Sample日本語版.ipynb

{
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://github.com/cometscome/TightBinding.jl/blob/master/Sample日本語版.ipynb\n",
    "# TightBinding.jl\n",
    "サンプルコード\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2D 正方格子"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General`\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m git-repo `https://github.com/JuliaRegistries/General.git`\n",
      "\u001b[?25l\u001b[2K\u001b[?25h\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m git-repo `https://github.com/cometscome/TightBinding.jl`\n",
      "\u001b[?25l\u001b[2K\u001b[?25h\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.1/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.1/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "] add https://github.com/cometscome/TightBinding.jl\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Possible hoppings\n",
      "(1,1), x:-1//1, y:-1//1\n",
      "(1,1), x:-1//1, y:0//1\n",
      "(1,1), x:-1//1, y:1//1\n",
      "(1,1), x:0//1, y:-1//1\n",
      "(1,1), x:0//1, y:0//1\n",
      "(1,1), x:0//1, y:1//1\n",
      "(1,1), x:1//1, y:-1//1\n",
      "(1,1), x:1//1, y:0//1\n",
      "(1,1), x:1//1, y:1//1\n"
     ]
    }
   ],
   "source": [
    "using TightBinding\n",
    "#２次元格子を設定\n",
    "la2 = set_Lattice(2,[[1,0],[0,1]])\n",
    "#原子を追加\n",
    "add_atoms!(la2,[0,0])\n",
    "#可能なホッピングベクトルを表示\n",
    "show_neighbers(la2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#6 (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#ホッピングを追加\n",
    "t = 1.0\n",
    "#x 方向\n",
    "add_hoppings!(la2,-t,1,1,[1,0])\n",
    "#y 方向\n",
    "add_hoppings!(la2,-t,1,1,[0,1])\n",
    "#ハミルトニアンの作成\n",
    "ham2 = hamiltonian_k(la2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[4.0 3.94545 3.78327 3.5179 3.15656 2.70913 2.18779 1.60678 0.981942 0.330317 -0.330317 -0.981942 -1.60678 -2.18779 -2.70913 -3.15656 -3.5179 -3.78327 -3.94545 -4.0]\n"
     ]
    }
   ],
   "source": [
    "#エネルギーを計算できる\n",
    "        kmin = [-π,-π]\n",
    "        kmax = [0.0,0.0]\n",
    "        nk = 20\n",
    "        vec_k,energies = calc_band(kmin,kmax,nk,la2,ham2)\n",
    "        println(energies)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#バンド構造をプロットしてみる\n",
    "#プロットしたいk点を作成するためにklinesを設定\n",
    "        klines = set_Klines()\n",
    "#プロットしたいk点をlineとして設定\n",
    "        kmin = [0,0] #kの最小値\n",
    "        kmax = [π,0] #kの最大値\n",
    "        #kminからkmaxまでのlineがklinesに追加される\n",
    "        add_Kpoints!(klines,kmin,kmax,\"(0,0)\",\"(pi,0)\")\n",
    "\n",
    "        kmin = [π,0]\n",
    "        kmax = [π,π]\n",
    "        add_Kpoints!(klines,kmin,kmax,\"(pi,0)\",\"(pi,pi)\")\n",
    "\n",
    "        kmin = [π,π]\n",
    "        kmax = [0,0]\n",
    "        add_Kpoints!(klines,kmin,kmax,\"(pi,pi)\",\"(pi,0)\")\n",
    "\n",
    "#バンド構造のプロット\n",
