{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spin frustration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.interactive import printing\n",
    "printing.init_printing(use_latex = True)\n",
    "\n",
    "import numpy as np\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Triangular copper structure](frustrated.png \"Triangular copper\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two equal A-B interactions and one A-A interaction.\n",
    "The local spins are $S_{a1}$, $S_{a2}$ ($S_{a1}$ = $S_{a2}$ = $S_{a}$) and $S_{b}$. Interactions parameters are $J$ and $J'$.\n",
    "\n",
    "The spin Hamiltonian in zero field is:\n",
    "\n",
    "$$H = -J(S_{a1} \\dot S_{b} + S_{a2} \\dot S_{b}) - J'(S_{a1} \\dot S_{a2})$$\n",
    "\n",
    "Negrecting anisotropic interactions:\n",
    "\n",
    "$$H = -J(S_{a1} \\dot S_{b} + S_{a2} \\dot S_{b} + S_{a1} \\dot S_{a2}) - (J'-J)(S_{a1} \\dot S_{a2})$$\n",
    "\n",
    "Or:\n",
    "\n",
    "$$H = -\\frac{J}{2}(S^{2} - S_{a1}^{2} - S_{a2}^{2} - S_{b}^{2}) - \\frac{J'-J}{2}(S'^{2} - S_{a1}^{2} - S_{a2}^{2})$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Assignment of all symbolic variables\n",
    "Sa = sp.Symbol(\"Sa\")\n",
    "Sa1 = sp.Symbol(\"Sa1\")\n",
    "Sa2 = sp.Symbol(\"Sa2\")\n",
    "Sb = sp.Symbol(\"Sb\")\n",
    "J = sp.Symbol(\"J\")\n",
    "J_prime = sp.Symbol(\"J\\'\")\n",
    "S_prime = Sa1 + Sa2\n",
    "S = S_prime + Sb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{J}{2} \\left(- Sa_{1}^{2} - Sa_{2}^{2} - Sb^{2} + \\left(Sa_{1} + Sa_{2} + Sb\\right)^{2}\\right) - \\left(- \\frac{J}{2} + \\frac{J'}{2}\\right) \\left(- Sa_{1}^{2} - Sa_{2}^{2} + \\left(Sa_{1} + Sa_{2}\\right)^{2}\\right)$$"
      ],
      "text/plain": [
       "    ⎛     2      2     2                   2⎞                                 \n",
       "  J⋅⎝- Sa₁  - Sa₂  - Sb  + (Sa₁ + Sa₂ + Sb) ⎠   ⎛  J   J'⎞ ⎛     2      2     \n",
       "- ─────────────────────────────────────────── - ⎜- ─ + ──⎟⋅⎝- Sa₁  - Sa₂  + (S\n",
       "                       2                        ⎝  2   2 ⎠                    \n",
       "\n",
       "           \n",
       "         2⎞\n",
       "a₁ + Sa₂) ⎠\n",
       "           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Phythonic version of the Hamiltonian (Here it's showed expanded)\n",
