Ising/MC_and_Ising.ipynb

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   "source": [
    "# Monte Carlo methods and Application to Ising Model"
   ]
  },
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   "source": [
    "                Yingjie Wang\n",
    "\n",
    "                Apr. 19,  2022"
   ]
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   "source": [
    "## 1. Introduction to the Monte Carlo Method"
   ]
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   "source": [
    "### 1.1 Background"
   ]
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   "source": [
    "#### What's the problem?"
   ]
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   "source": [
    "- In canonical ensemble, everything can be calculated from:\n",
    "\t$$Z = \\operatorname{tr}\\left(\\mathrm{e}^{- \\beta \\hat{H}}\\right) = \\sum_{s} \\mathrm{e}^{- \\beta E_s},$$"
   ]
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   "source": [
    "- but it's hard to calculate in general!"
   ]
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   "cell_type": "markdown",
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   "source": [
    "- You may say that, OK let's do that numerically, but running over the states is still hard"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "#### What's Monte Carlo Method?"
   ]
  },
  {
   "cell_type": "markdown",
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    "tags": []
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   "source": [
    "It's hard to define generally..."
   ]
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   "source": [
    "Here we refer to the *Metropolis Monte Carlo Algorithm*, which will be defined later."
   ]
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   "cell_type": "markdown",
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   "source": [
    "#### How Monte Carlo Solve the problem?"
   ]
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    "- Every quantity can be calculated by\n",
    "\t$$\\left\\langle A\\right\\rangle = \\sum_s A(s) \\frac{\\mathrm{e}^{-\\beta E_s}}{Z},$$\n",
    "\tso we can randomly select states according to the probability\n",
    "\t$$p(s) = \\frac{\\mathrm{e}^{-\\beta E_s}}{Z}$$\n",
    "\tand calculate the average on the sample states, instead of run over all states"
   ]
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   "source": [
    "- The point is: overall constant factor is not matter in the Metropolis algorithm!\n",
    "- Knowing that\n",
    "\t$$p(s) \\propto \\mathrm{e}^{- \\beta E_s}$$\n",
    "\tis enough. Good News!"
   ]
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   "source": [
    "### 1.2 The Metropolis algorithm"
   ]
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   "source": [
    "Suppose that we want a dataset $\\{ x_i \\}$ that match some probability distribution $p(x)$."
   ]
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   "source": [
    "1. Start from any initial value $x_0$;"
   ]
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   "source": [
    "2. Do the following loop:\n",
    "\t1. suppose we already have $x_i$. Move to a test value $x^{\\prime}_{i+1}$ for $x_{i+1}$ randomly from $x_i$ according to any symmetric distribution $T(x_{i+1},x_i)$ (but must make sure that $x^{\\prime}_{i+1} \\neq x_i$);\n",
    "\t2. accept the change according to the probability $A(x^\\prime_j,x_i) = \\min(1, p(x^\\prime_j)/p(x_i))$. If we accept the move, then $x_{i_1} = x^\\prime_{i+1}$; else we throw $x^\\prime_j$ and take $x_{i+1} = x_i$.\n",
    "   \t**Note that the algorithm only depends on** $p(y)/p(x)$**, any overall factor doesn't matter!**"
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   "source": [
    "3. After many loops, the distribution of $\\{ x_i \\}$ will converge to $\\propto p(x)$."
   ]
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   "source": [
    "4. Then throw all $x_i$ generated before, just leave the last one, then do more loops to collect enough samples."
   ]
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   "source": [
    "#### Example: coding time"
   ]
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   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "rng = np.random.default_rng()"
   ]
  },
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   "source": [
    "def p(x):\n",
    "\treturn np.exp(-x*x/2)\n",
    "def metro_norm(stepnum: int, dx: float) -> np.array:\n",
    "\tx = np.empty(stepnum)\n",
    "\tx[0] = -10\n",
    "\tfor i in range(stepnum-1):\n",
    "\t\ty = x[i] + (rng.random()-0.5)*dx\n",
    "\t\tif rng.random() < p(y)/p(x[i]):\n",
    "\t\t\tx[i+1] = y\n",
    "\t\telse:\n",
    "\t\t\tx[i+1] = x[i]\n",
    "\treturn x\n",
    "xlist = metro_norm(10000,0.5)"
   ]
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   "source": [
    "# plot\n",
    "plt.subplot(2, 2, 1) \n",
    "plt.scatter(range(10000),xlist,s=1)\n",
    "plt.subplot(2, 2, 2)\n",
    "plt.hist(xlist[3000:],density=True)\n",
    "plt.plot(np.arange(-3,3,step=0.1),p(np.arange(-3,3,step=0.1))/np.sqrt(2*np.pi))\n",
    "%config InlineBackend.figure_format='svg'\n",
    "plt.show()"
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   "source": [
    "only 10000 run in $0.6 s$, but still not bad, right?"
