demo.ipynb

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    "# Demo\n",
    "## Vortex dynamics\n",
    "\n",
    "In this example, we are going to simulate vortex core dynamics. After creating a vortex structure, we are first going to displace it by applying an external magnetic field. We will then turn off the external field, and compute the time-development of the system, and then be able to see the dynamics of the vortex core.\n",
    "\n",
    "The sample is a two-dimensional Permalloy disk sample with $r=50 \\,\\text{nm}$ edge length and $10\\,\\text{nm}$ thickness. Its energy equation consists of ferromagnetic exchange, Zeeman, and demagnetisation energy terms:\n",
    "    \n",
    "$$E = \\int_{V} \\left[-A\\mathbf{m}\\cdot\\nabla^{2}\\mathbf{m} - \\mu_{0}M_\\text{s}\\mathbf{m}\\cdot\\mathbf{H} + w_\\text{d}\\right] \\text{d}V,$$\n",
    "\n",
    "where $A = 13 \\,\\text{pJ}\\,\\text{m}^{-1}$ is the exchange energy constant, $M_\\text{s} = 8 \\times 10^{5} \\,\\text{A}\\,\\text{m}^{-1}$ magnetisation saturation, $w_\\text{d}$ demagnetisation energy density, $\\mathbf{H}$ an external magnetic field, and $\\mathbf{m}=\\mathbf{M}/M_\\text{s}$ the normalised magnetisation field.\n",
    "\n",
    "The magnetisation dynamics is governed by the Landau-Lifshitz-Gilbert equation consisting of precession and damping terms:\n",
    "\n",
    "$$\\frac{\\partial\\mathbf{m}}{\\partial t} = -\\frac{\\gamma_{0}}{1+\\alpha^{2}}\\mathbf{m}\\times\\mathbf{H}_\\text{eff} - \\frac{\\gamma_{0}\\alpha}{1+\\alpha^{2}}\\mathbf{m}\\times(\\mathbf{m}\\times\\mathbf{H}_\\text{eff}),$$\n",
    "\n",
    "where $\\gamma_{0} = 2.211 \\times 10^{5} \\,\\text{m}\\,\\text{A}^{-1}\\,\\text{s}^{-1}$ and $\\alpha = 0.05$ is the Gilbert damping.\n",
    "\n",
    "The (initial) magnetisation field is a vortex state, whose magnetisation at each point $(x, y, z)$ in the sample can be represented as $(m_{x}, m_{y}, m_{z}) = (-cy, cx, 0.1)$, with $c = 10^{-9} \\text{m}^{-1}$."
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   "source": [
    "# Some initial configurations\n",
    "%config InlineBackend.figure_formats = ['svg']  # output matplotlib plots as SVG\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "pd.options.display.max_rows = 5\n",
    "pd.options.display.float_format = \"{:,.2e}\".format"
   ]
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   "source": [
    "## System initialisation\n",
    "\n",
    "The Ubermag code for defining the micromagnetic system is:"
   ]
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   "source": [
    "import discretisedfield as df\n",
    "import micromagneticmodel as mm\n",
    "\n",
    "# Geometry\n",
    "r = 50e-9  # Radius of the thin nano magnetic disk (m)\n",
    "thickness = 10e-9  # sample thickness (m)\n",
    "\n",
    "# Material (Permalloy) parameters\n",
    "Ms = 8e5  # saturation magnetisation (A/m)\n",
    "A = 13e-12  # exchange energy constant (J/m)\n",
    "\n",
    "# Dynamics (LLG equation) parameters\n",
    "gamma0 = mm.consts.gamma0  # gyromagnetic ratio (m/As)\n",
    "alpha = 0.05  # Gilbert damping\n",
    "\n",
    "system = mm.System(name=\"vortex_dynamics\")\n",
    "\n",
    "# Energy equation. We omit Zeeman energy term, because H=0.\n",
    "system.energy = mm.Exchange(A=A) + mm.Demag()\n",
    "\n",
    "# Dynamics equation\n",
    "system.dynamics = mm.Precession(gamma0=gamma0) + mm.Damping(alpha=alpha)\n",
    "\n",
    "\n",
    "# initial magnetisation state\n",
    "def m_init(point):\n",
    "    x, y, _ = point\n",
    "    c = 1e9  # (1/m)\n",
    "    return (-c * y, c * x, 0.1)\n",
    "\n",
    "\n",
    "# Defining the geometry of the material as a circular disk\n",
    "def Ms_func(point):\n",
    "    x, y, _ = point\n",
    "    if x**2 + y**2 <= r**2:\n",
    "        return Ms\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "\n",
    "# Sample's centre is placed at origin\n",
    "region = df.Region(p1=(-r, -r, -thickness / 2), p2=(r, r, thickness / 2))\n",
    "mesh = df.Mesh(region=region, cell=(5e-9, 5e-9, 10e-9))\n",
    "\n",
    "system.m = df.Field(mesh, nvdim=3, value=m_init, norm=Ms_func, valid=\"norm\")"
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   "source": [
    "The system object is now defined and we can investigate some of its properties:"
   ]
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      "text/latex": [
       "$- A \\mathbf{m} \\cdot \\nabla^{2} \\mathbf{m}-\\frac{1}{2}\\mu_{0}M_\\text{s}\\mathbf{m} \\cdot \\mathbf{H}_\\text{d}$"
      ],
      "text/plain": [
       "Exchange(A=1.3e-11) + Demag()"
