Analytic continuation of Green's functions with a neural network

TL;DR

A convolutional neural network is used to obtain the spectral density for a given imaginary time Green's function and it is found that the network outperforms MaxEnt when presented data close to the training set.

Abstract

An important problem in many-body physics is to reconstruct the spectral density from the imaginary-time domain Green's function. Typically, the imaginary-time Green's function is generated by Monte Carlo methods. As the one-point fermionic kernel diverges exponentially for large frequencies, numerical noise generically causes instabilities. We use a convolutional neural network to obtain the spectral density for a given imaginary time Green's function. The network is trained by data which we generate using random Gaussians. We improve the training data set available by including collision centers for the Gaussians rather than employing uniformly distributed Gaussians. Our network is constructed in such a way that its output fulfills positive semidefiniteness. We compare the results of our network with results of the Maximum Entropy method (MaxEnt), a standard method for the same reconstruction problem for the spectral density. This comparison is performed for three different cases, namely our Gaussian based test data as well as two physical models, the 1d Hubbard model showing spin-charge separation, and the two-dimensional SSH model in the self-consistent Born approximation. We find that the network outperforms MaxEnt when presented data close to the training set. For the physical models considered, MaxEnt recognizes physical features more precisely as compared to our network prediction. While it is hard to improve MaxEnt, the quality of the network depends on the training data set which can be systematically enhanced and improved.

Figures (4)

• Figure 1: Scheme of the layers in the convolution (a), the dense (b), and deconvolution (c) parts of the neural networks. In (a), the top row denotes the layer type, the first value $k$ describes the kernel size, the second value $c$ the output channel size for this layer, and the optional third value $s$ the stride if different to one. A similar notation applies to (c), where convolutional transposed layers are utilized. For the dense part described in (b) the corresponding layer sizes are shown.
• Figure 2: Exemplary predictions for artificially generated data. Each example includes the true (generated) spectral density function as well as predictions by MaxEnt and the neural network. Systematic differences between the predictions of MaxEnt and the neural network can be seen in the prediction of heights and positions of peaks, as well as MaxEnt inducing oscillations around sharp peaks.
• Figure 3: Predicted spectral density functions for the 1D Hubbard model using the neural network \ref{['fig:performance_physical_results:comp_2_MaxEnt:spin-charge_a']} and MaxEnt \ref{['fig:performance_physical_results:comp_2_MaxEnt:spin-charge_b']}. Here we consider an $L=46$ site chain at $U/t=4$ and $\frac{t}{k_bT} = 10$ at half filling. The MaxEnt plot is taken from ALF_v2.4. While in both images spin-charge separation is discernible, the network produces spurious stratification of the image.
• Figure 4: Fig. \ref{['fig:SSH_a']} shows the neural network prediction for the spectral density of the 2D SSH model. Fig. \ref{['fig:SSH_b']} is reproduced from Fig. 12(a) of Goetz21. The data presented here Goetz21 corresponds to the half-filling case at $\omega_0 = \sqrt{\frac{k}{m}} = 1.0$, $k_b T = 1/40$, $t = 1.85$ and $g = 1.5$. $\boldsymbol{k}$ runs over a path in the Brillouin zone, from $(0,0)$ to $(0,\pi)$ to $(\pi,\pi)$ and back to $(0,0)$. The frequency ranges from $-10$ to $10$ in steps of $0.01$.