dgamore.greens_function#

Single-particle Green’s function. GreensFunction builds the momentum-dependent interacting Green’s function \(G_{ab}(k, \nu) = [(\imath\nu + \mu)\delta_{ab} - \varepsilon_{ab}(k) - \Sigma_{ab}(k, \nu)]^{-1}\) from a SelfEnergy, the band dispersion \(\varepsilon(k)\) and the chemical potential \(\mu\), and derives the filling, occupation, kinetic and (Galitskii-Migdal) potential energies. The module-level helpers adjust \(\mu\) to a target filling via a Newton root search. Moment-corrected asymptotic sums are used so the finite Matsubara box does not bias the energies/filling.

Functions

`get_total_fill`(mu, ek, sigma_mat, beta, smom0)	Returns the total filling calculated from the self-energy, \(\mu\), kinetic Hamiltonian and more.
`root_fun`(mu, target_filling, ek, sigma_mat, ...)	Residual function used to find a new chemical potential \(\mu\) via Newton's method: the difference between the current filling and the target filling.
`update_mu`(mu0, target_filling, ek, ...[, logger])	Updates the chemical potential to match the target filling by using Newton's method to find the optimal \(\mu\).

class dgamore.greens_function.GreensFunction(mat: ndarray, sigma: SelfEnergy = None, ek: ndarray = None, full_niv_range: bool = True, calc_filling: bool = True, has_compressed_q_dimension: bool = False, nk: tuple = (1, 1, 1), beta: float = None, mu: float = None)[source]#

Bases: TwoPoint

The single-particle Green’s function \(G_{ab}(k, \nu) = [(\imath\nu + \mu)\delta_{ab} - \varepsilon_{ab}(k) - \Sigma_{ab}(k, \nu)]^{-1}\). Built from a SelfEnergy, the band dispersion \(\varepsilon(k)\) and the chemical potential \(\mu\); on top of the two-point orbital bookkeeping inherited from LocalTwoPoint it adds the Dyson construction (local and momentum-resolved) and the derived quantities — filling, occupation matrices, kinetic and (Galitskii-Migdal) potential energy — all using moment-corrected asymptotic Matsubara sums so the finite frequency box does not bias the result.

Parameters:

mat (ndarray)
sigma (SelfEnergy)
ek (ndarray)
full_niv_range (bool)
calc_filling (bool)
has_compressed_q_dimension (bool)
nk (tuple)
beta (float)
mu (float)

static create_g_loc(siw: SelfEnergy, ek: ndarray, beta: float, mu: float, calc_filling: bool = True) → GreensFunction[source]#

Builds a local (k-summed) Green’s function from a self-energy and band dispersion.

Parameters:

siw (SelfEnergy) – The SelfEnergy \(\Sigma\).
ek (ndarray) – Band dispersion \(\varepsilon(k)\).
beta (float) – Inverse temperature \(\beta\).
mu (float) – Chemical potential \(\mu\).
calc_filling (bool) – If True, compute the filling/occupation (exposed via the n/occ/occ_k properties).

Returns:

The local GreensFunction.

Return type:

GreensFunction

property ek: ndarray#

The band dispersion stored on this object.

Returns:: The band dispersion \(\varepsilon(k)\) as a numpy array.

get_ekin() → float[source]#

Returns the kinetic energy from the band dispersion and the k-resolved occupation, \(E_{kin} = \sum_{\sigma \vec{k} ab} \varepsilon(\vec{k})_{ab}\, n(\vec{k})_{ba}\).

Returns:: The kinetic energy per site.
Return type:: float

get_epot(niv_asympt: int = 50000) → float[source]#

Computes the moment-corrected Galitskii-Migdal potential energy,

\[E_{pot} = \sum_k \mathrm{Tr}[\Sigma_\infty \rho_k] + \frac{1}{\beta} \sum_{k,\nu} \mathrm{Tr}[(\Sigma - \Sigma_\infty) G] + \frac{1}{\beta} \big[\textstyle\sum_{\mathrm{big}} - \sum_{\mathrm{box}}\big]\, \mathrm{Tr}[(\Sigma_{\mathrm{mod}} - \Sigma_\infty) G_{\mathrm{mod}}],\]

i.e. the exact Hartree term, the in-box correlation part, and the analytic \(1/\nu^2\) tail. Here \(\Sigma_{\mathrm{mod}} - \Sigma_\infty = -\Sigma_1/(\imath\nu)\) and \(G_{\mathrm{mod}} = [\imath\nu + \mu - \varepsilon_k - \Sigma_\infty]^{-1}\). The model subtraction cancels the \(1/\nu^2\) tail of the correlation sum (remainder \(\sim 1/\nu^4\)), while the large sum supplies the part beyond the stored box.

Parameters:: niv_asympt (int) – Number of positive fermionic frequencies used for the asymptotic (“big”) tail sum.
Returns:: The potential energy per site.
Return type:: float

get_fill_nonlocal() → tuple[float, ndarray, ndarray][source]#

Computes the filling and occupation from the momentum-resolved Green’s function, using the analytic density-matrix of the model (moment) Green’s function plus the box correction to accelerate convergence.