      ]
     },
     "execution_count": 3,
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     "output_type": "execute_result"
    }
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   "source": [
    "system.energy"
   ]
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      "text/latex": [
       "$-\\frac{\\gamma_{0}}{1 + \\alpha^{2}} \\mathbf{m} \\times \\mathbf{H}_\\text{eff}-\\frac{\\gamma_{0} \\alpha}{1 + \\alpha^{2}} \\mathbf{m} \\times (\\mathbf{m} \\times \\mathbf{H}_\\text{eff})$"
      ],
      "text/plain": [
       "Precession(gamma0=221276.14872118403) + Damping(alpha=0.05)"
      ]
     },
     "execution_count": 4,
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    "system.dynamics"
   ]
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    "## Energy minimisation\n",
    "To carry out micromagnetic simulation, we need to use a micromagnetic calulator. We are going to use OOMMF for this.\n",
    "We can now relax the system in the absence of external magnetic field using energy minimisation driver (`MinDriver`):"
   ]
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running OOMMF (ExeOOMMFRunner)[2024/08/09 18:15]... (0.4 s)\n"
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   ],
   "source": [
    "import oommfc as oc  # Micromagnetic Calculator\n",
    "\n",
    "md = oc.MinDriver()\n",
    "md.drive(system)\n",
    "\n",
    "system.m.orientation.sel(\"z\").mpl()"
   ]
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   "source": [
    "## Displacement with magnetic field\n",
    "\n",
    "Now, we have a relaxed vortex state, with its core at the centre of the sample. As the next step, we want to add an external magnetic field $H=3.4 \\times 10^{4}\\,\\text{Am}^{-1}$ in the positive $x$-direction to displace the vortex core. We do that by adding the Zeeman energy term to the energy equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running OOMMF (ExeOOMMFRunner)[2024/08/09 18:15]... (0.5 s)\n"
     ]
    },
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   "source": [
    "H = (3.4e4, 0, 0)  # an external magnetic field (A/m)\n",
    "\n",
    "system.energy += mm.Zeeman(H=H)\n",
    "\n",
    "md.drive(system)\n",
    "system.m.orientation.sel(\"z\").mpl()"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Free relaxation\n",
    "\n",
    "The vortex core is now displaced in the positive $y$-direction. As the last step, we are going to turn off the external magnetic field and simulate dynamics using `TimeDriver`. We are going to run simulation for $20\\,\\text{ns}$ and save the magnetisation in $500$ steps."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
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       "model_id": "9b2bae448a8841dfaedaeb5d1dc76300",
       "version_major": 2,
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      "text/plain": [
       "Running OOMMF (ExeOOMMFRunner):   0%|          | 0/500 files written [00:00]"
      ]
     },
     "metadata": {},
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    },
    {
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     "output_type": "stream",
     "text": [
      "Running OOMMF (ExeOOMMFRunner)[2024/08/09 18:15] took 36.6 s\n"
     ]
    }
   ],
   "source": [
    "system.energy -= mm.Zeeman(H=H)\n",
    "\n",
    "td = oc.TimeDriver()\n",
    "td.drive(system, t=20e-9, n=500, verbose=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The final magnetisation state shows that the vortex core has moved back to the sample's centre."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
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   "source": [
    "## Data analysis\n",
    "\n",
    "### Spatially averaged data\n",
    "The table with scalar data saved during the simulation. Each row corresponds to one of the 500 saved configurations. We only show selected columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "tags": [
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   },
   "outputs": [
    {
     "data": {
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       "      <th>E</th>\n",
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       "         t       mx       my       mz        E\n",
       "0 4.00e-11 4.18e-01 5.03e-02 1.82e-02 3.94e-18\n",
       "1 8.00e-11 4.27e-01 1.92e-01 1.07e-02 3.89e-18\n",
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   "source": [
    "We can now plot the average $m_{x}$, $m_{y}$ and $m_{z}$ values as taken from the table as a function of time to give us an idea of the vortex core position."
