Spring_semester_second_session.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "from ipywidgets import interact\n",
    "import scipy.sparse as scsp\n",
    "import scipy.sparse.linalg as scspl\n",
    "from topobp import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Topological insulators II/02: Majorana fermions and brading"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:** Berry-phase calculation for four-site y-turn braiding.\n",
    "\n",
    "Some helper functions you may use for the braiding protocol:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fOn(t,n):\n",
    "    '''\n",
    "    A piecewise function ...\n",
    "    '''\n",
    "    return ( 1*((t>=n)&(t<n+1)) + 0*((t<n)&(t>=n+1))); \n",
    "\n",
    "def fSwitchOn(t,n):\n",
    "    return (t-n)*fOn(t,n);\n",
    "\n",
    "def fSwitchOff(t,n):\n",
    "    return fOn(t,n)*(1-fSwitchOn(t,n));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us explore them graphically"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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h+jzptveYbc/PKynoGmTLCj3TdJ/3PBnK1F8ecp504TMp6Bqogn7iiRGjcfhF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\nc2l76EIIIcIhY+8U9fLmpLA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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0a44754ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tvec = linspace(0,6,600)\n",
    "fOn(tvec,1)\n",
    "plot(tvec,fOn(tvec,1), label='square pulse',lw=2)\n",
    "plot(tvec,fSwitchOn(tvec,2),label='switch on',lw=2)\n",
    "plot(tvec,fSwitchOff(tvec,3),label='switch off',lw=2)\n",
    "ylim(0,1.5)\n",
    "grid()\n",
    "legend();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Hint:* You can use predifined functions in `topobp.py`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First let us define four fermionic annihilation $\\hat{c}_i$ operators:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# construct the c operators; Nsp stands for the number of single-particle states, \n",
    "# that is 4 for our 4-site T-junction\n",
    "Nsp = 4;\n",
    "c = fermion_Fock_matrices(Nsp,dense=True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then the projectors for the even and odd fermion number subspaces:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# construct the projectors on the even and odd subpsaces\n",
    "PEven , POdd =  even_odd_Fock_operators(Nsp,dense=True)\n",
    "#Identity matrix \n",
    "IdentityMatrix=PEven+POdd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We shall also nead the functions that describe how different hopping and pairing amplitudes change as we vary time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# define the functions that govern the braiding protocol\n",
    "def ft23(t):\n",
    "    return fSwitchOff(t,0) + fSwitchOn(t,3) + fOn(t,4) + fOn(t,5);           \n",
    "def ft34(t): \n",
    "    return fOn(t, 0) + fOn(t, 1) + fSwitchOff(t, 2) + fSwitchOn(t, 5);\n",
    "def ft13(t): \n",
    "    return fSwitchOn(t, 1) + fOn(t, 2) + fOn(t, 3) + fSwitchOff(t, 4);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we have the $\\hat{c}_i$ operators we can build up a parameter dependent Hamiltonian that performs the braiding:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# construct the adiabatic Fock space Hamiltonian for the braiding procedure\n",
    "def Braid_Fock_Ham(t):\n",
    "    ham=ft23(t)*(c[1].H*c[2] + c[2].H*c[1] + c[1].H*c[2].H + c[2]*c[1]) +\\\n",
    "        ft34(t)*(c[2].H*c[3] + c[3].H*c[2] + c[2].H*c[3].H + c[3]*c[2]) +\\\n",
    "        ft13(t)*(c[0].H*c[2] + c[2].H*c[0] + 1.0j*c[0].H*c[2].H - 1.0j*c[2]*c[0]) +\\\n",
    "        (1-ft23(t))*c[1].H*c[1]+(1-ft34(t))*c[3].H*c[3]+(1-ft13(t))*c[0].H*c[0];\n",
    "    return ham"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we are ready to perform the braiding. Below we write a simple for loop and calculate the Berry's phase for the even and odd parity groundstate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# loop for time points\n",
    "# set the initial value of the even and odd projector products as the identity matrix\n",
    "ProjectorProductEven = IdentityMatrix;\n",
    "ProjectorProductOdd = IdentityMatrix;\n",
    "taumin = 0.0;\n",
    "taumax = 6.0;\n",
    "taunum=300;\n",
    "tran=linspace(taumin,taumax,taunum,endpoint=False)\n",
    "for t in tran:\n",
    "    [val,vec]=eigh(Braid_Fock_Ham(t))\n",
    "    GSProjector=vec[:,0]*vec[:,0].H+vec[:,1]*vec[:,1].H\n",
    "    ## construct the projectors of the even and odd ground states and update the projector product\n",
    "    ProjectorProductEven = ProjectorProductEven*(PEven*GSProjector*PEven)\n",
    "    ProjectorProductOdd = ProjectorProductOdd*(POdd*GSProjector*POdd)\n",
    "\n",
    "# end of loop for time points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we can check the values of the two calculated Berry's phases:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Berry's phase for the even subspace: 0.785398163397\n",
      "Berry's phase for the odd subspace: -0.785398163397\n",
      "Numeeric value of $\\pi/4$ 0.7853981633974483\n"
     ]
    }
   ],
   "source": [
    "print(r\"Berry's phase for the even subspace:\",angle(ProjectorProductEven.trace()[0,0]))\n",
    "print(r\"Berry's phase for the odd subspace:\",angle(ProjectorProductOdd.trace()[0,0]))\n",
    "print(r'Numeeric value of $\\pi/4$',pi/4)"
   ]
  }
 ],
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