   ]
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   "source": [
    "## 2 Metropolis algorithms for statistical mechanics"
   ]
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   "metadata": {
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   "source": [
    "### 2.1 In canonical ensemble"
   ]
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   "cell_type": "markdown",
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   "source": [
    "$x_i \\rightarrow s_i, \\ p(x_i) \\rightarrow p(s_i) = \\mathrm{e}^{-\\beta E}$"
   ]
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   "cell_type": "markdown",
   "metadata": {
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    },
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   },
   "source": [
    "The accepting probability is now $p(s_j)/p(s_i) = \\mathrm{e}^{- \\beta \\Delta E}$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "- Everything is almost the same, do loops to generate a list of states."
   ]
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   "source": [
    "- Calculate $\\left< A \\right>$ on the sample list"
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   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   "source": [
    "#### Toy model: Ideal gas"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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    "tags": []
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   "source": [
    "- Suppose that we have $N$ particles in $V=L^3$ box."
   ]
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  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
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   "source": [
    "- For each particle, the energy eigenstates are $\\left| \\vec{n} \\right> = \\left| n_1 \\right>\\left| n_2 \\right>\\left| n_3 \\right>$ with the eigenvalue $E_{\\vec{n}} = \\frac{\\pi^2 \\hbar^2}{2mL^2} \\left( n_1^2 + n_2^2 + n_3^2 \\right)$."
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   "metadata": {
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   "source": [
    "- Take $\\hbar = m = L = k = 1$ for convenient."
   ]
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   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "- The steps:\n",
    "  - We create a list `n` with shape $N \\times 3$ to present the state.\n",
    "  - Randomly choose one particle $i \\in [0,N-1]$ and one direction $j \\in [0,2]$, then randomly add $\\pm 1$ to `n[i,j]`\n",
    "  - The change of energy is $\\Delta E = \\frac{\\pi^2}{2} (\\pm 2 n_{i,j} + 1)$\n",
    "  - If `rand()`$< \\mathrm{e}^{- E/T}$, then accept the move. Otherwise, keep `n[i,j]` unchanged.\n",
    "  - Let the computer repeat!"
   ]
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   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "subslide"
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   "source": [
    "#### The code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
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     "start_time": "2022-04-19T14:21:44.162791Z"
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   "outputs": [],
   "source": [
    "from rich.progress import track\n",
    "def elist(T, N=1000, movenum = 2000000):\n",
    "\tenergy = np.pi*np.pi*3*N/2\n",
    "\tnlist = np.ones((N,3))\n",
    "\tenergylist = np.empty(movenum)\n",
    "\tfor move in track(range(movenum)):\n",
    "\t\ti,j = rng.integers(N),rng.integers(3)\n",
    "\t\tx = rng.integers(2)\n",
    "\t\tif x == 1 or nlist[i,j] > 1:\n",
    "\t\t\tdE = (2*nlist[i,j]*(2*x-1) + 1)*np.pi*np.pi/2\n",
    "\t\t\tif rng.random() < np.exp(-dE/T):\n",
    "\t\t\t\tnlist[i,j] += 2*x - 1\n",
    "\t\t\t\tenergy += dE\n",
    "\t\tenergylist[move] = energy\n",
    "\treturn energylist/N"
   ]
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   "source": [