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   "cell_type": "code",
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    "system.table.mpl(y=[\"mx\", \"my\", \"mz\"])"
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   "source": [
    "### Spatially resolved data\n",
    "Finally, we are going to have a look at the magnetisation field at different time-steps using `micromagneticdata`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>drive_number</th>\n",
       "      <th>date</th>\n",
       "      <th>time</th>\n",
       "      <th>driver</th>\n",
       "      <th>adapter</th>\n",
       "      <th>n_threads</th>\n",
       "      <th>t</th>\n",
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       "      <th>0</th>\n",
       "      <td>0</td>\n",
       "      <td>2024-08-09</td>\n",
       "      <td>18:15:22</td>\n",
       "      <td>MinDriver</td>\n",
       "      <td>oommfc</td>\n",
       "      <td>None</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>2024-08-09</td>\n",
       "      <td>18:15:23</td>\n",
       "      <td>MinDriver</td>\n",
       "      <td>oommfc</td>\n",
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       "      <td>2024-08-09</td>\n",
       "      <td>18:15:23</td>\n",
       "      <td>TimeDriver</td>\n",
       "      <td>oommfc</td>\n",
       "      <td>None</td>\n",
       "      <td>2.00e-08</td>\n",
       "      <td>5.00e+02</td>\n",
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      ],
      "text/plain": [
       "   drive_number        date      time      driver adapter n_threads        t  \\\n",
       "0             0  2024-08-09  18:15:22   MinDriver  oommfc      None      NaN   \n",
       "1             1  2024-08-09  18:15:23   MinDriver  oommfc      None      NaN   \n",
       "2             2  2024-08-09  18:15:23  TimeDriver  oommfc      None 2.00e-08   \n",
       "\n",
       "         n  \n",
       "0      NaN  \n",
       "1      NaN  \n",
       "2 5.00e+02  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import micromagneticdata as mdata\n",
    "\n",
    "data = mdata.Data(system.name)\n",
    "data.info"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To interactively inspect the time dependent magnetisation, we use `data[-1]` to refer to the last drive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "tags": [
     "nbval-ignore-output"
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   "outputs": [
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i < inline_js.length; i++) {\n\ttry {\n          inline_js[i].call(root, root.Bokeh);\n\t} catch(e) {\n\t  if (!reloading) {\n\t    throw e;\n\t  }\n\t}\n      }\n      // Cache old bokeh versions\n      if (Bokeh != undefined && !reloading) {\n\tvar NewBokeh = root.Bokeh;\n\tif (Bokeh.versions === undefined) {\n\t  Bokeh.versions = new Map();\n\t}\n\tif (NewBokeh.version !== Bokeh.version) {\n\t  Bokeh.versions.set(NewBokeh.version, NewBokeh)\n\t}\n\troot.Bokeh = Bokeh;\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    }\n    root._bokeh_is_initializing = false\n  }\n\n  function load_or_wait() {\n    // Implement a backoff loop that tries to ensure we do not load multiple\n    // versions of Bokeh and its dependencies at the same time.\n    // In recent versions we use the root._bokeh_is_initializing flag\n    // to determine whether there is an ongoing attempt to initialize\n    // bokeh, however for backward compatibility we also try to ensure\n    // that we do not start loading a newer (Panel>=1.0 and Bokeh>3) version\n    // before older versions are fully initialized.\n    if (root._bokeh_is_initializing && Date.now() > root._bokeh_timeout) {\n      root._bokeh_is_initializing = false;\n      root._bokeh_onload_callbacks = undefined;\n      console.log(\"Bokeh: BokehJS was loaded multiple times but one version failed to initialize.\");\n      load_or_wait();\n    } else if (root._bokeh_is_initializing || (typeof root._bokeh_is_initializing === \"undefined\" && root._bokeh_onload_callbacks !== undefined)) {\n      setTimeout(load_or_wait, 100);\n    } else {\n      root._bokeh_is_initializing = true\n      root._bokeh_onload_callbacks = []\n      var bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n      if (!reloading && !bokeh_loaded) {\n\troot.Bokeh = undefined;\n      }\n      load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n\trun_inline_js();\n      });\n    }\n  }\n  // Give older versions of the autoload script a head-start to ensure\n  // they initialize before we start loading newer version.\n  setTimeout(load_or_wait, 100)\n}(window));",
