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dft_tools/doc/reference/c++/gf/tail.rst
Priyanka Seth 71fa498833 API change for tail fitters
python fit_tail, replace_by_tail ==> fit_tail_depr, replace_by_tail_depr
c++ set_tail_from_fit ==> fit_tail
2014-10-01 10:52:32 +02:00

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.. highlight:: c
.. _gf_tail:
High frequency tail
===========================
Definition
----------------------
The tail of a Green's function is defined as the behavior of the Green's
function :math:`G` at large Matsubara frequencies, namely
.. math:: \mathbf{G}(i\omega_n) \stackrel {=}{\infty} \mathbf{a}_{-1}\cdot i\omega_n + \mathbf{a}_{0} +\mathbf{a}_{1}\cdot \frac{1}{ i\omega_n} +\mathbf{a}_{2}\cdot \frac{1}{ (i\omega_n)^2} +\dots
Generically, the tail is parametrized by matrix-valued coefficients
:math:`\mathbf{a}_{i}` (of size :math:`N_1\times N_2`\ )
.. math:: t = \sum_{i=o_{min}}^{o_{max}} \mathbf{a}_i (i\omega_n)^{-i}
Implementation
--------------
In TRIQS, the tail is implemented as an object ``tail``. Here is a simple example of use:
.. triqs_example:: ./tail_0.cpp
Fitting the tail of a Green's function
---------------------------------------
Given an imaginary-frequency Green's function, one can compute the moments of its high-frequency tail with the function ``fit_tail``:
.. triqs_example:: ./tail_1.cpp
The full documentation of ``fit_tail`` is :doc:`here<fit_tail>`.
API
****
Here are the main methods of the ``tail`` class:
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| Member | Description | Type |
+=================================+=========================================================================================+==========================+
| data() | 3-dim array of the coefficients: ``data(i,n,m)`` :math:`=(\mathbf{a}_{i+o_{min}})_{nm}` | data_view_type |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| mask_view() | 2-dim (:math:`N_1 \times N_2`) array of the maximum non-zero indices | mask_view_type |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| order_min() | minimum order | long |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| order_max() | maximum order | long |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| size() | first dim of data() | size_t |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| shape() | shape of data() | shape_type |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| smallest_nonzeros() | order of the smallest_nonzero coefficient | long |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| is_decreasing_at_infinity() | true if the tail is decreasing at infinity | bool |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| operator() (int n) | matrix_valued coefficient :math:`(\mathbf{a}_i)_{nm}` | mv_type |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| get_or_zero (int n) | matrix_valued coefficient :math:`(\mathbf{a}_i)_{nm}` | const_mv_type |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
| evaluate(dcomplex const &omega) | value of the tail at frequency omega | arrays::matrix<dcomplex> |
+---------------------------------+-----------------------------------------------------------------------------------------+--------------------------+
The tail is DefaultConstructible, H5Serializable and BoostSerializable.