DEHam/notebooks/t_J_Model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. t-J Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. t-J Hamiltonian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### a. Basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $t-J$ Hamiltonian with a S=1/2 spin-model is constructed in a basis of $\\mathcal{R}^3$ given by the vector B as follows:\n",
    "\n",
    "$$\n",
    "B=\n",
    "  \\begin{bmatrix}\n",
    "     0 \\\\\n",
    "     \\uparrow \\\\\n",
    "     \\downarrow\n",
    "  \\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### b.) Operators"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Hamiltonian is constructed using two types of operators.\n",
    "\n",
    "1. The particle creation and annihilation operators $\\hat{C}^\\dagger,\\hat{C}$\n",
    "2. The spin ladder operators $\\hat{S}^+, \\hat{S}^-, \\hat{S}^z$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrix form of the five operators in the above basis is given as follows:\n",
    "\n",
    "$$\n",
    "\\hat{C}^{\\dagger}_{\\uparrow}= \\left( \\hat{C}_{\\uparrow} \\right)^\\dagger=\n",
    "  \\begin{bmatrix}\n",
    "     0 & 0 & 0 \\\\\n",
    "     1 & 0 & 0 \\\\\n",
    "     0 & 0 & 0\n",
    "  \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\hat{C}^{\\dagger}_{\\downarrow}= \\left( \\hat{C}_{\\downarrow} \\right)^\\dagger=\n",
    "  \\begin{bmatrix}\n",
    "     0 & 0 & 0 \\\\\n",
    "     0 & 0 & 0 \\\\\n",
    "     1 & 0 & 0\n",
    "  \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\hat{S}^{z}\\ \\ \\ \\  =\\ \\ \\ \\ \\ \n",
    "  \\begin{bmatrix}\n",
    "     0 & 0 & 0 \\\\\n",
    "     0 & \\frac{1}{2} & 0 \\\\\n",
    "     0 & 0 & -\\frac{1}{2}\n",
    "  \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\hat{S}^{+}_{\\downarrow}= \\left( \\hat{S}^- \\right)^\\dagger=\n",
    "  \\begin{bmatrix}\n",
    "     0 & 0 & 0 \\\\\n",
    "     0 & 0 & 0 \\\\\n",
    "     0 & 1 & 0\n",
    "  \\end{bmatrix}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### c.) Hamiltonian "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Nearest Neighbor coupling**\n",
    "\n",
    "The $t-J$ Hamiltonian is written in the above basis with the above five operators and the two parameters $t$ which governs the kinetic energy of the particles and the magnetic-coupling parameter $J$ which governs the coupling of the localized spins on neighboring sites.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{H_{t-J}} = \\sum^{N-1}_i \\left\\{ t_{i,i+1} \\sum_{\\sigma} \\left(\\hat{C}^{\\dagger}_{\\sigma,i}\\cdot\\hat{C}_{\\sigma,i+1} + h.c. \\right) + J_{i,i+1} \\left( \\frac{\\hat{S}^+_{i}\\cdot\\hat{S}^-_{i+1} + \\hat{S}^-_{i}\\cdot\\hat{S}^+_{i+1}}{2} + \\hat{S}^z_i\\cdot\\hat{S}^z_{i+1} \\right)\\right\\}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$h.c.$ stands for Hermitian conjugate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. 2D Lattice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.) Labeling\n",
    "Here a 4x4 lattice is shown with periodic boundary conditions where each site and bond is labelled."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib as mplt\n",
    "import networkx as nx\n",
    "import numpy as np\n",
    "\n",
    "Lx = 4\n",
    "Ly = 4\n",
    "\n",
    "draw2DLattice(Lx,Ly)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.) Generating input file \n",
    "\n",
    "In this work, we will be using the [DEHam](https://github.com/v1j4y/DEHam) code to solve the Hamiltonian using Exact Diagonalization method. \n",
    "\n",
    "The input file for the code is as follows:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "  1. The DEHam program requires an input file which \n",
