{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import sympy\n", "import qsymm" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1.0 \\alpha k_{z} + \\frac{0.5 \\hbar^{2} k_{x}^{2}}{m} + \\frac{0.5 \\hbar^{2} k_{y}^{2}}{m} + \\frac{0.5 \\hbar^{2} k_{z}^{2}}{m} & 1.0 \\alpha k_{x} - 1.0 i \\alpha k_{y}\\\\1.0 \\alpha k_{x} + 1.0 i \\alpha k_{y} & - 1.0 \\alpha k_{z} + \\frac{0.5 \\hbar^{2} k_{x}^{2}}{m} + \\frac{0.5 \\hbar^{2} k_{y}^{2}}{m} + \\frac{0.5 \\hbar^{2} k_{z}^{2}}{m}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1.0*alpha*k_z + 0.5*hbar**2*k_x**2/m + 0.5*hbar**2*k_y**2/m + 0.5*hbar**2*k_z**2/m, 1.0*alpha*k_x - 1.0*I*alpha*k_y],\n", "[ 1.0*alpha*k_x + 1.0*I*alpha*k_y, -1.0*alpha*k_z + 0.5*hbar**2*k_x**2/m + 0.5*hbar**2*k_y**2/m + 0.5*hbar**2*k_z**2/m]])" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ham = (\"hbar^2 / (2 * m) * (k_x**2 + k_y**2 + k_z**2) * eye(2) +\" +\n", " \"alpha * sigma_x * k_x + alpha * sigma_y * k_y + alpha * sigma_z * k_z\")\n", "# Convert to standard monomials form\n", "H = qsymm.Model(ham)\n", "H.tosympy()" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "cubic_group = qsymm.groups.cubic()" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "sg, cg = qsymm.symmetries(H, cubic_group)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "48 0\n" ] } ], "source": [ "print(len(sg), len(cg))" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$R\\left(-\\frac{\\pi}{2}, [1 0 0]\\right) \\mathcal{T}$" ], "text/plain": [ "R(-π/2, [1 0 0]) T" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$R\\left(\\frac{\\pi}{2}, [1 0 0]\\right) \\mathcal{T}$" ], "text/plain": [ "R(π/2, [1 0 0]) T" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$R\\left(\\pi, [1 0 0]\\right) \\mathcal{T}$" ], "text/plain": [ "R(π, [1 0 0]) T" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$R\\left(\\pi, [1 1 0]\\right) \\mathcal{T}$" ], "text/plain": [ "R(π, [1 1 0]) T" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from IPython.display import display\n", "\n", "for i in range(1, 5):\n", " display(sg[-i])" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle U H(\\mathbf{k})^* U^{-1} = H(-R\\mathbf{k}) \\\\R = R\\left(\\pi, [0 1 0]\\right)\\\\U = \\begin{bmatrix} 1.+0.j&-0.+0.j\\\\-0.+0.j& 1.+0.j\\\\\\end{bmatrix}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from IPython.display import Math\n", "\n", "display(Math(qsymm.groups.pretty_print_pge(sg[28], latex=True, full=True)))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C4 = qsymm.rotation(1/4, [1, 0, 0])\n", "C3 = qsymm.rotation(1/3, [1, 1, 1])\n", "TR = qsymm.time_reversal(3)\n", "\n", "set(sg) == qsymm.groups.generate_group({C3, C4, TR})" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "ham_degenerate = \"kron(eye(2), \" + ham + \")\"\n", "H_degenerate = qsymm.Model(ham_degenerate)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "sg, cg = qsymm.symmetries(H_degenerate, cubic_group)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\left[ H(\\mathbf{{k}}), L \\right] = 0 \\\\L = \\begin{bmatrix}1.