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"source": [
"# Fermions on the square lattice & perfect nesting\n",
"\n",
"This tutorial is the first in a series of tutorials on two-particle response were we will use TRIQS and the [**Two-Particle Response-Function toolbox (TPRF)**](https://triqs.github.io/tprf/latest/) to compute;\n",
"\n",
"1. the **non-interacting Green's function** of fermions on the square lattice with nearest-neighbour hopping \n",
" and study the Fermi surface [01]\n",
"\n",
"2. the **non-interacting two-particle response**, also called the susceptibility [03]\n",
"\n",
"3. the **Random-Phase Approximation (RPA)** susceptibility for weak interactions, \n",
" studying the anti-ferromagnetic divergence at ($\\pi,\\pi)$ [05]\n",
"\n",
"4. the **Two-Particle Self Conistent (TPSC)** susceptibility\n",
" and show that it satisfies the Pauli principle, while RPA does not [07]\n",
" \n",
"5. and show that TPSC obeys the **Mermin-Wagner theorem**,\n",
" since its spin susceptibility does not diverge at finite temperature [09]\n"
]
},
{
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"execution": {
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"source": [
"%matplotlib inline\n",
"%config InlineBackend.figure_format = 'svg'\n",
"from triqs.plot.mpl_interface import plt\n",
"\n",
"import numpy as np"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Square lattice with nearest-neighbour hopping\n",
"\n",
"The square lattice with nearest-neighbour hopping $t$ appeared in earier tutorials, were the dispersion relation \n",
"\n",
"\\begin{equation}\n",
" \\epsilon(\\mathbf{k})=-2t(\\cos{k_x}+\\cos{k_y}),\n",
"\\end{equation}\n",
"\n",
"was computed using TRIQS in more than one way.\n",
"\n",
"However, in TRIQS and TPRF there are a number of helper routines for lattice models that simplifies the study of general tight-binding models. Here we will therefore use these standard routines.\n",
"\n",
"Insead of constructing $\\epsilon(\\mathbf{k})$ directly in momentum space we construct a real-space tight binding lattice Hamitonian $H(\\mathbf{r})$ using [the `TBLattice` class](https://triqs.github.io/triqs/latest/documentation/python_api/triqs.lattice.tight_binding.TBLattice.html?highlight=tblattice#triqs.lattice.tight_binding.TBLattice) corresponding to the square lattice with nearest neighbour hopping $t=1$."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"execution": {
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{
"name": "stdout",
"output_type": "stream",
"text": [
"Warning: could not identify MPI environment!\n",
"Tight Binding Hamiltonian on Bravais Lattice with dimension 2, units \n",
"[[1,0,0]\n",
" [0,1,0]\n",
" [0,0,1]], n_orbitals 1\n",
"with hoppings [\n",
" [1,0] : \n",
"[[(-1,0)]]\n",
" [-1,0] : \n",
"[[(-1,0)]]\n",
" [0,1] : \n",
"[[(-1,0)]]\n",
" [0,-1] : \n",
"[[(-1,0)]] ]\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Starting serial run at: 2023-08-29 11:09:00.866052\n"
]
}
],
"source": [
"from triqs.lattice.tight_binding import TBLattice\n",
"\n",
"t = 1.0 # nearest-neigbhour hopping amplitude\n",
"\n",
"H_r = TBLattice( \n",
" units=[\n",
" (1,0,0), # basis vector in the x-direction \n",
" (0,1,0), # basis vector in the y-direction\n",
" ],\n",
" hoppings={\n",
" (+1,0) : [[-t]], # hopping in the +x direction\n",
" (-1,0) : [[-t]], # hopping in the -x direction\n",
" (0,+1) : [[-t]], # hopping in the +y direction\n",
" (0,-1) : [[-t]], # hopping in the -y direction\n",
" })\n",
"\n",
