Physically interpretable approximations of many-body spectral functions

The rational function approximation provides a natural and interpretable representation of response functions such as the many-body spectral functions. We apply the vector fitting (VFIT) algorithm to fit a variety of spectral functions calculated from the Holstein model of electron-phonon interactions. We show that the resulting rational functions are highly efficient in their fitting of sharp features in the spectral functions, and could provide a means to infer physically relevant information from a spectral data set. The position of the peaks in the approximated spectral function are determined by the location of poles in the complex plane. In addition, we developed a variant of VFIT that incorporates regularization to improve the quality of fits. With this procedure, we demonstrate it is possible to achieve accurate spectral function fits that vary smoothly as a function of physical conditions.

Figure 1

A sample of Holstein spectral functions $A\left(\omega \right)$ with (a) varying phonon frequencies $\mathrm{\Omega }$ or (b) varying coupling strengths $\lambda$.

Figure 2

To improve the quality of spectral function fits, we incorporated regularization into the VFIT algorithm, as described in the main text. The left and right columns show two different fitting tasks. Each spectral function was fit using five poles, and their complex conjugates. Top row: The original VFIT algorithm may converge to a bad local minimum of the objective function. Middle row: VFIT with a fixed regularization strength of $\gamma ={10}^{-3}$ converges more reliably, but the fit is suboptimal. Bottom row: Annealing the regularization strength $\gamma$ to zero yields the best fits.

Figure 3

The mean absolute error (MAE) of spectral function fits at coupling strength $\lambda =1$ and averaged over phonon frequencies $\mathrm{\Omega }=0.5$–3. Each spectral function was fit using five poles and their complex conjugates. The $x$ axis counts the number of independent real parameters; for rational fits, these are the real and imaginary parts of the poles and residues in Eq. (2), not counting complex conjugate pairs. The line denotes the median of the MAE for spectral functions from $\mathrm{\Omega }=0.5$ to 3 for a fixed $\lambda =1$ and the shaded confidence bands indicate the region between the 25th and 75th percentile. For this data set of spectral functions, the rational approximation needs fewer parameters than the Chebyschev polynomial expansion.

Figure 4

Each spectral function was fit using five poles and their complex conjugates. (a) The mean absolute error of rational approximations for each value of $\lambda$ averaged over $\mathrm{\Omega }=0.5$–3. (b) The mean absolute error of rational approximations for each value of $\mathrm{\Omega }$ averaged over $\lambda =0.1$–1. The shaded regions in both (a) and (b) represent the range of the mean absolute error between the 25th and 75th percentile. These results are largely independent of the initial guess for the fitting parameters.

Figure 5

(a) Typical spectral function fits. Each spectral function was fit using five poles and their complex conjugates. The three spectral functions were selected according to their MAE. From top to bottom, these are the fits with errors in the 75th, 50th, and 25th percentile. Each spectral function and its fit were shifted horizontally and vertically for improved visibility. The bottom $x$ axis corresponds to the fit of the 25th percentile and the top $x$ axis corresponds to the fit of the 75th percentile (b), (c) Errors for the 75th and 25th percentile fits, respectively.

Figure 6

Visualization of the fitted simple pole expansion for spectral functions with $\lambda =1.0$ and varying $\mathrm{\Omega }$. Left: Positions of the five poles in the negative half of the complex plane. The real and imaginary axes control the peak location and width, respectively. The marker size indicates the magnitude of the corresponding residues, which control the spectral weight and the shape of each peak. Right: The reconstructed contribution of each pole to the spectral fit; for purely real residues, these contributions would be exactly Lorentzian.

Figure 7

(a) The mean absolute error of rational approximations with two poles and their complex conjugates for each value of $\lambda$ averaged over $\mathrm{\Omega }=0.5$–3. (b) The mean absolute error of rational approximations for each value of $\mathrm{\Omega }$ averaged over $\lambda =0.1$–0.33. The shaded regions in both (a) and (b) represent the range of the mean absolute error between the 25th and 75th percentile.

Figure 8

Visualization of the fitted simple pole expansion for spectral functions with $\mathrm{\Omega }=3.0$ and varying $\lambda$. The left and right columns should be interpreted analogously to those in Fig. 6.

Figure 9

Fitting a spectral function under increasing noise. Each spectral function was fit using four poles and their complex conjugates. Gaussian square noise was used with mean 0 and increasing variance as seen in (a)–(f) given by $\sigma$.