pi-flux.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from math import pi, sqrt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hopp_vec(Mx,My,vec,px,py):\n",
    "    '''generates MxM on-slice hopping_matg matrix cubic lattice\n",
    "     vec= hopping vector\n",
    "     py=1 PBC along y otherwise OBC\n",
    "     px=1 PBC along x otherwise OBC\n",
    "    ''' \n",
    "    \n",
    "    hopping_mat=np.zeros(shape=(Mx*My,Mx*My),dtype=complex)\n",
    "    for i_y in range(My):\n",
    "        for i_x in range(Mx):\n",
    "            i_px=i_x+vec[0]\n",
    "            i_py=i_y+vec[1]\n",
    "            if ((i_px>=Mx or i_px<0) and i_py< My and i_py>=0):\n",
    "                if i_px>=Mx :\n",
    "                    hx=px\n",
    "                else:\n",
    "                    hx=px.conjugate()\n",
    "                i_px=i_px%Mx\n",
    "                hopping_mat[i_px+i_py*Mx,i_x+i_y*Mx]=hx\n",
    "            elif ((i_py>=My or i_py<0) and i_px< Mx and i_px>=0):\n",
    "                if i_py>=My :\n",
    "                    hy=py\n",
    "                else:\n",
    "                    hy=py.conjugate()\n",
    "                    \n",
    "                i_py=i_py%My\n",
    "                hopping_mat[i_px+i_py*Mx,i_x+i_y*Mx]=hy\n",
    "                    \n",
    "            elif (i_px>=Mx or i_px<0) and (i_py>=My or i_py<0):\n",
    "                if i_px>=Mx:\n",
    "                    hx=px\n",
    "                else:\n",
    "                    hx=px.conjugate()\n",
    "    \n",
    "                if i_py>=My:\n",
    "                    hy=py\n",
    "                else:\n",
    "                    hy=py.conjugate()\n",
    "                i_px=i_px%Mx\n",
    "                i_py=i_py%My\n",
    "                hopping_mat[i_px+i_py*Mx,i_x+i_y*Mx]=hx*hy\n",
    "                \n",
    "            elif i_px>=0 and i_px<Mx and i_py>=0 and i_py<My :\n",
    "                hopping_mat[i_px+i_py*Mx,i_x+i_y*Mx]=1\n",
    "    return hopping_mat\n",
    "\n",
    "def wf_gen(Mx,My,N_pt,r,t,m,px,py):\n",
    "    '''\n",
    "    returns eigenvalues and eigenvectors of pi-flux model\n",
    "    Mx x My sites on a square lattice\n",
    "    N_pt: total number of particles (choose half filling for insulating phase)\n",
    "    t: nearest neighbor hopping\n",
    "    r: next nearest neighbor hopping (delta in arXiv:1412.5597 notation)\n",
    "    m: onsite +/- potential\n",
    "    px: complex phase of hopping across vertical edge\n",
    "    py: complex phase of hopping across horizontal edge    \n",
    "    '''\n",
    "    tx=t\n",
    "    ty=t    \n",
    "    x_sign=np.diag(np.power(-1,np.mod(np.arange(0,Mx*My),Mx)))\n",
    "    y_sign=np.diag(np.power(-1,np.floor(np.arange(0,Mx*My)/Mx)))\n",
    "    H_x=tx*hopp_vec(Mx,My,[1,0],px,py)\n",
    "    H_y=ty*np.dot(x_sign,hopp_vec(Mx,My,[0,1],px,py))\n",
    "    H_xy_nn=-1j*r*np.dot(x_sign,hopp_vec(Mx,My,[1,1],px,py))\n",
    "    H_Mxy_nn=1j*r*np.dot(x_sign,hopp_vec(Mx,My,[-1,1],px,py))\n",
    "    H_dd=m*np.dot(y_sign,x_sign)  \n",
    "    H_t=H_x+H_y+H_xy_nn+H_Mxy_nn\n",
    "    H_t=H_t+np.transpose(H_t.conjugate())+H_dd      \n",
    "    evals,evecs=np.linalg.eigh(H_t)\n",
    "    evals=evals.real\n",
    "    evecs=evecs[:,evals.argsort()]\n",
    "    evals.sort()\n",
    "    evecs=evecs[:,0:N_pt]\n",
    "    evals=evals[0:N_pt]\n",
    "    return evals,evecs\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### system dims\n",
    "Lx= 10\n",
    "Ly= 10\n",
    "### bdy conditions\n",
    "px=1\n",
    "py=1\n",
    "N_pt = int(Lx*Ly/2) # number of particles\n",
    "### hopping parameters\n",
    "t = 2\n",
    "r = 1\n",
    "m = 0\n",
    "\n",
    "evals, evecs= wf_gen(Lx,Ly,N_pt,r,t,m,px,py)\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(evals,'o')\n",
    "plt.ylabel('eigenvalues')\n",
    "plt.show()\n",
    "\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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