triqs.mesh.meshes.MeshChebyshev

class triqs.mesh.meshes.MeshChebyshev

Bases: object

Chebyshev imaginary time mesh type.

A Chebyshev imaginary time mesh is defined by its size \(N \geq 0\), an inverse temperature \(\beta > 0\) and its particle statistics. It stores function values at Chebyshev collocation points and uses barycentric interpolation for evaluation at arbitrary imaginary times, providing exponential convergence for smooth functions.

A Chebyshev imaginary time mesh 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.

• An index \(n\) is mapped to the corresponding value \(\tau\) by the function \(\tau(n) = \frac{\beta}{2} (x_n + 1)\) where \(x_n = \cos\left(\frac{(2n + 1) \pi}{2N}\right)\) is the Chebyshev point of the first kind.

Green’s function containers that are based on a Chebyshev imaginary time mesh store the function values at the discrete time points \(\tau(n)\), i.e. \(f_n = f(\tau(n))\), and use barycentric interpolation to evaluate the function at an arbitrary imaginary time \(\tau \in [0, \beta]\):

\[f(\tau) \approx \frac{\sum_{n=0}^{N-1} \frac{w_n}{x - x_n} f_n}{\sum_{n=0}^{N-1} \frac{w_n}{x - x_n}}\]

where \(x = 2\tau/\beta - 1\) is the scaled coordinate and \(w_n\) are the barycentric weights.

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Dispatched C++ constructor(s).

[1] ()

[2] (beta: float, stat: Statistic ("Fermion" | "Boson"), N: int)

[1] Default constructor constructs an empty mesh.

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[2] Construct a Chebyshev mesh on \([0, \beta]\) with \(N\) collocation points.

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Parameters:

betafloat

Inverse temperature \(\beta > 0\).

statStatistic (“Fermion” | “Boson”)

Particle statistics.

Nint

Number of Chebyshev points \(N > 0\).

Attributes

beta	Get the inverse temperature \(\beta\).
inv_beta_x2	Get the precomputed \(2 / \beta\) for fast tau -> [-1, 1] mapping.
mesh_hash	Get the hash value of the mesh.
points	Access to Chebyshev points scaled to [0, beta].
points_standard	Access to Chebyshev points on [-1, 1].
statistic	Get the particle statistics.
weights	Access to barycentric weights.

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 \(n(d)\).
to_value	Map an index \(n\) to its corresponding value \(\tau_n\).
values	Get the values of all mesh points in a mesh.