|
| 1 | +{ |
| 2 | + "cells": [ |
| 3 | + { |
| 4 | + "cell_type": "markdown", |
| 5 | + "id": "a5a75eb6", |
| 6 | + "metadata": {}, |
| 7 | + "source": [ |
| 8 | + "## Self-consistent spectral unfolding\n", |
| 9 | + "\n", |
| 10 | + " by M.Süzen\n", |
| 11 | + " (c) 2025\n", |
| 12 | + "\n", |
| 13 | + "Prerequisite to this tutorial is finishing the `wigner_semicircle.ipynb` and \n", |
| 14 | + "`wigner_dyson_spacing.ipynb` lectures first as a background. \n", |
| 15 | + "\n", |
| 16 | + "Spectral unfolding appears in quantum mechanical description of atomic systems \n", |
| 17 | + "from random matrix theory perspective. The core idea is to remove local fluctuations \n", |
| 18 | + "in the analysis of the spectra, and obtain a new `unfolded spectra`. \n", |
| 19 | + "\n", |
| 20 | + "In this lecture notes we will understand. \n", |
| 21 | + "\n", |
| 22 | + "* What does unfolding entails. \n", |
| 23 | + "* A robust way of doing this in detail. This is something \n", |
| 24 | + " called `self-consistent spectral unfolding`\n", |
| 25 | + "* How to just use `leymosun`'s tools to unfold a spectra: \n", |
| 26 | + " We will use `leymosun`" |
| 27 | + ] |
| 28 | + }, |
| 29 | + { |
| 30 | + "cell_type": "markdown", |
| 31 | + "id": "04206576", |
| 32 | + "metadata": {}, |
| 33 | + "source": [ |
| 34 | + "## Needed components \n", |
| 35 | + "\n", |
| 36 | + "Load all tools from `leymosun` that we would use" |
| 37 | + ] |
| 38 | + }, |
| 39 | + { |
| 40 | + "cell_type": "code", |
| 41 | + "execution_count": null, |
| 42 | + "id": "e37e3d19", |
| 43 | + "metadata": {}, |
| 44 | + "outputs": [], |
| 45 | + "source": [ |
| 46 | + "from leymosun.spectral import unfold_spectra, empirical_spectral_density, nnsd\n", |
| 47 | + "from leymosun.gaussian import goe \n", |
| 48 | + "import numpy as np\n", |
| 49 | + "import matplotlib.pyplot as plt\n", |
| 50 | + "import leymosun \n", |
| 51 | + "leymosun.__version__" |
| 52 | + ] |
| 53 | + }, |
| 54 | + { |
| 55 | + "cell_type": "markdown", |
| 56 | + "id": "e8edeb89", |
| 57 | + "metadata": {}, |
| 58 | + "source": [ |
| 59 | + "## Definition of spectral unfolding statistically \n", |
| 60 | + "\n", |
| 61 | + "The idea of unfolding is removing the fluctuations in spectra locally. \n", |
| 62 | + "Here, locally means in the vicinity of any given eigenvalues (singular values). \n", |
| 63 | + " This can be measured by nearest-neighbor spacing values $\\Delta e_{i}$, fluctuations. \n", |
| 64 | + " Then the mean fluctuation over $N$ sorted eigenvalues \n", |
| 65 | + " $\\Delta e_{i} = |e_{i}-e_{i-1}|$ should be close to $1.0$ \n", |
| 66 | + "\n", |
| 67 | + "If we `transformed` the empirical density $\\rho(e)$, via an `unfolding` procedure, \n", |
| 68 | + "that yields this mean value, $\\frac{1}{N} \\sum_{i=1}^{N} \\Delta e_{i} \\rightarrow 1.0$. \n", |
| 69 | + "It is important that it isn't smoothen as in smoothing the data, it is thought as of \n", |
| 70 | + "unfolding a rough paper or transformation or mapping, $P(x)$. $P(x)$ here is simply \n", |
| 71 | + "ranked of the sorted eigenvalues and similar to the concept of \n", |
| 72 | + "{\\it density of states} in the `unfolded` `raw` case.\n" |
| 73 | + ] |
| 74 | + }, |
| 75 | + { |
| 76 | + "cell_type": "markdown", |
| 77 | + "id": "53f23d7a", |
| 78 | + "metadata": {}, |
| 79 | + "source": [ |
| 80 | + "## Polynomial Unfolding: Self-consistent approach \n", |
| 81 | + "\n", |
| 82 | + "The approach starts with fitting a $n$ degree polynomial, \n", |
| 83 | + "$P(e) = \\sum_{k=0}^{k=n} a_{k} e^{k}$ to eigenvalues $e_{i}$. \n", |
| 84 | + "Then, we check which degree leads mean fluctuation of $1.0$. \n", |
| 85 | + "Essentially, we fit different degree polynomials on $(y, e_{i})$, \n", |
| 86 | + "where by $y$ is the rank order. This is what implemented in \n", |
| 87 | + "`leymosun`'s `unfold_spectra` functionality. " |
| 88 | + ] |
| 89 | + }, |
| 90 | + { |
| 91 | + "cell_type": "markdown", |
| 92 | + "id": "c89caa13", |
| 93 | + "metadata": {}, |
| 94 | + "source": [ |
| 95 | + "## Example: Raw and Unfolded spectrum\n", |
| 96 | + "\n", |
| 97 | + "In this example we will generate spectra from GOE and plot \n", |
| 98 | + "$(y, e_{i})$, where $e_{i}$. Let's just use ensemble of size 1. \n", |
| 99 | + "We choose a smaller matrix to see how unfolded eigenvalues looks \n", |