+0.j&0.+0.j&0.+0.j&0.+0.j\\\\0.+0.j&1.+0.j&0.+0.j&0.+0.j\\\\ 0.+0.j& 0.+0.j&-1.+0.j& 0.+0.j\\\\ 0.+0.j& 0.+0.j& 0.+0.j&-1.+0.j\\\\\\end{bmatrix} \\\\\\\\\\left[ H(\\mathbf{{k}}), L \\right] = 0 \\\\L = \\begin{bmatrix}0.+0.j&0.+0.j&1.+0.j&0.+0.j\\\\0.+0.j&0.+0.j&0.+0.j&1.+0.j\\\\1.+0.j&0.+0.j&0.+0.j&0.+0.j\\\\0.+0.j&1.+0.j&0.+0.j&0.+0.j\\\\\\end{bmatrix} \\\\\\\\\\left[ H(\\mathbf{{k}}), L \\right] = 0 \\\\L = \\begin{bmatrix}0.+0.j&0.+0.j&0.+1.j&0.+0.j\\\\0.+0.j&0.+0.j&0.+0.j&0.+1.j\\\\0.-1.j&0.+0.j&0.+0.j&0.+0.j\\\\0.+0.j&0.-1.j&0.+0.j&0.+0.j\\\\\\end{bmatrix} \\\\$" ], "text/plain": [ "[\n", "[H(k), L] = 0\n", "L = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]\n", " [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]\n", " [ 0.+0.j 0.+0.j -1.+0.j 0.+0.j]\n", " [ 0.+0.j 0.+0.j 0.+0.j -1.+0.j]],\n", " \n", "[H(k), L] = 0\n", "L = [[0.+0.j 0.+0.j 1.+0.j 0.+0.j]\n", " [0.+0.j 0.+0.j 0.+0.j 1.+0.j]\n", " [1.+0.j 0.+0.j 0.+0.j 0.+0.j]\n", " [0.+0.j 1.+0.j 0.+0.j 0.+0.j]],\n", " \n", "[H(k), L] = 0\n", "L = [[0.+0.j 0.+0.j 0.+1.j 0.+0.j]\n", " [0.+0.j 0.+0.j 0.+0.j 0.+1.j]\n", " [0.-1.j 0.+0.j 0.+0.j 0.+0.j]\n", " [0.+0.j 0.-1.j 0.+0.j 0.+0.j]]]" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cg" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1\n" ] } ], "source": [ "ham_ph = \"kron(sigma_z, \" + ham + \")\"\n", "H_ph = qsymm.Model(ham_ph)\n", "sg, cg = qsymm.symmetries(H_ph, cubic_group)\n", "print(len(cg))" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\left[ H(\\mathbf{{k}}), L \\right] = 0 \\\\L = \\begin{bmatrix}-0.5+0.j& 0. +0.j& 0. +0.j& 0. +0.j\\\\ 0. +0.j&-0.5+0.j& 0. +0.j& 0. +0.j\\\\0. +0.j&0. +0.j&0.5+0.j&0. +0.j\\\\0. +0.j&0. +0.j&0. +0.j&0.5+0.j\\\\\\end{bmatrix} \\\\$" ], "text/plain": [ "[\n", "[H(k), L] = 0\n", "L = [[-0.5+0.j 0. +0.j 0. +0.j 0. +0.j]\n", " [ 0. +0.j -0.5+0.j 0. +0.j 0. +0.j]\n", " [ 0. +0.j 0. +0.j 0.5+0.j 0. +0.j]\n", " [ 0. +0.j 0. +0.j 0. +0.j 0.5+0.j]]]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cg" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\frac{3 \\alpha k_{z}}{2} + \\frac{\\hbar^{2} k_{x}^{2}}{2 m} + \\frac{\\hbar^{2} k_{y}^{2}}{2 m} + \\frac{\\hbar^{2} k_{z}^{2}}{2 m} & \\frac{\\sqrt{3} \\alpha k_{x}}{2} - \\frac{\\sqrt{3} i \\alpha k_{y}}{2} & 0 & 0\\\\\\frac{\\sqrt{3} \\alpha k_{x}}{2} + \\frac{\\sqrt{3} i \\alpha k_{y}}{2} & \\frac{\\alpha k_{z}}{2} + \\frac{\\hbar^{2} k_{x}^{2}}{2 m} + \\frac{\\hbar^{2} k_{y}^{2}}{2 m} + \\frac{\\hbar^{2} k_{z}^{2}}{2 m} & \\alpha k_{x} - i \\alpha k_{y} & 0\\\\0 & \\alpha k_{x} + i \\alpha k_{y} & - \\frac{\\alpha k_{z}}{2} + \\frac{\\hbar^{2} k_{x}^{2}}{2 m} + \\frac{\\hbar^{2} k_{y}^{2}}{2 m} + \\frac{\\hbar^{2} k_{z}^{2}}{2 m} & \\frac{\\sqrt{3} \\alpha k_{x}}{2} - \\frac{\\sqrt{3} i \\alpha k_{y}}{2}\\\\0 & 0 & \\frac{\\sqrt{3} \\alpha k_{x}}{2} + \\frac{\\sqrt{3} i \\alpha k_{y}}{2} & - \\frac{3 \\alpha k_{z}}{2} + \\frac{\\hbar^{2} k_{x}^{2}}{2 m} + \\frac{\\hbar^{2} k_{y}^{2}}{2 m} + \\frac{\\hbar^{2} k_{z}^{2}}{2 m}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[3*alpha*k_z/2 + hbar**2*k_x**2/(2*m) + hbar**2*k_y**2/(2*m) + hbar**2*k_z**2/(2*m), sqrt(3)*alpha*k_x/2 - sqrt(3)*I*alpha*k_y/2, 0, 0],\n", "[ sqrt(3)*alpha*k_x/2 + sqrt(3)*I*alpha*k_y/2, alpha*k_z/2 + hbar**2*k_x**2/(2*m) + hbar**2*k_y**2/(2*m) + hbar**2*k_z**2/(2*m), alpha*k_x - I*alpha*k_y, 0],\n", "[ 0, alpha*k_x + I*alpha*k_y, -alpha*k_z/2 + hbar**2*k_x**2/(2*m) + hbar**2*k_y**2/(2*m) + hbar**2*k_z**2/(2*m), sqrt(3)*alpha*k_x/2 - sqrt(3)*I*alpha*k_y/2],\n", "[ 0, 0, sqrt(3)*alpha*k_x/2 + sqrt(3)*I*alpha*k_y/2, -3*alpha*k_z/2 + hbar**2*k_x**2/(2*m) + hbar**2*k_y**2/(2*m) + hbar**2*k_z**2/(2*m)]])" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "J_x, J_y, J_z = qsymm.groups.spin_matrices(3/2)\n", "ham_rashba = (\n", " \"hbar^2 / (2 * m) * (k_x**2 + k_y**2 + k_z**2) * eye(4) +\" +\n", " \"alpha * J_x * k_x + alpha * J_y * k_y + alpha * J_z * k_z\"\n", ")\n", "H_rashba = qsymm.Model(ham_rashba, locals=dict(J_x=J_x, J_y=J_y, J_z=J_z))\n", "H_rashba.tosympy(nsimplify=True)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "sg, cg = qsymm.symmetries(H_rashba, cubic_group)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "set(sg) == qsymm.groups.generate_group({C4, C3, TR})" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "hex_group_2D = qsymm.groups.hexagonal()" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle m \\left(\\cos{\\left(k_{x} \\right)} + \\cos{\\left(- \\frac{k_{x}}{2} + \\frac{\\sqrt{3} k_{y}}{2} \\right)} + \\cos{\\left(\\frac{k_{x}}{2} + \\frac{\\sqrt{3} k_{y}}{2} \\right)}\\right)$" ], "text/plain": [ "m*(cos(k_x) + cos(-k_x/2 + sqrt(3)*k_y/2) + cos(k_x/2 + sqrt(3)*k_y/2))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "ham_tri = 'm * (cos(k_x) + cos(1/2*k_x + sqrt(3)/2*k_y) + cos(-1/2*k_x + sqrt(3)/2*k_y))'\n", "display(qsymm.sympify(ham_tri))" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "H_tri = qsymm.Model(ham_tri, momenta=['k_x', 'k_y'])" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "sg, cg = qsymm.symmetries(H_tri, hex_group_2D)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "set(sg) == qsymm.groups.hexagonal(ph=False)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\alpha k_{z} + \\frac{\\hbar^{2} \\left(k_{x}^{2} + k_{y}^{2} + k_{z}^{2}\\right)}{2 m} & \\alpha k_{x} - i \\alpha k_{y}\\\\\\alpha k_{x} + i \\alpha k_{y} & - \\alpha k_{z} + \\frac{\\hbar^{2} \\left(k_{x}^{2} + k_{y}^{2} + k_{z}^{2}\\right)}{2 m}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[alpha*k_z + hbar**2*(k_x**2 + k_y**2 + k_z**2)/(2*m), alpha*k_x - I*alpha*k_y],\n", "[ alpha*k_x + I*alpha*k_y, -alpha*k_z + hbar**2*(k_x**2 + k_y**2 + k_z**2)/(2*m)]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "display(qsymm.sympify(ham))\n", "pg, cg = qsymm.symmetries(H, continuous_rotations=True, prettify=True)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$e^{-i\\phi L}H(\\mathbf{k})e^{i\\phi L} = H(R_{\\phi}\\mathbf{k}) \\\\R_{\\phi} = R\\left(\\phi, [0 0 1]\\right)\\\\L = \\begin{bmatrix}0.5+0.j&0. +0.j\\\\ 0. +0.j&-0.5+0.j\\\\\\end{bmatrix} \\\\\\\\e^{-i\\phi L}H(\\mathbf{k})e^{i\\phi L} = H(R_{\\phi}\\mathbf{k}) \\\\R_{\\phi} = R\\left(\\phi, [ 0 -1 0]\\right)\\\\L = \\begin{bmatrix}0.+0.j &0.+0.5j\\\\0.-0.5j&0.+0.j \\\\\\end{bmatrix} \\\\\\\\e^{-i\\phi L}H(\\mathbf{k})e^{i\\phi L} = H(R_{\\phi}\\mathbf{k}) \\\\R_{\\phi} = R\\left(\\phi, [1 0 0]\\right)\\\\L = \\begin{bmatrix}0. +0.j&0.5+0.j\\\\0.5+0.j&0. +0.j\\\\\\end{bmatrix} \\\\$" ], "text/plain": [ "[\n", "exp(-i ϕ L)⋅H(k)⋅exp(i ϕ L) = H(R_ϕ⋅k)\n", "R_ϕ = R(ϕ, [0 0 1])\n", "L = [[ 0.5+0.j 0. +0.j]\n", " [ 0. +0.j -0.5+0.j]],\n", " \n", "exp(-i ϕ L)⋅H(k)⋅exp(i ϕ L) = H(R_ϕ⋅k)\n", "R_ϕ = R(ϕ, [ 0 -1 0])\n", "L = [[0.+0.j 0.+0.5j]\n", " [0.-0.5j 0.+0.j ]],\n", " \n", "exp(-i ϕ L)⋅H(k)⋅exp(i ϕ L) = H(R_ϕ⋅k)\n", "R_ϕ = R(ϕ, [1 0 0])\n", "L = [[0. +0.j 0.5+0.j]\n", " [0.5+0.j 0. +0.j]]]" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cg" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [], "source": [ "ham_kp = 'Matrix([[hbar**2*k_x*gamma_0*k_x/(2*m_0)+hbar**2*k_y*gamma_0*k_y/(2*m_0)+hbar**2*k_z*gamma_0*k_z/(2*m_0)+E_0+E_v,0,-sqrt(2)*P*k_x/2-sqrt(2)*I*P*k_y/2,sqrt(6)*P*k_z/3,sqrt(6)*P*k_x/6-sqrt(6)*I*P*k_y/6,0,-sqrt(3)*P*k_z/3,-sqrt(3)*P*k_x/3+sqrt(3)*I*P*k_y/3],[0,hbar**2*k_x*gamma_0*k_x/(2*m_0)+hbar**2*k_y*gamma_0*k_y/(2*m_0)+hbar**2*k_z*gamma_0*k_z/(2*m_0)+E_0+E_v,0,-sqrt(6)*P*k_x/6-sqrt(6)*I*P*k_y/6,sqrt(6)*P*k_z/3,sqrt(2)*P*k_x/2-sqrt(2)*I*P*k_y/2,-sqrt(3)*P*k_x/3-sqrt(3)*I*P*k_y/3,sqrt(3)*P*k_z/3],[-sqrt(2)*k_x*P/2+sqrt(2)*I*k_y*P/2,0,-hbar**2*k_x*gamma_1*k_x/(2*m_0)-hbar**2*k_x*gamma_2*k_x/(2*m_0)-I*hbar**2*k_x*k_y/(2*m_0)-3*I*hbar**2*k_x*kappa*k_y/(2*m_0)-hbar**2*k_y*gamma_1*k_y/(2*m_0)-hbar**2*k_y*gamma_2*k_y/(2*m_0)+I*hbar**2*k_y*k_x/(2*m_0)+3*I*hbar**2*k_y*kappa*k_x/(2*m_0)-hbar**2*k_z*gamma_1*k_z/(2*m_0)+hbar**2*k_z*gamma_2*k_z/m_0+E_v,sqrt(3)*hbar**2*k_x*gamma_3*k_z/(2*m_0)+sqrt(3)*hbar**2*k_x*k_z/(6*m_0)+sqrt(3)*hbar**2*k_x*kappa*k_z/(2*m_0)-sqrt(3)*I*hbar**2*k_y*gamma_3*k_z/(2*m_0)-sqrt(3)*I*hbar**2*k_y*k_z/(6*m_0)-sqrt(3)*I*hbar**2*k_y*kappa*k_z/(2*m_0)+sqrt(3)*hbar**2*k_z*gamma_3*k_x/(2*m_0)-sqrt(3)*I*hbar**2*k_z*gamma_3*k_y/(2*m_0)-sqrt(3)*hbar**2*k_z*k_x/(6*m_0)+sqrt(3)*I*hbar**2*k_z*k_y/(6*m_0)-sqrt(3)*hbar**2*k_z*kappa*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_z*kappa*k_y/(2*m_0),sqrt(3)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(3)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)-sqrt(3)*hbar**2*k_y*gamma_2*k_y/(2*m_0)-sqrt(3)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),0,-sqrt(6)*hbar**2*k_x*gamma_3*k_z/(4*m_0)-sqrt(6)*hbar**2*k_x*k_z/(