    "   has the topology of the Hamiltonian and the various parameters\n",
    "   as explained below in a sample inputfile:\n",
    "\n",
    "```python\n",
    "8     \t\t\t\t\t\t\t\t# The number of orbitals (total)\n",
    "1     \t\t\t\t\t\t\t\t# The largest number of non-zero elements per row (Multiple of Ndet)\n",
    "1     \t\t\t\t\t\t\t\t# The total number of processors used in parallel (Multiple of Ndet)\n",
    "1     \t\t\t\t\t\t\t\t# The number of holes\n",
    "0     \t\t\t\t\t\t\t\t# The isz (ms-1/2) value\n",
    "true     \t# Restrict the hole to the 1'st (i.e. half of natom) Family of states. *false* for no restrictions\n",
    "1,2,3,1,2,3,4,5,6,7\t\t\t\t\t# (L1) The topology of the system is specified here\n",
    "2,3,4,8,7,6,5,6,7,8\t\t\t\t\t# (L2) first and second line contain the two sites linked\n",
    "1,1,1,2,2,2,2,3,3,3\t\t\t\t\t# (ltype) third line contains the type of link (1 for t or J, 2 for K and 3 for none)\n",
    ".1430,-0.20,0.0000\t\t\t\t\t# The three types of links this line gives Jz, Kz\n",
    ".1430,-0.20,0.0000\t\t\t\t\t# The three types of links this line gives Jx,y; Kx,y\n",
    "-1.00,0.0,0.00\t\t\t\t\t\t# This line gives t(i,i+1), t(i,i+2), t(i,i+3)\n",
    "0.,0.,0.,0.,0.,0.,0.,0.,0. \t\t    # Energy of each orbital + additional energy\n",
    "2                   \t\t\t\t# The total number of roots\n",
    "1                   \t\t\t\t# I   The position of the first\n",
    "1                   \t\t\t\t# I   SBox\n",
    "1                   \t\t\t\t# I\n",
    "1                   \t\t\t\t# I\n",
    "1                   \t\t\t\t# II  The positions of the second\n",
    "1                   \t\t\t\t# II  SBox\n",
    "1                   \t\t\t\t# II\n",
    "1                   \t\t\t\t# II\n",
    "1                   \t\t\t\t# III\n",
    "1                   \t\t\t\t# III The positions of the third\n",
    "1                   \t\t\t\t# III SBox\n",
    "1                   \t\t\t\t# III\n",
    "1                   \t\t\t\t# positio of the hole\n",
    "0                   \t\t\t\t# fix the position of the first hole during the CI\n",
    "0                   \t\t\t\t# fix the position of the second hole during the CI\n",
    "0                   \t\t\t\t# Print the wavefunction. It is stored in the FIL666 file after the run\n",
    "```\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### a.) Script for $t-J$ input file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]\n",
      "[2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4]\n",
      "[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]\n"
     ]
    }
   ],
   "source": [
    "l1, l2, ltype = getInput(Lx,Ly)\n",
    "print(l1)\n",
    "print(l2)\n",
    "print(ltype)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.) Misc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Draw Lattice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "jupyter": {
     "source_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Function that draws a 2D Lattice\n",
    "\"\"\"\n",
    "def draw2DLattice(Lx,Ly):\n",
    "    n=Lx+2\n",
    "    m=Ly+2\n",
    "\n",
    "    G = nx.grid_2d_graph(n,m)  # Lx, Ly grid\n",
    "\n",
    "    pos = dict( (n, n) for n in G.nodes() );\n",
    "    #labels = Dict( ((i, j), i + (Ly-j) * (Lx+1) ) for (i,j) in G.nodes() );\n",
    "    #edges=Dict()\n",
    "    #for (i,edge) in enumerate(G.edges)\n",
    "    #    edges[edge]=i\n",
    "    #    print(i,\"\\t\",edge,\"\\n\")\n",
    "    #end\n",
    "\n",
    "\n",
    "    for j in range(0,m-1):\n",
    "        G.remove_edge((0,j),(0,j+1))\n",
    "        G.remove_edge((n-1,j),(n-1,j+1))\n",
    "\n",