    "H = (-J/2)*(S**2 - Sa1**2 - Sa2**2 - Sb**2) - ((J_prime - J)/2)*(S_prime**2 - Sa1**2 - Sa2**2)\n",
    "display(H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Relative energies in Zero Field:\n",
    "$$E(S,S')=-\\frac{J}{2}S(S+1) - \\frac{J'-J}{2}S'(S'+1)$$\n",
    "With:\n",
    "\n",
    "$$S' = S_{a1} + S_{a2}$$\n",
    "$$S = S' + S_{b}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAxcAAAAZCAYAAABDwtPRAAAABHNCSVQICAgIfAhkiAAAC0FJREFUeJztnXusHUUdxz+9LZTSiwWqYhWsJVqUoJSAYE1oDgUkaFRE/xFTPSI+IhERSQgkyA1ENMFHBQXTSKmA+ACkpDEoEh8IRIQ+kMSqleQWFJRW0KIUKe31j99szt49M7Ozs4+zp+f3SU7uvbszu/Od+c3vd2dnds5MFEVRFEXZm3kXMAvYPuiCKIqiKIqiKIoyvMwGnge+NeiCKIqi1MGNwNPA3EEXRGk1nwamgLNz0h1r0n0s4Jpqe0oeoXbXVor0h2FD+288HcQultV0ffXDw4e2mVIFjcace83NlmaOHwfsAS7w5O0AtwCTwAvADmALsBb4HDBWbVEHSgfV6tL6HcSGjg649h3AU8C4J43a3nQ6qFab1iJ21xQfAK4BfoOUfQq42ZM+pD8AHALsBq42f88HzjH5/wLsBP4N3IcEjkHaREj/LYsrbsH0umpzPYFdxxXAHxzpq7KDKvxwB/VLTWrVNitGh71fa9F4A+ExpxRjpkAvAftnzt0N/AuYY8k3E1iFCHkO+BFwFXAd8HOkIf9aT5EbR7Xma92IBLVZAfc43lz/Ek8atT1Btfq1FrG7pthET8Nm8p19SH8A+IRJd5L5+1Pm7yeB7wFfAlYj/WYKuA2YEaWgPL7+WwW+uAXT66rN9eTScT/wWUeequygjB9WvzQYrdpmYYyS1qLxBsJjTineaG7y+8zxxcjod5Uj3xdMvjuBeZbzBwOnVFTGsnSRsnYi86vWHjats4EXgQcL3GczsBVxAlnU9nqo1h5ZrTF2F0KXcjpOAt6A/EPXIczZ+/pDwl3Iy71JmuXAu+l/+vYq4HFz3/cXKHdV5PXfKnDFrYR0XdVZT13K2YpNx1xk5uEgR54q7SDWDw+TX4Jy7dQ2rdpm+YyS1ph4A2ExpxQfMoVZnTn+ZXP8ZEueGYhzm0IcWBHOQp6u/BkZaT0LPAR8tOB1itAlvuFUaz7HmTzXAkuA24F/Av9FnqIcZclzmclzmuWc2p6gWv3E2F0IXcr9w5imQ5iz9/UHkAD5P+CGwPteYq53TWD6KvH136pwxS0oVldl66lLOVux6Tgd+K4jfdV2EOOHh80vQXw7tVGrtpmfUdKapUP44KLPjqpeI3as+flw5vgpyLrO31ryvAxZ5wmwq8C9DkCc5uHI+rBvAj8GFiHO9aIC12oK1ZpPYkMLken8KeB6YANwKvAL4MBMnvvNz1Mt11PbE1Srnxi7ayu+/gCyLem+SDuGkNThS2UKFYmv/1aFK25BsboaZD2BXccxwLcd6au2gxg/rH4pnzq1apv5GSWtZciLOaX5NRKUj08dm4s4o0c9+Z4w+TYgL40dGnCvcewjyQXIKPGPAdeIoUu5UaFq9ZOsbdyOBMY0N2Jf2zfPHP9d5rja3nRUq5sYuwuhS/MzF67+kHAr8B9gv4B7zkL6j28mpC5C+m8V2OJWQmhdVVFPXcrZik+HjartINYPD5NfgnLt1Dat2mb5jJLWNB3CZy7yYk4pkpfJdjHdWS02N73bk/ftyM4UU6nPVuSJSzbQh/AnZOu0OuhSruFUq1/rwyadbTvQE8y571vO7QT+njmmtjcd1erWGmt3eXRpfnAB9v4A4pufQ/6xDOEr5p4/CUxfJSH9tyyuuAXF6qqKeuoSbys+HTbqsoMYPzxMfgnKtVMbtWqb+RklrWk6hMcbcMccQLbYmirwSd80eZlsY+aaS83xH+YUbAzZh/uLyLZ3u0y+PcDHLekPAi5FpuyeRabv0mXbkEq7DFiH7IAxhWy3FcIkxepjTeB169R6MbKubwewDdEdsl58MlBjnVr3QdYAT2LfsWehyXuH5dzf6J+uV9vrR22vX2sZu2tCR0KHcGdv6w8A7zHXOCvgGueZtJuRlxZDmCQ+hmQJ6b9l7+eKWxBeV03U05qc6/