Generated documentation for tprf/unstable

jenkins-TRIQS-tprf-unstable-871 fec7ca47d7872727fe17f3734acd0ce8740c0d9c

tprf/unstable/_sources/cpp2rst_generated/triqs_tprf/gw_sigma.rst.txt

@@ -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})`

tprf/unstable/_sources/cpp2rst_generated/triqs_tprf/lindhard_chi00.rst.txt

@@ -129,4 +129,20 @@ Parameters
 Returns
 ^^^^^^^
 
-real frequency generalized Lindhard susceptibility in the particle-hole channel :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})`
+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).

tprf/unstable/_sources/cpp2rst_generated/triqs_tprf/solve_rpa_PH.rst.txt

@@ -65,4 +65,4 @@ Parameters
 Returns
 ^^^^^^^
 
-RPA suceptibility :math:`\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, \omega)`
+RPA suceptibility :math:`\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)`

tprf/unstable/cpp2rst_generated/triqs_tprf/gw_sigma.html

@@ -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>
 

tprf/unstable/cpp2rst_generated/triqs_tprf/lindhard_chi00.html

@@ -798,7 +798,26 @@ <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>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>
+<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>
 </section>
 </section>
 

tprf/unstable/cpp2rst_generated/triqs_tprf/solve_rpa_PH.html

@@ -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">\(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, \omega)\)</span></p>
+<p>RPA suceptibility <span class="math notranslate nohighlight">\(\chi_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\)</span></p>
 </section>
 </section>