    "calc_band_plot(klines,la2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## グラフェン"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Possible hoppings\n",
      "(1,1), x:-1//1, y:-1//1\n",
      "(1,2), x:-2//3, y:-2//3\n",
      "(2,2), x:-1//1, y:-1//1\n",
      "(1,1), x:-1//1, y:0//1\n",
      "(1,2), x:-2//3, y:1//3\n",
      "(2,2), x:-1//1, y:0//1\n",
      "(1,1), x:-1//1, y:1//1\n",
      "(1,2), x:-2//3, y:4//3\n",
      "(2,2), x:-1//1, y:1//1\n",
      "(1,1), x:0//1, y:-1//1\n",
      "(1,2), x:1//3, y:-2//3\n",
      "(2,2), x:0//1, y:-1//1\n",
      "(1,1), x:0//1, y:0//1\n",
      "(1,2), x:1//3, y:1//3\n",
      "(2,2), x:0//1, y:0//1\n",
      "(1,1), x:0//1, y:1//1\n",
      "(1,2), x:1//3, y:4//3\n",
      "(2,2), x:0//1, y:1//1\n",
      "(1,1), x:1//1, y:-1//1\n",
      "(1,2), x:4//3, y:-2//3\n",
      "(2,2), x:1//1, y:-1//1\n",
      "(1,1), x:1//1, y:0//1\n",
      "(1,2), x:4//3, y:1//3\n",
      "(2,2), x:1//1, y:0//1\n",
      "(1,1), x:1//1, y:1//1\n",
      "(2,2), x:1//1, y:1//1\n"
     ]
    }
   ],
   "source": [
    "#基本並進ベクトル\n",
    "a1 = [sqrt(3)/2,1/2]\n",
    "a2= [0,1]\n",
    "#格子を設定\n",
    "la = set_Lattice(2,[a1,a2])\n",
    "#原子を追加\n",
    "add_atoms!(la,[1/3,1/3])\n",
    "add_atoms!(la,[2/3,2/3])\n",
    "show_neighbers(la)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{TightBinding.Hopping,1}:\n",
       " TightBinding.Hopping(-1.0, [1, 2], [0.333333, 0.333333]) \n",
       " TightBinding.Hopping(-1.0, [1, 2], [-0.666667, 0.333333])\n",
       " TightBinding.Hopping(-1.0, [1, 2], [0.333333, -0.666667])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#ホッピングを設定\n",
    "        t = 1.0\n",
    "        add_hoppings!(la,-t,1,2,[1/3,1/3])\n",
    "        add_hoppings!(la,-t,1,2,[-2/3,1/3])\n",
    "        add_hoppings!(la,-t,1,2,[1/3,-2/3])"
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   "cell_type": "code",
   "execution_count": 10,
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     "text": [
      "Plot the 2D lattice structure\n"
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#結晶構造を見る\n",
    "plot_lattice_2d(la)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#バンド構造を見る\n",
    "        klines = set_Klines()\n",
    "        kmin = [0,0]\n",
    "        kmax = [2π/sqrt(3),0]\n",
    "        add_Kpoints!(klines,kmin,kmax,\"G\",\"K\")\n",
    "\n",
    "        kmin = [2π/sqrt(3),0]\n",
    "        kmax = [2π/sqrt(3),2π/3]\n",
    "        add_Kpoints!(klines,kmin,kmax,\"K\",\"M\")\n",
    "\n",
    "        kmin = [2π/sqrt(3),2π/3]\n",
    "        kmax = [0,0]\n",
    "        add_Kpoints!(klines,kmin,kmax,\"M\",\"G\")\n",
    "calc_band_plot(klines,la)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# グラフェンナノリボン\n",
    "周期境界条件と端あり境界条件の場合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{TightBinding.Kpoints,1}:\n",
       " TightBinding.Kpoints([-3.14159], Irrational{:π}[π = 3.1415926535897...], 20, \"-pi\", \"pi\")"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#すでに原子とホッピングは設定済み。\n",
