diff --git a/doc/reference/c++/gf/fourier.rst b/doc/reference/c++/gf/fourier.rst index 1e6fc101..0703acd3 100644 --- a/doc/reference/c++/gf/fourier.rst +++ b/doc/reference/c++/gf/fourier.rst @@ -37,10 +37,10 @@ It will be used to calculate the (inverse) Fourier transform, in real/imaginary The DFT transforms of a sequence of :math:`N` complex numbers :math:`f_0...,f_{N-1}` into a sequence of :math:`N` complex numbers :math:`\tilde f_0...,\tilde f_{N-1}` according to the formula: :label: _DFT - .. math:: \tilde f_m = \sum_{k=0}^{N-1} f_k e^{-i 2 \pi m k / N}. + .. math:: \tilde f_k = \sum_{n=0}^{N-1} f_n e^{-i 2 \pi k n / N}. The inverse DFT formula is :label: _inv_DFT - .. math:: f_k = \frac{1}{N} \sum_{m=0}^{N-1} \tilde f_m e^{i 2 \pi m k / N}. + .. math:: f_n = \frac{1}{N} \sum_{k=0}^{N-1} \tilde f_k e^{i 2 \pi k n / N}. @@ -54,17 +54,15 @@ The times are :math:`t_k=t_{min}+k\delta t` and the frequencies :math:`\omega_m= By approximating Eq. :ref:`TF_R` by .. math:: \tilde G(\omega_m) = \delta t \sum_{k=0}^{N_t} G(t_k) e^{i\omega_m t_k}, -we recognize a DFT (Eq. :ref:`DFT`). To calculate it using FFTW, we first need to prepare the input: - .. math:: f_k = G(t_k) e^{i \omega_{min}t_k}, -then to do the DFT and finally to modify the output to obtain :math:`\tilde G(\omega_m)` as - .. math:: \tilde G(\omega_m) = \delta t \tilde f_m e^{i t_{min}(\omega_m-\omega_{min})}. +we recognize an inverse DFT (Eq. :ref:`inv_DFT`). To calculate it using FFTW, we first need to prepare the input :math:`\tilde f_k`, then to do the DFT and finally to modify the output to obtain :math:`\tilde G(\omega_m)` using the two formulas: + .. math:: \tilde f_k = G(t_k) e^{i \omega_{min}t_k}, + .. math:: \tilde G(\omega_m) = \delta t f_m e^{i t_{min}(\omega_m-\omega_{min})}. Similarly, the inverse transformation is obtained by approximating Eq. :ref:`eq_inv_TF_R` by .. math:: G(t_k)=\frac{\delta\omega}{2\pi}\sum_{m=0}^{N_\omega} \tilde G(\omega_m)e^{-i\omega_m t_k}, -we recognize an inverse DFT (Eq. :ref:`inv_DFT`). To calculate it using FFTW, we first need to prepare the input: - .. math:: \tilde f_m = \tilde G(\omega_m) e^{-i t_{min}\omega_m}, -then to do the inverse DFT and finally to modify the output to obtain :math:`G(t_k)` as - .. math:: G(t_k) = \frac{1}{N_t \delta t}f_k e^{-i \omega_{min}(t_k-t_{min})}, +we recognize a DFT (Eq. :ref:`DFT`). To calculate it using FFTW, we first need to prepare the input :math:`f_m`, then to do the inverse DFT and finally to modify the output to obtain :math:`G(t_k)`: + .. math:: f_m = \tilde G(\omega_m) e^{-i t_{min}\omega_m}, + .. math:: G(t_k) = \frac{1}{N_t \delta t}\tilde f_k e^{-i \omega_{min}(t_k-t_{min})}. @@ -74,24 +72,55 @@ Implementation in imaginary time/frequency using FFTW The imaginary time mesh parameters are :math:`\beta` and :math:`N_\tau`, plus a tag ``half_bins``, ``full_bins`` or ``without_last``. In the ``full_bins`` case, one point of the time GF has to be removed for the fourier transform. -From these parameters, we deduce :math:`\delta\tau=\beta/N_\tau` +From these parameters, we deduce :math:`\delta\tau=\beta/N_\tau`. -CHAPTER NOT FINISHED !!!! It seems that only real GF's in time are considered (w_n is always >0)... - -For the imaginary frequency mesh, they are :math:`n_{min}`, :math:`\beta` and :math:`N_\omega`. +For the imaginary frequency mesh, the