gftool.density_iw

gftool.density_iw(iws, gf_iw, beta, weights=1.0, moments=(1.0,), n_fit=0)

Calculate the number density of the Green’s function gf_iw at finite temperature beta.

This function can be used for fermionic Matsubara frequencies matsubara_frequencies, as well as fermionic Padé frequencies pade_frequencies.

Parameters:

iws, gf_iw(…, N_iw) complex np.ndarray

Positive Matsubara frequencies \(iω_n\) (or Padé \(iz_p\)) and the Green’s function at these frequencies.

betafloat

The inverse temperature \(beta = 1/k_B T\).

weights(…, N_iw) float np.ndarray, optional

Residues of the frequencies with respect to the residues of the Matsubara frequencies 1/beta (default: 1.0). For Padé frequencies this needs to be provided.

moments(…, M) float array_like, optional

Moments of the high-frequency expansion, where G(z) = np.sum(moments / z**np.arange(N)) for large z.

n_fitint, optional

Number of additionally to moments fitted moments. If Padé frequencies are used, this is typically not necessary (default: 0).

Returns:

float

The number density of the given Green’s function gf_iw.

See also

matsubara_frequencies

Method generating Matsubara frequencies iws.

pade_frequencies

Method generating Padé frequencies iws with weights.

Examples

>>> BETA = 50
>>> iws = gt.matsubara_frequencies(range(1024), beta=BETA)

Example Green’s function

>>> np.random.seed(0)  # to have deterministic results
>>> poles = 2*np.random.random(10) - 1  # partially filled
>>> residues = np.random.random(10); residues = residues / np.sum(residues)
>>> pole_gf = gt.basis.PoleGf(poles=poles, residues=residues)
>>> gf_iw = pole_gf.eval_z(iws)
>>> exact = pole_gf.occ(BETA)
>>> exact
0.17858151698239388

Numerical calculation of the occupation number, using Matsubara frequency

>>> occ = gt.density_iw(iws, gf_iw, beta=BETA)
>>> m2 = pole_gf.moments([1, 2])  # additional high-frequency moment
>>> m2
array([1.        , 0.30839757])
>>> occ_m2 = gt.density_iw(iws, gf_iw, beta=BETA, moments=m2)
>>> occ_fit2 = gt.density_iw(iws, gf_iw, beta=BETA, n_fit=1)
>>> exact, occ, occ_m2, occ_fit2
(0.17858151..., 0.17934437..., 0.17858150..., 0.17858198...)
>>> abs(occ - exact), abs(occ_m2 - exact), abs(occ_fit2 - exact)
(0.00076286..., 8.18...e-09, 4.72...e-07)

using more accurate Padé frequencies

>>> izp, rp = gt.pade_frequencies(100, beta=BETA)
>>> gf_izp = pole_gf.eval_z(izp)
>>> occ_izp = gt.density_iw(izp, gf_izp, beta=BETA, weights=rp)
>>> occ_izp
0.17858151...
>>> abs(occ_izp - exact) < 1e-12
True