    "#プロットにつかうk点のlineを設定\n",
    "klines = set_Klines()\n",
    "kmin = [-π]\n",
    "kmax = [π]\n",
    "add_Kpoints!(klines,kmin,kmax,\"-pi\",\"pi\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "\n",
    "#周期境界条件を導入する方向\n",
    "direction = 1\n",
    "#周期境界条件\n",
    "TightBinding.calc_band_plot_finite(klines,la,direction,periodic=true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#基本並進ベクトルに垂直な壁を導入することができる。\n",
    "direction = 1\n",
    "#Open boundary condition\n",
    "TightBinding.calc_band_plot_finite(klines,la,direction,periodic=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 鉄系超伝導体の模型 2軌道模型\n",
    "2軌道模型[S. Rachu et al. Phys. Rev. B 77, 220503(R) (2008)].を作ってみよう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling TightBinding [4482633c-c459-11e8-019e-737cadda7745]\n",
      "└ @ Base loading.jl:1186\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.45"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using TightBinding\n",
    "la = set_Lattice(2,[[1,0],[0,1]]) #正方格子\n",
    "#2軌道模型を考える。\n",
    "add_atoms!(la,[0,0]) #dxz 軌道\n",
    "add_atoms!(la,[0,0]) #dyz 軌道\n",
    "#ホッピング\n",
    "t1 = -1.0\n",
    "t2 = 1.3\n",
    "t3 = -0.85\n",
    "t4 = t3\n",
    "μ = 1.45\n",
    "\n",
    "#dxz\n",
    "add_hoppings!(la,-t1,1,1,[1,0])\n",
    "add_hoppings!(la,-t2,1,1,[0,1])\n",
    "add_hoppings!(la,-t3,1,1,[1,1])\n",
    "add_hoppings!(la,-t3,1,1,[1,-1])\n",
    "\n",
    "#dyz\n",
    "add_hoppings!(la,-t2,2,2,[1,0])\n",
    "add_hoppings!(la,-t1,2,2,[0,1])\n",
    "add_hoppings!(la,-t3,2,2,[1,1])\n",
    "add_hoppings!(la,-t3,2,2,[1,-1])\n",
    "\n",
    "#between dxz and dyz\n",
    "add_hoppings!(la,-t4,1,2,[1,1])\n",
    "add_hoppings!(la,-t4,1,2,[-1,-1])\n",
    "add_hoppings!(la,t4,1,2,[1,-1])\n",
    "add_hoppings!(la,t4,1,2,[-1,1])\n",
    "\n",
    "#Chemical potentials\n",
    "set_μ!(la,μ) #set the chemical potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#バンド構造を見る。\n",
    "klines = set_Klines()\n",
    "kmin = [0,0]\n",
    "kmax = [π,0]\n",
    "add_Kpoints!(klines,kmin,kmax,\"(0,0)\",\"(pi,0)\")\n",
    "\n",
    "kmin = [π,0]\n",
    "kmax = [π,π]\n",
    "add_Kpoints!(klines,kmin,kmax,\"(pi,0)\",\"(pi,pi)\")\n",
    "\n",
    "kmin = [π,π]\n",
    "kmax = [0,0]\n",
    "add_Kpoints!(klines,kmin,kmax,\"(pi,pi)\",\"(0,0)\")\n",
    "\n",
    "pls = calc_band_plot(klines,la)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.30741 -0.067435; -0.067435 1.2387]\n"
     ]
    }
   ],
   "source": [
    "\n",
    "ham = hamiltonian_k(la) #ハミルトニアンは \"ham([kx,ky])\"で呼び出せる\n",
    "kx = 0.1\n",
    "ky = 0.2\n",
    "hamk = ham([kx,ky]) #ham はk=[kx,ky]の関数\n",
    "println(hamk)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 鉄系超伝導体の模型 5軌道模型\n",