"print(H_r)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The real-space Hamiltonian $H(\\mathbf{r})$ can both construct a discretized momentum mesh and evaluate the dispersion $\\epsilon(\\mathbf{k})$ on a given momentum mesh, by"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"execution": {
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"shell.execute_reply": "2023-08-29T09:09:00.888364Z"
}
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Greens Function with mesh Brillouin Zone Mesh with linear dimensions (128 128 1)\n",
" -- units = \n",
"[[0.0490874,0,0]\n",
" [0,0.0490874,0]\n",
" [0,0,6.28319]]\n",
" -- brillouin_zone: Brillouin Zone with 2 dimensions and reciprocal matrix \n",
"[[6.28319,0,0]\n",
" [0,6.28319,0]\n",
" [0,0,6.28319]] and target_shape (1, 1): \n",
"\n"
]
}
],
"source": [
"n_k = 128\n",
"kmesh = H_r.get_kmesh(n_k=n_k)\n",
"e_k = H_r.fourier(kmesh)\n",
"print(e_k)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since the square lattice is two-dimensional it is possible to visualize the dispersion using a color plot.\n",
"\n",
"Here is an example that plots $\\epsilon(\\mathbf{k})$ in the entire Brillouin zone as well as the shape of the Fermi surface at $\\omega = 0$ (dotted line)."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"execution": {
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"shell.execute_reply": "2023-08-29T09:09:01.067063Z"
},
"scrolled": false
},
"outputs": [
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"source": [
"k = np.linspace(-np.pi, np.pi, num=100)\n",
"kx, ky = np.meshgrid(k, k)\n",
"\n",
"e_k_interp = np.vectorize(lambda kx, ky : e_k((kx, ky, 0)).real)(kx, ky)\n",
"\n",
"plt.figure()\n",
"\n",
"plt.pcolormesh(kx, ky, e_k_interp, rasterized=True, cmap='RdBu')\n",
"plt.colorbar().ax.set_ylabel(r'$\\epsilon(\\mathbf{k})$'); \n",
"\n",
"plt.contour(kx, ky, e_k_interp, levels=[0], linestyles='dotted')\n",
"\n",
"plt.xlabel(r'$k_x$'); plt.ylabel(r'$k_y$');\n",
"k_ticks, k_labels = [-np.pi, 0, np.pi], [r\"$-\\pi$\", r\"0\", r\"$\\pi$\"]\n",
"plt.xticks(k_ticks, k_labels); plt.yticks(k_ticks, k_labels);\n",
"plt.axis('square');\n",
"\n",
"# -- High-symmetry path G-X-M-G\n",
"\n",
"pts = [\n",
" (0., 0., r'$\\Gamma$', 'w', 'bottom', 'right'), \n",
" (np.pi, 0., r'$X$', 'k', 'center', 'left'),\n",
" (np.pi, np.pi, r'$M$', 'r', 'bottom', 'left'),\n",
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"\n",
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]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Momentum dependent quantities can also be visualized along high-symmetry paths in the Brillouin zone, see above for the high-symmetry points $\\Gamma$, $X$ and $M$ of the square lattice. \n",
"\n",
"Here is an example that plots the dispersion $\\epsilon(\\mathbf{k})$ along thepath $\\Gamma - X - M - \\Gamma$ in k-space using [the `triqs_tprf.lattice_utils.k_space_path` function](https://triqs.github.io/tprf/unstable/reference/python_reference.html#triqs_tprf.lattice_utils.k_space_path)."
]
},
{
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"shell.execute_reply": "2023-08-29T09:09:01.126033Z"
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"source": [
"G = [0.0, 0.0, 0.0]\n",
"X = [0.5, 0.0, 0.0]\n",
"M = [0.5, 0.5, 0.0]\n",
"\n",
"path = [(G, X), (X, M), (M, G)]\n",
"\n",
"from triqs_tprf.lattice_utils import k_space_path\n",
"\n",
"k_vecs, k_plot, k_ticks = k_space_path(path, num=32, bz=H_r.bz)\n",
"\n",
"e_k_interp = np.vectorize(lambda k : e_k(k).real, signature='(n)->()')\n",
"\n",
"plt.plot(k_plot, e_k_interp(k_vecs))\n",
"plt.xticks(k_ticks, labels=[r'$\\Gamma$', '$X$', '$M$', r'$\\Gamma$'])\n",
"plt.ylabel(r'$\\epsilon(\\mathbf{k})$')\n",
