SciPost Submission Page
Compactness of quantics tensor train representations of local imaginary-time propagators
by Haruto Takahashi, Rihito Sakurai, Hiroshi Shinaoka
Submission summary
| Authors (as registered SciPost users): | Hiroshi Shinaoka |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2403.09161v1 (pdf) |
| Date submitted: | 2024-03-18 05:33 |
| Submitted by: | Shinaoka, Hiroshi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Space-time dependence of imaginary-time propagators, vital for \textit{ab initio} and many-body calculations based on quantum field theories, has been revealed to be compressible using Quantum Tensor Trains (QTTs) [Phys. Rev. X {\bf 13}, 021015 (2023)]. However, the impact of system parameters, like temperature, on data size remains underexplored. This paper provides a comprehensive numerical analysis of the compactness of local imaginary-time propagators in QTT for one-time/-frequency objects and two-time/-frequency objects, considering truncation in terms of the Frobenius and maximum norms. We employ random pole models to study worst-case scenarios. The numerical analysis reveals several key findings. For one-time/-frequency objects and two-time/-frequency objects, the bond dimensions saturate at low temperatures, especially for truncation in terms of the Frobenius norm. We provide counting-number arguments for the saturation of bond dimensions for the one-time/-frequency objects, while the origin of this saturation for two-time/-frequency objects remains to be clarified. This paper's findings highlight the critical need for further research on the selection of truncation methods, tolerance levels, and the choice between imaginary-time and imaginary-frequency representations in practical applications.
Current status:
Reports on this Submission
Strengths
1. Of possible broad interest
Weaknesses
1. Requires additional details about range of applicability and physical rationale
Report
The manuscript provides valuable insight into the compactness of local imaginary-time propagators in quantics tensor train format. The work is written with the quantum field theory community in mind, but given its focus on tensor trains could have relatively broad readership in mathematics, physics, chemistry, etc. with changes. It would also improve the paper to clarify not only the findings, but the key breakthrough they wish to highlight (with regards to literature in SVD/TCI, etc.).
Requested changes
1. It would be helpful to provide more information (and physical intuition) to explain why the numerical results for the models expected here are expected to be more broadly applicable.
2. The paper could have a broader impact by further referencing the extensive work being done outside of quantum field theory with imaginary-time propagators. Connection to MPS would also help.
3. Can a statement be made about how the reshaping order is chosen in the general case (i.e., in applications where there are concerns that QTT rank depends significantly on ordering)? Are there additional calculations to support whether reordering significantly affects the presented results?
4. It would be beneficial to provide a rationale for why the Bethe-Salpeter equation is an important future direction.
5. When Figure 9 is mentioned, it would help to provide an explanation for the dip near 1000, which is distinct from all other models presented.
Recommendation
Ask for minor revision
Report
The authors investigate the quantics tensor train (QTT) representation of local imaginary-time and -frequency propagators. They consider one- and two-time/frequency objects. They find that - in certain cases - the considered objects are compressible with a bond dimension that remains finite even for inverse temperature going to infinity.
At the present stage I do not believe that the paper has enough substance to warrant publication in SciPost Physics. There are two main reasons:
(1) The paper remains on a superficial level. Whenever an interesting observation is made a thorough analysis thereof is delegated to future work. Particularly for two-frequency objects, Figs. 7 and 8, it seems that the Frobenius-norm approach has small bond dimension but large errors (the color plot for the absolute error looks similar to that of the absolute values) while the maximum-norm approach gives small errors in the color plots but has bond dimensions growing indefinitely. The question whether two-frequency objects are QTT compressible or not - crucial for the scope of the paper - remains unclear.
(2) Generally it seems insufficient to me to consider random pole models to judge the compressibility of many-body functions. Particularly in the field of analytic continuation it is a common problem that new methods are presented with excellent results on pole models and later turn out not remotely as successful for models encountered in 'real life'. Therefore it is advisable to study realistic models too. Here these could be obtained e.g. in dynamical mean-field theory with quantum Monte Carlo solvers. Moreover one should also consider non Fermi liquid scenarios, possibly using simple but typical analytic expression.