    "K.Kurokiらが提案した鉄系超伝導体の5軌道模型[K. Kuroki et al., Phys. Rev. Lett. 101, 087004 (2008)]を作ってみよう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.96"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using TightBinding\n",
    "la = set_Lattice(2,[[1,0],[0,1]])\n",
    "add_atoms!(la,[0,0])\n",
    "add_atoms!(la,[0,0])\n",
    "add_atoms!(la,[0,0])\n",
    "add_atoms!(la,[0,0])\n",
    "add_atoms!(la,[0,0])\n",
    "\n",
    "tmat = [\n",
    "-0.7    0 -0.4  0.2 -0.1\n",
    "-0.8    0    0    0    0\n",
    " 0.8 -1.5    0    0 -0.3\n",
    "   0  1.7    0    0 -0.1\n",
    "-3.0    0    0 -0.2    0\n",
    "-2.1  1.5    0    0    0\n",
    " 1.3    0  0.2 -0.2    0\n",
    " 1.7    0    0  0.2    0\n",
    "-2.5  1.4    0    0    0\n",
    "-2.1  3.3    0 -0.3  0.7\n",
    " 1.7  0.2    0  0.2    0\n",
    " 2.5    0    0  0.3    0\n",
    " 1.6  1.2 -0.3 -0.3 -0.3\n",
    "   0    0    0 -0.1    0\n",
    " 3.1 -0.7 -0.2    0    0\n",
    "]\n",
    "tmat = 0.1.*tmat\n",
    "imap = zeros(Int64,5,5)\n",
    "count = 0\n",
    "for μ=1:5\n",
    "    for ν=μ:5\n",
    "        count += 1\n",
    "        imap[μ,ν] = count\n",
    "    end\n",
    "end\n",
    "Is = [1,-1,-1,1,1,1,1,-1,-1,1,-1,-1,1,1,1]\n",
    "σds = [1,-1,1,1,-1,1,-1,-1,1,1,1,-1,1,-1,1]\n",
    "tmat_σy = tmat[:,:]\n",
    "tmat_σy[imap[1,2],:] = -tmat[imap[1,3],:]\n",
    "tmat_σy[imap[1,3],:] = -tmat[imap[1,2],:]\n",
    "tmat_σy[imap[1,4],:] = -tmat[imap[1,4],:]\n",
    "tmat_σy[imap[2,2],:] = tmat[imap[3,3],:]\n",
    "tmat_σy[imap[2,4],:] = tmat[imap[3,4],:]\n",
    "tmat_σy[imap[2,5],:] = -tmat[imap[3,5],:]\n",
    "tmat_σy[imap[3,3],:] = tmat[imap[2,2],:]\n",
    "tmat_σy[imap[3,4],:] = tmat[imap[2,4],:]\n",
    "tmat_σy[imap[3,5],:] = -tmat[imap[2,5],:]\n",
    "tmat_σy[imap[4,5],:] = -tmat[imap[4,5],:]\n",
    "\n",
    "hoppingmatrix = zeros(Float64,5,5,5,5)\n",
    "hops = [-2,-1,0,1,2]\n",
    "hopelements = [[1,0],[1,1],[2,0],[2,1],[2,2]]\n",
    "\n",
    "for μ = 1:5\n",
    "    for ν=μ:5\n",
    "        for ii=1:5\n",
    "            ihop = hopelements[ii][1]\n",
    "            jhop = hopelements[ii][2]\n",
    "            #[a,b],[a,-b],[-a,-b],[-a,b],[b,a],[b,-a],[-b,a],[-b,-a]\n",
    "\n",
    "            #[a,b]\n",
    "            i = ihop +3\n",
    "            j = jhop +3\n",
    "            hoppingmatrix[μ,ν,i,j]=tmat[imap[μ,ν],ii]\n",
    "            #[a,-b] = σy*[a,b] [1,1] -> [1,-1]\n",
    "            if jhop != 0\n",
    "                i = ihop +3\n",
    "                j = -jhop +3\n",
    "                hoppingmatrix[μ,ν,i,j]=tmat_σy[imap[μ,ν],ii]\n",
    "            end\n",
    "\n",
    "            if μ != ν\n",
    "                #[-a,-b] = I*[a,b] [1,1] -> [-1,-1],[1,0]->[-1,0]\n",
    "                i = -ihop +3\n",
    "                j = -jhop +3\n",
    "                hoppingmatrix[μ,ν,i,j]=Is[imap[μ,ν]]*tmat[imap[μ,ν],ii]\n",
    "                #[-a,b] = I*[a,-b] = I*σy*[a,b]  #[2,0]->[-2,0]\n",
    "                if jhop != 0\n",