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else if (comm !== null) {\n    comm.send({event_type: 'delete', 'id': id});\n  }\n  delete PyViz.plot_index[id];\n  if ((window.Bokeh !== undefined) & (id in window.Bokeh.index)) {\n    var doc = window.Bokeh.index[id].model.document\n    doc.clear();\n    const i = window.Bokeh.documents.indexOf(doc);\n    if (i > -1) {\n      window.Bokeh.documents.splice(i, 1);\n    }\n  }\n}\n\n/**\n * Handle kernel restart event\n */\nfunction handle_kernel_cleanup(event, handle) {\n  delete PyViz.comms[\"hv-extension-comm\"];\n  window.PyViz.plot_index = {}\n}\n\n/**\n * Handle update_display_data messages\n */\nfunction handle_update_output(event, handle) {\n  handle_clear_output(event, {cell: {output_area: handle.output_area}})\n  handle_add_output(event, handle)\n}\n\nfunction register_renderer(events, OutputArea) {\n  function append_mime(data, metadata, element) {\n    // create a DOM node to render to\n    var toinsert = this.create_output_subarea(\n    metadata,\n    CLASS_NAME,\n    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      "application/vnd.holoviews_load.v0+json": ""
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<style>*[data-root-id],\n",
       "*[data-root-id] > * {\n",
       "  box-sizing: border-box;\n",
       "  font-family: var(--jp-ui-font-family);\n",
       "  font-size: var(--jp-ui-font-size1);\n",
       "  color: var(--vscode-editor-foreground, var(--jp-ui-font-color1));\n",
       "}\n",
       "\n",
       "/* Override VSCode background color */\n",
       ".cell-output-ipywidget-background:has(\n",
       "    > .cell-output-ipywidget-background > .lm-Widget > *[data-root-id]\n",
       "  ),\n",
       ".cell-output-ipywidget-background:has(> .lm-Widget > *[data-root-id]) {\n",
       "  background-color: transparent !important;\n",
       "}\n",
       "</style>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
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     "data": {
      "application/vnd.holoviews_exec.v0+json": "",
      "text/html": [
       "<div id='p1002'>\n",
       "  <div id=\"e817d17f-4247-4b3f-b220-f27753e7884c\" data-root-id=\"p1002\" style=\"display: contents;\"></div>\n",
       "</div>\n",
       "<script type=\"application/javascript\">(function(root) {\n",
       "  var docs_json = {\"71eee380-b348-49f5-8139-f2430ce449af\":{\"version\":\"3.4.2\",\"title\":\"Bokeh Application\",\"roots\":[{\"type\":\"object\",\"name\":\"panel.models.browser.BrowserInfo\",\"id\":\"p1002\"},{\"type\":\"object\",\"name\":\"panel.models.comm_manager.CommManager\",\"id\":\"p1003\",\"attributes\":{\"plot_id\":\"p1002\",\"comm_id\":\"ef5f6fc671ab4ed8b3ed3798d7948ddf\",\"client_comm_id\":\"e81f12194da8415ab61fab41a077f519\"}}],\"defs\":[{\"type\":\"model\",\"name\":\"ReactiveHTML1\"},{\"type\":\"model\",\"name\":\"FlexBox1\",\"properties\":[{\"name\":\"align_content\",\"kind\":\"Any\",\"default\":\"flex-start\"},{\"name\":\"align_items\",\"kind\":\"Any\",\"default\":\"flex-start\"},{\"name\":\"flex_direction\",\"kind\":\"Any\",\"default\":\"row\"},{\"name\":\"flex_wrap\",\"kind\":\"Any\",\"default\":\"wrap\"},{\"name\":\"gap\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"justify_content\",\"kind\":\"Any\",\"default\":\"flex-start\"}]},{\"type\":\"model\",\"name\":\"FloatPanel1\",\"properties\":[{\"name\":\"config\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"contained\",\"kind\":\"Any\",\"default\":true},{\"name\":\"position\",\"kind\":\"Any\",\"default\":\"right-top\"},{\"name\":\"offsetx\",\"kind\":\"Any\",\"default\":null},{\"name\":\"offsety\",\"kind\":\"Any\",\"default\":null},{\"name\":\"theme\",\"kind\":\"Any\",\"default\":\"primary\"},{\"name\":\"status\",\"kind\":\"Any\",\"default\":\"normalized\"}]},{\"type\":\"model\",\"name\":\"GridStack1\",\"properties\":[{\"name\":\"mode\",\"kind\":\"Any\",\"default\":\"warn\"},{\"name\":\"ncols\",\"kind\":\"Any\",