l02KjLDmL9cBt9MNTTp9vmg7XN8mlbmyVMBmqMiTkdwuMNZOwoG0gfQ7bVCuXJ1O+udas7zc+8pyl7kD267zV/HwysBFYgT02SPegB3oKMqA9BpmF+ADyDNPgi4MPAI6lrzzV/34C8qBnKSvrXWS8B3ousp5vMnNsUeN06tXaQl1IfQl5Guhy4BzjS5HPRBq1vRtYAr8P+j9FC83Or5dwceraWoLbXj9pev9YydpemLh0x2PoDwPuQXbHynkCfC3wD+X6Ek/G3X5oyMSRLSP8tez/f+xYhdRVbT1Xbik+HjbrsINYPt9EHQz19um0+WNssn7a1WcIwxJzSfA2p3E9mjr/aHL8v4pqH0RtxzU8dfwQJKB1LnstN+nMd1yw6Cs7SNdew3bsMdWgFWfu3G9lesChdmtV6jjn2eUe+r5vz78wcH0M6/2OZ42p7YYy67cXaXQhdqtPRIexJkqs/zETeKbkrJ//55j6PAq8sXMrqKNN/Q3HFrZC6qrqeusTbikuHjbrsoGo/3EYfDPX4pkH5YG2zeNoYN2Ewy6L67KjK3aJcT06eQqZ6joi45gLz8xl6I7jDkJHhr8wnzYH0tvlaH3G/QVKX1gOQdg59otYENq3QsyHb1PsCJHBuoX8d6BHIiD87KlfbC2PUbS/W7tqKqz8sQ4Kgb3nXRchgahOy53mda4PzKNN/Q3HFrby6alM9QbGZi7rsoGo/rD64fh+sbRbPKMXNPPrsqKrBxQzkxZYX6d9dYAqZSno58PrMuRXIdme2b/w8nN6Xt6ykN+X0Qur8Pqn085H1gYciSxvqnP6JYVBaV5rzdW7lmCVGK/QC5AeRadGEcWTHntnI+t/s0pW3mZ+/zBxX2xPU9gSX1li7ayuu/nAm8nTpTke+S5G97dcjS2C211K6cHz9twp8cctXV22rJ58OG3XZQVE/rD5YGKQP1jZz09Y2ayN9dmR7eTGGxchoaz3i4LLcjnyr52nIW/cJZyNTL48jhvwEEtAXI406E1mLdmUqzzZkz/nlyLfp3oOMIE9HhO1B1ocWWYfbBIPQehXylOpEZJqtKWK0zkLWvm8E9kc611rkH7szkWna84GfWu73DkSfLViq7ant+bSWsbsmOMN8oLe14VJ6L+ZtBy7M5HH1hzOAB4B/WO7zEWTKfjeyL/t5ljSTFH8JvSyu/lsFvrjlqqs21lNe/M1Slx0U9cPqgwfvg7XN3LS1zeokJt6A345KsQIZubm+uXNfZIuqBzPHTwCuQJzYVqTCn0e+0XA1IsrGK5AnituQ7fQeQF6SOcaU43pPWQe17r1prV9FgseRBcuZpktzWo8291oFvAb5wpkd5vMzTxnmIS8RrXWcV9tT2/NpjbW7ULqUW/86gX/3j8lMeld/eKtJf0Hkfabon+ZvAlf/rQJX3PLV1QT11VOXOFvJi79p6rKDGD88jD4Y4tqpjT5Y28xPG9sshC7xMWeCYvEG8u2oFLeQ/8LjxSZNzL7AVVKFobadq6nGSIeBzyBteqInjdpec4yS7bURV3+40hxf1HiJylNX/3XFrWGrq5D4m1CXtmHxw+qDe2ibtYdhjpshdhRFstZzCzJN5GI/ZPS3ruoCBDCObM+1BKmEC83vrx1AWermWuSp63JkWiv5jA+yUDUxB9la8racdGp7zTBKttdGfP1hM+1b0xxKHf3XF7eGqa5C429CHdra7ofVB/f7YG2z9jDMcTPUjoIZQ6ZPbkKmQ3YSNmpZBlzG9Jcmm6CDfYpnTcPlaALXdNbEAMtUF29CdL0uIK3aXv2Mku21kSL9Ydioov/Gxq220TYdbffDHdQHT2TSaZu1h2GOm5XHnKOQF1OeBm5F1i0riqIoSlvZW+LW3qJDURRFURRFURRFURRFURRFURRFURRFURRFURRFURRFURRFURRFURRlZPg/xb+akE86YisAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{J}{2} \\left(Sa_{1} + Sa_{2} + Sb\\right) \\left(Sa_{1} + Sa_{2} + Sb + 1\\right) - \\left(- \\frac{J}{2} + \\frac{J'}{2}\\right) \\left(Sa_{1} + Sa_{2}\\right) \\left(Sa_{1} + Sa_{2} + 1\\right)$$"
      ],
      "text/plain": [
       "  J⋅(Sa₁ + Sa₂ + Sb)⋅(Sa₁ + Sa₂ + Sb + 1)   ⎛  J   J'⎞                        \n",
