Concepts¶
A Green function is simply a function, which has:
a domain for its variable(s) (e.g. Matsubara/real time/frequencies, Legendre coefficients).
a target space, i.e. the value of the Green function which can be:
- a scalar (double, complex)
- a matrix,
- another Green function (See below, currying Green functions … REF … ).
In this section, we define the general concepts for these objects.
First, we need to distinguish the domain on which the function is defined from its representation in a computer, which we call a mesh.
Note
“mesh” should be understood here in a general and abstract way, as the representation of the domain in the computer. In most cases, it is indeed a real mesh on a domain (e.g. a Brillouin zone), but the set of Legendre coefficients is also a mesh in our sense.
We will therefore now formally define the concept for domain, for mesh, the notion of pure function on a domain (i.e. a mathematical Green function) and the notion of function on a grid.
Domain¶
- Purpose : The domain of definition of a function. It is a mathematical definition of the domain, and does not contain any mesh, or details on its representation in a computer.
- Refines: RegularType.
- Definition:
| Elements | Comment |
|---|---|
| point_t | Type of element in the domain (int, int, double, k_vector, …) as in the call of a function over this domain. |
- Examples :
- Matsubara time
- Matsubara frequencies (boson/fermion): in this case, point_t is matsubara_freq, a simple type containing (n, beta, statistics).
- Real frequencies
- Real time
- Brillouin zone
- Cartesian product of previous domains to build multi-variable functions.
PureFunctionOnDomain¶
- Purpose :
- A mathematical (pure) function from a domain to a target space.
- it has a domain of definition
- it can be called on any point of the domain, as a pure function, i.e. without any side effect.
- Refines :
- Definition:
| Elements | Comment |
|---|---|
| domain_t const & domain() const | Returns the domain (deduced as domain_t) |
| operator (domain_t::point_t) const | Calling for all elements of the Domain (including infty if it is in the domain… |
- NB: Note that the return type of the function is NOT part of the concept, it has to be deduced by the compiler (using C++11 decltype, std::result_of, eg..).
Note
Probably domain_t should also be deduced from the return type of domain … TO BE CORRECTED
Mesh¶
- Purpose : A mesh over a domain, and more generally the practical representation of the domain in a computer. It does not really need to be a mesh: e.g. if the function is represented on a polynomial basis, it is the parameters of this representation (max number of coordinates, e.g.)
- Refines: RegularType, H5-serializable, Printable.
- Definition:
| Elements | Comment |
|---|---|
| domain_t | Type of the Domain represented, modeling the Domain concept |
| domain_t const & domain() const | Access to the domain |
| index_t | Type of indices of a point on the grid. Typically a tuple of long or a long |
| long size() const | The number of points in the mesh. |
| domain_t::point_t index_to_point(index_t) const | From the index of a mesh point, compute the corresponding point in the domain |
| long index_to_linear(index_t const &) const | Flattening the index of the mesh into a contiguous linear index |
| mesh_point_t | A type modeling MeshPoint concept (see below). |
| mesh_point_t operator[](index_t const & index ) const | From an index, return a mesh_point_t containing this a ref to this mesh and the index. |
| mesh_pt_generator<mesh_t> const_iterator | A generator of all the mesh points. |
| const_iterator begin()/end() const cbegin()/cend() const | Standard access to iterator on the mesh Standard access to iterator on the mesh |
| Free functions | Comment |
|---|---|
| void foreach (mesh_t, F) | If F is a function of synopsis auto F( mesh_t::mesh_point_t) it calls F for each point on the mesh, in arbitrary order |
MeshPoint¶
- Purpose : Abstraction of a point on a mesh. A little more than a ref to the mesh and a index.
- Refines: CopyConstructible.
- Definition:
| Elements | Comment |
|---|---|
| mesh_t | Type of the mesh |
| mesh_t const * m | A pointer to the mesh to which the point belongs. |
| mesh_t::index_t index | The index of the point |
| mesh_point_t( mesh_t const &, index_t const &) | Constructor: a mesh point at the given index |
| mesh_point_t( mesh_t const &) | Constructor: the first mesh point |
| mesh_t::index_t [const &,] index() const | The index corresponding to the point |
| size_t linear_index() const | The linear index of the point (same as m->index_to_linear(index()) |
| void advance() | Advance to the next point on the mesh (used by iterators). |
| void at_end() | Is the point at the end of the grid |
| void reset() | Reset the mesh point to the first point |
| cast_t operator cast_t() const | == mesh_t::domain_t::point_t implicit cast to the corresponding domain point |
For one dimensional mesh, we also require that the MeshPoint implement the basic arithmetic operations using the cast.
- Discussion:
A MeshPoint is just an index of a point on the mesh, and containers like gf can easily be overloaded for this type to have a direct access to the grid (Cf [] operator of gf).
However, since the MeshPoint can be implicitely casted into the domain point, simple expression like
g[p] = 1/ (p +2)
make sense and fill the corresponding point wiht the evaluation of 1/ (p+2) in the domain.
As a result, because iterating on a mesh result in a series of object modelling MeshPoint, one can write naturally
// example of g, a Green function in Matsubara frequencies w
for (auto w: g.mesh())
g[w] = 1/(w + 2)
// This runs overs the mesh, and fills the function with 1/(w+2)
// In this expression, w is casted to the domain_t::point_t, here a complex<double>
// which allows to evaluate the function