From b51ac1edbdec24fa10028888b14734d2c5d4ed54 Mon Sep 17 00:00:00 2001 From: Bas Nijholt Date: Tue, 2 May 2017 23:20:31 +0200 Subject: [PATCH] Add missing minus sign in equation, closes #192 --- w1_topointro/1D.ipynb | 42 +++++++++++++++++++++++++++++++----------- 1 file changed, 31 insertions(+), 11 deletions(-) diff --git a/w1_topointro/1D.ipynb b/w1_topointro/1D.ipynb index 51c991d..0429b0b 100644 --- a/w1_topointro/1D.ipynb +++ b/w1_topointro/1D.ipynb @@ -3,7 +3,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "import sys\n", @@ -92,7 +94,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "MoocVideo(\"U84MzZm9Gbo\", src_location='1.2-intro')" @@ -182,7 +186,7 @@ "\n", "Starting from this Hamiltonian, the unpaired Majorana regime is the special point $\\Delta=t$ and $\\mu=0$, while the completely trivial regime of isolated fermions is $\\Delta=t=0$ and $\\mu\\neq 0$.\n", "\n", - "As we learned just before, it is useful to write down the above superconducting Hamiltonian in the Bogoliubov-de Gennes formalism $H = \\tfrac{1}{2} C^\\dagger H_\\textrm{BdG} C$, with $C$ a column vector containing all creation and annihilation operators, $C=(c_1, \\dots, c_N, c_1^\\dagger, \\dots, c^\\dagger_N)^T$. The $2N\\times 2N$ matrix $H_\\textrm{BdG}$ can be written in a compact way by using Pauli matrices $\\tau$ in particle and hole space, and denoting with $\\left|n\\right\\rangle$ a column basis vector $(0,\\dots,1,0,\\dots)^T$ corresponding to the $n$-th site of the chain. In this way, we have for instance that $C^\\dagger\\,\\tau_z\\,\\left|n\\right\\rangle\\left\\langle n\\right|\\,C = 2c_n^\\dagger c_n$. The Bogoliubov-de Gennes Hamiltonian is then given by\n", + "As we learned just before, it is useful to write down the above superconducting Hamiltonian in the Bogoliubov-de Gennes formalism $H = \\tfrac{1}{2} C^\\dagger H_\\textrm{BdG} C$, with $C$ a column vector containing all creation and annihilation operators, $C=(c_1, \\dots, c_N, c_1^\\dagger, \\dots, c^\\dagger_N)^T$. The $2N\\times 2N$ matrix $H_\\textrm{BdG}$ can be written in a compact way by using Pauli matrices $\\tau$ in particle and hole space, and denoting with $\\left|n\\right\\rangle$ a column basis vector $(0,\\dots,1,0,\\dots)^T$ corresponding to the $n$-th site of the chain. In this way, we have for instance that $C^\\dagger\\,\\tau_z\\,\\left|n\\right\\rangle\\left\\langle n\\right|\\,C = 2c_n^\\dagger c_n-1$. The Bogoliubov-de Gennes Hamiltonian is then given by\n", "\n", "$$H_{BdG}=-\\sum_n \\mu \\tau_z\\left|n\\right\\rangle\\left\\langle n\\right|-\\sum_n \\left[(t\\tau_z+i\\Delta\\tau_y)\\,\\left|n\\right\\rangle\\left\\langle n+1 \\right| + \\textrm{h.c.}\\right].$$\n", "\n", @@ -205,7 +209,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "def plot_wf(syst, wf1, wf2, lstyle='-', lcolor='b'):\n", @@ -276,7 +282,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "question = (\"But wait! What happens if we remove the last Majorana site of a Kitaev chain\"\n", @@ -359,7 +367,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "mus = np.arange(-3, 3, 0.25)\n", @@ -490,7 +500,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "mus = np.arange(-3, 3, 0.25)\n", @@ -548,7 +560,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "syst = kitaev_chain(L=25, periodic=True)\n", @@ -578,7 +592,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "question = (\"What will happen if we take a 100 site Kitaev chain in the topological phase \" \n", @@ -599,7 +615,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "MoocVideo(\"wiHPQlEha6g\", src_location='1.2-summary')" @@ -608,7 +626,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "MoocDiscussion('Questions', 'Bulk-edge correspondence in the Kitaev chain')" -- GitLab