mesh parameters are :math:`\beta`, :math:`n_{min}` and :math:`N_{\omega_n}`. From them, we deduce :math:`\delta\omega=\frac{2\pi}{\beta}`. + The Fourier transform requires :math:`N_\omega=N_\tau`. The times are :math:`\tau_k=\tau_{min}+k\delta\tau` and the frequencies :math:`\omega_n=\omega_{min}+n\delta \omega`. :math:`\tau_{min}` is either 0 or :math:`\delta\tau/2` depending on the mesh kind. -:math:`\omega_{min}` is either :math:`\frac{2\pi(n_{min}+1)}{\beta}` or :math:`\frac{2\pi n_{min}}{\beta}` depending on the statistic. +:math:`\omega_{min}` is either :math:`\frac{2\pi(n_{min}+1)}{\beta}` or :math:`\frac{2\pi n_{min}}{\beta}` depending on the statistics. We approximate the TF and its inverse by .. math:: \tilde G(i\omega_n) = \delta\tau \sum_{k=0}^{N_\tau} G(\tau_k)e^{i\omega_n \tau_k} - .. math:: G(\tau_k) = \sum_{n=0}^{N_\tau} \frac{1}{\beta} \tilde G(i\omega_n)e^{-i\omega_n \tau_k} + .. math:: G(\tau_k) = \sum_{n=n_{min}}^{N_\tau} \frac{1}{\beta} \tilde G(i\omega_n)e^{-i\omega_n \tau_k} We use for the TF: - .. math:: f_k = G(\tau_k) e^{i \omega_{min}\tau_k}, - .. math:: \tilde G(i\omega_m) = \frac{\beta}{N_\tau} \tilde f_m e^{i \tau_{min}(\omega_m-\omega_{min})}. + .. math:: \tilde f_k = G(\tau_k) e^{i \omega_{min}\tau_k}, + .. math:: \tilde G(i\omega_n) = \frac{\beta}{N_\tau} f_n e^{i \tau_{min}(\omega_n-\omega_{min})}. +and for the inverse TF: + .. math:: f_m = \frac{1}{\beta}\tilde G(i\omega_n) e^{-i t_{min}\omega_n}, + .. math:: G(t_k) = \tilde f_k e^{-i \omega_{min}(\tau_k-\tau_{min})}, + + +Special case of real functions in time for fermions +---------------------------------------------------- + +In this case, :math:`G(i\omega_n)=conj(G(i\omega_n))` and we only store the values of :math:`G(i\omega_n)` for :math:`\omega_n > 0`. +The Eq. :ref:`inv_DFT_I` becomes: + + :label: _inv_TF_I_real_fermion + .. math:: G(\tau)=\sum_{n=0}^\infty \frac{2}{\beta} \tilde G(i\omega_n)\cos(\omega_n \tau) + +The inverse TF formulas are in this case + .. math:: f_m = \frac{2}{\beta}\tilde G(i\omega_n) e^{-i t_{min}\omega_n}, + .. math:: G(t_k) = \tilde f_k e^{-i \omega_{min}(\tau_k-\tau_{min})}, + +Special case of real functions in time for bosons +-------------------------------------------------- + +In this case, :math:`G(i\omega_n)=conj(G(i\omega_n))` and we only store the values of :math:`G(i\omega_n)` for :math:`\omega_n \ge 0`. +The Eq. :ref:`inv_DFT_I` becomes: + + :label: _inv_TF_I_real_bosons + .. math:: G(\tau)=\frac{1}{\beta} \tilde G(0)+\sum_{n=1}^\infty \frac{2}{\beta} \tilde G(i\omega_n)\cos(\omega_n \tau) + +The inverse TF formulas are in this case + .. math:: f_0 = \frac{1}{\beta}\tilde G(0), + .. math:: f_m = \frac{2}{\beta}\tilde G(i\omega_n) \cos(t_{min}\omega_n), + .. math:: G(t_k) = \tilde f_k e^{-i \omega_{min}(\tau_k-\tau_{min})}, + Effect of a TF on the tail =========================== diff --git a/doc/reference/c++/gf/meshes.rst b/doc/reference/c++/gf/meshes.rst index 3527b957..05256a9d 100644 --- a/doc/reference/c++/gf/meshes.rst +++ b/doc/reference/c++/gf/meshes.rst @@ -25,9 +25,63 @@ Briefly, if we want to describe a function on an interval: We then have to be careful for example when we fourier transform the function (to not take twice the same point). +How to access to a mesh point with its index +--------------------------------------------- -The four basic