    "data = elist(1000)\n",
    "plt.plot(data)\n",
    "plt.show()"
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   "source": [
    "##### Vectorized code"
   ]
  },
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   "source": [
    "def e_vec(T, N=1000, movenum = 1000000):\n",
    "\tNT = T.size\n",
    "\tenergy = np.pi*np.pi*3*N/2\n",
    "\tnlist = np.ones((N,3,NT))\n",
    "\tenergylist = np.empty((movenum, NT))\n",
    "\tfor move in track(range(movenum)):\n",
    "\t\ti,j = rng.integers(N),rng.integers(3)\n",
    "\t\tx = rng.integers(2)\n",
    "\t\tvalidlist = ([x == 1] | (nlist[i,j] > 1))\n",
    "\t\tdE = (2*nlist[i,j]*(2*x-1) + 1)*np.pi*np.pi/2\n",
    "\t\tboollist = rng.random() < np.exp(-dE/T)\n",
    "\t\tnlist[i,j] += (2*x - 1)*validlist*boollist\n",
    "\t\tenergy += dE*validlist*boollist\n",
    "\t\tenergylist[move] = energy\n",
    "\telist = np.mean(energylist[movenum//2:],axis=0)\n",
    "\treturn elist/N"
   ]
  },
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   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
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       "<Figure size 640x480 with 1 Axes>"
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   ],
   "source": [
    "tlist = np.arange(100,1000)\n",
    "elist = e_vec(tlist)\n",
    "plt.scatter(tlist,elist, s=1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When $T$ is a large number (prepare to $\\frac{\\hbar^2}{kmL^2}$), $e(T)$ recover the straight line $e = \\frac{3}{2} T$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## 3 Ising model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's apply the MC method to a specific system. Here I take Ising model for example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "#### What's it for?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Ising model is used to explain ferromagnet.\n",
    "\n",
    "![menu](img/mag.png \"Why magnet \\\"store\\\" magnetic field?\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "Questions:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Why $M \\neq 0$ when $\\vec{B} = 0$ and $T$ is lower than some value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Why there's a critical temperature"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "#### What's Ising model's answer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "In most ordinary materials the magnetic dipoles of the atoms have a random orientation. In effect this non-specific distribution results in no overall macroscopic magnetic moment. However in certain cases, such as iron, a magnetic moment is produced as a result of a preferred alignment of the atomic spins."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Ising model suppose that there's a short-range interaction term in the Hamiltonian:\n",
    "$$H = - J \\sum_{<i,j>} \\sigma_{i} \\sigma_j - B \\sum_i \\sigma_i, \\quad J>0,$$\n",
    "where $\\sum_{<i,j>}$ means that sum over nearby spin pairs, and $\\sigma$ is the $z$-direction spin operator."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But since $\\sigma_i$ commute with each other, the Hamiltonian just acts as number. We will treat this model as a \"classical\" system and treat $\\sigma_i$ as a variable taking it's value of $\\pm 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "The existing of $J$ term can explain the difference between ferromagnetism behavior and paramagnetism behavior."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### 3.1 Ising model on 2D finite square lattice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "We introduce the periodic boundary condition on a $N \\times N$ lattice, so that each spin always interacts with 4 spins."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Thus the model is defined on a torus.\n",
    "\n",
