triqs.mesh.meshes.MeshLegendre

class triqs.mesh.meshes.MeshLegendre

Bases: object

Legendre mesh type.

A Legendre mesh is defined by the number of Legendre polynomials \(N\) used in the series expansion, an inverse temperature \(\beta > 0\) and its particle statistics.

A Legendre mesh is an index-only mesh (no value attached to each point) and has the following properties:

• Each mesh point is identified by a unique index \(n \in \{0, 1, \ldots, N-1\}\).

• An index \(n\) is mapped to the corresponding data index \(d\) by the identity function \(d(n) = n\) and vice versa.

• There is no explicit value associated with a mesh point, i.e. it carries no value at each mesh point.

Green’s function containers that are based on a Legendre mesh store the coefficients \(f_n\) of the generalized Fourier series expansion of a function \(f(\tau)\) in terms of Legendre polynomials, which is defined on the imaginary time axis. To evaluate the function at an arbitrary imaginary time \(\tau \in [0, \beta]\), the GF container sums up the Fourier series to the maximum degree \(N-1\):

\[f(\tau) \approx \sum_{n=0}^{N-1} \frac{\sqrt{2n + 1}}{\beta} f_n P_n(x(\tau)) \; ,\]

where \(P_n\) denotes the Legendre polynomial of degree \(n\), \(x(\tau) = 2\tau / \beta - 1\) is a linear map from \([0, \beta]\) to \([-1, 1]\) and

\[f_n = \sqrt{2n + 1} \int_0^\beta d\tau \, f(\tau) P_n(x(\tau)) \; ,\]

are the expansion coefficients.

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Dispatched C++ constructor(s).

[1] ()

[2] (beta: float, statistic: Statistic ("Fermion" | "Boson"), max_n: int)

[1] Default constructor constructs an empty mesh.

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[2] Construct a mesh of Legendre polynomials with degrees \(n = 0, 1, \ldots, N - 1\) on the interval \([0, \beta]\) and the given particle statistics.

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Parameters:

betafloat

Inverse temperature \(\beta > 0\).

statisticStatistic (“Fermion” | “Boson”)

Particle statistics.

max_nint

Size of the mesh, i.e. the number of Legendre polynomial used in the series expansion.

Attributes

beta	Get the inverse temperature \(\beta\).
mesh_hash	Get the hash value of the mesh.
statistic	Get the particle statistics.

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).
is_index_valid	Check if an index \(n\) is valid.
to_data_index	Map an index \(n \in \{0, 1, \ldots, N-1\}\) to its corresponding data index \(d(n)\).
to_index	Map a data index \(d \in \{0, 1, \ldots, N-1\}\) to the corresponding index \(n(d)\).