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following we will re-purpose these visualization scripts to study the one-particle and two-particle Green's functions of the square lattice model."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Non-interacting lattice Green's function\n",
"\n",
"Given the dispersion $\\epsilon(\\mathbf{k})$ the non-interacting Green's function $G_0(i\\omega_n, \\mathbf{k})$ is given by\n",
"\n",
"\\begin{equation}\n",
" G_0(i\\omega_n, \\mathbf{k}) = \\frac{1}{i\\omega_n - \\epsilon(\\mathbf{k})}\n",
" \\, .\n",
"\\end{equation}\n",
"\n",
"As shown in the Basic Tutorial it is of course possible to compute $G_0$ using a loop over frequency and momentum:\n",
"\n",
"```python\n",
"from triqs.gf import Gf, MeshImFreq, MeshProduct\n",
"\n",
"wmesh = MeshImFreq(beta=2.5, S='Fermion', n_max=128)\n",
"wkmesh = MeshProduct(wmesh, kmesh)\n",
"g0_wk = Gf(mesh=wkmesh, target_shape=e_k.target_shape)\n",
"\n",
"for w, k in wkmesh: \n",
" g0_wk[w, k] = 1/(w - e_k[k])\n",
"```\n",
"\n",
"However, TPRF has Dyson equation solvers that are OpenMP+MPI parallell and all implemented in C++, see [the TPRF documentation](https://triqs.github.io/tprf/latest/reference/python_reference.html#lattice-green-s-functions). Here we will use these fast routines!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### <i class=\"fa fa-gear fa-x\" style=\"color: #186391\"></i> Exercise 1:\n",
"\n",
"Use [`triqs_tprf.lattice.lattice_dyson_g0_wk`](https://triqs.github.io/tprf/latest/reference/python_reference.html#triqs_tprf.lattice.lattice_dyson_g0_wk) to compute $G_0(i\\omega_n, \\mathbf{k})$ at inverse temperature $\\beta = 2.5$ using a [fermionic `MeshImFreq` frequency mesh](https://triqs.github.io/triqs/latest/documentation/python_api/triqs.gf.meshes.MeshImFreq.html?highlight=meshimfreq#triqs.gf.meshes.MeshImFreq) with 128 Matsubara frequencies and name the resulting Green's function `g0_wk`.\n",
"\n",
"Check the properties of `g0_wk` by printing it, i.e.\n",
"```python\n",
"print(g0_wk)\n",
"```"
]
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"name": "stdout",
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"text": [
"Greens Function with mesh Imaginary Freq Mesh with beta = 2.5, statistic = Fermion, n_iw = 32, positive_only = false, Brillouin Zone Mesh with linear dimensions (128 128 1)\n",
" -- units = \n",
"[[0.0490874,0,0]\n",
" [0,0.0490874,0]\n",
" [0,0,6.28319]]\n",
" -- brillouin_zone: Brillouin Zone with 2 dimensions and reciprocal matrix \n",
"[[6.28319,0,0]\n",
" [0,6.28319,0]\n",
" [0,0,6.28319]] and target_shape (1, 1): \n",
"\n"
]
}
],
"source": [
"# Write your code here\n",
"\n",
"from triqs.gf import MeshImFreq\n",
"wmesh = MeshImFreq(beta=2.5, S='Fermion', n_iw=32)\n",
"\n",
"from triqs_tprf.lattice import lattice_dyson_g0_wk\n",
"g0_wk = lattice_dyson_g0_wk(mu=0., e_k=e_k, mesh=wmesh)\n",
"\n",
"print(g0_wk)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Questions**\n",
"\n",
"- How many meshes does the Green's function have?\n",
"- What is the k-space discretization?\n",
"- How is the reciprocal basis vectors of the Brillouin zone related to the lattice (`units`) vectors of the tight binding lattice `H_r` above?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### <i class=\"fa fa-gear fa-x\" style=\"color: #186391\"></i> Exercise 2:\n",
"\n",
"Save the Green's function `g0_wk` into a hdf5 file named `g0_wk.h5` using [`h5.HDFArchive`](https://triqs.github.io/triqs/latest/documentation/manual/triqs/hdf5/ref.html), so that we can read `g0_wk` from file in the following tutorials. Do the same with the dispersion `e_k` and save it to a file with the name `e_k.h5`."
