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<title>A tight-binding model on a square lattice &mdash; TRIQS 3.2.0 documentation</title>
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Expand Up @@ -1758,7 +1758,7 @@ <h2>Dynamical mean-field theory<a class="headerlink" href="#Dynamical-mean-field
\Big( \sum_k \frac{1}{i \omega_n + \mu - \epsilon_k - \Sigma_\mathrm{imp}} \Big)^{-1}
+ \Sigma_\mathrm{imp}\]</div>
<p>We solve the quantum impurity for this new <span class="math notranslate nohighlight">\(G_0\)</span> and loop until convergence</p>
<p><img alt="773e4031f9834ef090f8fd8b8a210731" class="no-scaled-link" src="../../../../_images/selfcons.png" style="width: 40%;" /></p>
<p><img alt="0d5ab08bddbe41f8b37a96fcbd0595aa" class="no-scaled-link" src="../../../../_images/selfcons.png" style="width: 40%;" /></p>
</section>
<section id="Bethe-lattice-DMFT">
<h2>Bethe lattice DMFT<a class="headerlink" href="#Bethe-lattice-DMFT" title="Link to this heading"></a></h2>
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<section id="General-reminder:-Anderson-impurity-model-and-CTHYB-solver">
<h1>General reminder: Anderson impurity model and CTHYB solver<a class="headerlink" href="#General-reminder:-Anderson-impurity-model-and-CTHYB-solver" title="Link to this heading"></a></h1>
<p>In the Anderson impurity model, we decompose the full lattice problem into an interacting site (‘impurity’) hybridised to a bath:</p>
<p><img alt="971a464aa04646f3885176c5e608ea2b" src="../../../../_images/dmft_bath_impurity.png" /></p>
<p><img alt="a571c4edfd8741eca75e4ca0895825be" src="../../../../_images/dmft_bath_impurity.png" /></p>
<dl>
<dt>with the Hamiltonian :nbsphinx-math:<a href="#id1"><span class="problematic" id="id2">`</span></a>begin{align*}</dt><dd><blockquote>
<div><p>H = &amp; color{red}{H_{rm imp}} + color{darkgreen}{H_{rm hyb}} + color{blue}{H_{rm bath}} \
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Expand Up @@ -1723,7 +1723,7 @@ <h1>Valence-Bond DMFT solution of the Hubbard model<a class="headerlink" href="#
<p>.</p>
<p>In the following, we use <span class="math notranslate nohighlight">\(U/t=10\)</span> and <span class="math notranslate nohighlight">\(t'/t=-0.3\)</span>, which are values commonly used for modeling hole-doped cuprates in a single-band framework. All energies (and temperatures) are expressed in units of <span class="math notranslate nohighlight">\(D=4t=1\)</span>, and the doping is denoted by <span class="math notranslate nohighlight">\(\delta\)</span>.</p>
<p>We subdivide the Brillouin Zone into a minimal set of two patches of <strong>equal</strong> area <span class="math notranslate nohighlight">\(P_+\)</span> (even) and <span class="math notranslate nohighlight">\(P_-\)</span> (odd).</p>
<p><img alt="69c55740c29a4283b5999ceee1cbd8a9" class="no-scaled-link" src="../../../../_images/vb-patching.png" style="width: 240px; height: 180px;" /></p>
<p><img alt="db9b42a1c8224c20b61f2bf9713205aa" class="no-scaled-link" src="../../../../_images/vb-patching.png" style="width: 240px; height: 180px;" /></p>
<p><span class="math notranslate nohighlight">\(P_+\)</span> is a central square centered at momentum <span class="math notranslate nohighlight">\((0,0)\)</span> and containing the nodal region; the complementary region <span class="math notranslate nohighlight">\(P_{-}\)</span> extends to the edge of the BZ and contains in particular the antinodal region and the <span class="math notranslate nohighlight">\((\pi,\pi)\)</span> momentum.</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[2]:
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Expand Up @@ -1769,7 +1769,7 @@ <h2>Dynamical mean-field theory<a class="headerlink" href="#Dynamical-mean-field
\Big( \sum_k \frac{1}{i \omega_n + \mu - \epsilon_k - \Sigma_\mathrm{imp}} \Big)^{-1}
+ \Sigma_\mathrm{imp}\]</div>
<p>We solve the quantum impurity for this new <span class="math notranslate nohighlight">\(G_0\)</span> and loop until convergence</p>
