triqs.mesh.meshes.MeshDLR

class triqs.mesh.meshes.MeshDLR

Bases: object

Discrete Lehmann representation (DLR) mesh type.

A DLR mesh is defined by the inverse temperature \(\beta > 0\), the particle statistics, a DLR energy cutoff \(\omega_{\text{max}}\), an error tolerance \(\epsilon\) and a boolean flag specifying if the mesh should be symmetric around \(\omega = 0\).

A DLR mesh has the following properties:

Each mesh point is identified by a unique index \(l \in \{0, 1, \ldots, N-1\}\).
The size of the mesh \(N\) depends on \(\beta\) and the choice of \(\omega_{\text{max}}\) and \(\epsilon\). It is equal to the DLR rank \(r\) and the number of DLR basis functions \(K(\tau, \omega_l)\) or \(K(i\omega_n, \omega_l)\).
An index \(l\) is mapped to the corresponding data index \(d\) by the identity function \(d(l) = l\) and vice versa.
An index \(l\) is mapped to the corresponding value \(\omega_l\), where \(\omega_l\) is the l-th DLR frequency.

Green’s function containers that are based on a DLR mesh store the coefficients \(f_l\) of the discrete Lehmann representation of a function \(f(\tau)\) or \(f(i\omega_n)\). To evaluate the function at an arbitrary imaginary time \(\tau \in [0, \beta]\) or at a specific Matsubara frequency \(i\omega_n\), the GF container calculates the DLR approximation of the function via

\[f(\tau) \approx \sum_{l=0}^{N-1} \frac{e^{-\omega_l \tau}}{1 + e^{-\omega_l \beta}} f_l \; ,\]

or

\[f(i\omega_n) \approx \sum_{l=0}^{N-1} \frac{1}{i\omega_n + \omega_l} f_l \; .\]

Dispatched C++ constructor(s).

[1] ()

[2] (beta: float,
     statistic: Statistic ("Fermion" | "Boson"),
     w_max: float,
     eps: float,
     symmetrize: bool = True)

[3] (m: MeshDLRImTime)

[4] (m: MeshDLRImFreq)

[1] Default constructor constructs an empty mesh.

[2] Construct a DLR mesh with a given energy cutoff \(\omega_{\text{max}}\) and error tolerance \(\epsilon\) .

It builds the DLR frequencies \(\omega_l\) from \(\Lambda = \omega_{\text{max}} \beta\) and the error tolerance \(\epsilon\), then constructs the imaginary-time and imaginary-frequency DLR operator tables from them.

[3, 4] Construct a DLR mesh from another DLR type mesh.

Parameters:

betafloat: Inverse temperature \(\beta > 0\).
statisticStatistic (“Fermion” | “Boson”): Particle statistics.
w_maxfloat: DLR energy cutoff \(\omega_{\text{max}} = \Lambda / \beta\).
epsfloat: Error tolerance \(\epsilon\).
symmetrizebool: Whether to choose the DLR frequencies symmetrically around \(\omega = 0\).
mMeshDLRImTime, MeshDLRImFreq: Other mesh.

Attributes

`beta`	Get the inverse temperature \(\beta\).
`dlr_freq`	Get the array of DLR frequencies \(\omega_l\).
`eps`	Get the DLR error tolerance \(\epsilon\).
`mesh_hash`	Get the hash value of the mesh.
`statistic`	Get the particle statistics.
`symmetrize`	Is the mesh symmetric around \(\omega = 0\)?
`w_max`	Get the DLR energy cutoff \(\omega_{\text{max}} = \Lambda / \beta\).

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 \(l\) is valid.
`to_data_index`	Map an index \(l \in \{0, 1, \ldots, N-1\}\) to its corresponding data index \(d(l)\).
`to_index`	Map a data index \(d \in \{0, 1, \ldots, N-1\}\) to the corresponding index \(l(d)\).
`to_value`	Map an index \(l \in \{0, 1, \ldots, N-1\}\) to its corresponding value \(\omega_l\).
`values`	Get the values of all mesh points in a mesh.