    "![torus](img/torus.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "We'll also assuming that\n",
    "$ B = 0, \\quad J=1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "#### 3.1.1 the Analytical Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just post to show it...\n",
    "\n",
    "I get the partition function using mathematica for $N=8$ lattice:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "$$ Z(\\beta) =  \\mathrm{e}^{2N^2 \\beta} \\tilde{Z}(\\mathrm{e}^{-2 \\beta}),$$\n",
    "where $\\tilde{Z}(x)$ is a polynomial:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "$$ \\tilde{Z}(x) = 2 x^{128}+128 x^{124}+256 x^{122}+4672 x^{120}+17920 x^{118}+145408 x^{116}+712960 x^{114}+4274576 x^{112}+22128384 x^{110}+118551552 x^{108}+610683392 x^{106}+3150447680 x^{104}+16043381504 x^{102}+80748258688 x^{100}+396915938304 x^{98}+1887270677624 x^{96}+8582140066816 x^{94}+36967268348032 x^{92}+149536933509376 x^{90}+564033837424064 x^{88}+1971511029384704 x^{86}+6350698012553216 x^{84}+18752030727310592 x^{82}+50483110303426544 x^{80}+123229776338119424 x^{78}+271209458049836032 x^{76}+535138987032308224 x^{74}+941564975390477248 x^{72}+1469940812209435392 x^{70}+2027486077172296064 x^{68}+2462494093546483712 x^{66}+2627978003957146636 x^{64}+2462494093546483712 x^{62}+2027486077172296064 x^{60}+1469940812209435392 x^{58}+941564975390477248 x^{56}+535138987032308224 x^{54}+271209458049836032 x^{52}+123229776338119424 x^{50}+50483110303426544 x^{48}+18752030727310592 x^{46}+6350698012553216 x^{44}+1971511029384704 x^{42}+564033837424064 x^{40}+149536933509376 x^{38}+36967268348032 x^{36}+8582140066816 x^{34}+1887270677624 x^{32}+396915938304 x^{30}+80748258688 x^{28}+16043381504 x^{26}+3150447680 x^{24}+610683392 x^{22}+118551552 x^{20}+22128384 x^{18}+4274576 x^{16}+712960 x^{14}+145408 x^{12}+17920 x^{10}+4672 x^8+256 x^6+128 x^4+2 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "OK, I think you already don't want to see any more equation like this, let's move to the MC method..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "#### 3.1.2 Python Time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "- Use a table `s[i,j]` to save spins;\n",
    "- Set any initial state;\n",
    "- Randomly choose `i,j` to flip the spin;\n",
    "- $\\Delta E = -J(\\text{nearby sum}) \\times \\Delta s_{ij}  = 2J(\\text{nearby sum})*s_{ij}$;\n",
    "- When `random()` ${}< \\mathrm{e}^{-\\Delta E/T}$, accept the flip;\n",
    "- Let the computer repeat!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "##### Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-19T14:30:13.513885Z",
     "start_time": "2022-04-19T14:30:13.507179Z"
    }
   },
   "outputs": [],
   "source": [
    "def gqlist(N):\n",
    "\tif N==2:\n",
    "\t\treturn [2, 0, 12, 0, 2]\n",
    "\telif N== 4:\n",
    "\t\treturn [2,0,32,64,424,1728,6688,13568,20524,13568,6688,1728,424,64,32,0,2]\n",
    "\telif N== 8:\n",
    "\t\treturn [2, 0, 128, 256, 4672, 17920, 145408, 712960, 4274576, 22128384, 118551552, 610683392, 3150447680, 16043381504, 80748258688, 396915938304, 1887270677624, 8582140066816, 36967268348032, 149536933509376, 564033837424064, 1971511029384704, 6350698012553216, 18752030727310592, 50483110303426544, 123229776338119424, 271209458049836032, 535138987032308224, 941564975390477248, 1469940812209435392, 2027486077172296064, 2462494093546483712, 2627978003957146636, 2462494093546483712, 2027486077172296064, 1469940812209435392, 941564975390477248, 535138987032308224, 271209458049836032, 123229776338119424, 50483110303426544, 18752030727310592, 6350698012553216, 1971511029384704, 564033837424064, 149536933509376, 36967268348032, 8582140066816, 1887270677624, 396915938304, 80748258688, 16043381504, 3150447680, 610683392, 118551552, 22128384, 4274576, 712960, 145408, 17920, 4672, 256, 128, 0, 2]\n",
