triqs.mesh.meshes.MeshCycLat

class triqs.mesh.meshes.MeshCycLat

Bases: object

Cyclic lattice mesh type for Bravais lattices with Born-von Karman periodic boundary conditions.

A cyclic lattice mesh is defined by an underlying Bravais lattice and the number of unit cells in each of the three dimensions, \(N_1\), \(N_2\) and \(N_3\), which define the supercell and the periodic boundary conditions (PBC) \(f(\mathbf{R}^{\mathbf{n}}) = f(\mathbf{R}^{\mathbf{n}} + \mathbf{T}^{\mathbf{m}})\). Here, \(\mathbf{T}^{\mathbf{m}} = \sum_{i=1}^3 \mathbf{a}_i N_i m_i = \tilde{\mathbf{A}} \mathbf{m}\) is a supercell translation vector, \(\mathbf{a}_i\) is a Bravais lattice basis vector in direction \(i\) and \(\mathbf{m} = (m_1, m_2, m_3)\) with \(m_i \in \mathbb{Z}\).

It has the following properties:

Each mesh point is identified by
- a unique index in the supercell \(\mathbf{n} = (n_1, n_2, n_3)\), where \(0 \leq n_i < N_i\), and
- an infinite set of indices due to the periodic boundary conditions, i.e. \(\{ \tilde{\mathbf{n}} = \mathbf{n} + \mathbf{N} \mathbf{m} : \mathbf{N} \mathbf{m} = (N_1 m_1, N_2 m_2, N_3 m_3) \in \mathbb{Z}^3 \}\).
The size of the mesh is \(N = N_1 \, N_2 \, N_3\), i.e. the total number of unit cells in the supercell.
An index \(\mathbf{n}\) is mapped to the corresponding data index \(d\) by the function \(d(\mathbf{n}) = n_3 + N_3 (n_2 + N_2 n_1) = n_3 + n_2 N_3 + n_1 N_2 N_3\). The inverse map is \(\mathbf{n}(d) = (\lfloor d / s_1 \rfloor, \lfloor (d \mod s_1) / s_2 \rfloor, (d \mod s_1) \mod s_2 )\), where \(s_1 = N_2 N_3\) and \(s_2 = N_3\).
An index \(\mathbf{n}\) is mapped to the corresponding value (lattice point) \(\mathbf{R}(\mathbf{n}) \equiv \mathbf{R}^{\mathbf{n}} = \sum_{i=1}^3 \mathbf{a}_i n_i = \mathbf{A} \mathbf{n}\), where \(\mathbf{a}_i\) is the Bravais lattice basis vectors in direction \(i\). The inverse map is then \(\mathbf{n}(\mathbf{R}^{\mathbf{n}}) = \mathbf{A}^{-1} \mathbf{R}^{\mathbf{n}} = \mathbf{n}\).

Green’s function containers that are based on a cyclic lattice mesh store the function values at the discrete lattice points \(\mathbf{R}^{\mathbf{n}}\), i.e. \(f_{\mathbf{n}} = f(\mathbf{R}^{\mathbf{n}})\). Because of the PBC, the container only has to store values for indices with \(0 \leq n_i < N_i\). To evaluate the function at an arbitrary lattice point, it is first mapped to the supercell using the PBC and then the corresponding function value is returned.

Dispatched C++ constructor(s).

[1] (bl: BravaisLattice, dims: [int, len = 3])

[2] (bl: BravaisLattice, L: int)

[3] (L1: int = 1, L2: int = 1, L3: int = 1)

[1] Construct a cyclic lattice mesh on a Bravais lattice with the given supercell dimensions.

[2] Construct a cyclic lattice mesh on a Bravais lattice with a cubic supercell.

[3] Construct a cyclic lattice mesh on a cubic Bravais lattice with \(a = 1\) and the given supercell dimensions.

Parameters:

blBravaisLattice: Bravais lattice object representing the underlying Bravais lattice.
dims[int, len = 3]: Number of unit cells in the supercell along each of the three dimensions, i.e. \((N_1, N_2, N_3)\) .
Lint: Number of unit cells in the supercell along each of the three dimensions.
L1int: Number of unit cells in the supercell along the first dimension.
L2int: Number of unit cells in the supercell along the second dimension.
L3int: Number of unit cells in the supercell along the third dimension.

Attributes

`dims`	Get the number of unit cells in each of the three dimensions.
`lattice`	Get the underlying Bravais lattice.
`mesh_hash`	Get the hash value of the mesh.
`units`	Get the matrix \(\mathbf{A}^T\) containing the basis vectors of the Bravais lattice in its rows.

Methods

`__call__`(args, *kwargs)	Call self as a function.
`copy`	Get a copy of a mesh (for Python bindings).
`copy_from`	Copy one mesh into another (for Python bindings).
`index_modulo`	Map an arbitrary index \(\tilde{\mathbf{n}}\) to the unique index \(\mathbf{n}\) in the supercell.
`is_index_valid`	Check if an index \(\mathbf{n}\) is valid, i.e. corresponds to a unit cell/lattice point in the.
`to_data_index`	Map an index \(\mathbf{n}\) to its corresponding data index \(d(\mathbf{n})\).
`to_index`	Map a data index \(d \in \{0, 1, \ldots, N-1\}\) to the corresponding index \(\mathbf{n}(d)\)
`to_value`	Map an index \(\mathbf{n}\) to its corresponding lattice point \(\mathbf{R}^{\mathbf{n}}\).
`values`	Get the values of all mesh points in a mesh.