| 100 | + "close up as stair function. We compute eigenvalues and plot the \n", |
| 101 | + "sorted eigenvalues. " |
| 102 | + ] |
| 103 | + }, |
| 104 | + { |
| 105 | + "cell_type": "code", |
| 106 | + "execution_count": null, |
| 107 | + "id": "8bfdd8ba", |
| 108 | + "metadata": {}, |
| 109 | + "outputs": [], |
| 110 | + "source": [ |
| 111 | + "A = goe(50)\n", |
| 112 | + "eigenvalues, _, _ = empirical_spectral_density([A])\n", |
| 113 | + "eigenvalues = sorted(eigenvalues[0]) # as this was ensemble and raw sort them\n", |
| 114 | + "y= np.arange(1,len(eigenvalues))" |
| 115 | + ] |
| 116 | + }, |
| 117 | + { |
| 118 | + "cell_type": "code", |
| 119 | + "execution_count": null, |
| 120 | + "id": "ac75ed3c", |
| 121 | + "metadata": {}, |
| 122 | + "outputs": [], |
| 123 | + "source": [ |
| 124 | + "plt.stairs(y, eigenvalues)\n", |
| 125 | + "plt.title(\"Raw Spectra \")\n", |
| 126 | + "plt.xlabel(\"Eigenvalue Locations\")\n", |
| 127 | + "plt.ylabel(\"Density of States\")\n" |
| 128 | + ] |
| 129 | + }, |
| 130 | + { |
| 131 | + "cell_type": "markdown", |
| 132 | + "id": "0ecc958f", |
| 133 | + "metadata": {}, |
| 134 | + "source": [ |
| 135 | + "Let's compute the `mean fluctuations`, which must be far from 1.0. " |
| 136 | + ] |
| 137 | + }, |
| 138 | + { |
| 139 | + "cell_type": "code", |
| 140 | + "execution_count": null, |
| 141 | + "id": "feddb728", |
| 142 | + "metadata": {}, |
| 143 | + "outputs": [], |
| 144 | + "source": [ |
| 145 | + "np.mean(np.diff(eigenvalues))" |
| 146 | + ] |
| 147 | + }, |
| 148 | + { |
| 149 | + "cell_type": "markdown", |
| 150 | + "id": "f10187e9", |
| 151 | + "metadata": {}, |
| 152 | + "source": [ |
| 153 | + "Let's now unfold the spectrum with a polynomial self-consistent approach, \n", |
| 154 | + "here we use only up to a degree $10$ and do not remove outliers as this is a too small matrix \n", |
| 155 | + "for demonstration purposes. Then we see that `mean fluctuations` are close to $1.0$" |
| 156 | + ] |
| 157 | + }, |
| 158 | + { |
| 159 | + "cell_type": "code", |
| 160 | + "execution_count": null, |
| 161 | + "id": "9c777290", |
| 162 | + "metadata": {}, |
| 163 | + "outputs": [], |
| 164 | + "source": [ |
| 165 | + "unfolded_spec, _, _ = unfold_spectra(eigenvalues=eigenvalues, iqr=False, deg_max=20)\n", |
| 166 | + "np.mean(np.diff(unfolded_spec)) # ~1.0" |
| 167 | + ] |
| 168 | + }, |
| 169 | + { |
| 170 | + "cell_type": "markdown", |
| 171 | + "id": "b8630a13", |
| 172 | + "metadata": {}, |
| 173 | + "source": [ |
| 174 | + "Let's now compare local fluctuations over density of states." |
| 175 | + ] |
| 176 | + }, |
| 177 | + { |
| 178 | + "cell_type": "code", |
| 179 | + "execution_count": null, |
| 180 | + "id": "24d017c8", |
| 181 | + "metadata": {}, |
| 182 | + "outputs": [], |
| 183 | + "source": [ |
| 184 | + "plt.stairs(np.cumsum(np.diff(eigenvalues))/y, label= \"Raw spectrum\")\n", |
| 185 | + "plt.stairs(np.cumsum(np.diff(unfolded_spec))/y, label=\"Unfolded spectrum \")\n", |
| 186 | + "plt.title(\"Effect of Spectral Unfolding \")\n", |
| 187 | + "plt.ylabel(\"Eigenvalue Mean Fluctuations\")\n", |
| 188 | + "plt.xlabel(\"Density of States\")\n", |
| 189 | + "plt.xlim([1,49])\n", |
| 190 | + "plt.legend()" |
| 191 | + ] |
| 192 | + }, |
| 193 | + { |
| 194 | + "cell_type": "markdown", |
| 195 | + "id": "281cc145", |
| 196 | + "metadata": {}, |
| 197 | + "source": [ |
| 198 | + "**Exercise** Repeat the analysis for larger ensemble, $M=50$ and $N=500$. Plot with uncertainty quantification. Use `leymosun`'s `bootstrap_observed_matrix_ci` utility." |
| 199 | + ] |
| 200 | + } |
| 201 | + ], |
| 202 | + "metadata": { |
| 203 | + "kernelspec": { |
| 204 | + "display_name": "Python 3", |
| 205 | + "language": "python", |
| 206 | + "name": "python3" |
| 207 | + }, |
| 208 | + "language_info": { |
| 209 | + "codemirror_mode": { |
| 210 | + "name": "ipython", |
| 211 | + "version": 3 |
| 212 | + }, |
| 213 | + "file_extension": ".py", |
| 214 | + "mimetype": "text/x-python", |
| 215 | + "name": "python", |
| 216 | + "nbconvert_exporter": "python", |
| 217 | + "pygments_lexer": "ipython3", |
| 218 | + "version": "3.11.6" |
| 219 | + } |
| 220 | + }, |
| 221 | + "nbformat": 4, |
| 222 | + "nbformat_minor": 5 |
| 223 | +} |
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