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Efficient construction of the Feynman-Vernon influence functional as matrix product states
by Chu Guo, Ruofan Chen
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Submission summary
| Authors (as registered SciPost users): | Ruofan Chen |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2402.14350v4 (pdf) |
| Date accepted: | 2024-09-03 |
| Date submitted: | 2024-08-09 04:45 |
| Submitted by: | Chen, Ruofan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
The time-evolving matrix product operator (TEMPO) method has become a very competitive numerical method for studying the real-time dynamics of quantum impurity problems. For small impurities, the most challenging calculation in TEMPO is to construct the matrix product state representation of the Feynman-Vernon influence functional. In this work we propose an efficient method for this task, which exploits the time-translationally invariant property of the influence functional. The required number of matrix product state multiplication in our method is almost independent of the total evolution time, as compared to the method originally used in TEMPO which requires a linearly scaling number of multiplications. The accuracy and efficiency of this method are demonstrated for the Toulouse model and the single impurity Anderson model.
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List of changes
1.Added some descriptive sentences according to the referee's comments
2. Added an appendix about explicit QUAPI expressions
3. Added a figure about OSEE
Published as SciPost Phys. Core 7, 063 (2024)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-8-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2402.14350v4, delivered 2024-08-22, doi: 10.21468/SciPost.Report.9635
Report
I thank the authors for their thorough responses and revisions. They have noticeably amended their discussion of their method in terms of entanglement content of the MPS, along with adding a discussion contrasting a memory truncation in the spirit of QUAPI/TEMPO against their zipup algorithm. These additions, alongside minor modifications, have clarified some of my previous remarks.
A few more remarks:
1- Is eq (4) correct as written? I would have expected there to be $\bar{a}$ GVs.
2- I understand that the memory truncation as presented in Appendix D is incompatible with the zipup algorithm. My suggestion of a memory truncation and its potential to improve computational costs was aimed at the level of the IF, not the ADT. I imagine, for example, that the partial IF (eq (8)) would be truncated to $\Delta k_{max}$ terms. Each of these terms would be multiplied into the IF as it is being constructed. As a further approximation to the MPS compression, singular value truncations would occur only over those $\Delta k_{max}$ sites. This would be similar in spirit to Jorgenson and Pollock's construction in Phys. Rev. Lett. 123, 240602 (2019). This would avoid the problem pointed out by the authors that the ADT would need to be explicitly constructed. In my mind this would have been a better comparison to the authors' TTI IF method, as far as computational cost is concerned.
3- While it is nice to have information on the OSEE, what would be better is if the figure showed either how the OSEE in the MPS changes with the max bond dimension chi, or if it showed the entanglement spectrum explicitly.
As these are only minor comments, I am happy to recommend publication of this manuscript in SciPost Phys Core.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2024-8-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2402.14350v4, delivered 2024-08-14, doi: 10.21468/SciPost.Report.9596
Report
Dear SciPost Team,
thank you for forwarding the 3rd revised manuscript by Guo et al. on the Efficient construction of the Feynman-Vernon influence functional as matrix product states (2402.14350v4).
The authors addressed all concerns up to the following minor suggestions. Since at the same time the paper was transferred to SciPost Physics Core, I thus recommend publication.
Remaining comments:
The newly added Fig. 4 is very useful, but immediately raises further questions since the computed OSSE is extremely small! (Is the small OSEE a generic feature also away from Toulouse?)
Values OSEE < 0.005 suggest that the underlying operator is nearly a product operator. A bond dimension of chi=100 may be much of an overkill for this.
From the presented data one may speculate that OSEE->0 for dt->0; one may compare this to a Trotter MPO for exp(-iH*dt) which also becomes the identity product operator for dt->0; is this a sensible analogy?
A crucial extra piece of information therefore is the SVD spectrum that underlies the OSEE: how quickly does it decay? if it decays strongly, a much smaller chi may have sufficed for the same accuracy.
The inset to Fig. 4 shows `absolute error', of what quantity?
It appears that the smaller error for smaller dt is a systematic time-descretization error that is already completely converged for given chi.
following (B3-B4)
> where the upper indices i_k,j_k are actual powers.
this is misleading now, since this does not concern the tensor A.
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Ruofan Chen on 2024-09-10 [id 4754]
(in reply to Report 1 on 2024-08-14)
We thank the referee for the refereeing and comments. Our response to the comments are shown below.
The referee writes:
The newly added Fig. 4 is very useful, but immediately raises further questions since the computed OSSE is extremely small! (Is the small OSEE a generic feature also away from Toulouse?) Values OSEE < 0.005 suggest that the underlying operator is nearly a product operator. A bond dimension of chi=100 may be much of an overkill for this. From the presented data one may speculate that OSEE->0 for dt->0; one may compare this to a Trotter MPO for exp(-iHdt) which also becomes the identity product operator for dt->0; is this a sensible analogy? A crucial extra piece of information therefore is the SVD spectrum that underlies the OSEE: how quickly does it decay? if it decays strongly, a much smaller chi may have sufficed for the same accuracy.
Our response: We thank the referee for the comment. Yes the OSEE is extremely small, and this is a generic feature determined solely by the hybridization function, independent the bare impurity Hamiltonian. The small OSEE also means that the entanglement spectrum of the MPS-IF decays very fast. And indeed bond dimension 100 is an overkill, which is only used for the purpose of Fig.4 to illustrate the growth of OSEE. For most simulations in this work, bond dimension 50 is already more than enough (for the similation of the transport problem, we have used a larger dt, and in this case a larger bond dimension 160 is used). A rough explanation of this is that the MPS-IF is similar to a high-temperature thermal state if we take F as a long-range Hamiltonian, but a more quantitive theoretical explanation is currently in lack. The fact that has been observed also in the bosonic case is that for larger dt one needs a larger bond dimension, we guess the referee could be right that OSEE->0 for dt->0. We would also like to stress that the small singular values of the MPS-IF are very important in our practice, as if we throw away singular values with a moderate threshold 10^-6, we can see that the accuracy can be affacted, and this is the reason that we essentially only use the bond dimension to compress the MPS.
The referee writes:
The inset to Fig. 4 shows `absolute error', of what quantity?
Our response: It is the absolute error of retarded Green's function, as shown in Fig. 3. We would add some sentence in the final version to describe it.
The referee writes:
following (B3-B4) where the upper indices i_k,j_k are actual powers. this is misleading now, since this does not concern the tensor A.
Our response: Thanks for the reminder. We would change it to
where the upper indices i_k, j_k of GV \xi_k are actual powers.
in the final version.
Author: Ruofan Chen on 2024-09-10 [id 4755]
(in reply to Report 2 on 2024-08-22)We thank the referee for the refereeing and comments. Our response to the comments are shown below.
The referee writes:
Our response: Yes, it is correct. The ket \ket{a} indicates \bar{a}, so usually the bar is not shown explicitly.
The referee writes:
Our response: Thanks for the suggestions, and we have considered this kind of memory truncation earlier. However, since the most expensive part is due to the large bond dimension and global SVD truncation is in principle more numerical stable than truncation within Δkmax terms, we decide not to employ such a memory truncation at this time to reduce the source of errors. The memory truncation in GTEMPO may be explored in the future.
The referee writes:
Our response: Thanks for the suggestion. However, we have already used a large bond dimension, which is already more than necessary, in the figure to avoid the inaccuracy of OSEE. Therefore the OSEE plotted is expected to be accurate enough, and an additional figure is not necessary.