12*m_0)-sqrt(6)*hbar**2*k_x*kappa*k_z/(4*m_0)+sqrt(6)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)+sqrt(6)*I*hbar**2*k_y*k_z/(12*m_0)+sqrt(6)*I*hbar**2*k_y*kappa*k_z/(4*m_0)-sqrt(6)*hbar**2*k_z*gamma_3*k_x/(4*m_0)+sqrt(6)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)+sqrt(6)*hbar**2*k_z*k_x/(12*m_0)-sqrt(6)*I*hbar**2*k_z*k_y/(12*m_0)+sqrt(6)*hbar**2*k_z*kappa*k_x/(4*m_0)-sqrt(6)*I*hbar**2*k_z*kappa*k_y/(4*m_0),-sqrt(6)*hbar**2*k_x*gamma_2*k_x/(2*m_0)+sqrt(6)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)+sqrt(6)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(6)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0)],[sqrt(6)*k_z*P/3,-sqrt(6)*k_x*P/6+sqrt(6)*I*k_y*P/6,sqrt(3)*hbar**2*k_x*gamma_3*k_z/(2*m_0)-sqrt(3)*hbar**2*k_x*k_z/(6*m_0)-sqrt(3)*hbar**2*k_x*kappa*k_z/(2*m_0)+sqrt(3)*I*hbar**2*k_y*gamma_3*k_z/(2*m_0)-sqrt(3)*I*hbar**2*k_y*k_z/(6*m_0)-sqrt(3)*I*hbar**2*k_y*kappa*k_z/(2*m_0)+sqrt(3)*hbar**2*k_z*gamma_3*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_z*gamma_3*k_y/(2*m_0)+sqrt(3)*hbar**2*k_z*k_x/(6*m_0)+sqrt(3)*I*hbar**2*k_z*k_y/(6*m_0)+sqrt(3)*hbar**2*k_z*kappa*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_z*kappa*k_y/(2*m_0),-hbar**2*k_x*gamma_1*k_x/(2*m_0)+hbar**2*k_x*gamma_2*k_x/(2*m_0)-I*hbar**2*k_x*k_y/(6*m_0)-I*hbar**2*k_x*kappa*k_y/(2*m_0)-hbar**2*k_y*gamma_1*k_y/(2*m_0)+hbar**2*k_y*gamma_2*k_y/(2*m_0)+I*hbar**2*k_y*k_x/(6*m_0)+I*hbar**2*k_y*kappa*k_x/(2*m_0)-hbar**2*k_z*gamma_1*k_z/(2*m_0)-hbar**2*k_z*gamma_2*k_z/m_0+E_v,hbar**2*k_x*k_z/(3*m_0)+hbar**2*k_x*kappa*k_z/m_0-I*hbar**2*k_y*k_z/(3*m_0)-I*hbar**2*k_y*kappa*k_z/m_0-hbar**2*k_z*k_x/(3*m_0)+I*hbar**2*k_z*k_y/(3*m_0)-hbar**2*k_z*kappa*k_x/m_0+I*hbar**2*k_z*kappa*k_y/m_0,sqrt(3)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(3)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)-sqrt(3)*hbar**2*k_y*gamma_2*k_y/(2*m_0)-sqrt(3)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),-sqrt(2)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(2)*I*hbar**2*k_x*k_y/(6*m_0)-sqrt(2)*I*hbar**2*k_x*kappa*k_y/(2*m_0)-sqrt(2)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(2)*I*hbar**2*k_y*k_x/(6*m_0)+sqrt(2)*I*hbar**2*k_y*kappa*k_x/(2*m_0)+sqrt(2)*hbar**2*k_z*gamma_2*k_z/m_0,3*sqrt(2)*hbar**2*k_x*gamma_3*k_z/(4*m_0)-sqrt(2)*hbar**2*k_x*k_z/(12*m_0)-sqrt(2)*hbar**2*k_x*kappa*k_z/(4*m_0)-3*sqrt(2)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)+sqrt(2)*I*hbar**2*k_y*k_z/(12*m_0)+sqrt(2)*I*hbar**2*k_y*kappa*k_z/(4*m_0)+3*sqrt(2)*hbar**2*k_z*gamma_3*k_x/(4*m_0)-3*sqrt(2)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)+sqrt(2)*hbar**2*k_z*k_x/(12*m_0)-sqrt(2)*I*hbar**2*k_z*k_y/(12*m_0)+sqrt(2)*hbar**2*k_z*kappa*k_x/(4*m_0)-sqrt(2)*I*hbar**2*k_z*kappa*k_y/(4*m_0)],[sqrt(6)*k_x*P/6+sqrt(6)*I*k_y*P/6,sqrt(6)*k_z*P/3,sqrt(3)*hbar**2*k_x*gamma_2*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)-sqrt(3)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