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Identifying Quantum Phase Transitions with Minimal Prior Knowledge by Unsupervised Learning

by Mohamad Ali Marashli, Ho Lai Henry Lam, Hamam Mokayed, Fredrik Sandin, Marcus Liwicki, Ho-Kin Tang, Wing Chi Yu

Submission summary

Authors (as registered SciPost users): Wing Chi Yu
Submission information
Preprint Link: scipost_202409_00005v1  (pdf)
Date submitted: 2024-09-06 10:11
Submitted by: Yu, Wing Chi
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Computational
Approach: Computational

Abstract

In this work, we proposed a novel approach for identifying quantum phase transitions in one-dimensional quantum many-body systems using AutoEncoder (AE), an unsupervised machine learning technique, with minimal prior knowledge. The training of the AEs is done with reduced density matrix (RDM) data obtained by Exact Diagonalization (ED) across the entire range of the driving parameter and thus no prior knowledge of the phase diagram is required. With this method, we successfully detect the phase transitions in a wide range of models with multiple phase transitions of different types, including the topological and the Berezinskii-Kosterlitz-Thouless transitions by tracking the changes in the reconstruction loss of the AE. The learned representation of the AE is used to characterize the physical phenomena underlying different quantum phases. Our methodology demonstrates a new approach to studying quantum phase transitions with minimal knowledge, small amount of needed data, and produces compressed representations of the quantum states.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
In refereeing

Reports on this Submission

Report #1 by Anonymous (Referee 1) on 2024-10-11 (Invited Report)

Report

The article is a correct extension of Kottmann et al. ([19] in the manuscript). Results are presented in a clear manner, and bibliography on previous work seems to be addressed properly. If the remarks bellow are correctly address by the authors I recommend the publication of this manuscript in SciPost Core instead of SciPost Physics, since the results do not "provide details on groundbreaking results obtained in any (sub)specialization of the field."
Limitations of the method proposed are not properly addressed.

Remarks:

1. In Fig. 3, caption and corresponding text, could you emphasize more that you are working with the entanglement spectrum input?

2. In page 7 the authors claim

" There are three main distinctive regions corresponding to the three phases, and the transitions are captured by the abrupt changes in the loss gradient near ∆ = −1 and 1."

But in Fig. 4 the transition is smooth at \Delta = + 1. This seems to happen also in Fig. 3 inset a) of [19]. So in both cases the transition is not sharply signaled. Even if this transition is typically difficult to see, actually your Fig. 3 has a clear cusp at \Delta = + 1, so apparently the combination of your method of training and the entanglement spectrum as input actually also helps validating the results using the whole RDM as input. I think a comment on this can be made.

3. When changing from entanglement spectrum as input to the reduced density matrix you are enlarging the size from n eigenvalues to n**2 matrix entries. This is a drawback of your approach and it should be mentioned in the manuscript. Any possibility (or absence of it) of taking advantage of sparsity in writing the RDM as input should be mentioned.

4. Results on Fig. 9 are very close to the results of Ref. [35], but there are larger discrepancies seen in the cases of Fig. 5 and Fig. 6 and Ref [37]. These discrepancies are not well addressed and some clarifications on the level of precision of Ref. 37 should be made if it is considered as ground truth. Is there any way of improving your results or reducing the discrepancies? (Enlarging system size, changing the architecture, enlarging the network, adding more points for the Haldane phase, etc?)

5. Any comment on why Fig. 9 b) presents a peak or a valley for both transitions?

6. In Fig. 9.a U is in [1,5], whereas in Ref. [35] their phase diagram (fig. 7b) has U in [-5,5]. Their "PS" area is left aside in your manuscript without any comment on it. Please justify properly why you left it aside or add it for completeness.

7. Same as last comment applies for your Fig. 6 and the phase diagram of the S=1 XXZ model in Ref. [37], where the region with negative \Delta even displays another phase called "XY2". Please justify properly why you don't study this region or add it for completeness.

8. At the end in the conclusions you claim that you could study higher dimensional systems. Nonetheless if you need the reduced density matrix as input, that means that you cannot go to any reasonable size to study a 2D or 3D system, right? Even in the present manuscript the system lengths are extremely short, which is a mayor problem of your approach.
Please justify on the algorithmic scalability of your setup to seriously consider this perspective.

Best regards.

Recommendation

Accept in alternative Journal (see Report)

  • validity: -
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