.. highlight:: c Green functions ################# Here are the concepts and techniques related to the Green functions. All Green functions are implemented as a gf and gf_view where : * Descriptor contains everything specific to the Green function (number of variables, domain of definition, interpolation techniques, etc...). * The gf/gf_view class implements the generic part (hdf5, view mechanism, interaction with clef library, etc...). We first present the concept of the Descriptor, and the concepts of its elements. In a second part, we present a more detailed description of gf/gf_view. Pure functions on domains ============================== This formalize a mathematical function : * 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. Domain concept ------------------------------------------------- * **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** : CopyConstructible, DefaultContructible, EqualComparable, BoostSerializable, H5-serializable. * **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. In particular, in | | | Matsubara, it is a complex. | +----------------------------------------------------------------------------+---------------------------------------------------------------------+ * **Examples** : * Matsubara frequencies (boson/fermion) * Matsubara time * Real frequencies * Real time * Brillouin zone PureFunctionOnDomain ----------------------- * **Purpose** : A function from a domain to a target space. * **Refines** : * **Definition** : +--------------------------------------+----------------------------------------------------------+ | Elements | Comment | +======================================+==========================================================+ | domain_t | Type of the Domain represented, modelling Domain concept | +--------------------------------------+----------------------------------------------------------+ | domain_t const & domain() const | Returns the domain | +--------------------------------------+----------------------------------------------------------+ | 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..). Functions stored on a mesh ================================ A special kind of function is those stored on a mesh. The mesh can be regular or not, multi-dimensionnal. We first define the concept of a mesh, and then the concept of a function discretized over that mesh. Mesh concept ------------------------------------------------- * **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** : CopyConstructible, DefaultContructible, EqualComparable, BoostSerializable, H5-serializable. * **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 | +--------------------------------------------------------------+-------------------------------------------------------------------------------+ | size_t 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 | +--------------------------------------------------------------+-------------------------------------------------------------------------------+ | size_t index_to_linear(index_t const &) const | Flattening the index of the mesh into a contiguous linear index | +--------------------------------------------------------------+-------------------------------------------------------------------------------+ | mesh_point_t | 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 iterator | A generator of all the mesh points. | +--------------------------------------------------------------+-------------------------------------------------------------------------------+ | iterator begin()/end() const | Standard access to iterator on the mesh | +--------------------------------------------------------------+-------------------------------------------------------------------------------+ | friend bool operator == (mesh_t const& M1, mesh_t const &M2) | Comparison between meshes | +--------------------------------------------------------------+-------------------------------------------------------------------------------+ * **Examples** : Some domains and the corresponding possible meshes. +-----------------------------------------------------+--------------------------------------------------------+ | Domain Type | What does the corresponding mesh type contain ? | +=====================================================+========================================================+ | Matsubara frequencies | Nmax, the max number of Matsubara Freq. | +-----------------------------------------------------+--------------------------------------------------------+ | Matsubara time | The time slicing is on a mesh, or the Legendre mesh is | | | we store the function with Legendre representation. | +-----------------------------------------------------+--------------------------------------------------------+ | Real frequencies | Parameters of the mesh in frequency | +-----------------------------------------------------+--------------------------------------------------------+ | Real time | | +-----------------------------------------------------+--------------------------------------------------------+ | Brillouin zone | parameters of the mesh over the BZ, symmetry ? | +-----------------------------------------------------+--------------------------------------------------------+ MeshPoint concept ------------------------------------------------- * **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 | +------------------------------------------------+-----------------------------------------------------------------------------+ | 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 | type of the corresponding domain point | | operator cast_t() const | cast to the corresponding domain point | +------------------------------------------------+-----------------------------------------------------------------------------+ | Implements the basic operations on the domain | Only for non composite mesh | | by using