Expanded the notebooks

DipoleAsymmetricCluster.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -11,7 +11,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -38,6 +38,41 @@
     "from scipy.interpolate import interp1d"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\begin{align}\n",
+    "H_{jk} = &D_{jk}(\\hat{z}\\otimes\\hat{z})-\\frac{D_{jk}}{2}(1+\\eta)(\\hat{x}\\otimes\\hat{x})-\\frac{D_{jk}}{2}(1-\\eta)(\\hat{y}\\otimes\\hat{y}) \\\\\n",
+    "= &D_{jk}(\\hat{z}\\otimes\\hat{z})-\\frac{D_{jk}}{2}(\\mathbb{I}-\\hat{z}\\otimes\\hat{z})+\\frac{D_{jk}}{2}\\eta(\\hat{y}\\otimes\\hat{y}-\\hat{x}\\otimes\\hat{x}) \\\\\n",
+    "= &\\frac{3}{2}D_{jk}(1+\\frac{\\eta}{3})(\\hat{z}\\otimes\\hat{z}) -\\frac{D_{jk}}{2}(1+\\eta)\\mathbb{I} +D\\eta\\hat{y}\\otimes\\hat{y}\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Taking only the secular terms:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\mathbf{S}_j(\\hat{z}\\otimes\\hat{z})\\mathbf{S}_k \\approx &\\alpha_z^2S_j^{x}S_k^{x}+\\beta_z^2S_j^{y}S_k^{y}+\\gamma_z^2S_j^{z}S_k^{z} \\\\\n",
+    "= &\\gamma_z^2 S_j^{z}S_k^{z} + \\frac{\\alpha_z^2}{4}(S_j^{+}+S_j^{-})(S_k^{+}+S_k^{-})-\\frac{\\beta_z^2}{4}(S_j^{+}-S_j^{-})(S_k^{+}-S_k^{-}) \\\\\n",
+    "\\approx & \\gamma_z^2 S_j^{z}S_k^{z}+ \\frac{1}{4}(\\alpha_z^2+\\beta_z^2)(S_j^{+}S_k^{-}+S_j^{-}S_k^{+}) \\\\\n",
+    "= & \\gamma_z^2 S_j^{z}S_k^{z}+ \\frac{1}{2}(1-\\gamma_z^2)(\\mathbf{S}_j\\mathbf{S}_k-S_j^{z}S_k^{z}) \\\\\n",
+    "\\approx & (\\frac{3}{2}\\gamma_z^2-1)S_j^{z}S_k^{z}\n",
+    "\\end{align} \n",
+    "$$\n",
+    "\n",
+    "The last line is possible because we know that $\\mathbf{S}_j\\mathbf{S}_k$ commutes with the operators we care about and so there's no need to retain it."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -66,7 +101,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 99,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -84,7 +119,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 317,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -97,7 +132,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 318,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -118,7 +153,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 319,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -129,7 +164,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 320,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -192,7 +227,7 @@
     "                    continue\n",
     "                Dc = self.Dc[tuple(sorted((i,j)))]\n",
     "                d, eta = Dc[0], Dc[2]\n",
-    "                B2[i,j] = d**2*(9/4*(1+eta/3)**2+eta**2-1.5*(1+eta/3)*eta)\n",
+    "                B2[i,j] = d**2*(9/4+3/4*eta**2)\n",
     "                \n",
     "        return np.sum(I[:,None]*(I[None,:]+1.0)*B2/15.0)/len(I)\n",
     "    \n",
@@ -225,7 +260,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 321,
+   "execution_count": 15,
    "metadata": {
     "scrolled": false
    },
@@ -234,8 +269,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "389.95707534565497\n",
-      "390.0473783142472\n"
+      "448.9132444963204\n",
+      "448.9802129801245\n"
      ]
     }
    ],
@@ -251,16 +286,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 323,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "487.6554808675555\n",
-      "487.6390165519275\n",
-      "487.3474151656362\n"
+      "500.99114755088885\n",
+      "501.0207115070033\n",
+      "499.6272631756669\n"
      ]
     }
    ],