f0e; stroke: #ff7f0e\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 50.14375 232.759219 \n",
       "L 50.14375 10.999219 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 384.94375 232.759219 \n",
       "L 384.94375 10.999219 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 50.14375 232.759219 \n",
       "L 384.94375 232.759219 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 50.14375 10.999219 \n",
       "L 384.94375 10.999219 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"legend_1\">\n",
       "    <g id=\"patch_7\">\n",
       "     <path d=\"M 324.248438 48.355469 \n",
       "L 377.94375 48.355469 \n",
       "Q 379.94375 48.355469 379.94375 46.355469 \n",
       "L 379.94375 17.999219 \n",
       "Q 379.94375 15.999219 377.94375 15.999219 \n",
       "L 324.248438 15.999219 \n",
       "Q 322.248438 15.999219 322.248438 17.999219 \n",
       "L 322.248438 46.355469 \n",
       "Q 322.248438 48.355469 324.248438 48.355469 \n",
       "z\n",
       "\" style=\"fill: #ffffff; opacity: 0.8; stroke: #cccccc; stroke-linejoin: miter\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_17\">\n",
       "     <path d=\"M 326.248438 24.097656 \n",
       "L 336.248438 24.097656 \n",
       "L 346.248438 24.097656 \n",
       "\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "     <g>\n",
       "      <use xlink:href=\"#m5ec4cb5270\" x=\"336.248438\" y=\"24.097656\" style=\"fill: #1f77b4; stroke: #1f77b4\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_17\">\n",
       "     <!-- NCA -->\n",
       "     <g transform=\"translate(354.248438 27.597656) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-4e\" d=\"M 628 4666 \n",
       "L 1478 4666 \n",
       "L 3547 763 \n",
       "L 3547 4666 \n",
       "L 4159 4666 \n",
       "L 4159 0 \n",
       "L 3309 0 \n",
       "L 1241 3903 \n",
       "L 1241 0 \n",
       "L 628 0 \n",
       "L 628 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-43\" d=\"M 4122 4306 \n",
       "L 4122 3641 \n",
       "Q 3803 3938 3442 4084 \n",
       "Q 3081 4231 2675 4231 \n",
       "Q 1875 4231 1450 3742 \n",
       "Q 1025 3253 1025 2328 \n",
       "Q 1025 1406 1450 917 \n",
       "Q 1875 428 2675 428 \n",
       "Q 3081 428 3442 575 \n",
       "Q 3803 722 4122 1019 \n",
       "L 4122 359 \n",
       "Q 3791 134 3420 21 \n",
       "Q 3050 -91 2638 -91 \n",
       "Q 1578 -91 968 557 \n",
       "Q 359 1206 359 2328 \n",
       "Q 359 3453 968 4101 \n",
       "Q 1578 4750 2638 4750 \n",
       "Q 3056 4750 3426 4639 \n",
       "Q 3797 4528 4122 4306 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-41\" d=\"M 2188 4044 \n",
       "L 1331 1722 \n",
       "L 3047 1722 \n",
       "L 2188 4044 \n",
       "z\n",
       "M 1831 4666 \n",
       "L 2547 4666 \n",
       "L 4325 0 \n",
       "L 3669 0 \n",
       "L 3244 1197 \n",
       "L 1141 1197 \n",
       "L 716 0 \n",
       "L 50 0 \n",
       "L 1831 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-4e\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-43\" transform=\"translate(74.804688 0)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-41\" transform=\"translate(144.628906 0)\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"line2d_18\">\n",
       "     <path d=\"M 326.248438 38.775781 \n",
       "L 336.248438 38.775781 \n",
       "L 346.248438 38.775781 \n",
       "\" style=\"fill: none; stroke: #ff7f0e; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "     <g>\n",
       "      <use xlink:href=\"#mfad4838c3f\" x=\"336.248438\" y=\"38.775781\" style=\"fill: #ff7f0e; stroke: #ff7f0e\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_18\">\n",
       "     <!-- OCA -->\n",
       "     <g transform=\"translate(354.248438 42.275781) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-4f\" d=\"M 2522 4238 \n",
       "Q 1834 4238 1429 3725 \n",
       "Q 1025 3213 1025 2328 \n",
       "Q 1025 1447 1429 934 \n",
       "Q 1834 422 2522 422 \n",
       "Q 3209 422 3611 934 \n",
       "Q 4013 1447 4013 2328 \n",
       "Q 4013 3213 3611 3725 \n",
       "Q 3209 4238 2522 4238 \n",
       "z\n",
       "M 2522 4750 \n",
       "Q 3503 4750 4090 4092 \n",
       "Q 4678 3434 4678 2328 \n",
       "Q 4678 1225 4090 567 \n",
       "Q 3503 -91 2522 -91 \n",
       "Q 1538 -91 948 565 \n",
       "Q 359 1222 359 2328 \n",
       "Q 359 3434 948 4092 \n",
       "Q 1538 4750 2522 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-4f\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-43\" transform=\"translate(78.710938 0)\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-41\" transform=\"translate(148.535156 0)\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p1e73bd6e21\">\n",
       "   <rect x=\"50.14375\" y=\"10.999219\" width=\"334.8\" height=\"221.76\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(nca.U, nca.d, 'o-', label='NCA')\n",
    "plt.plot(oca.U, oca.d, 'o-', label='OCA')\n",
    "\n",
    "plt.ylabel(r'$\\langle n_{\\uparrow} n_{\\downarrow} \\rangle$')\n",
    "plt.xlabel('U')\n",
    "plt.ylim(bottom=0, top=0.12)\n",
    "plt.legend(loc='best');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91b02ba5-6e6d-488a-90ff-22e96f9ccea1",
   "metadata": {},
   "source": [
    "This reproduces the results in Fig. 4a in [M. Eckstein, P. Werner, Phys. Rev. B 82, 115115 (2010)](https://doi.org/10.1103/PhysRevB.82.115115)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.10.17"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}