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Generated documentation for tprf/3.3.x
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jenkins-TRIQS-tprf-3.3.x-13 fec7ca47d7872727fe17f3734acd0ce8740c0d9c
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flatiron-jenkins committed Jul 5, 2024
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Expand Up @@ -51,4 +51,4 @@ Parameters
Returns
^^^^^^^

g_fk : Green's function :math:`G_{ab}(\omega, \mathbf{k}) + G_{ab}(\mathbf{k})`.
g_wk : Green's function :math:`G_{ab}(i\omega_n, \mathbf{k}) + G_{ab}(\mathbf{k})`.
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Expand Up @@ -148,4 +148,4 @@ Parameters
Returns
^^^^^^^

GW self-energy :math:`\Sigma_{ab}(i\omega_n, \mathbf{k})`
Static GW self-energy (Fock) :math:`\Sigma_{ab}(\mathbf{k})`
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Expand Up @@ -129,20 +129,4 @@ Parameters
Returns
^^^^^^^

generalized Lindhard susceptibility in the particle-hole channel :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})`

.. math::
\sum_{\bar{a}b} U_{i\bar{a}}(\mathbf{k}) \epsilon_{\bar{a}b}(\mathbf{k}) U^\dagger_{bj} (\mathbf{k})
= \delta_{ij} \epsilon_{\mathbf{k}, i}
.. note::
The analytic formula is sub-optimal in terms of performance for higher temperatures. The evaluation
scales as :math:`\mathcal{O}(N_k^2)` which is worse than computing the bubble explicitly in imaginary
time, with scaling :math:`\mathcal{O}(N_k N_\tau \log(N_k N_\tau)` for :math:`N_k \gg N_\tau`.

.. note::
Care must be taken when evaluating the fermionic Matsubara frequency sum of the
product of two simple poles. By extending the sum to an integral over the complex
plane the standard expression for the Lindhard response is obtained when the
poles are non-degenerate. The degenerate case produces an additional frequency independent
contribution (the last term on the last row).
real frequency generalized Lindhard susceptibility in the particle-hole channel :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})`
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Expand Up @@ -65,4 +65,4 @@ Parameters
Returns
^^^^^^^

RPA suceptibility :math:`\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)`
RPA suceptibility :math:`\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, \omega)`
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Expand Up @@ -725,7 +725,7 @@ <h2>Parameters<a class="headerlink" href="#parameters" title="Link to this headi
</section>
<section id="returns">
<h2>Returns<a class="headerlink" href="#returns" title="Link to this heading"></a></h2>
<p>g_fk : Green’s function <span class="math notranslate nohighlight">\(G_{ab}(\omega, \mathbf{k}) + G_{ab}(\mathbf{k})\)</span>.</p>
<p>g_wk : Green’s function <span class="math notranslate nohighlight">\(G_{ab}(i\omega_n, \mathbf{k}) + G_{ab}(\mathbf{k})\)</span>.</p>
</section>
</section>

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2 changes: 1 addition & 1 deletion tprf/3.3.x/cpp2rst_generated/triqs_tprf/gw_sigma.html
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Expand Up @@ -803,7 +803,7 @@ <h2>Parameters<a class="headerlink" href="#parameters" title="Link to this headi
</section>
<section id="returns">
<h2>Returns<a class="headerlink" href="#returns" title="Link to this heading"></a></h2>
<p>GW self-energy <span class="math notranslate nohighlight">\(\Sigma_{ab}(i\omega_n, \mathbf{k})\)</span></p>
<p>Static GW self-energy (Fock) <span class="math notranslate nohighlight">\(\Sigma_{ab}(\mathbf{k})\)</span></p>
</section>
</section>

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21 changes: 1 addition & 20 deletions tprf/3.3.x/cpp2rst_generated/triqs_tprf/lindhard_chi00.html
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Expand Up @@ -798,26 +798,7 @@ <h2>Parameters<a class="headerlink" href="#parameters" title="Link to this headi
</section>
<section id="returns">
<h2>Returns<a class="headerlink" href="#returns" title="Link to this heading"></a></h2>
<p>generalized Lindhard susceptibility in the particle-hole channel <span class="math notranslate nohighlight">\(\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})\)</span></p>
<blockquote>
<div><div class="math notranslate nohighlight">
\[\sum_{\bar{a}b} U_{i\bar{a}}(\mathbf{k}) \epsilon_{\bar{a}b}(\mathbf{k}) U^\dagger_{bj} (\mathbf{k})
= \delta_{ij} \epsilon_{\mathbf{k}, i}\]</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The analytic formula is sub-optimal in terms of performance for higher temperatures. The evaluation
scales as <span class="math notranslate nohighlight">\(\mathcal{O}(N_k^2)\)</span> which is worse than computing the bubble explicitly in imaginary
time, with scaling <span class="math notranslate nohighlight">\(\mathcal{O}(N_k N_\tau \log(N_k N_\tau)\)</span> for <span class="math notranslate nohighlight">\(N_k \gg N_\tau\)</span>.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Care must be taken when evaluating the fermionic Matsubara frequency sum of the
product of two simple poles. By extending the sum to an integral over the complex
plane the standard expression for the Lindhard response is obtained when the
poles are non-degenerate. The degenerate case produces an additional frequency independent
contribution (the last term on the last row).</p>
</div>
</div></blockquote>
<p>real frequency generalized Lindhard susceptibility in the particle-hole channel <span class="math notranslate nohighlight">\(\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})\)</span></p>
</section>
</section>

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2 changes: 1 addition & 1 deletion tprf/3.3.x/cpp2rst_generated/triqs_tprf/solve_rpa_PH.html
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Expand Up @@ -744,7 +744,7 @@ <h2>Parameters<a class="headerlink" href="#parameters" title="Link to this headi
</section>
<section id="returns">
<h2>Returns<a class="headerlink" href="#returns" title="Link to this heading"></a></h2>
<p>RPA suceptibility <span class="math notranslate nohighlight">\(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\)</span></p>
<p>RPA suceptibility <span class="math notranslate nohighlight">\(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, \omega)\)</span></p>
</section>
</section>

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2 changes: 1 addition & 1 deletion tprf/3.3.x/searchindex.js

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