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https://github.com/triqs/dft_tools
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579368f24b
- lazy_fourier and co --> fourier - ex fourier --> make_gf_from_fourier to make a new gf - = fourier (g) works only iif lhs is a view, like scalar. - updated python (commented fourier method).
82 lines
2.3 KiB
ReStructuredText
82 lines
2.3 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 and example
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======================
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**Synopsis** ::
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lazy_object fourier (gf<imfreq,Target,Opt> const &)
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lazy_object fourier (gf_const_view<imfreq,Target,Opt> const &)
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lazy_object inverse_fourier (gf<imfreq,Target,Opt> const &)
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lazy_object inverse_fourier (gf_const_view<imfreq,Target,Opt> const &)
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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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Example
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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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gt() = inverse_fourier(gw); // fills the *View* with the contents of the FFT.
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// NB : the mesh must have the same size.
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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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Note that :
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* the LHS of the = must be a view, since the RHS can not compute the domain of the function
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(how many points to use ?).
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* In the make_gf_from_fourier function, choice is explicitly made to generate a new gf with the same number of points in the mesh.
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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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*
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.. toctree::
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:maxdepth: 1
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fourier_impl_notes
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