(3)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),-hbar**2*k_x*k_z/(3*m_0)-hbar**2*k_x*kappa*k_z/m_0-I*hbar**2*k_y*k_z/(3*m_0)-I*hbar**2*k_y*kappa*k_z/m_0+hbar**2*k_z*k_x/(3*m_0)+I*hbar**2*k_z*k_y/(3*m_0)+hbar**2*k_z*kappa*k_x/m_0+I*hbar**2*k_z*kappa*k_y/m_0,-hbar**2*k_x*gamma_1*k_x/(2*m_0)+hbar**2*k_x*gamma_2*k_x/(2*m_0)+I*hbar**2*k_x*k_y/(6*m_0)+I*hbar**2*k_x*kappa*k_y/(2*m_0)-hbar**2*k_y*gamma_1*k_y/(2*m_0)+hbar**2*k_y*gamma_2*k_y/(2*m_0)-I*hbar**2*k_y*k_x/(6*m_0)-I*hbar**2*k_y*kappa*k_x/(2*m_0)-hbar**2*k_z*gamma_1*k_z/(2*m_0)-hbar**2*k_z*gamma_2*k_z/m_0+E_v,-sqrt(3)*hbar**2*k_x*gamma_3*k_z/(2*m_0)+sqrt(3)*hbar**2*k_x*k_z/(6*m_0)+sqrt(3)*hbar**2*k_x*kappa*k_z/(2*m_0)+sqrt(3)*I*hbar**2*k_y*gamma_3*k_z/(2*m_0)-sqrt(3)*I*hbar**2*k_y*k_z/(6*m_0)-sqrt(3)*I*hbar**2*k_y*kappa*k_z/(2*m_0)-sqrt(3)*hbar**2*k_z*gamma_3*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_z*gamma_3*k_y/(2*m_0)-sqrt(3)*hbar**2*k_z*k_x/(6*m_0)+sqrt(3)*I*hbar**2*k_z*k_y/(6*m_0)-sqrt(3)*hbar**2*k_z*kappa*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_z*kappa*k_y/(2*m_0),3*sqrt(2)*hbar**2*k_x*gamma_3*k_z/(4*m_0)-sqrt(2)*hbar**2*k_x*k_z/(12*m_0)-sqrt(2)*hbar**2*k_x*kappa*k_z/(4*m_0)+3*sqrt(2)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)-sqrt(2)*I*hbar**2*k_y*k_z/(12*m_0)-sqrt(2)*I*hbar**2*k_y*kappa*k_z/(4*m_0)+3*sqrt(2)*hbar**2*k_z*gamma_3*k_x/(4*m_0)+3*sqrt(2)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)+sqrt(2)*hbar**2*k_z*k_x/(12*m_0)+sqrt(2)*I*hbar**2*k_z*k_y/(12*m_0)+sqrt(2)*hbar**2*k_z*kappa*k_x/(4*m_0)+sqrt(2)*I*hbar**2*k_z*kappa*k_y/(4*m_0),sqrt(2)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(2)*I*hbar**2*k_x*k_y/(6*m_0)-sqrt(2)*I*hbar**2*k_x*kappa*k_y/(2*m_0)+sqrt(2)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(2)*I*hbar**2*k_y*k_x/(6*m_0)+sqrt(2)*I*hbar**2*k_y*kappa*k_x/(2*m_0)-sqrt(2)*hbar**2*k_z*gamma_2*k_z/m_0],[0,sqrt(2)*k_x*P/2+sqrt(2)*I*k_y*P/2,0,sqrt(3)*hbar**2*k_x*gamma_2*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)-sqrt(3)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(3)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),-sqrt(3)*hbar**2*k_x*gamma_3*k_z/(2*m_0)-sqrt(3)*hbar**2*k_x*k_z/(6*m_0)-sqrt(3)*hbar**2*k_x*kappa*k_z/(2*m_0)-sqrt(3)*I*hbar**2*k_y*gamma_3*k_z/(2*m_0)-sqrt(3)*I*hbar**2*k_y*k_z/(6*m_0)-sqrt(3)*I*hbar**2*k_y*kappa*k_z/(2*m_0)-sqrt(3)*hbar**2*k_z*gamma_3*k_x/(2*m_0)-sqrt(3)*I*hbar**2*k_z*gamma_3*k_y/(2*m_0)+sqrt(3)*hbar**2*k_z*k_x/(6*m_0)+sqrt(3)*I*hbar**2*k_z*k_y/(6*m_0)+sqrt(3)*hbar**2*k_z*kappa*k_x/(2*m_0)+sqrt(3)*I*hbar**2*k_z*kappa*k_y/(2*m_0),-hbar**2*k_x*gamma_1*k_x/(2*m_0)-hbar**2*k_x*gamma_2*k_x/(2*m_0)+I*hbar**2*k_x*k_y/(2*m_0)+3*I*hbar**2*k_x*kappa*k_y/(2*m_0)-hbar**2*k_y*gamma_1*k_y/(2*m_0)-hbar**2*k_y*gamma_2*k_y/(2*m_0)-I*hbar**2*k_y*k_x/(2*m_0)-3*I*hbar**2*k_y*kappa*k_x/(2*m_0)-hbar**2*k_z*gamma_1*k_z/(2*m_0)+hbar**2*k_z*gamma_2*k_z/m_0+E_v,sqrt(6)*hbar**2*k_x*gamma_2*k_x/(2*m_0)+sqrt(6)