Spring_semester_first_session.ipynb

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     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
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   "source": [
    "%pylab inline\n",
    "% config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "from ipywidgets import interact\n",
    "\n",
    "import scipy.sparse as scsp\n",
    "import scipy.sparse.linalg as scspl"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Topological insulators II/01: Superconductivity and the Kitaev model"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Many-body approach\n",
    "\n",
    "We first construct annihilation operators $\\hat{c}_i$ acting on the Fock space of fermions in the binary sequence basis. "
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   "source": [
    "def fermion_Fock_matrices(NN=3):\n",
    "    '''\n",
    "    Returns list of 2^NN X 2^NN sparse matrices, \n",
    "    representing fermionic annihilation operators\n",
    "    acting on the Fock space of NN fermions.\n",
    "    '''\n",
    "    l=list(map(lambda x: list(map(int,list(binary_repr(x,NN)))),arange(0,2**NN)))\n",
    "    ll=-(-1)**cumsum(l,axis=1)\n",
    "    AA=(array(l)*array(ll))[:,::-1]\n",
    "    cc=[]\n",
    "    for p in range(NN):\n",
    "        cc.append(scsp.dia_matrix((AA[:,p],array([2**p])),shape=(2**NN,2**NN), dtype='d'))\n",
    "    return cc"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Some more handy simple Fock space operators are defined below. First the total fermion number operator:\n",
    "$$ \\hat{N}=\\sum_p \\hat{c}^\\dagger_p\\hat{c}_p$$"
   ]
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   "execution_count": 3,
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   "source": [
    "def particle_number_Fock_operator(NN=3):\n",
    "    '''\n",
    "    Returns particle number operator in Fock space of NN fermions\n",
    "    '''\n",
    "    return scsp.dia_matrix((list(map(lambda x:bin(x).count(\"1\"),arange(0,2**NN))),[0]),\n",
    "                                   shape=(2**NN,2**NN))"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the fermion particle number parity operator \n",
    "$$\\hat{P}=(-1)^{\\hat{N}}$$"
   ]
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   "execution_count": 4,
   "metadata": {
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   "outputs": [],
   "source": [
    "def parity_Fock_operator(NN=3):\n",
    "    '''\n",
    "    Returns particle number parity operator in Fock space of NN fermions\n",
    "    '''\n",
    "    return scsp.dia_matrix((list(map(lambda x:1-2*mod(bin(x).count(\"1\"),2),arange(0,2**NN))),[0]),\n",
    "                                   shape=(2**NN,2**NN))"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Single site s-wave superconductor \n",
    "$$ \n",
    "\\hat{H}=\\mu\\left(\\hat{c}^\\dagger_\\uparrow\\hat{c}_\\uparrow+\\hat{c}^\\dagger_\\downarrow\\hat{c}_\\downarrow\\right)\n",
    "+\\Delta\\left(\\hat{c}^\\dagger_\\uparrow\\hat{c}^\\dagger_\\downarrow + \\hat{c}_\\downarrow\\hat{c}_\\uparrow \\right)\n",
    "$$"
   ]
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  {
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   "execution_count": 5,
   "metadata": {
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   "outputs": [],
   "source": [
    "def Singe_Site_Superconductor_Fock_Ham(cc,mu,Delta,**kwargs):\n",
    "    '''\n",
    "    Returns Fock space representation of the Hamiltonian \n",
    "    of the single site s-wave superconductor.\n",