]
},
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"source": [
"# Write your code here\n",
"\n",
"from h5 import HDFArchive\n",
"\n",
"with HDFArchive(\"g0_wk.h5\", \"w\") as R:\n",
" R['g0_wk'] = g0_wk\n",
" \n",
"with HDFArchive(\"e_k.h5\", \"w\") as R:\n",
" R['e_k'] = e_k"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Fermi surface nesting\n",
"\n",
"We will now study the Fermi surface of Fermions on the square lattice with nearest neighbour hopping, which has a special property called *perfect nesting*. A Fermi surface is said to be *nested* if parts of the Fermi surface map to each other by a single momentum vector $\\mathbf{Q}$, called the *nesting vector*.\n",
"\n",
"For a non-interacting system the Fermi surface is the surface in k-space defined by\n",
"\n",
"$$ \\epsilon(\\mathbf{k}) - \\mu = 0 \\, ,$$\n",
"\n",
"where $\\mu$ is the chemical potential. In terms of the spectral function $A(\\omega, \\mathbf{k})$ this corresponds to large values of $A$ at $\\omega=0$\n",
"\n",
"$$ A(\\omega = 0, \\mathbf{k}) = \\frac{1}{\\pi} \\text{Im} \n",
"\\left[ \\frac{1}{ 0 - \\epsilon(\\mathbf{k}) + \\mu - i\\delta } \\right] \\gg 1 \\, ,$$\n",
"\n",
"which also generalizes to interacting systems."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### <i class=\"fa fa-gear fa-x\" style=\"color: #186391\"></i> Exercise 3:\n",
"\n",
"Make a color plot of the zero-frequency spectral function $A(k, \\omega=0)$ over the Brillouin zone, using the approximation\n",
"\n",
"$$ A(k, \\omega=0) \\approx -\\frac{1}{\\pi} \\text{Im}[ G_0(\\mathbf{k}, i\\omega_0) ] \\, ,$$\n",
"\n",
"where we neglect the fact that the first fermionic Matsubara frequency $i\\omega_0$ is not exactly $0$.\n",
"\n",
"The right hand side can be evaluated using the Triqs Green's function `g0_wk` and the interpolation feature:\n",
"\n",
"```python\n",
"n = 0 # Matsubara frequency index\n",
"kx, ky, kz = 0., 0., 0.\n",
"k_vec = (kx, ky, kz)\n",
"g0_wk(n, k_vec)\n",
"```"
]
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"# Write your code here\n",
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"A_inv_k = np.vectorize(lambda kx, ky: (1 / g0_wk( 0, (kx, ky, 0) )).real)\n",
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"**Hint**: Re-purpose the code for the color plot of $\\epsilon(\\mathbf{k})$ above.\n",
"\n",
"**Questions**\n",
"\n",
" * How can we see from the plot that the Fermi surface is nested?\n",
" * What is the nesting vector?\n",
" * Actually the Fermi surface is **perfectly** nested. What do you think is the difference between *nesting* and *perfect nesting*?"
]
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"source": [
"### <i class=\"fa fa-gear fa-x\" style=\"color: #186391\"></i> Exercise 4:\n",
"\n",
"For non-interacting systems the Fermi surface can also be observed in the Fermionic density distribution in k-space $\\rho(\\mathbf{k})$, which can be computed from the Matsubara Green's function $G_0(i\\omega_n, \\mathbf{k})$.\n",
"\n",
"In TRIQS this is available through the `.density()` method of Matsubara frequency Green's functions. For the lattice Green's function it can be evaluated for a given k-vector using\n",
"\n",
"```python\n",
"kx, ky, kz = 0., 0., 0.\n",
"k_vec = (kx, ky, kz)\n",
"rho_k = g0_wk(all, k_vec).density().real\n",
"```\n",
"\n",
"Plot the momentum distribution $\\rho(\\mathbf{k})$ along the Brillouin zone high-symmetry path $\\Gamma - X - M - \\Gamma$."
]
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"source": [
"rho_k_interp = np.vectorize(lambda k: g0_wk(all, k).density(), signature='(n)->()')\n",
"\n",
"plt.plot(k_plot, rho_k_interp(k_vecs).real)\n",
"plt.xticks(k_ticks, labels=[r'$\\Gamma$', '$X$', '$M$', r'$\\Gamma$'])\n",
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},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Hint**: Re-purpose the code above plotting $\\epsilon(\\mathbf{k})$ along the same path.\n",
"\n",
"**Questions**\n",
"\n",
"- What is the value of $\\rho(\\mathbf{k})$ at the Fermi surface?\n",
"- What is the sign of $\\epsilon(\\mathbf{k})$ in the regions of k-space where $\\rho(\\mathbf{k}) \\approx 1$ and $\\approx 0$, respectively?\n",
"- How is this related to the Fermi distribution function $$f(\\epsilon) = \\frac{1}{1 + e^{\\beta \\epsilon}}$$ plotted below?"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
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"iopub.status.idle": "2023-08-29T09:09:01.825794Z",
"shell.execute_reply": "2023-08-29T09:09:01.825547Z"
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