    "def E_theory(T,N=16):\n",
    "\tx = np.exp(-2/T)\n",
    "\tZtitle = 0\n",
    "\tglist = gqlist(N)\n",
    "\tfor i in range(N*N+1):\n",
    "\t\tZtitle += glist[i]*np.power(x,2*i)\n",
    "\tthE = 0\n",
    "\tfor i in range(N*N+1):\n",
    "\t\tthE += i*(glist[i]*np.power(x,2*i)/(Ztitle*N*N))\n",
    "\tthE *= 4\n",
    "\tthE -= 2\n",
    "\treturn thE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "from rich.progress import track\n",
    "from rich.console import Console\n",
    "from rich.table import Table\n",
    "def Ising2D_E(T,moves,N=16):\n",
    "\tJ = 1\n",
    "\tB = 0\n",
    "\n",
    "\tslist = np.ones((N,N), dtype=int)\n",
    "\tenergylist = np.empty(moves)\n",
    "\tenergy = - 2*J * N * N\n",
    "\tacp = 0\n",
    "\n",
    "\tfor step in track(range(moves)):\n",
    "\t\t[i,j] = rng.integers(N,size=2)\n",
    "\t\tdE = 2*J*(slist[(i-1)%N,j] + slist[(i+1)%N,j] + slist[i,(j-1)%N] + slist[i,(j+1)%N])*slist[i,j]\n",
    "\t\tif rng.random() < np.exp(-dE/T):\n",
    "\t\t\tacp += 1\n",
    "\t\t\tslist[i,j] *= -1\n",
    "\t\t\tenergy += dE\n",
    "\t\tenergylist[step] = energy\n",
    "\teth = E_theory(N=N,T=T)\n",
    "\temc = np.mean(energylist[moves//2:])/(N*N)\n",
    "\n",
    "\ttable = Table(title=\"energy of 2D Ising\")\n",
    "\ttable.add_column(\"T\", style = \"green\")\n",
    "\ttable.add_column(\"theoretical\", style=\"cyan\")\n",
    "\ttable.add_column(\"Monte Calor\", style=\"magenta\")\n",
    "\ttable.add_column(\"persentage error\", style = \"blue\")\n",
    "\ttable.add_row(str(T),str(eth), str(emc), str(100*(emc-eth)/eth)+\"%\")\n",
    "\n",
    "\tconsole = Console()\n",
    "\tconsole.print(table)\n",
    "\treturn energylist/(N*N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
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       "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\"><span style=\"font-style: italic\">                       energy of 2D Ising                       </span>\n",
       "┏━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┓\n",
       "┃<span style=\"font-weight: bold\"> T </span>┃<span style=\"font-weight: bold\"> theoretical         </span>┃<span style=\"font-weight: bold\"> Monte Calor </span>┃<span style=\"font-weight: bold\"> persentage error     </span>┃\n",
       "┡━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━┩\n",
       "│<span style=\"color: #008000; text-decoration-color: #008000\"> 5 </span>│<span style=\"color: #008080; text-decoration-color: #008080\"> -0.4283680589440815 </span>│<span style=\"color: #800080; text-decoration-color: #800080\"> -0.41901625 </span>│<span style=\"color: #000080; text-decoration-color: #000080\"> -2.1831247098893214% </span>│\n",
       "└───┴─────────────────────┴─────────────┴──────────────────────┘\n",
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       "┡━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━┩\n",
       "│\u001b[32m \u001b[0m\u001b[32m5\u001b[0m\u001b[32m \u001b[0m│\u001b[36m \u001b[0m\u001b[36m-0.4283680589440815\u001b[0m\u001b[36m \u001b[0m│\u001b[35m \u001b[0m\u001b[35m-0.41901625\u001b[0m\u001b[35m \u001b[0m│\u001b[34m \u001b[0m\u001b[34m-2.1831247098893214%\u001b[0m\u001b[34m \u001b[0m│\n",
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   "source": [
    "elist = Ising2D_E(5, 100000, N=8)\n",
    "plt.plot(elist)\n",
    "%config InlineBackend.figure_format='svg'\n",
    "plt.show()"
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   "source": [
    "##### (1) Energy"
   ]
  },
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     "end_time": "2022-04-19T14:29:47.508655Z",
     "start_time": "2022-04-19T14:29:47.292419Z"
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   "outputs": [],
   "source": [
    "from scipy.stats import chi2,halfnorm,t\n",
    "\n",
    "def Ising2D_E_vec(T,moves,N=16):\n",