       "- ─────────────────────────────────────── - ⎜- ─ + ──⎟⋅(Sa₁ + Sa₂)⋅(Sa₁ + Sa₂ \n",
       "                     2                      ⎝  2   2 ⎠                        \n",
       "\n",
       "    \n",
       "+ 1)\n",
       "    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Pythonic version of the energy equation\n",
    "E = (-J/2)*(S*(S + 1)) - ((J_prime -J)/2)*(S_prime*(S_prime + 1))\n",
    "display(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- J Sa_{1} Sb - J Sa_{2} Sb - \\frac{J Sb^{2}}{2} - \\frac{J Sb}{2} - \\frac{J' Sa_{1}^{2}}{2} - J' Sa_{1} Sa_{2} - \\frac{J' Sa_{1}}{2} - \\frac{J' Sa_{2}^{2}}{2} - \\frac{J' Sa_{2}}{2}$$"
      ],
      "text/plain": [
       "                           2                2                               2 \n",
       "                       J⋅Sb    J⋅Sb   J'⋅Sa₁                 J'⋅Sa₁   J'⋅Sa₂  \n",
       "-J⋅Sa₁⋅Sb - J⋅Sa₂⋅Sb - ───── - ──── - ─────── - J'⋅Sa₁⋅Sa₂ - ────── - ─────── \n",
       "                         2      2        2                     2         2    \n",
       "\n",
       "        \n",
       "  J'⋅Sa₂\n",
       "- ──────\n",
       "    2   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# And the expanded version of the equation:\n",
    "display(sp.expand(E))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**S'** varies by an integer from 0 to $2S_{a}$ and for every **S'** value **S** varies by an integer from $|S'-S_{b}|$ to $S'+S_{b}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S = [0.5 1.5] e S' = [0. 1.]\n"
     ]
    }
   ],
   "source": [
    "# Entre valores numéricos para os spins\n",
    "Sa = 1/2\n",
    "Sb = 1/2\n",
    "Sa1 = Sa\n",
    "Sa2 = Sa\n",
    "S_prime = np.arange(0, 2*Sa + 1, 1)\n",
    "Sba = np.amin(np.absolute(S_prime - Sb))\n",
    "Sbb = np.amax(S_prime + Sb)\n",
    "S = np.arange(np.amin(np.absolute(S_prime - Sb)), np.amax(S_prime + Sb + 1), 1)\n",
    "print(\"S = {} e S' = {}\".format(S, S_prime))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E(0.5,0.0)\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 0.375 J$$"
      ],
      "text/plain": [
       "-0.375⋅J"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E(0.5,1.0)\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$0.625 J - 1.0 J'$$"
      ],
      "text/plain": [