\"default\":null},{\"name\":\"nrows\",\"kind\":\"Any\",\"default\":null},{\"name\":\"allow_resize\",\"kind\":\"Any\",\"default\":true},{\"name\":\"allow_drag\",\"kind\":\"Any\",\"default\":true},{\"name\":\"state\",\"kind\":\"Any\",\"default\":[]}]},{\"type\":\"model\",\"name\":\"drag1\",\"properties\":[{\"name\":\"slider_width\",\"kind\":\"Any\",\"default\":5},{\"name\":\"slider_color\",\"kind\":\"Any\",\"default\":\"black\"},{\"name\":\"value\",\"kind\":\"Any\",\"default\":50}]},{\"type\":\"model\",\"name\":\"click1\",\"properties\":[{\"name\":\"terminal_output\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"debug_name\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"clears\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"FastWrapper1\",\"properties\":[{\"name\":\"object\",\"kind\":\"Any\",\"default\":null},{\"name\":\"style\",\"kind\":\"Any\",\"default\":null}]},{\"type\":\"model\",\"name\":\"NotificationAreaBase1\",\"properties\":[{\"name\":\"js_events\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"position\",\"kind\":\"Any\",\"default\":\"bottom-right\"},{\"name\":\"_clear\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"NotificationArea1\",\"properties\":[{\"name\":\"js_events\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"notifications\",\"kind\":\"Any\",\"default\":[]},{\"name\":\"position\",\"kind\":\"Any\",\"default\":\"bottom-right\"},{\"name\":\"_clear\",\"kind\":\"Any\",\"default\":0},{\"name\":\"types\",\"kind\":\"Any\",\"default\":[{\"type\":\"map\",\"entries\":[[\"type\",\"warning\"],[\"background\",\"#ffc107\"],[\"icon\",{\"type\":\"map\",\"entries\":[[\"className\",\"fas fa-exclamation-triangle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]},{\"type\":\"map\",\"entries\":[[\"type\",\"info\"],[\"background\",\"#007bff\"],[\"icon\",{\"type\":\"map\",\"entries\":[[\"className\",\"fas fa-info-circle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]}]}]},{\"type\":\"model\",\"name\":\"Notification\",\"properties\":[{\"name\":\"background\",\"kind\":\"Any\",\"default\":null},{\"name\":\"duration\",\"kind\":\"Any\",\"default\":3000},{\"name\":\"icon\",\"kind\":\"Any\",\"default\":null},{\"name\":\"message\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"notification_type\",\"kind\":\"Any\",\"default\":null},{\"name\":\"_destroyed\",\"kind\":\"Any\",\"default\":false}]},{\"type\":\"model\",\"name\":\"TemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"BootstrapTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"TemplateEditor1\",\"properties\":[{\"name\":\"layout\",\"kind\":\"Any\",\"default\":[]}]},{\"type\":\"model\",\"name\":\"MaterialTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"copy_to_clipboard1\",\"properties\":[{\"name\":\"fill\",\"kind\":\"Any\",\"default\":\"none\"},{\"name\":\"value\",\"kind\":\"Any\",\"default\":null}]}]}};\n",
       "  var render_items = [{\"docid\":\"71eee380-b348-49f5-8139-f2430ce449af\",\"roots\":{\"p1002\":\"e817d17f-4247-4b3f-b220-f27753e7884c\"},\"root_ids\":[\"p1002\"]}];\n",
       "  var docs = Object.values(docs_json)\n",
       "  if (!docs) {\n",
       "    return\n",
       "  }\n",
       "  const py_version = docs[0].version.replace('rc', '-rc.').replace('.dev', '-dev.')\n",
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       "    var Bokeh = get_bokeh(root)\n",
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       "\t  root_el.id = root_el.id + '-rendered'\n",
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       "            // Ensure JupyterLab does not capture keyboard shortcuts\n",
       "            // see: https://jupyterlab.readthedocs.io/en/4.1.x/extension/notebook.html#keyboard-interaction-model\n",
       "\t    child.setAttribute('data-lm-suppress-shortcuts', 'true')\n",
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       "    if (root.Bokeh === undefined) {\n",
       "      return null\n",
       "    } else if (root.Bokeh.version !== py_version) {\n",
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      ]
     },
     "metadata": {
      "application/vnd.holoviews_exec.v0+json": {
       "id": "p1002"
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
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       "model_id": "2e681e004b2c48cc8c0f1b10a09a38bd",