    "\tJ = 1\n",
    "\tB = 0\n",
    "\tNT=T.size\n",
    "\n",
    "\tslist = np.ones((N,N,NT), dtype=int)\n",
    "\tenergylist = np.empty((moves,NT))\n",
    "\tenergy = - 2*J * N * N\n",
    "\n",
    "\tfor step in track(range(moves)):\n",
    "\t\t[i,j] = rng.integers(N,size=2)\n",
    "\t\tdE = 2*J*(slist[(i-1)%N,j] + slist[(i+1)%N,j] + slist[i,(j-1)%N] + slist[i,(j+1)%N])*slist[i,j]\n",
    "\t\tboollist = rng.random() < np.exp(-dE/T)\n",
    "\t\tslist[i,j] *= 1-2*boollist\n",
    "\t\tenergy += dE*boollist\n",
    "\t\tenergylist[step] = energy\n",
    "\tvalidElist = energylist[moves//2:]/(N*N)\n",
    "\temc = np.mean(validElist,axis=0)\n",
    "\tS=np.std(validElist, ddof=1, axis=0)\n",
    "\thalfn = validElist.size\n",
    "\terrbar = S*t.isf(0.05,halfn-1)/np.sqrt(halfn)\n",
    "\treturn emc,errbar"
   ]
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   "source": [
    "N = 8"
   ]
  },
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   "execution_count": 11,
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     "end_time": "2022-04-19T14:30:32.853919Z",
     "start_time": "2022-04-19T14:30:24.259306Z"
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       "Output()"
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   "source": [
    "T=np.arange(0.5, 5, step=0.1)\n",
    "N=8\n",
    "elist, errbar = Ising2D_E_vec(T, 200000, N=N)\n",
    "if N in [2,4,8]:\n",
    "\tthElist = E_theory(T,N=N)\n",
    "\tplt.plot(T,thElist)\n",
    "plt.errorbar(T, elist, yerr=errbar, fmt=\"None\")\n",
    "plt.scatter(T,elist, marker=\"x\", c=\"r\", zorder=2.5)\n",
    "plt.title(\"E/N^2 vs T\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "##### (2) Magnetization $M$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ M = \\left< \\sum_i \\sigma_i \\right> $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-19T14:31:11.370171Z",
     "start_time": "2022-04-19T14:31:11.363989Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "from scipy.stats import chi2,halfnorm,t\n",
    "def Ising2D_M_vec(T,moves=1000000, N=16):\n",
    "\tJ = 1\n",
    "\tB = 0\n",
    "\tNT=T.size\n",
    "\n",
    "\tslist = np.ones((N,N,NT), dtype=int)\n",
    "\tMlist = np.empty((moves, NT))\n",
    "\tM = N*N\n",
    "\n",
    "\tfor step in track(range(moves)):\n",
    "\t\t[i,j] = rng.integers(N,size=2)\n",
    "\t\tdE = 2*J*(slist[(i-1)%N,j] + slist[(i+1)%N,j] + slist[i,(j-1)%N] + slist[i,(j+1)%N])*slist[i,j]\n",
    "\t\tboollist = rng.random() < np.exp(-dE/T)\n",
    "\t\tslist[i,j] *= 1-2*boollist\n",
    "\t\tM += 2*slist[i,j]*boollist\n",
    "\t\tMlist[step] = M\n",
    "\t# eth = -J*(1+2*(2*(np.tanh(2*J/T)**2)-1)*ellipk(2*np.tanh(2*J/T)/np.cosh(2*J/T)))/np.tanh(2*J/T)\n",
    "\tvalidMlist = Mlist[moves//2:]/(N*N)\n",
    "\tMmc = np.mean(validMlist,axis=0)\n",
    "\tS=np.std(validMlist, ddof=1, axis=0)\n",
    "\thalfn = validMlist.size\n",
    "\terrbar = S*t.isf(0.00135,halfn-1)/np.sqrt(halfn)\n",
    "\treturn Mmc,errbar"
   ]
  },
  {
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   "execution_count": 17,
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     "end_time": "2022-04-19T14:31:44.065707Z",
     "start_time": "2022-04-19T14:31:14.334542Z"
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     "slide_type": "fragment"
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   "source": [
    "T=np.arange(0.5, 5, step=0.1)\n",
    "Tt= np.arange(0.5,5,step = 0.02)\n",
    "N=16\n",
    "Mlist, errbar = Ising2D_M_vec(T, 1000000, N=N)\n",
    "a = 1-np.power(np.sinh(2/Tt),-4)\n",
    "thMlist = np.power(a*(a>0),1/8)\n",
    "plt.plot(Tt,thMlist)\n",
    "plt.errorbar(T, Mlist, yerr=errbar, marker=\"x\")\n",
    "plt.title(\"M/N^2 vs T\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Metropolis MC method works not very well around the critical temp!"
