mirror of
https://github.com/triqs/dft_tools
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bdac3e159c
- little details : code cleaning, clang formatting, along with documentation writing for c++ gf. - separated the mesh in small class for better doc. - work on documentation : reorganize specialisation, ...
117 lines
3.9 KiB
ReStructuredText
117 lines
3.9 KiB
ReStructuredText
.. highlight:: c
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Fourier transforms
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###################
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The Fourier transforms from real and imaginary frequencies to times, and inverse, are currently implemented,
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along with the analogous transformation from the Legendre expansion to imaginary time and frequencies.
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Synopsis
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=========
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**Synopsis** ::
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auto fourier (gf<imfreq,Target,Opt> const &)
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auto fourier (gf_view<imfreq,Target,Opt> const &)
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auto fourier (gf_const_view<imfreq,Target,Opt> const &)
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auto inverse_fourier (gf<imfreq,Target,Opt> const &)
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auto inverse_fourier (gf_view<imfreq,Target,Opt> const &)
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auto inverse_fourier (gf_const_view<imfreq,Target,Opt> const &)
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gf<imfreq, Target, Opt> make_gf_from_fourier(gf<imtime, Target, Opt> const&);
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gf<imfreq, Target, Opt> make_gf_from_fourier(gf_view<imtime, Target, Opt> const&);
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gf<imfreq, Target, Opt> make_gf_from_fourier(gf_const_view<imtime, Target, Opt> const&);
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gf<imtime, Target, Opt> make_gf_from_inverse_fourier(gf<imfreq, Target, Opt> const&);
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gf<imtime, Target, Opt> make_gf_from_inverse_fourier(gf_view<imfreq, Target, Opt> const&);
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gf<imtime, Target, Opt> make_gf_from_inverse_fourier(gf_const_view<imfreq, Target, Opt> const&);
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**fourier, inverse_fourier**
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The fourier/inverse_fourier functions do **not** perform the Fourier transformation,
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but returns a small lazy object (basically saying "Fourier Transform of XXX"),
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which is then used in an assignment of a *view* of a gf.
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The reason is the following: when putting e.g. a Fourier transform of a function in time, say gt,
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into a Green function in frequencies, say gw, we want to say something like::
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gw = fourier(gt); // ??? (1)
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However, if the fourier function performs the transformation, how could it know the details
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of the mesh of gw ? That information is not available when calling *fourier*.
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Since *fourier* returns a small lazy object, the library can then rewrite (1) internally into something like ::
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call_the_fourier_implementation(gt, gw);
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where all the information about the mesh of gw is now available to the implementation.
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Moreover, since fourier(gt) does not possess a domain (for the same reason), (1)
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makes no sense : RHS of gf assignment requires a domain (cf concepts).
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We therefore use *a view* as LHS::
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gw() = fourier(gt); // correct usage.
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**make_gf_from_fourier, make_gf_from_inverse_fourier**
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In the case where we want to create a *new* container from the fourier transform of gt,
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we can use the function make_gf_from_fourier, in which choice is explicitly made to generate a new gf with the same number of points in the mesh.
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(Cf example below).
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DOC TO BE FINISHED.
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Example
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=========
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.. compileblock::
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#include <triqs/gfs.hpp>
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using namespace triqs::gfs;
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int main() {
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double beta =1, a=1;
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int N=10000;
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auto gw = gf<imfreq> {{beta, Fermion, N}, {1,1}};
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auto gt = gf<imtime> {{beta, Fermion, N}, {1,1}};
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triqs::clef::placeholder<0> om_;
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gw (om_) << 1/(om_-a);
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// fills a full *view* of gt with the contents of the FFT
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// NB : the mesh of gt *must* have the same size as the mesh of gw.
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gt() = inverse_fourier(gw);
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// make a new fresh gf, with the same size mesh, from the FFT of gt
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auto gw2 = make_gf_from_fourier(gt);
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}
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Convention
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===========
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For real time/frequency:
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.. math:: \tilde G(\omega)=\int_{-\infty}^\infty dt G(t)e^{i\omega t}
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.. math:: G(t)=\int_{-\infty}^\infty \frac{d\omega}{2\pi} \tilde G(\omega)e^{-i\omega t}
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For Matsubara (imaginary) time/frequency:
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.. math:: \tilde G(i\omega_n)=\int_{0}^\beta d\tau G(t)e^{i\omega_n \tau}
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.. math:: G(\tau)=\sum_{n=-\infty}^\infty \frac{1}{\beta} \tilde G(i\omega_n)e^{-i\omega_n \tau}
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The :math:`\omega_n`'s are :math:`\frac{(2n+1)\pi}{\beta}` for fermions, :math:`\frac{2n\pi}{\beta}` for bosons (as :math:`G(\tau+\beta)=-G(\tau)` for fermions, :math:`G(\tau)` for bosons).
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.. toctree::
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:maxdepth: 1
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fourier_impl_notes
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