{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import sympy\n", "\n", "import qsymm" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# C3 rotational symmetry - invariant under 2pi/3\n", "C3 = qsymm.rotation(1/3, spin=1/2)\n", "\n", "# Time reversal\n", "TR = qsymm.time_reversal(2, spin=1/2)\n", "\n", "# Mirror symmetry in x\n", "Mx = qsymm.mirror([1, 0], spin=1/2)\n", "\n", "symmetries = [C3, TR, Mx]" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "dim = 2 # Momenta along x and y\n", "total_power = 3 # Maximum power of momenta" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0],\n", "[0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{x} + k_{y}\\\\- i k_{x} + k_{y} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_x + k_y],\n", "[-I*k_x + k_y, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} & 0\\\\0 & k_{x}^{2} + k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2, 0],\n", "[ 0, k_x**2 + k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{x}^{3} + k_{x}^{2} k_{y} + i k_{x} k_{y}^{2} + k_{y}^{3}\\\\- i k_{x}^{3} + k_{x}^{2} k_{y} - i k_{x} k_{y}^{2} + k_{y}^{3} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_x**3 + k_x**2*k_y + I*k_x*k_y**2 + k_y**3],\n", "[-I*k_x**3 + k_x**2*k_y - I*k_x*k_y**2 + k_y**3, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- \\frac{k_{x}^{3}}{3} + k_{x} k_{y}^{2} & 0\\\\0 & \\frac{k_{x}^{3}}{3} - k_{x} k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[-k_x**3/3 + k_x*k_y**2, 0],\n", "[ 0, k_x**3/3 - k_x*k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "qsymm.hamiltonian_generator.check_symmetry(family, symmetries)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# Spatial inversion\n", "pU = np.array([\n", " [1.0, 0.0, 0.0, 0.0],\n", " [0.0, -1.0, 0.0, 0.0],\n", " [0.0, 0.0, 1.0, 0.0],\n", " [0.0, 0.0, 0.0, -1.0],\n", "])\n", "pS = qsymm.inversion(2, U=pU)\n", "\n", "# Time reversal\n", "trU = np.array([\n", " [0.0, 0.0, -1.0, 0.0],\n", " [0.0, 0.0, 0.0, -1.0],\n", " [1.0, 0.0, 0.0, 0.0],\n", " [0.0, 1.0, 0.0, 0.0],\n", "])\n", "trS = qsymm.time_reversal(2, U=trU)\n", "\n", "# Rotation\n", "phi = 2.0 * np.pi / 4.0 # Impose 4-fold rotational symmetry\n", "rotU = np.array([\n", " [np.exp(-1j*phi/2), 0.0, 0.0, 0.0],\n", " [0.0, np.exp(-1j*3*phi/2), 0.0, 0.0],\n", " [0.0, 0.0, np.exp(1j*phi/2), 0.0],\n", " [0.0, 0.0, 0.0, np.exp(1j*3*phi/2)],\n", "])\n", "rotS = qsymm.rotation(1/4, U=rotU)\n", "\n", "symmetries = [rotS, trS, pS]" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "dim = 2\n", "total_power = 2" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 1, 0],\n", "[0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, 1, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & k_{x} + i k_{y} & 0 & 0\\\\k_{x} - i k_{y} & 0 & 0 & 0\\\\0 & 0 & 0 & - k_{x} + i k_{y}\\\\0 & 0 & - k_{x} - i k_{y} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, k_x + I*k_y, 0, 0],\n", "[k_x - I*k_y, 0, 0, 0],\n", "[ 0, 0, 0, -k_x + I*k_y],\n", "[ 0, 0, -k_x - I*k_y, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{x} - k_{y} & 0 & 0\\\\- i k_{x} - k_{y} & 0 & 0 & 0\\\\0 & 0 & 0 & i k_{x} + k_{y}\\\\0 & 0 & - i k_{x} + k_{y} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_x - k_y, 0, 0],\n", "[-I*k_x - k_y, 0, 0, 0],\n", "[ 0, 0, 0, I*k_x + k_y],\n", "[ 0, 0, -I*k_x + k_y, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & k_{x}^{2} + k_{y}^{2} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2, 0, 0, 0],\n", "[ 0, 0, 0, 0],\n", "[ 0, 0, k_x**2 + k_y**2, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & k_{x}^{2} + k_{y}^{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & k_{x}^{2} + k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, k_x**2 + k_y**2, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, k_x**2 + k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "# Spatial inversion\n", "pU = np.diag([1, -1, 1, -1])\n", "pS = qsymm.inversion(3, pU)\n", "\n", "# Time reversal\n", "trU = np.array([\n", " [0.0, 0.0, -1.0, 0.0],\n", " [0.0, 0.0, 0.0, -1.0],\n", " [1.0, 0.0, 0.0, 0.0],\n", " [0.0, 1.0, 0.0, 0.0],\n", "])\n", "trS = qsymm.time_reversal(3, trU)\n", "\n", "# Rotation\n", "phi = 2.0 * np.pi / 3.0 # Impose 3-fold rotational symmetry\n", "rotU = np.array([\n", " [np.exp(1j*phi/2), 0.0, 0.0, 0.0],\n", " [0.0, np.exp(1j*phi/2), 0.0, 0.0],\n", " [0.0, 0.0, np.exp(-1j*phi/2), 0.0],\n", " [0.0, 0.0, 0.0, np.exp(-1j*phi/2)],\n", "])\n", "rotS = qsymm.rotation(1/3, axis=[0, 0, 1], U=rotU)\n", "\n", "symmetries = [rotS, trS, pS]" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "dim = 3\n", "total_power = 2" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 1, 0],\n", "[0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, 1, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & k_{x} + i k_{y}\\\\0 & 0 & k_{x} + i k_{y} & 0\\\\0 & k_{x} - i k_{y} & 0 & 0\\\\k_{x} - i k_{y} & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, k_x + I*k_y],\n", "[ 0, 0, k_x + I*k_y, 0],\n", "[ 0, k_x - I*k_y, 0, 0],\n", "[k_x - I*k_y, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & i k_{x} - k_{y}\\\\0 & 0 & i k_{x} - k_{y} & 0\\\\0 & - i k_{x} - k_{y} & 0 & 0\\\\- i k_{x} - k_{y} & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, I*k_x - k_y],\n", "[ 0, 0, I*k_x - k_y, 0],\n", "[ 0, -I*k_x - k_y, 0, 0],\n", "[-I*k_x - k_y, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & k_{z} & 0 & 0\\\\k_{z} & 0 & 0 & 0\\\\0 & 0 & 0 & - k_{z}\\\\0 & 0 & - k_{z} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, k_z, 0, 0],\n", "[k_z, 0, 0, 0],\n", "[ 0, 0, 0, -k_z],\n", "[ 0, 0, -k_z, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{z} & 0 & 0\\\\- i k_{z} & 0 & 0 & 0\\\\0 & 0 & 0 & i k_{z}\\\\0 & 0 & - i k_{z} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_z, 0, 0],\n", "[-I*k_z, 0, 0, 0],\n", "[ 0, 0, 0, I*k_z],\n", "[ 0, 0, -I*k_z, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{z}^{2} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & k_{z}^{2} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_z**2, 0, 0, 0],\n", "[ 0, 0, 0, 0],\n", "[ 0, 0, k_z**2, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & k_{z}^{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & k_{z}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, k_z**2, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, k_z**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & k_{x}^{2} + k_{y}^{2} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2, 0, 0, 0],\n", "[ 0, 0, 0, 0],\n", "[ 0, 0, k_x**2 + k_y**2, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & k_{x}^{2} + k_{y}^{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & k_{x}^{2} + k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, k_x**2 + k_y**2, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, k_x**2 + k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0],\n", "[0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{z} & k_{x} - i k_{y}\\\\k_{x} + i k_{y} & - k_{z}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ k_z, k_x - I*k_y],\n", "[k_x + I*k_y, -k_z]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} + k_{z}^{2} & 0\\\\0 & k_{x}^{2} + k_{y}^{2} + k_{z}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2 + k_z**2, 0],\n", "[ 0, k_x**2 + k_y**2 + k_z**2]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 3D real space rotation generators\n", "L = qsymm.groups.L_matrices(3)\n", "# Spin-1/2 matrices\n", "S = qsymm.groups.spin_matrices(1/2)\n", "# Spin-3/2 spin matrices\n", "J = qsymm.groups.spin_matrices(3/2)\n", "# Continuous rotation generators\n", "symmetries = [qsymm.ContinuousGroupGenerator(l, s) for l, s in zip(L, S)]\n", "\n", "dim = 3\n", "total_power = 2\n", "\n", "family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0],\n", "[0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} + k_{z}^{2} & 0\\\\0 & k_{x}^{2} + k_{y}^{2} + k_{z}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2 + k_z**2, 0],\n", "[ 0, k_x**2 + k_y**2 + k_z**2]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "symmetries = [qsymm.ContinuousGroupGenerator(l, s) for l, s in zip(L, S)]\n", "# Add inversion\n", "symmetries.append(qsymm.PointGroupElement(-np.eye(3), False, False, np.eye(2)))\n", "dim = 3\n", "total_power = 2\n", "family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0, 0, 0],\n", "[0, 1, 0, 0],\n", "[0, 0, 1, 0],\n", "[0, 0, 0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\sqrt{3} k_{z} & k_{x} - i k_{y} & 0 & 0\\\\k_{x} + i k_{y} & \\frac{\\sqrt{3} k_{z}}{3} & \\frac{2 \\sqrt{3} k_{x}}{3} - \\frac{2 \\sqrt{3} i k_{y}}{3} & 0\\\\0 & \\frac{2 \\sqrt{3} k_{x}}{3} + \\frac{2 \\sqrt{3} i k_{y}}{3} & - \\frac{\\sqrt{3} k_{z}}{3} & k_{x} - i k_{y}\\\\0 & 0 & k_{x} + i k_{y} & - \\sqrt{3} k_{z}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[sqrt(3)*k_z, k_x - I*k_y, 0, 0],\n", "[k_x + I*k_y, sqrt(3)*k_z/3, 2*sqrt(3)*k_x/3 - 2*sqrt(3)*I*k_y/3, 0],\n", "[ 0, 2*sqrt(3)*k_x/3 + 2*sqrt(3)*I*k_y/3, -sqrt(3)*k_z/3, k_x - I*k_y],\n", "[ 0, 0, k_x + I*k_y, -sqrt(3)*k_z]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\frac{k_{x}^{2}}{4} + \\frac{k_{y}^{2}}{4} + k_{z}^{2} & \\frac{\\sqrt{3} k_{x} k_{z}}{2} - \\frac{\\sqrt{3} i k_{y} k_{z}}{2} & \\frac{\\sqrt{3} k_{x}^{2}}{4} - \\frac{\\sqrt{3} i k_{x} k_{y}}{2} - \\frac{\\sqrt{3} k_{y}^{2}}{4} & 0\\\\\\frac{\\sqrt{3} k_{x} k_{z}}{2} + \\frac{\\sqrt{3} i k_{y} k_{z}}{2} & \\frac{3 k_{x}^{2}}{4} + \\frac{3 k_{y}^{2}}{4} & 0 & \\frac{\\sqrt{3} k_{x}^{2}}{4} - \\frac{\\sqrt{3} i k_{x} k_{y}}{2} - \\frac{\\sqrt{3} k_{y}^{2}}{4}\\\\\\frac{\\sqrt{3} k_{x}^{2}}{4} + \\frac{\\sqrt{3} i k_{x} k_{y}}{2} - \\frac{\\sqrt{3} k_{y}^{2}}{4} & 0 & \\frac{3 k_{x}^{2}}{4} + \\frac{3 k_{y}^{2}}{4} & - \\frac{\\sqrt{3} k_{x} k_{z}}{2} + \\frac{\\sqrt{3} i k_{y} k_{z}}{2}\\\\0 & \\frac{\\sqrt{3} k_{x}^{2}}{4} + \\frac{\\sqrt{3} i k_{x} k_{y}}{2} - \\frac{\\sqrt{3} k_{y}^{2}}{4} & - \\frac{\\sqrt{3} k_{x} k_{z}}{2} - \\frac{\\sqrt{3} i k_{y} k_{z}}{2} & \\frac{k_{x}^{2}}{4} + \\frac{k_{y}^{2}}{4} + k_{z}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ k_x**2/4 + k_y**2/4 + k_z**2, sqrt(3)*k_x*k_z/2 - sqrt(3)*I*k_y*k_z/2, sqrt(3)*k_x**2/4 - sqrt(3)*I*k_x*k_y/2 - sqrt(3)*k_y**2/4, 0],\n", "[ sqrt(3)*k_x*k_z/2 + sqrt(3)*I*k_y*k_z/2, 3*k_x**2/4 + 3*k_y**2/4, 0, sqrt(3)*k_x**2/4 - sqrt(3)*I*k_x*k_y/2 - sqrt(3)*k_y**2/4],\n", "[sqrt(3)*k_x**2/4 + sqrt(3)*I*k_x*k_y/2 - sqrt(3)*k_y**2/4, 0, 3*k_x**2/4 + 3*k_y**2/4, -sqrt(3)*k_x*k_z/2 + sqrt(3)*I*k_y*k_z/2],\n", "[ 0, sqrt(3)*k_x**2/4 + sqrt(3)*I*k_x*k_y/2 - sqrt(3)*k_y**2/4, -sqrt(3)*k_x*k_z/2 - sqrt(3)*I*k_y*k_z/2, k_x**2/4 + k_y**2/4 + k_z**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\frac{3 k_{x}^{2}}{4} + \\frac{3 k_{y}^{2}}{4} & - \\frac{\\sqrt{3} k_{x} k_{z}}{2} + \\frac{\\sqrt{3} i k_{y} k_{z}}{2} & - \\frac{\\sqrt{3} k_{x}^{2}}{4} + \\frac{\\sqrt{3} i k_{x} k_{y}}{2} + \\frac{\\sqrt{3} k_{y}^{2}}{4} & 0\\\\- \\frac{\\sqrt{3} k_{x} k_{z}}{2} - \\frac{\\sqrt{3} i k_{y} k_{z}}{2} & \\frac{k_{x}^{2}}{4} + \\frac{k_{y}^{2}}{4} + k_{z}^{2} & 0 & - \\frac{\\sqrt{3} k_{x}^{2}}{4} + \\frac{\\sqrt{3} i k_{x} k_{y}}{2} + \\frac{\\sqrt{3} k_{y}^{2}}{4}\\\\- \\frac{\\sqrt{3} k_{x}^{2}}{4} - \\frac{\\sqrt{3} i k_{x} k_{y}}{2} + \\frac{\\sqrt{3} k_{y}^{2}}{4} & 0 & \\frac{k_{x}^{2}}{4} + \\frac{k_{y}^{2}}{4} + k_{z}^{2} & \\frac{\\sqrt{3} k_{x} k_{z}}{2} - \\frac{\\sqrt{3} i k_{y} k_{z}}{2}\\\\0 & - \\frac{\\sqrt{3} k_{x}^{2}}{4} - \\frac{\\sqrt{3} i k_{x} k_{y}}{2} + \\frac{\\sqrt{3} k_{y}^{2}}{4} & \\frac{\\sqrt{3} k_{x} k_{z}}{2} + \\frac{\\sqrt{3} i k_{y} k_{z}}{2} & \\frac{3 k_{x}^{2}}{4} + \\frac{3 k_{y}^{2}}{4}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 3*k_x**2/4 + 3*k_y**2/4, -sqrt(3)*k_x*k_z/2 + sqrt(3)*I*k_y*k_z/2, -sqrt(3)*k_x**2/4 + sqrt(3)*I*k_x*k_y/2 + sqrt(3)*k_y**2/4, 0],\n", "[ -sqrt(3)*k_x*k_z/2 - sqrt(3)*I*k_y*k_z/2, k_x**2/4 + k_y**2/4 + k_z**2, 0, -sqrt(3)*k_x**2/4 + sqrt(3)*I*k_x*k_y/2 + sqrt(3)*k_y**2/4],\n", "[-sqrt(3)*k_x**2/4 - sqrt(3)*I*k_x*k_y/2 + sqrt(3)*k_y**2/4, 0, k_x**2/4 + k_y**2/4 + k_z**2, sqrt(3)*k_x*k_z/2 - sqrt(3)*I*k_y*k_z/2],\n", "[ 0, -sqrt(3)*k_x**2/4 - sqrt(3)*I*k_x*k_y/2 + sqrt(3)*k_y**2/4, sqrt(3)*k_x*k_z/2 + sqrt(3)*I*k_y*k_z/2, 3*k_x**2/4 + 3*k_y**2/4]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "symmetries = [qsymm.ContinuousGroupGenerator(l, s) for l, s in zip(L, J)]\n", "dim = 3\n", "total_power = 2\n", "family = qsymm.continuum_hamiltonian(symmetries, dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "# Double spin-1/2 representation\n", "spin = [np.kron(s, np.eye(2)) for s in qsymm.groups.spin_matrices(1/2)]\n", "# C3 rotational symmetry\n", "C3 = qsymm.rotation(1/3, axis=[0, 0, 1], spin=spin)\n", "\n", "# Time reversal\n", "TR = qsymm.time_reversal(3, spin=spin)\n", "\n", "# Mirror x\n", "Mx = qsymm.mirror([1, 0, 0], spin=spin)\n", "\n", "# Inversion\n", "IU = np.kron(np.eye(2), np.diag([1, -1]))\n", "I = qsymm.inversion(3, IU)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 1, 0],\n", "[0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, 1, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, 1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & i k_{x} + k_{y}\\\\0 & 0 & i k_{x} + k_{y} & 0\\\\0 & - i k_{x} + k_{y} & 0 & 0\\\\- i k_{x} + k_{y} & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, I*k_x + k_y],\n", "[ 0, 0, I*k_x + k_y, 0],\n", "[ 0, -I*k_x + k_y, 0, 0],\n", "[-I*k_x + k_y, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{z} & 0 & 0\\\\- i k_{z} & 0 & 0 & 0\\\\0 & 0 & 0 & i k_{z}\\\\0 & 0 & - i k_{z} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_z, 0, 0],\n", "[-I*k_z, 0, 0, 0],\n", "[ 0, 0, 0, I*k_z],\n", "[ 0, 0, -I*k_z, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{z}^{2} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & k_{z}^{2} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_z**2, 0, 0, 0],\n", "[ 0, 0, 0, 0],\n", "[ 0, 0, k_z**2, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & k_{z}^{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & k_{z}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, k_z**2, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, k_z**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & k_{x}^{2} + k_{y}^{2} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2, 0, 0, 0],\n", "[ 0, 0, 0, 0],\n", "[ 0, 0, k_x**2 + k_y**2, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & k_{x}^{2} + k_{y}^{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & k_{x}^{2} + k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, k_x**2 + k_y**2, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, k_x**2 + k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "dim = 3\n", "total_power = 2\n", "family = qsymm.continuum_hamiltonian([C3, TR, Mx, I], dim, total_power, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & -1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & -1\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[1, 0, 0, 0],\n", "[0, -1, 0, 0],\n", "[0, 0, 1, 0],\n", "[0, 0, 0, -1]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & i k_{x} + k_{y}\\\\0 & 0 & i k_{x} + k_{y} & 0\\\\0 & - i k_{x} + k_{y} & 0 & 0\\\\- i k_{x} + k_{y} & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, I*k_x + k_y],\n", "[ 0, 0, I*k_x + k_y, 0],\n", "[ 0, -I*k_x + k_y, 0, 0],\n", "[-I*k_x + k_y, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{z}^{2} & 0 & 0 & 0\\\\0 & - k_{z}^{2} & 0 & 0\\\\0 & 0 & k_{z}^{2} & 0\\\\0 & 0 & 0 & - k_{z}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_z**2, 0, 0, 0],\n", "[ 0, -k_z**2, 0, 0],\n", "[ 0, 0, k_z**2, 0],\n", "[ 0, 0, 0, -k_z**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x}^{2} + k_{y}^{2} & 0 & 0 & 0\\\\0 & - k_{x}^{2} - k_{y}^{2} & 0 & 0\\\\0 & 0 & k_{x}^{2} + k_{y}^{2} & 0\\\\0 & 0 & 0 & - k_{x}^{2} - k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x**2 + k_y**2, 0, 0, 0],\n", "[ 0, -k_x**2 - k_y**2, 0, 0],\n", "[ 0, 0, k_x**2 + k_y**2, 0],\n", "[ 0, 0, 0, -k_x**2 - k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{z} & 0 & 0\\\\- i k_{z} & 0 & 0 & 0\\\\0 & 0 & 0 & i k_{z}\\\\0 & 0 & - i k_{z} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_z, 0, 0],\n", "[-I*k_z, 0, 0, 0],\n", "[ 0, 0, 0, I*k_z],\n", "[ 0, 0, -I*k_z, 0]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "identity_terms = qsymm.continuum_hamiltonian([qsymm.groups.identity(dim, 1)], dim, total_power)\n", "identity_terms = [\n", " qsymm.Model({\n", " key: np.kron(val, np.eye(4))\n", " for key, val in term.items()\n", " })\n", " for term in identity_terms\n", "]\n", "\n", "family = qsymm.hamiltonian_generator.subtract_family(family, identity_terms, prettify=True)\n", "qsymm.display_family(family)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "dim = 3\n", "total_power = 2\n", "no_c3_family = qsymm.continuum_hamiltonian([TR, Mx, I], dim, total_power, prettify=True)\n", "no_c3_family = qsymm.hamiltonian_generator.subtract_family(no_c3_family, identity_terms, prettify=True)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & k_{x} & 0 & 0\\\\k_{x} & 0 & 0 & 0\\\\0 & 0 & 0 & - k_{x}\\\\0 & 0 & - k_{x} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, k_x, 0, 0],\n", "[k_x, 0, 0, 0],\n", "[ 0, 0, 0, -k_x],\n", "[ 0, 0, -k_x, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & i k_{x} - k_{y}\\\\0 & 0 & i k_{x} - k_{y} & 0\\\\0 & - i k_{x} - k_{y} & 0 & 0\\\\- i k_{x} - k_{y} & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, I*k_x - k_y],\n", "[ 0, 0, I*k_x - k_y, 0],\n", "[ 0, -I*k_x - k_y, 0, 0],\n", "[-I*k_x - k_y, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{y} k_{z} & 0 & 0 & 0\\\\0 & - k_{y} k_{z} & 0 & 0\\\\0 & 0 & k_{y} k_{z} & 0\\\\0 & 0 & 0 & - k_{y} k_{z}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_y*k_z, 0, 0, 0],\n", "[ 0, -k_y*k_z, 0, 0],\n", "[ 0, 0, k_y*k_z, 0],\n", "[ 0, 0, 0, -k_y*k_z]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- k_{x}^{2} + k_{y}^{2} & 0 & 0 & 0\\\\0 & k_{x}^{2} - k_{y}^{2} & 0 & 0\\\\0 & 0 & - k_{x}^{2} + k_{y}^{2} & 0\\\\0 & 0 & 0 & k_{x}^{2} - k_{y}^{2}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[-k_x**2 + k_y**2, 0, 0, 0],\n", "[ 0, k_x**2 - k_y**2, 0, 0],\n", "[ 0, 0, -k_x**2 + k_y**2, 0],\n", "[ 0, 0, 0, k_x**2 - k_y**2]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & k_{z}\\\\0 & 0 & k_{z} & 0\\\\0 & k_{z} & 0 & 0\\\\k_{z} & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, k_z],\n", "[ 0, 0, k_z, 0],\n", "[ 0, k_z, 0, 0],\n", "[k_z, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & i k_{y} & 0 & 0\\\\- i k_{y} & 0 & 0 & 0\\\\0 & 0 & 0 & i k_{y}\\\\0 & 0 & - i k_{y} & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, I*k_y, 0, 0],\n", "[-I*k_y, 0, 0, 0],\n", "[ 0, 0, 0, I*k_y],\n", "[ 0, 0, -I*k_y, 0]])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "new_terms = qsymm.hamiltonian_generator.subtract_family(no_c3_family, family, prettify=True)\n", "qsymm.display_family(new_terms)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 1 & 0 & 0\\\\1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1\\\\0 & 0 & 1 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 1, 0, 0],\n", "[1, 0, 0, 0],\n", "[0, 0, 0, 1],\n", "[0, 0, 1, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & i\\\\0 & 0 & - i & 0\\\\0 & i & 0 & 0\\\\- i & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, 0, I],\n", "[ 0, 0, -I, 0],\n", "[ 0, I, 0, 0],\n", "[-I, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}k_{x} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & - k_{x} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[k_x, 0, 0, 0],\n", "[ 0, 0, 0, 0],\n", "[ 0, 0, -k_x, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & k_{x} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - k_{x}\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0, 0, 0],\n", "[0, k_x, 0, 0],\n", "[0, 0, 0, 0],\n", "[0, 0, 0, -k_x]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & i k_{x} & 0\\\\0 & 0 & 0 & 0\\\\- i k_{x} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[ 0, 0, I*k_x, 0],\n", "[ 0, 0, 0, 0],\n", "[-I*k_x, 0, 0, 0],\n", "[ 0, 0, 0, 0]])" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & i k_{x}\\\\0 & 0 & 0 & 0\\\\0 & - 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