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dev/assets/notebooks/ising2d.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "source": [
+    "# Classical Ising model\n",
+    "\n",
+    "This tutorial illustrates simulation of the classical 2D Ising model."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "using Sunny, Plots"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "Sunny expects a 3D `Crystal` unit cell. To model a square lattice, we\n",
+    "create an orthogonal unit cell where the $z$-spacing is distinct from the $x$\n",
+    "and $y$ spacing."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "a = 1\n",
+    "latvecs = lattice_vectors(a,a,10a,90,90,90)\n",
+    "crystal = Crystal(latvecs, [[0,0,0]])"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "Create a `System` of spins with linear size `L` in the $x$ and $y$\n",
+    "directions, and only one layer in the $z$ direction. The option `:dipole`\n",
+    "means that the system will store Heisenberg spins, as opposed to SU($N$)\n",
+    "coherent states. Polarize the initial spin configuration using\n",
+    "`polarize_spins!`. Following the Ising convention, we will restrict\n",
+    "these spins to the $z$-axis and give them magnitude $S=1$.\n",
+    "\n",
+    "By default, Sunny uses physical units, e.g. magnetic field in tesla. Here we\n",
+    "specify an alternative `Units` system, so that the Zeeman coupling\n",
+    "between the spin dipole $𝐬$ and an external field $𝐁$ has the dimensionless\n",
+    "form $-𝐁⋅𝐬$."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "L = 128\n",
+    "sys = System(crystal, (L,L,1), [SpinInfo(1, S=1, g=1)], :dipole, units=Units.theory, seed=0)\n",
+    "polarize_spins!(sys, (0,0,1))"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "Use `set_exchange!` to include a ferromagnetic Heisenberg interaction\n",
+    "along nearest-neighbor bonds. The `Bond` below connects two spins\n",
+    "displaced by one lattice constant in the $x$-direction. This interaction will\n",
+    "be propagated to all nearest-neighbors bonds in the system, consistent with\n",
+    "the symmetries of the square lattice."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "set_exchange!(sys, -1.0, Bond(1,1,(1,0,0)))"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "If an external field is desired, it can be set using\n",
+    "`set_external_field!`."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "B = 0\n",
+    "set_external_field!(sys, (0,0,B))"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "The critical temperature for the Ising model is known analytically."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "Tc = 2/log(1+√2)"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "Use a `LocalSampler` to perform `nsweeps` Monte Carlo sweeps. A sweep\n",
+    "consists of, on average, one trial update per spin in the system. Each\n",
+    "proposed update is accepted or rejected according to the Metropolis acceptance\n",
+    "probability. As its name suggests, the `propose_flip` function will\n",
+    "only propose pure spin flips, $𝐬 \\rightarrow -𝐬$."
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "nsweeps = 4000\n",
+    "sampler = LocalSampler(kT=Tc, propose=propose_flip)\n",
+    "for i in 1:nsweeps\n",
+    "    step!(sys, sampler)\n",
+    "end"
+   ],
+   "metadata": {},
+   "execution_count": null
+  },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "Plot the Ising spins by extracting the $z$-component of the dipoles"
+   ],
+   "metadata": {}
+  },
+  {
+   "outputs": [],
+   "cell_type": "code",
+   "source": [
+    "heatmap(reshape([s.z for s in sys.dipoles], (L,L)))"
+   ],
+   "metadata": {},
+   "execution_count": null
+  }
+ ],
+ "nbformat_minor": 3,
+ "metadata": {
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.9.3"
+  },
+  "kernelspec": {
+   "name": "julia-1.9",
+   "display_name": "Julia 1.9.3",
+   "language": "julia"
+  }
+ },
+ "nbformat": 4
+}