    "    '''\n",
    "    H=    mu*(cc[0].H*cc[0]+cc[1].H*cc[1]) \\\n",
    "      +Delta*(cc[0].H*cc[1].H+cc[1]*cc[0])\n",
    "    return H.todense()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:** Calculate the full many-body spectrum as function of $\\mu/\\Delta$, find the many-body groundstate for \n",
    "$\\mu/\\Delta=-5,0,5$"
   ]
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     "output_type": "display_data"
    }
   ],
   "source": [
    "muran=linspace(-5,5,100)\n",
    "fig=figsize(4,5)\n",
    "dat=[]\n",
    "cc=fermion_Fock_matrices(2)\n",
    "for mu in muran:\n",
    "    dat.append(eigvalsh(Singe_Site_Superconductor_Fock_Ham(cc,mu,1.)))\n",
    "plot(muran,dat,lw=3);\n",
    "#stuff below is only to make figure nice\n",
    "xticks(fontsize=16)\n",
    "yticks(fontsize=16)\n",
    "grid()\n",
    "xlabel(r'$\\mu/\\Delta$',fontsize=16);\n",
    "ylabel(r'$E_n$',fontsize=16);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-10.0990195136\n",
      "[[ 0.09853762]\n",
      " [ 0.        ]\n",
      " [ 0.        ]\n",
      " [ 0.99513333]]\n",
      "-1.0\n",
      "[[-0.70710678]\n",
      " [ 0.        ]\n",
      " [ 0.        ]\n",
      " [-0.70710678]]\n",
      "-0.0990195135928\n",
      "[[-0.99513333]\n",
      " [ 0.        ]\n",
      " [ 0.        ]\n",
      " [-0.09853762]]\n"
     ]
    }
   ],
   "source": [
    "for mu in [-5,0,5]:\n",
    "    val,vec=eigh(Singe_Site_Superconductor_Fock_Ham(fermion_Fock_matrices(2),mu,1.))\n",
    "    print (val[0])\n",
    "    print (vec[:,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kitaev model\n",
    "\n",
    "$$ \n",
    "\\hat{H}=\\mu\\sum_p\\hat{c}^\\dagger_p\\hat{c}_p+t\\left (\\sum_p\\hat{c}^\\dagger_{p+1}\\hat{c}_p+ \\mathrm{h.c.}\\right)+\n",
    "\\Delta\\left(\\sum_p\\hat{c}^\\dagger_{p+1}\\hat{c}^\\dagger_p+ \\mathrm{h.c.} \\right)\n",
    "$$\n",
    "\n",
    "\n",
    "**Task:** Write a routine that builds up the Kitaev model in Fock space, and calculate the spectrum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Kitaev_wire_BDG_Ham_Fock_Ham(cc,t,Delta,mu,**kwargs):\n",
    "    '''\n",
    "    Builds Kitaev wire Hamiltonian in Fock space.\n",
    "    '''\n",
    "    H=    t*sum(cc[p+1].H*cc[p]+cc[p].H*cc[p+1]   for p in range(len(cc)-1)) \\\n",
    "     +Delta*sum(cc[p+1]*cc[p]+cc[p].H*cc[p+1].H   for p in range(len(cc)-1)) \\\n",
    "        +mu*sum(cc[p].H*cc[p]                     for p in range(len(cc))) \n",
    "    return H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def playFock(N=10,Delta=0.2):\n",
    "    cc=fermion_Fock_matrices(N)\n",
    "    dat=[]\n",
    "    for u in uran:\n",
    "        val=scspl.eigsh(Kitaev_wire_BDG_Ham_Fock_Ham(cc,1,Delta,u),return_eigenvectors=False,k=50,which='SA')\n",
    "        dat.append(sort((val-val[-1])[:-1]))\n",
    "    plot(uran,dat,'r-',lw=2);\n",
    "    plot(uran,array(dat)[:,0],'r-',lw=2,label=r'$E^{\\mathrm{Fock}}_n-E^{\\mathrm{Fock}}_{GS}$');\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "source": [
    "#This lets you play with the Fock spectrum\n",
    "uran=linspace(-3,3,50)\n",
    "interact(playFock,N=(6,7),Delta=(0,2,0.1));\n",
    "xlabel(r'$\\mu$',fontsize=16);\n",
    "ylabel(r'$E^{\\mathrm{Fock}}_n-E^{\\mathrm{Fock}}_0$',fontsize=16);\n",
    "grid();"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Bogoliubov–de Gennes \"trick\"\n",
    "\n",
    "One can rewrite a generic superconductor many-body Hamiltonian \n",
    "\n",
    "$$\n",
    "\\hat{H} = \\sum_{\\alpha,\\beta} \\left( \n",
    "c_\\alpha^\\dagger h_{\\alpha,\\beta}\n",