       "0.625⋅J - 1.0⋅J'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E(1.5,1.0)\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAAZCAYAAADT59fvAAAABHNCSVQICAgIfAhkiAAABLVJREFUaIHt2mmsXVMUwPFfS1FiHiJBiKc1JjQSY9RrJWIeQnwxxBdTQghJP4gPbXwQQ6RSpCQdxJQgoaZSQmgMEREEoaWuaMyCos/8fFj75l2n59x77rnvvlvv3X9ys1/2sM5ae6+z99rrvE306TM6nIxN8V2vFekzftgc63FHrxXpM74YxDBmdvMhu2MxvsDvqGE+tq8g62SswFoMYQ0expE5fS8UxjX7/Z0zrtak/1dNdHs59cnTpVucjQVYiXXp+fd1IK/KWuXZfT0+aOy0aQdK5TGAV7ELluFDHIYrcQKOxvclZd2IOan/Y+Ls3gen4yxc4L+T+jbmFcg6BrOxvKD9JzGhWX4p6D8ZhwhHfafQgtHnOhws9FqL/TqQVWWtiuyejbs60KUlzwpvviJTf2uqX1hSzq5C+a+E4Y3MSrLWtKHXa2nMaTlttfRrh/2SvHfbHNcpszANk4wcP1V3rCprlWf3VuLFrHIilWLv9NBPhWc3srV4y35NirTi8CRrWUH7OvxcUq+Dkqy12CSnvaZ9xzo3yVzc5rjRZFB1x6q6Vnl2n4h7sg/ICu2E2alcgX8ybT/jFWyJI0rIWo0/xNa8U6ZtpjD++ZJ6XZLKRfJjLOJWcx6uFUfBLPlOWOfQVL5ZUoeNjaprlWf3DOVPokrcLLz5moL221P7ZSXlXSWM/gZ34wY8hN/EhGSPyDym4gfhUHsU9KnJD9zX4NiCMS+lPoeV0KFbDKq+Y1Vdq9J2j2bwvm0qfypor9dvV1LefLHoi3FRQ/3HWCocrhXnpOc9hc8L+iwRt6z3xdu6Ny7HxSLYP9J/A9XJ4i39y9jHWKNFlbVqy+7sUVjT+sre+GvnbZmUyuGS/efgEeFEA+K8P1TsJPfjphIyLk5lsxvLPLyAr0WS7z1cKoLYqZib6T9dHMXvid2ziJruzWW3yVursnZjwx3rkzKDGvii4e+6l2+b1xHbZPo1Y1CkGx7F1Q31b+FMrBLb+ELFt8MDcJQI2p8u8cwsC9Mzskm/svFVJ3PZbaqsVVtxZdaxjiunVy4fpXJ6Qfu0VK4qIeuUVL6Y07YebwgHm6HYscoE7c2oH7XZm1HZCe5kLrtNlbVqy7FG81ZYd4Ljc+RuLRJuQ3i9hKzNU7lzQXu9/o+C9i1wvgj+F5V4Xh71zHLWcf/vN0KqrVVP7a6SdBsQibcpDXXnGPmkslum/4nCYYawY4Ee56fxT7TQ90DskFO/p0h5DIsURJ1JIof2OzZrIbvbDCoXm+XNL+2tVc/tHhBB8LD4DHODCIyHxfab5wi11L5XQ91kPJfq14kE3I14XDjVsMg3FbEy9Tm1hb5zRRy0HHemZzwinHZY3CYbJ3LfVN+r3eoMcZlZimeSLp801N2SM6Zmw/mlvbXqtd2IfNESfCmOqs9wm/ydgWLDp4hc1uvCuf4Scc+TYgsvYv8k73PNk5xEnupB8Z3sR/yJb4VTX2DkdlSnvhMuaCG3W8zV/GZZyxlTkz+/lF+rXts97nlATPBJvVZkjJmodo8JM8RbvVrrnXA8MVHt7iqTxfFzr4i7hsS/4Ix3JqrdY8ZBRr5XPiz+F2oiMFHt7tOnT58+ffr02aj5FwFMi4yL0nYbAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- 0.875 J - 1.0 J'$$"
      ],
      "text/plain": [
       "-0.875⋅J - 1.0⋅J'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Iterativamente encontrando valores para as energias\n",
    "for i in S_prime:\n",
    "    S = np.arange(np.amin(np.absolute(i - Sb)), np.amax(i + Sb + 1), 1)\n",
    "    for j in S:\n",
    "        print(\"E({},{})\".format(j, i))\n",
    "        E = -(J/2)*(j*(j + 1)) - ((J_prime -J)/2)*(i*(i + 1))\n",
    "        display(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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