Gaussian-process-regression-based method for the localization of exceptional points in complex resonance spectra

TL;DR

An efficient machine learning algorithm to find EPs in the resonance spectra of excitons in cuprous oxide in external electric and magnetic fields by taking into account the complete valence band structure and central-cell corrections of the crystal.

Abstract

Resonances in open quantum systems depending on at least two controllable parameters can show the phenomenon of exceptional points (EPs), where not only the eigenvalues but also the eigenvectors of two or more resonances coalesce. Their exact localization in the parameter space is challenging, in particular in systems, where the computation of the quantum spectra and resonances is numerically very expensive. We introduce an efficient machine learning algorithm to find exceptional points based on Gaussian process regression (GPR). The GPR-model is trained with an initial set of eigenvalue pairs belonging to an EP and used for a first estimation of the EP position via a numerically cheap root search. The estimate is then improved iteratively by adding selected exact eigenvalue pairs as training points to the GPR-model. The GPR-based method is developed and tested on a simple low-dimensional matrix model and then applied to a challenging real physical system, viz., the localization of EPs in the resonance spectra of excitons in cuprous oxide in external electric and magnetic fields. The precise computation of EPs, by taking into account the complete valence band structure and central-cell corrections of the crystal, can be the basis for the experimental observation of EPs in this system.

Figures (10)

• Figure 1: (a) The ep $\kappa_+ = \mathrm{i}$ of the simple example \ref{['eq:EPSimpleExampleMatrix']} is encircled in the parameter space $\kappa$. Each point in the complex plane is described by the angle $\phi$ in Euler form as depicted in \ref{['eq:EPSimpleExampleCircularParametrization']}. (b) This leads to an exchange of the positions of the two eigenvalues in the complex energy plane. The eigenvalues calculated for each point on the circle are marked by the respective color of the color bar indicating the angle $\phi$, in order to illustrate the path of each eigenvalue and the associated permutation.
• Figure 2: The investigated five-dimensional matrix model is visualized with its respective energy and parameter plane. The ep, found via a two-dimensional root search on $p$ without a gpr model, is marked as a green cross. Due to its dimensionality, five eigenvalues are visible, and their course can be traced via the color bar, which represents the angle $\phi$, when the ep is encircled. The model shows one permutation, which indicates the existence of an ep. The shape of this permutation is complex because of the large radius and the fact that the ep is not at the center of the orbit.
• Figure 3: Illustration of the iterative process by means of the five-dimensional matrix model. (a) The points on the orbit around the ep, marked in blue, and their associated eigenvalues are used as initial training set. (b) The model slowly approaches the ep, marked as a green cross. After nine training steps the gpr model is converged and the Euclidean distance between the last model prediction and the exact ep is $d_\mathrm{e} = 5.526\times 10^{-6}$. Two different attempts are made to optimize convergence and reduce the number of exact diagonalizations. (c) First, an additional training point is added after the second iteration to explore the energy plane. For this purpose, the difference of the last two predictions is calculated and added to the second prediction. This leads to convergence after the third training step, i.e. after the fourth diagonalization (considering the additional point). Not only the number of diagonalizations is significantly reduced, but also the Euclidean distance to $d_\mathrm{e} = 1.342\times 10^{-6}$. (d) Similarly to the previous approach, an additional training point is added after the third iteration. This does not reduce the number of diagonalizations (nine training steps, ten diagonalizations) nor does it improve convergence ($d_\mathrm{e} = 5.537\times 10^{-6}$).
• Figure 4: Similarity measure employed for selecting the new eigenvalue pair in each iteration. The logarithmic plot displays the pair-discrepancy values defined in \ref{['eq:Toy5DDiscrepancy']}. The calculations are performed after the first training step for the model in \ref{['fig:Toy5DExampleEnergyKappa']}. A noticeable gap is observed between the smallest and second smallest pair-discrepancy values. This indicates that the eigenvalue pair with the lowest $c$ value is highly likely to correspond to the ep.
• Figure 5: The two convergence criteria, namely the eigenvalues of the covariance matrix and the eigenvalue difference, are shown for the model system. (a) The eigenvalues $\lambda_{\boldsymbol{K}}$ of the covariance matrix $\boldsymbol{K}$ are depicted for each training step. A drop is visible in the third iteration from order $\mathcal{O}\left(10^{-4}\right)$ to $\mathcal{O}\left(10^{-10}\right)$ compared to the previous step. This indicates an already seen training point that does not provide any new information. A clear deviation to the previous training step is visible. Thus an appropriate threshold value can be defined easily. (b) The eigenvalue difference of the two eigenvalues belonging to the ep is plotted over the number of training steps. It decreases strictly monotonically, verifying convergence of the iterative process. Defining a threshold value is not as straightforward as for the kernel eigenvalues, since no clear change is visible between the last two iterations.