    "                    i = -ihop +3\n",
    "                    j = jhop +3\n",
    "                    hoppingmatrix[μ,ν,i,j]=Is[imap[μ,ν]]*tmat_σy[imap[μ,ν],ii]\n",
    "                end\n",
    "            end\n",
    "            #[b,a],[b,-a],[-b,a],[-b,-a]\n",
    "            if jhop != ihop\n",
    "                #[b,a] = σd*[a,b]\n",
    "                i = jhop +3\n",
    "                j = ihop +3\n",
    "                hoppingmatrix[μ,ν,i,j]=σds[imap[μ,ν]]*tmat[imap[μ,ν],ii]\n",
    "                #[-b,a] = σd*σy*[a,b]\n",
    "                if jhop != 0\n",
    "                    i = -jhop +3\n",
    "                    j = ihop +3\n",
    "                    hoppingmatrix[μ,ν,i,j]=σds[imap[μ,ν]]*tmat_σy[imap[μ,ν],ii]\n",
    "                end\n",
    "\n",
    "                if μ != ν\n",
    "                    #[-b,-a] = σd*[-a,-b] = σd*I*[a,b]\n",
    "                    i = -jhop +3\n",
    "                    j = -ihop +3\n",
    "                    hoppingmatrix[μ,ν,i,j]=σds[imap[μ,ν]]*Is[imap[μ,ν]]*tmat[imap[μ,ν],ii]\n",
    "                    #[b,-a] = σd*[-a,b] = σd*I*[a,-b] = σd*I*σy*[a,b]  #[2,0]->[-2,0]\n",
    "                    if jhop != 0\n",
    "                        i = jhop +3\n",
    "                        j = -ihop +3\n",
    "                        hoppingmatrix[μ,ν,i,j]=σds[imap[μ,ν]]*Is[imap[μ,ν]]*tmat_σy[imap[μ,ν],ii]\n",
    "                    end\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "\n",
    "\n",
    "    end\n",
    "end\n",
    "\n",
    "for μ=1:5\n",
    "    for ν=μ:5\n",
    "        for i = 1:5\n",
    "            ih = hops[i]\n",
    "            for j = 1:5\n",
    "                jh = hops[j]\n",
    "                if hoppingmatrix[μ,ν,i,j] != 0.0                \n",
    "                    add_hoppings!(la,hoppingmatrix[μ,ν,i,j],μ,ν,[ih,jh])\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "end\n",
    "\n",
    "onsite = [10.75,10.96,10.96,11.12,10.62]\n",
    "set_onsite!(la,onsite)\n",
    "\n",
    "set_μ!(la,10.96) #set the chemical potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "#バンド図をプロットしよう。\n",
    "nk = 100\n",
    "klines = set_Klines()\n",
    "kmin = [0,0]\n",
    "kmax = [π,0]\n",
    "add_Kpoints!(klines,kmin,kmax,\"(0,0)\",\"(pi,0)\",nk=nk)\n",
    "\n",
    "kmin = [π,0]\n",
    "kmax = [π,π]\n",
    "add_Kpoints!(klines,kmin,kmax,\"(pi,0)\",\"(pi,pi)\",nk=nk)\n",
    "\n",
    "kmin = [π,π]\n",
    "kmax = [0,0]\n",
    "add_Kpoints!(klines,kmin,kmax,\"(pi,pi)\",\"(0,0)\",nk=nk)\n",
    "\n",
    "pls = calc_band_plot(klines,la)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "この図は[T. Nomura, J. Phys. Soc. Jpn. 78, 034716 (2009)]の図2と同じになっている。\n",
    "\n",
    "フェルミ面は"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "pls = plot_fermisurface_2D(la)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 波数表示のハミルトニアンから表面ハミルトニアンを構築する"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ハミルトニアン$H(k)$が与えられているときに、そのハミルトニアンを使った場合に表面に束縛状態が現れるかどうかをみてみる。\n",