   ]
  },
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   "source": [
    "##### (3) heat capacity $C$"
   ]
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   "source": [
    "$$ C = \\frac{\\partial \\left< E \\right>}{\\partial T} = \\frac{(\\Delta E)^2}{T^2} $$"
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     "slide_type": "fragment"
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   "outputs": [],
   "source": [
    "from scipy.stats import chi2,halfnorm,t\n",
    "def Ising2D_C_vec(T,moves=1000000,N=16):\n",
    "\tJ = 1\n",
    "\tB = 0\n",
    "\tNT=T.size\n",
    "\n",
    "\tslist = np.ones((N,N,NT), dtype=int)\n",
    "\tenergylist = np.empty((moves,NT))\n",
    "\tenergy = - 2*J * N * N\n",
    "\n",
    "\tfor step in track(range(moves)):\n",
    "\t\t[i,j] = rng.integers(N,size=2)\n",
    "\t\tdE = 2*J*(slist[(i-1)%N,j] + slist[(i+1)%N,j] + slist[i,(j-1)%N] + slist[i,(j+1)%N])*slist[i,j]\n",
    "\t\tboollist = rng.random() < np.exp(-dE/T)\n",
    "\t\tslist[i,j] *= 1-2*boollist\n",
    "\t\tenergy += dE*boollist\n",
    "\t\tenergylist[step] = energy\n",
    "\tvalidElist = energylist[moves//2:]/(N*N)\n",
    "\thalfn = validElist.size\n",
    "\tS=np.std(validElist, ddof=1, axis=0)\n",
    "\tcmc = S*S/(T*T)\n",
    "\talpha = 0.01\n",
    "\tc1 =chi2.isf(alpha/2, halfn - 1)\n",
    "\tc2 = chi2.isf(1-alpha/2, halfn - 1)\n",
    "\terrbar = np.array([(1-(halfn - 1)/c1)*cmc,((halfn-1)/c2 - 1)*cmc])\n",
    "\treturn cmc,errbar"
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   "source": [
    "T = np.arange(0.5, 10,step=0.2)\n",
    "C,errbar = Ising2D_C_vec(T)\n",
    "plt.errorbar(T, C, yerr=errbar, marker=\"x\")\n",
    "plt.title(\"C0 vs T\")\n",
    "plt.show()"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "##### (4) susceptibility $\\chi$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\chi=\\frac{\\partial M}{\\partial T}=\\frac{(\\Delta M)^{2}}{T}=\\frac{\\langle M^{2}\\rangle-\\langle M\\rangle^{2}}{T} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
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   },
   "outputs": [],
   "source": [
    "from scipy.stats import chi2\n",
    "def Ising2D_chi_vec(T,moves=1000000,N=16):\n",
    "\tJ = 1\n",
    "\tB = 0\n",
    "\tNT=T.size\n",
    "\n",
    "\tslist = np.ones((N,N,NT), dtype=int)\n",
    "\tMlist = np.empty((moves,NT))\n",
    "\tM = J * N * N\n",
    "\n",
    "\tfor step in track(range(moves)):\n",
    "\t\t[i,j] = rng.integers(N,size=2)\n",
    "\t\tdE = 2*J*(slist[(i-1)%N,j] + slist[(i+1)%N,j] + slist[i,(j-1)%N] + slist[i,(j+1)%N])*slist[i,j]\n",
    "\t\tboollist = rng.random() < np.exp(-dE/T)\n",
    "\t\tslist[i,j] *= 1-2*boollist\n",
    "\t\tM += 2*slist[i,j]*boollist\n",
    "\t\tMlist[step] = M\n",
    "\tvalidMlist = Mlist[moves//2:]/(N*N)\n",
    "\thalfn = validMlist.size\n",
    "\tS=np.std(validMlist, ddof=1, axis=0)\n",
    "\tchimc = S*S/t\n",
    "\talpha = 0.01\n",
    "\tc1 =chi2.isf(alpha/2, halfn - 1)\n",
    "\tc2 = chi2.isf(1-alpha/2, halfn - 1)\n",
    "\terrbar = np.array([(1-(halfn - 1)/c1)*chimc,((halfn-1)/c2 - 1)*chimc])\n",
    "\treturn chimc,errbar"
   ]
  },
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   "execution_count": 67,
   "metadata": {
    "slideshow": {
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   "source": [
    "T = np.arange(0.5, 6,step=0.2)\n",
    "chi,errbar = Ising2D_chi_vec(T)\n",
    "plt.errorbar(T, chi, yerr=errbar, marker=\"x\")\n",
    "plt.title(\"chi vs T\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "#### Run over T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Let $T$ vary slowly from high to low."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ising2D_runover_T.py\n",
    "# too long to show..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "![runT](img/runoverT.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "#### Run over $B$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We relax the $B=0$ condition, and fix $T=1$ (low temperature) --- Just like magnetize an iron block!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-19T15:09:04.866346Z",
     "start_time": "2022-04-19T15:08:58.734467Z"
    },
    "tags": []
   },
   "outputs": [
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       "Output()"
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    }
   ],
   "source": [
    "# run external Ising2D_runover_B.py\n",
    "%config InlineBackend.figure_format='svg'\n",
    "import Ising2D_runover_B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see the path dependence!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Conclusion"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "- The Metropolis MC is very fast"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "- It's also accuracy, except $T \\sim T_c$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "- The code can be easily reuse for another quantity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "- Depends on:\n",
    "  - `rng`, random number generator\n",
    "  - a good step $\\Delta x$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Thanks!"
   ]
  }
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