Nevanlinna Analytical Continuation

Nevanlinna interpolation

It is well known that the retarded Green's function, denoted as $G^{R}(\omega + i0^{+})$, and the Matsubara Green's function, denoted as $G(i\omega_n)$, can both be consistently represented as $G(z)$, where $z \in \mathbb{C} \backslash \mathbb{R}$. The Nevanlinna analytical continuation (NAC) method utilizes the fact that the negative fermionic Green's function, denoted as $f(z) = -G(z)$, belongs to the class of Nevanlinna functions. By applying the invertible M$\ddot{\text{o}}$bius transform $h(z) = (z-i)/(z+i)$ to the function value of $f(z)$, the Nevanlinna function is mapped in a one-to-one fashion to a contractive function $\theta(z) = h[f(z)]$. This contractive function $\theta(z)$ can be expressed in the form of a continued fraction expansion, and an iterative algorithm can be constructed accordingly. The recursion relation between two steps $\theta_j(z)$ and $\theta_{j+1}(z)$ is given by:

\[\begin{equation} \theta_j(z) = \frac{ \theta_{j+1}(z) + \gamma_j}{\gamma_j^*h_j(z) \theta_{j+1}(z) +1}. \end{equation}\]

In this equation, $h_j(z) = (z-Y_j)/(z+Y_j)$, $Y_j = i\omega_j$ represents the $j$-th Matsubara frequency used, and $\gamma_j = \theta_j(Y_j)$ represents the function value of the $j$-th contractive function at the point $Y_j$. The final expression of the recursive function $\theta(z)$ can be written as:

\[\begin{equation} \theta(z)[z;\theta_{N_s+1}(z)] = \frac{a(z)\theta_{N_s+1}(z) + b(z)}{c(z)\theta_{N_s+1}(z) + d(z)}, \end{equation}\]

where

\[\begin{equation} \left( \begin{matrix} a(z) & b(z) \\ c(z) & d(z) \end{matrix} \right) = \prod_{j=1}^{N_s} \left( \begin{matrix} h_j(z) & \gamma_j \\ \gamma_j^*h_j(z) & 1 \end{matrix} \right), \end{equation}\]

with $j$ increasing from left to right. Here $N_s$ is the overall iteration step, which is equivalent to the number of data points. After obtaining $\theta(z)$, one can immediately get the Green's function by an inverse $\rm{M\ddot{o}bius}$ transform as $G(z)= -h^{-1}[\theta(z)]$. Note that the Pick criterion should be fulfilled for the existence of the Nevanlinna interpolation.

Hardy basis functions

Additionally, it is worth noting that there is flexibility in choosing $\theta_{N_s+1}(z)$, which can be used to select the most desirable spectral function. In original reference for the NAC method, $\theta_{N_s+1}(z)$ is expanded in the Hardy basis and chosen in such a way that it achieves the smoothest possible spectral function. The loss function employed in this selection process is given by:

\[\begin{equation} \mathcal{L} = \left\vert 1-\int \frac{{\rm{d}} \omega}{2\pi} \rho_{\theta_{N_s+1}}(\omega) \right\vert^2 + \lambda \left\Vert \frac{{\rm{d}}^2 \rho_{\theta_{N_s+1}}(\omega)}{{\rm{d}} \omega^2} \right\Vert^2_F. \end{equation}\]

This loss function consists of two terms. The first term enforces the proper sum rule, while the second term incorporates the smoothness condition. $\lambda$ is an adjustable parameter. By preserving the ``Nevanlinna'' analytic structure of Green's functions, the NAC method automatically generates positive and normalized spectral functions. However, it is important to emphasize that the method is sensitive to noise, and either a large number of data points $N$ or a high Hardy order $H$ can potentially lead to numerical instabilities.

Bosonic cases

Although the NAC method has been extended to support the analytic continuation of matrix-valued Green's functions, it cannot be directly applied to bosonic systems in its original formalism. Quite recently, Nogaki et al. suggest an ingenious trick to work around this limitation. Their basic idea is to introduce an auxiliary fermionic function. Let us start with a bosonic Green's function $G(\tau)$ that satisfies the periodic condition $G(\tau+\beta) = G(\tau)$. One can construct an artificial anti-periodic fermionic Green's function $\tilde{G}(\tau)$ as follows:

\[\begin{equation} \tilde{G}(\tau) = \begin{cases} G(\tau) &\quad (0<\tau<\beta) \\ -G(\tau + \beta) &\quad (-\beta<\tau<0) \end{cases} \end{equation}\]

Clearly, this auxiliary fermionic Green's function exhibits the same value as the bosonic Green's function in the range $0<\tau<\beta$. It is easy to prove that the relation between the bosonic spectral function $\rho(\omega)$ and the auxiliary fermionic spectral function $\tilde{\rho}(\omega)$ is as follows:

\[\begin{equation} \rho(\omega) = \tilde{\rho}(\omega) \tanh(\beta\omega/2). \end{equation}\]

Furthermore, the sum rule for $\tilde{\rho}(\omega)$ is given by:

\[\begin{equation} \int_{-\infty}^{\infty} {\rm{d}} \omega~\tilde{\rho}(\omega) = G(\tau=0^+) + G(\tau=\beta-0^-). \end{equation}\]

Given $\tilde{G}(\tau)$, it is easy to construct $\tilde{G}(i\nu_n)$ via direct Fourier transformation, where $\nu_n = (2n+1)\pi/\beta$ are the fermionic Matsubara frequencies. Since

\[\begin{equation} \tilde{G}(i\nu_n) = \int_{-\infty}^{\infty} {\rm{d}} \omega \frac{\tilde{\rho}(\omega)}{i\nu_n - \omega}, \end{equation}\]

one can perform analytic continuation for $\tilde{G}(i\nu_n)$ via the standard NAC method to get $\tilde{\rho}(\omega)$. And then the bosonic spectral function $\rho(\omega)$ can be derived according to Eq.(6).

Relevant parameters

See [NevanAC] Block