meshes -===================== +.. compileblock:: + + #include + using namespace triqs::gf; + + int main() { + + //we construct a GF + double wmin = 0.0; + double wmax = 1.0; + int nw = 101; + auto Gw = make_gf(wmin, wmax, nw); + + //we print the mesh parameters and print te value of the 10th point + std::cout << "The kind of the mesh is " << Gw.mesh().kind() << std::endl; + std::cout << "The smallest mesh point value is w_min=" << Gw.mesh().x_min() << std::endl; + std::cout << "The largest mesh point value is w_max=" << Gw.mesh().x_max() << std::endl; + std::cout << "The number of mesh points is n=" << Gw.mesh().size() << std::endl; + std::cout << "Between two consecutive mesh points: delta=" << Gw.mesh().delta() << std::endl; + std::cout << "The 10th mesh point is w=" << Gw.mesh()[10] << std::endl; + + } + +How to access to a mesh point with a value +------------------------------------------- + +In this case, we look for the closest mesh point, but can need the distance of the value to the mesh point. +``windowing`` gives all these informations: + +.. compileblock:: + + #include + using namespace triqs::gf; + + int main() { + double wmin = 0.0; + double wmax = 1.0; + int nw=101; + auto Gw= make_gf(wmin, wmax, nw); + double w=0.25156; + size_t index; double wd; bool in; + std::tie(in, index, wd) = windowing ( Gw.mesh(), w); + std::cout << "Is the point w="<< w <<" in the mesh range ? " << in << std::endl; + if(in){ + std::cout << "The point before is the " << index << "th" << std::endl; + std::cout << "The position in the intervall is " << wd << std::endl; + } + } + + + + +The four basic linear meshes +============================ Real time @@ -60,7 +114,57 @@ Depending on what one needs, we can choose ``full_bins``, ``half_bins`` or ``w Matsubara frequency -------------------- -The domain is discrete. The Matsubara frequencies are :math:`\omega_n=\frac{(2n+1)\pi}{beta}` for fermions and :math:`\omega_n=\frac{2n\pi}{beta}` for bosons. +The domain is discrete. The Matsubara frequencies are :math:`\omega_n=\frac{(2n+1)\pi}{\beta}` for fermions and :math:`\omega_n=\frac{2n\pi}{\beta}` for bosons. + + + +Products of meshes +=================== + +We detail the case of a two mesh product, but what follows is true for any number of meshes. + +A mesh point can be labelled by a linear index, or by a tuple of indices. Each mesh point correspond to a point of the domain, which is a tuple of points of the subdomains. +We can navigate between these representations, through ``closest_mesh_pt``, ``get_closest_pt``, ``index_to_linear``,... + + +How to access to the closest mesh point +--------------------------------------- + +.. compileblock:: + + #include + using namespace triqs::gf; + + int main() { + double tmax = 1.0; + int nt = 101; + auto Gtt = make_gf(tmax, nt, triqs::arrays::make_shape(1,1)); + + //does not work for instance + //double t1 = 0.256, t2 = 0.758; + //Gtt(closest_mesh_pt(i1,i2)) = 1.5; + + } + + +How to access to a mesh point with its index +--------------------------------------------- + +.. compileblock:: + + #include + using namespace triqs::gf; + + int main() { + double tmax = 1.0; + int nt = 101; + auto Gtt = make_gf(tmax, nt, triqs::arrays::make_shape(1,1)); + + int i1 = 14, i2 = 86; + Gtt.on_mesh(i1, i2) = 1.8; + std::cout << Gtt.on_mesh(i1, i2)(0,0) << std::endl; + + }