<p><img alt="af3e4fb4dd8f46398845b82185fad884" class="no-scaled-link" src="../../../../../_images/selfcons1.png" style="width: 40%;" /></p>
<p><img alt="701d34ddf09846f9bbc870027ae97ee9" class="no-scaled-link" src="../../../../../_images/selfcons1.png" style="width: 40%;" /></p>
</section>
<section id="Bethe-lattice-DMFT">
<h2>Bethe lattice DMFT<a class="headerlink" href="#Bethe-lattice-DMFT" title="Link to this heading"></a></h2>
Expand Down Expand Up @@ -1884,7 +1884,7 @@ <h2>Visualizing the Mott transition<a class="headerlink" href="#Visualizing-the-
<section id="Comparison-with-the-literature">
<h2>Comparison with the literature<a class="headerlink" href="#Comparison-with-the-literature" title="Link to this heading"></a></h2>
<p>You can compare the result above with what can be found in the literature (review of Antoine Georges et al.)</p>
<p><img alt="ee074f5be56141c6917fb8583ffee021" class="no-scaled-link" src="../../../../../_images/mott.png" style="width: 30%;" /></p>
<p><img alt="5d22ae3e30814bd083610bf1ba8bf3c6" class="no-scaled-link" src="../../../../../_images/mott.png" style="width: 30%;" /></p>
</section>
</section>

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Expand Up @@ -1997,7 +1997,7 @@ <h2>Solution 6<a class="headerlink" href="#Solution-6" title="Link to this headi
</div>
</div>
<p>The result is completely wrong. This is because of the noise in the Monte Carlo data. One would have to make much longer runs in order to reduce the error bars. The Pade approximation can be used only on very accurate data. When the noise is still quite large, one has to use different analytical continuation methods, like MaxEnt, which produces the following spectral function:</p>
<p><img alt="b2473c79df404a8a807548edeb17bb24" src="../../../../../_images/maxent_Aw.png" /></p>
<p><img alt="bec834fda3a9410ebed8347424e9dc9b" src="../../../../../_images/maxent_Aw.png" /></p>
<p>Regardless of which package you use for MaxEnt, it is very important to remember that there are some important knobs with which one can play in MaxEnt that can substantially change the results, and so one must be very careful in its use!</p>
<blockquote>
<div><p>Exercise 7</p>
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Expand Up @@ -1727,7 +1727,7 @@ <h1>Valence-Bond DMFT solution of the Hubbard model<a class="headerlink" href="#
<p>.</p>
<p>In the following, we use <span class="math notranslate nohighlight">\(U/t=10\)</span> and <span class="math notranslate nohighlight">\(t'/t=-0.3\)</span>, which are values commonly used for modeling hole-doped cuprates in a single-band framework. All energies (and temperatures) are expressed in units of <span class="math notranslate nohighlight">\(D=4t=1\)</span>, and the doping is denoted by <span class="math notranslate nohighlight">\(\delta\)</span>.</p>
<p>We subdivide the Brillouin Zone into a minimal set of two patches of <strong>equal</strong> area <span class="math notranslate nohighlight">\(P_+\)</span> (even) and <span class="math notranslate nohighlight">\(P_-\)</span> (odd).</p>
<p><img alt="d686a1a5ead94b4789a5c372a1952ae1" class="no-scaled-link" src="../../../../../_images/vb-patching1.png" style="width: 240px; height: 180px;" /></p>
<p><img alt="054f8b4ebd074434a1b943ce9d8ab140" class="no-scaled-link" src="../../../../../_images/vb-patching1.png" style="width: 240px; height: 180px;" /></p>
<p><span class="math notranslate nohighlight">\(P_+\)</span> is a central square centered at momentum <span class="math notranslate nohighlight">\((0,0)\)</span> and containing the nodal region; the complementary region <span class="math notranslate nohighlight">\(P_{-}\)</span> extends to the edge of the BZ and contains in particular the antinodal region and the <span class="math notranslate nohighlight">\((\pi,\pi)\)</span> momentum.</p>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[1]:
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