Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting

1. Improved separation-of-variables algorithm for computing imaginary-time Feynman diagrams for quantum impurity models: speedup of several orders of magnitude relative to related previous work by the same authors.

2. Clear presentation of key ideas.

3. Several instructive benchmark examples.

In Eq. (8), I believe the upper limit should be the same for both summation, i.e. m = k [according to (2.1) of Ref. [45]).

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1. Constructing of a numerically efficient DMFT solver for practical calculations is an important goal.
2. The developed code, as described, is fast an effective.
3. The text is clear and convincing.

1. The code is not available so far.
2. The figures formt if poor.

1. Make code public.
2. Work on Figure style.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1. Clear demonstration that the AAA-based fitting combined with bilevel optimization improves computational performance relative to DLR.
2. Includes extensive benchmarks and comparisons, including tests on worst-case random spectra.
3. Promising applicability to multi-orbital impurity problems within DMFT frameworks.

1. Limited discussion of the physically reliable temperature range, especially in the low-temperature regime where strong-coupling expansions are known to develop artifacts.
2. Lack of quantitative comparison of other DMFT results to more established impurity solvers.
3. No demonstration of scaling or error behavior in larger Hilbert spaces (n >> 3).
4. The manuscript focuses heavily on numerical performance, with less discussion of the physical implications of the results.

Major:
1. While the authors emphasize the general applicability of their method to arbitrary multi-orbital systems, the benchmarks are mainly restricted to models with two or three orbitals. The scaling of computational cost and approximation error in larger Hilbert spaces (n >> 3) is not demonstrated, and the growth of the expansion length p in such cases remains a potential limitation, particularly at large β. For example, in Fig.2, the “random poles” benchmark shows a saturation of p relative to the DLR basis size which indicates that the advantage over DLR may diminish in more complex spectral structures. Additional discussion or data illustrating this scaling would be valuable.

2. While the authors convincingly show performance at β=10 (high temperature) and β=1000 (low temperature), it would be helpful to include results for intermediate inverse temperatures, e.g., β=50–200 in Fig. 1. These are typical parameter ranges in practical applications of DMFT and quantum embedding methods. A clearer illustration of how the required number of poles p scales in this regime would strengthen the assessment of the method’s practical efficiency. It seems authors calculated for β=40 , so adding this data to Fig.1 would make the presentation more consistent and complete.


3. Fig.5 and Fig.7 demonstrate that the proposed solver can be applied within a fully self-consistent DMFT loop in a multi-orbital system, I am concerned that the observed low-frequency errors (particularly near ωₙ=0) and the clear m-dependence of the self-energy could potentially lead to inaccuracies or instability in physically relevant quantities such as quasiparticle weights or magnetic ordering. It would be valuable if the authors could comment on this by comparing the physical observables obtained in their DMFT calculations (e.g., quasiparticle weights, Neel temperatures) to results from more established impurity solvers, such as CTQMC or NRG, in order to clarify the extent to which the observed m-dependence and low-frequency errors translate into deviations in physically relevant quantities.


4. As this is primarily a methodological contribution, it would be very helpful for users if the authors could provide explicit guidance on the practical temperature range where the method can be expected to yield not only numerically stable but also physically reliable results. Since it is well known that NCA (including OCA and higher-order diagrams) and related strong-coupling expansions develop artifacts in the low-temperature regime, clarifying whether higher-order diagrams sufficiently mitigate these issues, or whether a practical β limit should be recommended, would significantly strengthen the manuscript.

5. While the manuscript provides an impressive array of numerical benchmarks and performance comparisons, I feel that it would benefit significantly from a more detailed discussion of the physical implications of the observed behaviors , exemplified by the crossing of error curves in Fig.4 (right).


Minor:
1. In Fig.1, the plotting style could be improved for readability. The points and line thickness are disproportionately large relative to the figure size, making the error curves difficult to interpret. I recommend either increasing the overall figure dimensions or reducing the marker and line widths to improve clarity. In addition, the caption of Fig.1 currently does not explicitly label the three types of spectral densities used in each column (e.g., semicircular, sum of Gaussians, sum of δ-functions), which makes the figure less accessible to the reader. I recommend adding explicit labels in the caption to clarify which spectral shape corresponds to each panel.

Ask for major revision