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)-sqrt(6)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(6)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),-sqrt(6)*hbar**2*k_x*gamma_3*k_z/(4*m_0)-sqrt(6)*hbar**2*k_x*k_z/(12*m_0)-sqrt(6)*hbar**2*k_x*kappa*k_z/(4*m_0)-sqrt(6)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)-sqrt(6)*I*hbar**2*k_y*k_z/(12*m_0)-sqrt(6)*I*hbar**2*k_y*kappa*k_z/(4*m_0)-sqrt(6)*hbar**2*k_z*gamma_3*k_x/(4*m_0)-sqrt(6)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)+sqrt(6)*hbar**2*k_z*k_x/(12*m_0)+sqrt(6)*I*hbar**2*k_z*k_y/(12*m_0)+sqrt(6)*hbar**2*k_z*kappa*k_x/(4*m_0)+sqrt(6)*I*hbar**2*k_z*kappa*k_y/(4*m_0)],[-sqrt(3)*k_z*P/3,-sqrt(3)*k_x*P/3+sqrt(3)*I*k_y*P/3,-sqrt(6)*hbar**2*k_x*gamma_3*k_z/(4*m_0)+sqrt(6)*hbar**2*k_x*k_z/(12*m_0)+sqrt(6)*hbar**2*k_x*kappa*k_z/(4*m_0)-sqrt(6)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)+sqrt(6)*I*hbar**2*k_y*k_z/(12*m_0)+sqrt(6)*I*hbar**2*k_y*kappa*k_z/(4*m_0)-sqrt(6)*hbar**2*k_z*gamma_3*k_x/(4*m_0)-sqrt(6)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)-sqrt(6)*hbar**2*k_z*k_x/(12*m_0)-sqrt(6)*I*hbar**2*k_z*k_y/(12*m_0)-sqrt(6)*hbar**2*k_z*kappa*k_x/(4*m_0)-sqrt(6)*I*hbar**2*k_z*kappa*k_y/(4*m_0),-sqrt(2)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(2)*I*hbar**2*k_x*k_y/(6*m_0)-sqrt(2)*I*hbar**2*k_x*kappa*k_y/(2*m_0)-sqrt(2)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(2)*I*hbar**2*k_y*k_x/(6*m_0)+sqrt(2)*I*hbar**2*k_y*kappa*k_x/(2*m_0)+sqrt(2)*hbar**2*k_z*gamma_2*k_z/m_0,3*sqrt(2)*hbar**2*k_x*gamma_3*k_z/(4*m_0)+sqrt(2)*hbar**2*k_x*k_z/(12*m_0)+sqrt(2)*hbar**2*k_x*kappa*k_z/(4*m_0)-3*sqrt(2)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)-sqrt(2)*I*hbar**2*k_y*k_z/(12*m_0)-sqrt(2)*I*hbar**2*k_y*kappa*k_z/(4*m_0)+3*sqrt(2)*hbar**2*k_z*gamma_3*k_x/(4*m_0)-3*sqrt(2)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)-sqrt(2)*hbar**2*k_z*k_x/(12*m_0)+sqrt(2)*I*hbar**2*k_z*k_y/(12*m_0)-sqrt(2)*hbar**2*k_z*kappa*k_x/(4*m_0)+sqrt(2)*I*hbar**2*k_z*kappa*k_y/(4*m_0),sqrt(6)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(6)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)-sqrt(6)*hbar**2*k_y*gamma_2*k_y/(2*m_0)-sqrt(6)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),-hbar**2*k_x*gamma_1*k_x/(2*m_0)-I*hbar**2*k_x*k_y/(3*m_0)-I*hbar**2*k_x*kappa*k_y/m_0-hbar**2*k_y*gamma_1*k_y/(2*m_0)+I*hbar**2*k_y*k_x/(3*m_0)+I*hbar**2*k_y*kappa*k_x/m_0-hbar**2*k_z*gamma_1*k_z/(2*m_0)-Delta+E_v,hbar**2*k_x*k_z/(3*m_0)+hbar**2*k_x*kappa*k_z/m_0-I*hbar**2*k_y*k_z/(3*m_0)-I*hbar**2*k_y*kappa*k_z/m_0-hbar**2*k_z*k_x/(3*m_0)+I*hbar**2*k_z*k_y/(3*m_0)-hbar**2*k_z*kappa*k_x/m_0+I*hbar**2*k_z*kappa*k_y/m_0],[-sqrt(3)*k_x*P/3-sqrt(3)*I*k_y*P/3,sqrt(3)*k_z*P/3,-sqrt(6)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(6)*I*hbar**2*k_x*gamma_3*k_y/(2*m_0)+sqrt(6)*hbar**2*k_y*gamma_2*k_y/(2*m_0)-sqrt(6)*I*hbar**2*k_y*gamma_3*k_x/(2*m_0),3*sqrt(2)*hbar**2*k_x*gamma_3*k_z