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      "text/plain": [
       "BokehModel(combine_events=True, render_bundle={'docs_json': {'97dc65d0-0af3-489f-ae28-f2361a528c61': {'version…"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[-1].hv(\n",
    "    kdims=[\"x\", \"y\"], vdims=[\"x\", \"y\"], scalar_kw={\"cmap\": \"viridis\", \"clim\": (0, Ms)}\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now compute winding number using operators from `discretisedfield`:\n",
    "$$ S = \\frac{1}{4\\pi}\\iint q \\,\\,\\text{d}x\\text{d}y = \\frac{1}{4\\pi}\\iint\\mathbf{m}\\cdot\\left(\\frac{\\partial \\mathbf{m}}{\\partial x} \\times \\frac{\\partial \\mathbf{m}}{\\partial y~}\\right)\\text{d}x\\text{d}y$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.42290507])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import math\n",
    "\n",
    "m = system.m.orientation.sel(\"z\")\n",
    "S = m.dot(m.diff(\"x\").cross(m.diff(\"y\"))).integrate() / (4 * math.pi)\n",
    "S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The winding number is commonly used and there is a predefined function in `discretisedfield.tools`. To get more accurate results we use a different numerical method than just \"naively\" evaluating the integral."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5024556893362209"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.tools.topological_charge(system.m.sel(\"z\"), method=\"berg-luescher\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also plot the topological charge density in an interactive plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "380bc0b964a24d99b1d3755c964c0121",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "BokehModel(combine_events=True, render_bundle={'docs_json': {'24c4550b-78b2-4e13-a6d6-98d6d0ac7f4b': {'version…"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[-1].register_callback(\n",
    "    lambda f: df.tools.topological_charge_density(f.sel(\"z\"))\n",
    ").hv(kdims=[\"x\", \"y\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Trajectory of the vortex core\n",
    "\n",
    "We can compute the trajectory of the vortex core via the center of mass of the topological charge:\n",
    "$$\\mathbf{R} = \\frac{ \\int \\mathbf{r} \\rho(\\mathbf{r}) d^2\\mathbf{r}}{\\int \\rho(\\mathbf{r}) d^2\\mathbf{r}}. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([6.24115029e-12, 4.54330907e-12])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho = df.tools.topological_charge_density(system.m.sel(\"z\"))\n",
    "r = system.m.sel(\"z\").mesh.coordinate_field()\n",
    "R = (r * rho).integrate() / rho.integrate()\n",
    "R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we need to find the center of the vortex at each time step, this can be achieved by taking the data from last drive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_vortex_centre(drive):\n",
    "    x_coords = []\n",
    "    y_coords = []\n",
    "\n",
    "    r = drive[0].sel(\"z\").mesh.coordinate_field()\n",
    "\n",
    "    for m in drive:\n",
    "        tcd = df.tools.topological_charge_density(m.sel(\"z\"))\n",
    "        centre_of_mass = (r * tcd).integrate() / tcd.integrate()\n",
    "        x_coords.append(centre_of_mass[0])\n",
    "        y_coords.append(centre_of_mass[1])\n",
    "\n",
    "    return pd.DataFrame(\n",
    "        {\"t\": drive.table.data[\"t\"], \"pos x\": x_coords, \"pos y\": y_coords}\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "pos_pol_plus = compute_vortex_centre(data[-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now plot the vortex trajectory on top of the initial configuration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "nbval-ignore-output"
    ]
   },
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 800x600 with 2 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "data[-1][0].orientation.sel(\"z\").mpl(ax=ax, scalar_kw={\"clim\": (0, 1)})\n",
    "ax.plot(pos_pol_plus[\"pos x\"] * 1e9, pos_pol_plus[\"pos y\"] * 1e9, c=\"yellow\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, let us delete all simulation files:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "oc.delete(system)"
   ]
  }
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