Returns:: A tuple of (i) the total filling \(n\), (ii) the k-averaged occupation (shape [o1, o2]), and (iii) the k-resolved occupation (shape [kx, ky, kz, o1, o2]).
Return type:: tuple[float, ndarray, ndarray]

static get_g_full(siw: SelfEnergy, mu: float, ek: ndarray, beta: float)[source]#

Builds the full momentum-dependent Green’s function \(G(k, \nu) = [(\imath\nu + \mu) - \varepsilon(k) - \Sigma(k, \nu)]^{-1}\).

Parameters:

siw (SelfEnergy) – The SelfEnergy \(\Sigma\).
mu (float) – Chemical potential \(\mu\).
ek (ndarray) – Band dispersion \(\varepsilon(k)\).
beta (float) – Inverse temperature \(\beta\).

Returns:

The momentum-dependent GreensFunction (filling not recomputed).

get_g_wv(wn: ndarray, niv_cut: int) → ndarray[source]#

Returns the frequency-shifted Green’s function \(G_{ab}^{\nu - \omega}\) on a fermionic window of half width niv_cut, for the bosonic frequencies in wn.

Parameters:

wn (ndarray) – Array of bosonic Matsubara indices \(\omega\).
niv_cut (int) – Half width of the fermionic window \(\nu\).

Returns:

Array of shape [o1, o2, w, v].

Return type:

ndarray

property n: float#

The total filling computed for this Green’s function.

Returns:: The total filling \(n\), or None if the filling has not been computed.

property occ: ndarray#

The k-averaged occupation matrix.

Returns:: The k-averaged occupation (shape [o1, o2]), or None if it has not been computed.

property occ_k: ndarray#

The k-resolved occupation matrix.

Returns:: The k-resolved occupation (shape [kx, ky, kz, o1, o2]), or None if it has not been computed.

dgamore.greens_function.get_total_fill(mu: float, ek: ndarray, sigma_mat: ndarray, beta: float, smom0: ndarray) → float[source]#

Returns the total filling calculated from the self-energy, \(\mu\), kinetic Hamiltonian and more. Helper method for the root finding of \(\mu\) via a Newton method. A local model Green’s function built from the self-energy moment is subtracted to accelerate the Matsubara sum convergence. This is the cheap, purely local (k-summed) scalar variant used inside the \(\mu\) root search; GreensFunction.get_fill_nonlocal() is the k-resolved counterpart that additionally returns the occupation matrices. Both share the Fermi-Dirac density matrix via _fermi_dirac_density().

Parameters:

mu (float) – Chemical potential \(\mu\).
ek (ndarray) – Band dispersion \(\varepsilon(k)\), shape [kx, ky, kz, o1, o2].
sigma_mat (ndarray) – Self-energy array, shape [k, o1, o2, v].
beta (float) – Inverse temperature \(\beta\).
smom0 (ndarray) – Zeroth moment \(\Sigma_\infty\) of the self-energy, shape [o1, o2].

Returns:

The total filling (electron number) \(n\).

Return type:

float

dgamore.greens_function.root_fun(mu: float, target_filling: float, ek: ndarray, sigma_mat: ndarray, beta: float, smom0: ndarray) → float[source]#

Residual function used to find a new chemical potential \(\mu\) via Newton’s method: the difference between the current filling and the target filling.

Parameters:

mu (float) – Chemical potential \(\mu\).
target_filling (float) – Desired total filling.
ek (ndarray) – Band dispersion \(\varepsilon(k)\).
sigma_mat (ndarray) – Self-energy array, shape [k, o1, o2, v].
beta (float) – Inverse temperature \(\beta\).
smom0 (ndarray) – Zeroth moment \(\Sigma_\infty\) of the self-energy.

Returns:

The signed filling residual filling(mu) - target_filling.

Return type:

float

dgamore.greens_function.update_mu(mu0: float, target_filling: float, ek: ndarray, sigma_mat: ndarray, beta: float, smom0: ndarray, logger=None) → float[source]#

Updates the chemical potential to match the target filling by using Newton’s method to find the optimal \(\mu\). On failure to converge the starting value is returned unchanged.

Parameters:

mu0 (float) – Initial guess for the chemical potential.
target_filling (float) – Desired total filling.
ek (ndarray) – Band dispersion \(\varepsilon(k)\).
sigma_mat (ndarray) – Self-energy array, shape [k, o1, o2, v].
beta (float) – Inverse temperature \(\beta\).
smom0 (ndarray) – Zeroth moment \(\Sigma_\infty\) of the self-energy.
logger – Optional logger; if given, a failed root search is logged at debug level.

Returns:

The updated (real) chemical potential, or mu0 if the root search did not converge.

Raises:

ValueError – If the converged chemical potential has a non-negligible imaginary part.

Return type:

float