DLR tutorial

ModelDMFT/06-SYK_DLR.ipynb

@@ -0,0 +1,66 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "329ad08d",
+   "metadata": {},
+   "source": [
+    "# Self-consistent solution of the SYK equation using the Discrete Lehmann Representation\n",
+    "\n",
+    "The goal of this tutorial is to showcase the use of the discrete Lehmann representation (DLR) to solve the Dyson equation self-consistently (i.e., given an expression for the self-energy in terms of the Green's function), by implementing a simple self-consistent loop for the Sachdev-Ye-Kitaev (SYK) model. For background on the DLR and its use in TRIQS, we refer to the [TRIQS documentation](https://triqs.github.io/triqs/latest/userguide/python/tutorials/Basics/solutions/01s-Greens_functions.html#Compact-meshes-for-imaginary-time-/-frequency:-DLR-Green%E2%80%99s-function). We note that although this is not a DMFT calculation, some of the main steps (Fourier transform between imaginary time and Matsubara frequency, solving the Dyson equation) are similar to those in a DMFT calculation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fee0f10b",
+   "metadata": {},
+   "source": [
+    "### Constructing DLR Green's Functions\n",
+    "\n",
+    "We begin by setting various parameters, building a DLR imaginary frequency mesh, and creating Green's function and self-energy containers from this mesh. The DLR parameters are the desired accuracy, `dlr_error`, and the problem bandwidth or frequency cutoff, `w_max`. We also set some physical problem parameters, as well as a maximum number of iterations for the self-consistency, a self-consistency convergence tolerance, and a mixing parameter $\\alpha$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d5547d81",
+   "metadata": {},
+   "source": [
+    "### Exercise 1 \n",
+    "We first solve the $0+1$-dimensional (impurity) Sachdev-Ye-Kitaev (SYK) model self-consistently. The Hamiltonian is given by \n",
+    "$$H = \\frac{1}{(2N)^{3/2}}\\sum^N_{ijkl} J_{ijkl} c^\\dagger_i c^\\dagger_j c_k c_l$$\n",
+    "where the $J_{ijkl}$ are random Gaussian couplings between $N$ fermions with constant variance $J^2$.\n",
+    " This model has many interesting applications for non-Fermi liquids, or strange metals, and quantum criticality. You can read more about the model and its extensions [(here)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.94.035004). Here we are interested in a simple form of the self-energy in the large-$N$ limit of the model, given by\n",
+    "\\begin{equation}\n",
+    "\\Sigma(\\tau) = -J^2 G^2(\\tau)G(-\\tau).\n",
+    "\\end{equation}\n",
+    "Combining this self-energy with the Dyson equation yields a self-consistent set of equations. These were solved using the DLR with weighted fixed point iteration in Sec. VII of [(this paper)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.105.235115). Following that reference, it is your turn to write a self-consistent loop to solve for $G$.\n",
+    "\n",
+    "As shown in the paper, as we approach the $\\beta = \\infty$ limit, the result should begin to match the conformal solution of these equations. Try comparing to the conformal solution! We should already have quite good agreement at $\\beta = 1000$.\n",
+    "\\begin{equation}\n",
+    "G_c(\\tau)=-\\frac{\\pi^{1 / 4}}{\\sqrt{2 \\beta}}\\left(\\sin \\left(\\frac{\\pi \\tau}{\\beta}\\right)\\right)^{-1 / 2}\n",
+    "\\end{equation}"
+   ]
+  }
+ ],
+ "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": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}