More detailed criticism is as follows:
- In the first paragraph of 3.2 it is unclear how the grid of Matsubara frequencies is mapped to the bit representation. Which values does n take? How is the grid from -infinity to infinity mapped to an interval from 0 to 1?
- In Sec. 4 I do not understand the motivation for analyzing Eq. (12) and Eq. (14) separately. According to the Lehmann representation a generic two-frequency propagator is a superposition of Eq. (12) and Eq. (14). Moreover the authors did not address the presence of anomalous terms proportional to delta_{iOmega,0} where Omega is a bosonic Matsubara frequency.
- In the last paragraph of Sec. 4.3.2 the authors state that the two-frequency object G^FB has less information than G^FF but their explanation is based on one-frequency objects. It is unclear why properties of the latter carry over to the former.
- The color plots in Figs. 5-8 are not well formatted. It is unclear how to extract from these plots whether the QTT representation works or does not work. For instance in Fig. 6a the absolute values seem to have the same size as the absolute errors since both color bars go up to -2 (or even -1).
- In the last paragraph of Sec. 5 the authors mention the presence of constant terms in vertex functions. They state that the single-/multi-boson exchange (SBE/MBE) framework does not involve constant terms. I cannot follow here for two reasons: First the SBE/MBE framework does contain constant terms, namely the bosonic propagator contains the bare interaction and the Hedin vertex contains a term that equals unity. Second why is the SBE/MBE framework singled out here? There are many other frameworks that similarly have constant terms which similarly can be subtracted.
Author: Hiroshi Shinaoka on 2024-04-13 [id 4416]
(in reply to Report 1 on 2024-04-08)We greatly appreciate your efforts on reviewing the manuscript. Your comments are very useful to improve the readability of the manuscript, although we disagree with the conclusions that the paper has enough substance to warrant publication in SciPost Physics.
We are preparing a revised manuscript, but the SciPost platform allows the authors to communicate with a referee online interactively. Please let us address the main criticism before an extensive revision of the manuscript.
The scaling plots in Fig. 6-8 strongly indicate that the two-frequency objects are compressible. We agree that the colormaps are to be improved. We attached a revised version of Fig. 6 for a smaller cutoff of 1e-8 in this reply. This cutoff corresponds to 4-digit accuracy [Eq. (8)]. The new figure demonstrates the compressibility of the two-time object more clearly.
We believe that our non-trivial numerical finding will stimulate further (mathematical) studies. The bond dimensions remain constant at low T in the Frobenius norm case, leading to the data size scaling O(chi^2 log beta) = O(log beta). The results surprisingly indicate that the QTT approach superpasses the scaling of the conventional approach (i.e.,IR, O((log beta)^2).
The max-norm case is very singular; since the function is diverging, we should avoid using the max norm. Since TCI has been attracting more attention, this result is useful to other many-body theorists as a red flag.
Thank you for raising this important point. We should have explained the motivation of using the random-pole model in more detail.
We used the random-pole model to generate random imaginary-time data. In our analysis, we deal with only the imaginary-time data but never reconstruct the spectral functions. If the pole positions are chosen appropriately, we could reproduce an imaginary-time function generated by an any continuous spectral function precisely, according to previous studies on IR and DLR. This setup is clearly different from the numerical analytic continuation, where whether or not the original spectral function is discrete or continuous can make significant differences in the quality of the reconstruction of the spectral function.
Our random-pole model is more challenging than “real-life” data. We have already done extensive numerical analysis of real-life data from various numerical simulations such as, real-frequency spectral functions of Kondo impurity models, vertex functions from DFT+DMFT, nonequilibrium DMFT and more in PRX 13, 021015 (2023). However, analyzing data from numerical simulations is not enough to understand the intrinsic properties of the QTT representation. For example, QMC data include statistical noise, hindering analysis at low temperature. The Kondo spectral function is too simple and contains only a few peaks (Fig. 13 of the PRX). In contrast, our random pole model contains many poles, whose number increases at low temperature. This allows us to encode more and more information into the random pole model through many random coefficients. We will update the manuscript to stress these points, i.e., our test cases are much more challenging to QTT than real-life cases.
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two_svd_tau_rotated_cut864_8_5.pdf