    "    for i in range(0,n-1):\n",
    "        G.remove_edge((i,0),(i+1,0))\n",
    "        G.remove_edge((i,m-1),(i+1,m-1))\n",
    "\n",
    "\n",
    "\n",
    "    edgescenter = dict()\n",
    "    countedge = 1\n",
    "    for j in range(1,m-1):\n",
    "        for i in range(1,n-1):\n",
    "            edgescenter[((i,j),(i+1,j))] = countedge\n",
    "            countedge += 1\n",
    "\n",
    "\n",
    "    for j in range(1,m-1):\n",
    "        for i in range(1,n-1):\n",
    "            edgescenter[((i,j),(i,j+1))] = countedge\n",
    "            countedge += 1\n",
    "\n",
    "\n",
    "    labelscenter = dict()\n",
    "    countlabels = 1\n",
    "    for j in range(1,m-1):\n",
    "        for i in range(1,n-1):\n",
    "            labelscenter[(i,j)]=countlabels\n",
    "            countlabels+=1\n",
    "\n",
    "\n",
    "    color_map=[]\n",
    "    countcolor=0\n",
    "    for i in range(0,n):\n",
    "        for j in range(0,m):\n",
    "            if i==0 or j==0 or i==n-1 or j==m-1:\n",
    "                color_map.append(\"white\")\n",
    "            else:\n",
    "                if(countcolor%2 == 0):\n",
    "                    color_map.append(\"blue\")\n",
    "                else:\n",
    "                    color_map.append(\"red\")\n",
    "                countcolor+=1\n",
    "        countcolor+=1\n",
    "\n",
    "\n",
    "    nx.draw(G, pos=pos, labels=labelscenter, node_color=color_map, font_color=\"white\");\n",
    "    nx.draw_networkx_edge_labels(G,pos,edge_labels=edgescenter,font_color=\"black\");\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Generate Input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "jupyter": {
     "source_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Function that generates the input\n",
    "\"\"\"\n",
    "def getInput(Lx,Ly):\n",
    "    n=Lx+2\n",
    "    m=Ly+2\n",
    "\n",
    "\n",
    "    labelscenter = dict()\n",
    "    countlabels = 1\n",
    "    for j in range(1,m-1):\n",
    "        for i in range(1,n-1):\n",
    "            labelscenter[(i,j)]=countlabels\n",
    "            countlabels+=1\n",
    "\n",
    "    edgescenter = dict()\n",
    "    countedge = 1\n",
    "    for j in range(1,m-1):\n",
    "        for i in range(1,n-1):\n",
    "            edgescenter[((i,j),(i+1,j))] = countedge\n",
    "            countedge += 1\n",
    "    for j in range(1,m-1):\n",
    "        for i in range(1,n-1):\n",
    "            edgescenter[((i,j),(i,j+1))] = countedge\n",
    "            countedge += 1\n",
    "\n",
    "\n",
    "    l1 = []\n",
    "    l2 = []\n",
    "    itype = [1 for x in range(0,len(edgescenter))]\n",
    "\n",
    "    for x in edgescenter:\n",
    "        if Lx + 1 not in x[1]:\n",
    "            l1.append(labelscenter[x[0]])\n",
    "            l2.append(labelscenter[x[1]])\n",
    "        else:\n",
    "            l1.append(labelscenter[x[0]])\n",
    "            if x[1][0] > Lx:\n",
    "                l2.append(labelscenter[(x[1][0]-Lx,x[1][1])])\n",
    "            elif x[1][1] > Ly:\n",
    "                l2.append(labelscenter[(x[1][0],x[1][1]-Ly)])\n",
    "\n",
    "#    print(l1,\"\\n\")\n",
    "#    print(l2,\"\\n\")\n",
    "#    print(itype,\"\\n\")\n",
    "    return l1,l2,itype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9],\n",
       " [2, 3, 1, 5, 6, 4, 8, 9, 7, 4, 5, 6, 7, 8, 9, 1, 2, 3],\n",
       " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "getInput(3,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "header=r\"\"\"\n",
    "\n",
    "\"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
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