gftool.fourier.tau2iv

gftool.fourier.tau2iv(gf_tau, beta, fourier=<function tau2iv_ft_lin>)

Fourier transformation of the real Green’s function gf_tau.

Fourier transformation of a bosonic imaginary-time Green’s function to Matsubara domain. We assume a real Green’s function gf_tau, which is the case for commutator Green’s functions \(G_{AB}(τ) = ⟨A(τ)B⟩\) with \(A = B^†\). The Fourier transform gf_iv is then Hermitian. This function removes the discontinuity \(G_{AB}(β) - G_{AB}(0) = ⟨[A,B]⟩\).

TODO: if high-frequency moments are know, they should be stripped for increased accuracy.

Parameters

gf_tau(…, N_tau) float np.ndarray

The Green’s function at imaginary times \(τ \in [0, β]\).

betafloat

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

fourier{tau2iv_ft_lin, tau2iv_dft}, optional

Back-end to perform the actual Fourier transformation.

Returns

gf_iv(…, (N_iv + 1)/2) complex np.ndarray

The Fourier transform of gf_tau for non-negative bosonic Matsubara frequencies \(iν_n\).

See also

tau2iv_dft

Back-end: plain implementation using Riemann sum.

tau2iv_ft_lin

Back-end: Fourier integration using Filon’s method.

Examples

>>> BETA = 50
>>> tau = np.linspace(0, BETA, num=2049, endpoint=True)
>>> ivs = gt.matsubara_frequencies_b(range((tau.size+1)//2), beta=BETA)

>>> poles, weights = np.random.random(10), np.random.random(10)
>>> weights = weights/np.sum(weights)
>>> gf_tau = gt.pole_gf_tau_b(tau, poles=poles, weights=weights, beta=BETA)
>>> gf_ft = gt.fourier.tau2iv(gf_tau, beta=BETA)
>>> gf_tau.size, gf_ft.size
(2049, 1025)
>>> gf_iv = gt.pole_gf_z(ivs, poles=poles, weights=weights)

>>> import matplotlib.pyplot as plt
>>> __ = plt.plot(gf_iv.imag, label='exact Im')
>>> __ = plt.plot(gf_ft.imag, '--', label='DFT Im')
>>> __ = plt.plot(gf_iv.real, label='exact Re')
>>> __ = plt.plot(gf_ft.real, '--', label='DFT Re')
>>> __ = plt.legend()
>>> plt.show()

(png, pdf)

Accuracy of the different back-ends

>>> ft_lin, dft = gt.fourier.tau2iv_ft_lin, gt.fourier.tau2iv_dft
>>> gf_ft_lin = gt.fourier.tau2iv(gf_tau, beta=BETA, fourier=ft_lin)
>>> gf_dft = gt.fourier.tau2iv(gf_tau, beta=BETA, fourier=dft)
>>> __ = plt.plot(abs(gf_iv - gf_ft_lin), label='FT_lin')
>>> __ = plt.plot(abs(gf_iv - gf_dft), '--', label='DFT')
>>> __ = plt.legend()
>>> plt.yscale('log')
>>> plt.show()

(png, pdf)

The methods are resistant against noise:

>>> magnitude = 5e-6
>>> noise = np.random.normal(scale=magnitude, size=gf_tau.size)
>>> gf_ft_lin_noisy = gt.fourier.tau2iv(gf_tau + noise, beta=BETA, fourier=ft_lin)
>>> gf_dft_noisy = gt.fourier.tau2iv(gf_tau + noise, beta=BETA, fourier=dft)
>>> __ = plt.plot(abs(gf_iv - gf_ft_lin_noisy), '--', label='FT_lin')
>>> __ = plt.plot(abs(gf_iv - gf_dft_noisy), '--', label='DFT')
>>> __ = plt.axhline(magnitude, color='black')
>>> __ = plt.legend()
>>> plt.yscale('log')
>>> plt.show()

(png, pdf)