    "c_{\\beta} + \\frac{1}{2}c_\\alpha^\\dagger \\Delta_{\\alpha,\\beta}\n",
    "c_\\beta^\\dagger  + \\frac{1}{2}c_\\beta \\Delta_{\\alpha,\\beta}^\\ast\n",
    "c_\\alpha \\right)\n",
    "$$\n",
    "\n",
    "as \n",
    "\n",
    "$$\n",
    "\\hat{H} = \\frac{1}{2} \n",
    " \\begin{matrix}\\begin{pmatrix} c^\\dagger & c \\end{pmatrix}\\\\\\mbox{}\\end{matrix}\n",
    "\\mathcal{H}_{\\mathrm{BdG}}\n",
    "\\begin{pmatrix} c \\\\ c^\\dagger \\end{pmatrix} + \\frac{1}{2} \\mathrm{Tr} h \\mathbb{1}_{\\mathrm{Fock}}\n",
    "$$\n",
    "where we have introduced the Bogoliubov–de Gennes matrix as\n",
    "$$\n",
    "\\mathcal{H}_{\\mathrm{BdG}} = \\begin{pmatrix} \n",
    "h & \\Delta \\\\ \n",
    "-\\Delta^\\ast & -h^\\ast \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "The positive eigenvalues of the Bogoliubov–de Gennes matrix correspond to the single particle excitation spectrum of the full many-body Hamiltonian."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task:** Write a routine that calculates the Bogoliubov–de Gennes matrix of the Kitaev model, calculate the spectrum and compare the spectrum to the excitation spectrum of the many-body Hamiltonian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": [
    "#Define the Pauli matrices unit matrix and the zero matrix\n",
    "s0=matrix([[1,0],[0,1]])\n",
    "s1=matrix([[0,1],[1,0]])\n",
    "s2=matrix([[0,-1j],[1j,0]])\n",
    "s3=matrix([[1,0],[0,-1]])\n",
    "z2=zeros_like(s0);\n",
    "\n",
    "def Kitaev_wire_BDG_Ham(N,mu,t,Delta):\n",
    "    idL=eye(N); # identity matrix of dimension L\n",
    "    odL=diag(ones(N-1),1);# upper off diagonal matrix with ones of size L\n",
    "    U=mu*s3\n",
    "    T=-t*s3+1.0j*Delta*s2\n",
    "    return kron(idL,U)+kron(odL,T)+kron(odL,T).H"
   ]
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  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def playBdG(N=10,Delta=0.2):\n",
    "    dat=[]\n",
    "    for u in uran:\n",
    "        dat.append(eigvalsh(Kitaev_wire_BDG_Ham(N,u,1,Delta)))\n",
    "    plot(uran,dat,'k',lw=6);\n",
    "    plot(uran,array(dat)[:,0],'k',lw=6,label=r'$E^{\\mathrm{BdG}}_n$')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
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   ],
   "source": [
    "#This lets you play with the BdG spectrum\n",
    "uran=linspace(-3,3,100)\n",
    "interact(playBdG,N=(3,20),Delta=(0,2,0.1));\n",
    "xlabel(r'$\\mu$',fontsize=16);\n",
    "ylabel(r'$E^{\\mathrm{BdG}}_n$',fontsize=16);\n",
    "grid();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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   ],
   "source": [
    "figsize(4,5)\n",
    "uran=linspace(0,4,50)\n",
    "NN=6;Delta=0.9;\n",
    "playBdG(NN,Delta)\n",
    "playFock(NN,Delta)\n",
    "ylim(-6,6)\n",
    "xticks(linspace(min(uran),max(uran),5),fontsize=16)\n",
    "yticks(linspace(-6,6,5),fontsize=16)\n",
    "xlabel(r'$\\mu$',fontsize=16)\n",
    "grid()\n",
    "legend(fontsize=16,loc='lower right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Tasks to be done\n",
    "\n",
    "- Calculate parity of the groundstate of a Kitaev wire/circle of different length with the Fock-space reperesentation.\n",
    "- Investigate proximity effect in a two site spinfull model.\n",
    "- Implement Pfaffian and calculate groundstate parity via Bogoliubov–de Gennes approach.\n",
    "- Investigate effect of Coulomb interaction on the Kitaev wire. What happens with the zero mode?"
   ]
  }
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