gftool.polepade.continuation

gftool.polepade.continuation(z, fct_z, degree=-1, weight=None, moments=(), vandermond=<function polyvander>, rotate=None, real_asymp=True) → gftool.polepade.PadeApprox

Perform the Padé analytic continuation of (z, fct_z).

Parameters

z, fct_z(N_z) complex np.ndarray

Variable where function is evaluated and function values.

degreeint, optional

The difference of denominator and numerator degree. (default: -1) This determines how fct_z decays for large abs(z): fct_z → z**degree. For Green’s functions it typically is -1, for self-energies it typically is 0.

weight(N_z) float np.ndarray, optional

Weighting of the data points, for a known error σ this should be weight = 1./σ.

moments(N) float array_like

Moments of the high-frequency expansion, where f(z) = moments / z**np.arange(1, N+1) for large z. This only affects the calculated pade.residues, and constrains them to fulfill the moments.

Returns

padePadeApprox

Padé analytic continuation parametrized by pade.zeros, pade.poles and pade.residues.

Other Parameters

vandermondCallable, optional

Function giving the Vandermond matrix of the chosen polynomial basis. Defaults to simple polynomials.

rotatebool or None, optional

Whether to rotate the coordinate to calculated zeros and poles. (Default: rotate if z is purely imaginary)

real_asympbool, optional

Whether to consider only the real part of the asymptote, or treat it as complex number. Physically, to asymptote should typically be real. (Default: True)

Examples

>>> beta = 100
>>> iws = gt.matsubara_frequencies(range(512), beta=beta)
>>> gf_iw = gt.square_gf_z(iws, half_bandwidth=1)
>>> weight = 1./iws.imag  # put emphasis on low frequencies
>>> gf_pade = gt.polepade.continuation(iws, gf_iw, weight=weight, moments=[1.])

Compare the result on the real axis:

>>> import matplotlib.pyplot as plt
>>> ww = np.linspace(-1.1, 1.1, num=5000)
>>> __ = plt.plot(ww, gt.square_dos(ww, half_bandwidth=1))
>>> __ = plt.plot(ww, -1. / np.pi * gf_pade.eval_zeropole(ww).imag)
>>> __ = plt.plot(ww, -1. / np.pi * gf_pade.eval_polefct(ww).imag)
>>> plt.show()

(png, pdf)

Investigate the pole structure of the continuation:

>>> gf_pade.plot()
>>> plt.show()

(png, pdf)