    "例として、\n",
    "\n",
    "$$\n",
    "H(k) = A_x \\sin k_x \\sigma_x +  A_y \\sin k_y \\sigma_y + m(k) \\sigma_z\n",
    "$$\n",
    "$$\n",
    "m(k) = m_0 + 2m_{2x} (1-\\cos k_x)+2 m_{2y} (1-\\cos k_y)\n",
    "$$\n",
    "というトポロジカル絶縁体の模型を考える。ここで、パラメータを\n",
    "$$\n",
    "A_x = 1 \\\\\n",
    "A_y = 1 \\\\\n",
    "m_{2x} = 1 \\\\\n",
    "m_{2y} = m_{2x} \\\\\n",
    "m_0 = -2 m_{2x} \n",
    "$$\n",
    "とする。\n",
    "\n",
    "これを\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hk (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using TightBinding\n",
    "Ax = 1\n",
    "Ay = 1\n",
    "m2x = 1\n",
    "m2y = m2x\n",
    "m0 = -2*m2x\n",
    "m(k) = m0 + 2m2x*(1-cos(k[1]))+2m2y*(1-cos(k[2]))\n",
    "Hk(k) = Ax*sin(k[1]).*σx +  Ay*sin(k[2]).*σy + m(k).*σz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "として定義しておく。ここで、パウリ行列σx,σy,σz,σ0はTightBindingを読み込むと定義される。\n",
    "\n",
    "あとは、"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#2 (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norb = 2 # 軌道の数\n",
    "hamiltonian = surfaceHamiltonian(Hk,norb,numhop=3,L=32,kpara=\"kx\",BC=\"OBC\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "とすれば表面ハミルトニアンの出来上がりである。\n",
    "\n",
    "バンド図は"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "using LinearAlgebra\n",
    "nkx = 100\n",
    "kxs = range(-π,stop=π ,length=nkx)\n",
    "mat_e = zeros(Float64,nkx,32*2)\n",
    "for i=1:nkx\n",
    "    kx = kxs[i]\n",
    "    mat_h = hamiltonian(kx)\n",
    "    #println(mat_h)\n",
    "    \n",
    "    e,v = eigen(Matrix(mat_h))\n",
    "    #println(e)\n",
    "    mat_e[i,:] = real.(e[:])\n",
    "end\n",
    "plot(kxs,mat_e,labels=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "壁がない場合のバンド図は、PBCをセットすればよく"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#2 (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norb = 2 # 軌道の数\n",
    "hamiltonian_PBC = surfaceHamiltonian(Hk,norb,numhop=3,L=32,kpara=\"kx\",BC=\"PBC\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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       "  \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "using LinearAlgebra\n",
    "nkx = 100\n",
    "kxs = range(-π,stop=π ,length=nkx)\n",
    "mat_e = zeros(Float64,nkx,32*2)\n",
    "for i=1:nkx\n",
    "    kx = kxs[i]\n",
    "    mat_h = hamiltonian_PBC(kx)\n",
    "    #println(mat_h)\n",
    "    \n",
    "    e,v = eigen(Matrix(mat_h))\n",
    "    #println(e)\n",
    "    mat_e[i,:] = real.(e[:])\n",
    "end\n",
    "plot(kxs,mat_e,labels=\"\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "とすればよい。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.1.0"
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 "nbformat_minor": 2
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