the cast operation | | +------------------------------------------------+-----------------------------------------------------------------------------+ The MeshPoint mechanism --------------------------- A MeshPoint is just a storage of a reference to the mesh and the index of the point in a custom structure. The interest of having such a structure is that : * The gf function has a operator()(mesh_t::mesh_point_t) (see below) which is a direct access to the data on the grid. Hence if MP is a such a MeshPoint, g(MP) is equivalent to something like g.data_on_grid_at_index( MP.index) * MP however can be casted to a point in the domain and therefore it *is* a domain_t::point_t as well. As a result, g(MP) = 1/(MP + 2) makes senses iif it makes senses in the domain. * Moreover, because of the iterator on the mesh, one can write :: // 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 // which allows to evaluate the function FunctionOnMesh concept ----------------------------- * **Purpose** : A function from a domain to a target space, represented in a mesh. This function can only be evaluated on the mesh points. * **Refines** : BoostSerializable, H5-serializable. * **Definition** : +--------------------------------------------------+------------------------------------------------------------------------+ | Elements | Comment | +==================================================+========================================================================+ | mesh_t | Type of the mesh representing the domain. | +--------------------------------------------------+------------------------------------------------------------------------+ | mesh_t const & mesh() const | Returns the mesh. | +--------------------------------------------------+------------------------------------------------------------------------+ | auto operator ( grid_pt const &) const | Calling on a grid_pt gives direct access to the value on a grid point. | | auto & operator ( grid_pt const &) | Const and non const version. | +--------------------------------------------------+------------------------------------------------------------------------+ * **NB** : the result type of the () operator is deduced. Not needed in the concept. descriptors ====================================== A descriptor is a structure that contains everything specific to the Green function (number of variables, domain of definition, interpolation techniques, etc...). Descriptor concept --------------------------- * **Purpose** : * **Refines** : * **NB** : Pure static, does NOT contains any data. * **Definition** : +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | Elements | Comment | +====================================================================================+===============================================================================+ | struct tag {}; | A tag for the gf | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | mesh_t | Mesh for the gf, modeling Mesh concept | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | storage_t | The type of the storage of the data (array, vector, etc....) | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | singularity_t | Type of object storing the singularities of the gf. It is used e.g. in the | | | Fourier transformation, density computation, etc... For a simple g(omega), | | | g(t), it is typically a high frequency tail. For a more complex function | | | g(nu,nu'), it can be different. | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | symmetry_t | Type of the object storing the symmetry property of the Green function. It is | | | *nothing* by default. This type must be a value (DefaultConstructible, | | | CopyConstructible, Assignable) | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | target_indices_t | Type of the indices of the gf, typically array | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | static const int arity | Number of variable authorized in calling the gf (just for compile time check | | | and nice error message, it is not really necessary) | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | struct evaluator { auto operator()( mesh_t const &, DATA_t const &, S_t const &, | All the permitted const call of the gf ! (DATA_t defined below) with the | | Args&&... args) .... as many overload as necessary } | parenthesis operator The gf<...> function create such a struct, so it can | | | hold some data ... | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ | static std::string h5_name() | Name for hdf5 naming (attribute of the tree in which the gf is stored). | +------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+ * **Values vs Views** target_t, singularity_t, indices_t are expected to be *values*. The corresponding views, i.e., target_view_t, singularity_view_t, indices_view_t will be deduced from the value type, and replaced by the value_type if no view is available. * S_t is singularity_t or its corresponding view type (if it exists). The gf/gf_view class ==================== The gf/gf_view classes are generic Green function, templated on Descriptor. They handle : * view/non view aspect, copy, etc... * hdf5 interface * clef interface * the MeshPoint mechanism as explained above. * perfect forwarding of all other const call of operator() to Descriptor. Constructors are limited to a minimal number : * empty one for gf (value semantics). * copy construction, from gf and gf_view of the same descriptor. * construction from the data of the GF. The other custom construction are delegated to make_gf functions:: gf make_gf (Descriptor, my_favorite arguments). We use here the simplest dispatch using the fact that Descriptor is an empty struct, so we can dispath the make_gf. Example of use :: auto G = make_gf (matsubara_freq(), beta, Fermion, make_shape(2,2)); Doxygen documentation ====================== The :doxy:`full C++ documentation` is available here.