dyson.representations

Representations for Green’s functions and self-energies.

Both the Green’s function and self-energy can be represented in the frequency domain according to their Lehmann representation, for the Green’s function

\[G_{pq}(\omega) = \sum_{x} \frac{u_{px} u_{qx}^*}{\omega - \varepsilon_x},\]

where poles \(\varepsilon_x\) couple to the physical states of the system according to the Dyson orbitals \(u_{px}\). For the self-energy, the representation is given by

\[\Sigma_{pq}(\omega) = \sum_{k} \frac{v_{pk} v_{qk}^*}{\omega - \epsilon_k},\]

where \(v_{px}\) are the couplings between auxiliary states and the physical states, and \(\epsilon_k\) are the auxiliary state energies.

These two Lehmann representations can be relataed to each other via the Dyson equation, which can be written as an eigenvalue problem in the upfolded configuration space as

\[\begin{split}\begin{bmatrix} \boldsymbol{\Sigma}(\omega) & \mathbf{v} \\ \mathbf{v}^\dagger & \boldsymbol{\epsilon} \mathbf{I} \end{bmatrix} \begin{bmatrix} \mathbf{u} \\ \mathbf{w} \end{bmatrix} = \boldsymbol{\varepsilon} \begin{bmatrix} \mathbf{u} \\ \mathbf{w} \end{bmatrix}.\end{split}\]

The Lehmann representations of either the Green’s function or self-energy are contained in Lehmann objects, which is a simple container for the energies and couplings, along with a chemical potential. The Spectral representation provides a container for the full eigenspectrum (including \(\mathbf{w}\)), and can provide the Lehmann representation of both the Green’s function and self-energy.

>>> from dyson import util, quiet, FCI, Exact
>>> quiet()  # Suppress output
>>> mf = util.get_mean_field("H 0 0 0; H 0 0 1", "6-31g")
>>> fci = FCI.h.from_mf(mf)
>>> solver = Exact.from_expression(fci)
>>> result = solver.kernel()
>>> type(result)
<class 'dyson.representations.spectral.Spectral'>
>>> self_energy = result.get_self_energy()
>>> type(self_energy)
<class 'dyson.representations.lehmann.Lehmann'>
>>> greens_function = result.get_greens_function()
>>> type(greens_function)
<class 'dyson.representations.lehmann.Lehmann'>

Lehmann representations can be realised onto a subclass BaseGrid to provide a dynamic representation of the function, which is stored in a Dynamic object. This dynamic representation has varied formats, principally depending on the type of grid used, but also according to the so-called Reduction and Component of the representation. The Reduction enum encodes the format of the matrix, i.e. whether it is the full matrix, the diagonal part, or the trace. The Component enum encodes the numerical component of the matrix, i.e. whether it is the real or imaginary part, or the full complex matrix.

>>> from dyson.grids import GridRF
>>> grid = GridRF.from_uniform(-3.0, 3.0, 256, eta=1e-1)
>>> spectrum = grid.evaluate_lehmann(
...     greens_function, ordering="retarded", reduction="trace", component="imag"
... )
>>> type(spectrum)
<class 'dyson.representations.dynamic.Dynamic'>

The various solvers in solvers have different representations as their inputs and outputs.

Submodules

representation	Base class for representations.
lehmann	Container for a Lehmann representation.
spectral	Container for an spectral representation (eigenvalues and eigenvectors) of a matrix.
dynamic	Container for a dynamic representation.
enums	Enumerations for representations.