/(4*m_0)+sqrt(2)*hbar**2*k_x*k_z/(12*m_0)+sqrt(2)*hbar**2*k_x*kappa*k_z/(4*m_0)+3*sqrt(2)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)+sqrt(2)*I*hbar**2*k_y*k_z/(12*m_0)+sqrt(2)*I*hbar**2*k_y*kappa*k_z/(4*m_0)+3*sqrt(2)*hbar**2*k_z*gamma_3*k_x/(4*m_0)+3*sqrt(2)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)-sqrt(2)*hbar**2*k_z*k_x/(12*m_0)-sqrt(2)*I*hbar**2*k_z*k_y/(12*m_0)-sqrt(2)*hbar**2*k_z*kappa*k_x/(4*m_0)-sqrt(2)*I*hbar**2*k_z*kappa*k_y/(4*m_0),sqrt(2)*hbar**2*k_x*gamma_2*k_x/(2*m_0)-sqrt(2)*I*hbar**2*k_x*k_y/(6*m_0)-sqrt(2)*I*hbar**2*k_x*kappa*k_y/(2*m_0)+sqrt(2)*hbar**2*k_y*gamma_2*k_y/(2*m_0)+sqrt(2)*I*hbar**2*k_y*k_x/(6*m_0)+sqrt(2)*I*hbar**2*k_y*kappa*k_x/(2*m_0)-sqrt(2)*hbar**2*k_z*gamma_2*k_z/m_0,-sqrt(6)*hbar**2*k_x*gamma_3*k_z/(4*m_0)+sqrt(6)*hbar**2*k_x*k_z/(12*m_0)+sqrt(6)*hbar**2*k_x*kappa*k_z/(4*m_0)+sqrt(6)*I*hbar**2*k_y*gamma_3*k_z/(4*m_0)-sqrt(6)*I*hbar**2*k_y*k_z/(12*m_0)-sqrt(6)*I*hbar**2*k_y*kappa*k_z/(4*m_0)-sqrt(6)*hbar**2*k_z*gamma_3*k_x/(4*m_0)+sqrt(6)*I*hbar**2*k_z*gamma_3*k_y/(4*m_0)-sqrt(6)*hbar**2*k_z*k_x/(12*m_0)+sqrt(6)*I*hbar**2*k_z*k_y/(12*m_0)-sqrt(6)*hbar**2*k_z*kappa*k_x/(4*m_0)+sqrt(6)*I*hbar**2*k_z*kappa*k_y/(4*m_0),-hbar**2*k_x*k_z/(3*m_0)-hbar**2*k_x*kappa*k_z/m_0-I*hbar**2*k_y*k_z/(3*m_0)-I*hbar**2*k_y*kappa*k_z/m_0+hbar**2*k_z*k_x/(3*m_0)+I*hbar**2*k_z*k_y/(3*m_0)+hbar**2*k_z*kappa*k_x/m_0+I*hbar**2*k_z*kappa*k_y/m_0,-hbar**2*k_x*gamma_1*k_x/(2*m_0)+I*hbar**2*k_x*k_y/(3*m_0)+I*hbar**2*k_x*kappa*k_y/m_0-hbar**2*k_y*gamma_1*k_y/(2*m_0)-I*hbar**2*k_y*k_x/(3*m_0)-I*hbar**2*k_y*kappa*k_x/m_0-hbar**2*k_z*gamma_1*k_z/(2*m_0)-Delta+E_v]])'\n", "Hkp = qsymm.Model(ham_kp)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "96\n" ] } ], "source": [ "sg, cg = qsymm.symmetries(Hkp, cubic_group)\n", "print(len(sg))" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "{'hbar': 1} 96 0\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "{'hbar': 1, 'P': 0, 'Delta': 0} 96 7\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "{'hbar': 1, 'P': 0, 'Delta': 0, 'gamma_3': 0} 96 15\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "{'hbar': 1, 'gamma_2': 'gamma_1', 'gamma_3': 'gamma_1', 'k_z': 0} 32 1\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "{'hbar': 1, 'P': 0, 'Delta': 0, 'gamma_3': 0} 96 15\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "{'hbar': 1, 'Delta': 0, 'k_z': 0} 32 7\n" ] } ], "source": [ "kp_examples = [\n", " {'hbar': 1},\n", " {'hbar': 1, 'P': 0, 'Delta': 0},\n", " {'hbar': 1, 'P': 0, 'Delta': 0, 'gamma_3': 0},\n", " {'hbar': 1, 'gamma_2': 'gamma_1', 'gamma_3': 'gamma_1', 'k_z': 0},\n", " {'hbar': 1, 'P': 0, 'Delta': 0, 'gamma_3': 0},\n", " {'hbar': 1, 'Delta': 0, 'k_z': 0},\n", "]\n", "\n", "for subs in kp_examples:\n", " Hkp = qsymm.Model(ham_kp, locals=subs)\n", " sg, cg = qsymm.symmetries(Hkp, cubic_group)\n", " print(subs, len(sg), len(cg))" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }