Neural Quantum State Study of Fracton Models

1. The objectives and methods introduced in this study are well-justified

2. The here introduced correlation-enhanced RBM seems to be fast, reliable, quite general and easy to implement method.

3. The paper is well written and easy to follow with enough of additional information in appendices and online to reproduce the work.

1. The role of the hysteresis is overstated in the paper.

2. Other learning protocols and NQS can and should have been used the in search of lower variational energy or to test some of the statements.

3. The estimations of the critical field and the discussion of the type of the phase transition in unconvincing.

The work under consideration “Neural Quantum State Study of Fracton Models” employs Neural Quantum States (NQS) as variational wave functions within variational Monte Carlo (VMC) methods to explore the properties of strongly entangled fracton order and associated phase transitions. Motivated by unsatisfactory results from the standard restricted Boltzmann machine (and other typical NQS applications), the authors have developed a physically motivated extension known as the correlation-enhanced RBM (cRBM). Through the use of cRBM, the authors demonstrate that all three fracton models examined exhibit a first-order phase transition from a fracton phase to a trivial field-polarizing phase.

The objectives and methods introduced in this study are well-justified. The fractons under investigation necessitate long-range entanglement and a lattice dimensionality of three or higher, conditions that are a big challenge to standard techniques such as Tensor Networks. The authors effectively show that NQS can provide a solution in these cases. Overall, this is a very good study which, after addressing certain issues, could be confidently recommended for publication in SciPost Physics Core. However, to meet the standards of SciPost Physics as a flagship journal, the paper would need substantial enhancement.

Further comments and questions:

1- "NQS can be considered a subclass of variational Monte Carlo (VMC) [34] techniques.” VMC is a broad framework where the variational principle is used to approximate the ground state of quantum systems. S NQS, on the other hand, represent a clever variational wavefunction characterized by parameters that are optimized using machine learning techniques. So, in my understanding, NQSs aren't a subclass of VMC, but rather a type of wavefunction used within the VMC framework.

2- I am somewhat confused by the correlators. Admittedly, this confusion stems from my gaps in understanding fracton systems. Nevertheless, are the conditions imposed on the correlators satisfied for ANY ground states of the models, or only for some? Additionally, what is the difference between this technique and Group Convolutional Neural Networks?

3- Before Equation (14), the visible biases are set to zero. Why is this the case? As mentioned in the paper, visible biases are often critical for setting the correct sign structure of the wave function. Why can this aspect be ignored here?

4- Figure 9 is used to demonstrate the significant advantage of incorporating custom correlator features into RBM. However, this figure raises several questions. First, the alpha value of ¼ used here is very small. This is probably to keep the number of parameters approximately the same as in other networks compared to cRBM, where already 52 parameters suffice. Yet, this approach seems a bit self-serving, and the number of parameters for sRBM is still significantly smaller than for cRBM. So, how stable is this effect where other networks remain stuck in the initial local minimum for a significant number of training iterations? How does it depend on alpha, learning rate, and the distribution of initial parameters? I would guess that there is a learning protocol that could allow the RBM to learn the ground state as well. Or am I mistaken, and is the RBM used is too small to capture the exact ground state?

5- Figure 11 demonstrates that the transfer learning approach, where one starts at a high field and then gradually decreases it, has clear advantages over the other two protocols. However, since it is a variational method, why not retrace the steps in the opposite direction? That is, start with any "blue" point in Figure 11 that has a small relative error and perform a left-right sweep from there. This approach might be irrelevant for the 4x2x2 lattice where the agreement with the exact solution reaches the level of numerical (or Monte Carlo) precision for the right-left sweep. However, it could lower the energies for larger systems. And in the end, that is what the variational methods are all about. Such there-and-back techniques have previously been used for magnetic systems, for example, to retrace step changes in magnetization [SciPost Phys. Core 6, 088 (2023)]. The results could also enhance the understanding of the role of transfer learning here.

6- I think that the role of the hysteresis is overstated in the paper. The hysteresis is related to the details of the learning, there is no physical time involved. I therefore don’t know what is the significance of the statements like “... our work disclosed that the checkerboard model experiences a strong first-order phase transition with a large hysteresis when subjected to uniform magnetic fields.” , “ … the method detects hysteresis effects and provides insights into the nature of the underlying phase transitions…”. The hysteresis might completely disappear with different learning protocols or even different Monte Carlo sampling methods.

7- What is meant by “We estimate a critical field hcrit ≈ 0.445 from the intersection points of energy curves and the middle of the magnetization hysteresis.” The “hysteresis” is huge for L=8,6, actually the second branch is not even in the figure and it is not clear at all that the energies for systems with different lattice sizes cross each other. The magnetisation in Figure 13 seems to get more smooth with increasing lattice size and shifts to smaller values of h_x. On top of that the discontinuities in the relative energy contributions in Figure 14 also point to a much smaller critical value of h_x. Isn’t there a more reliable way to estimate the critical point from your data?

8- What is actually meant by “strong first order phase transition”? Is there some other way to test the order of the phase transition? Why not use some version of Binders cumulant?

9- The technique of splitting the network into one that encodes the phase and the other for amplitude is quite standard in the context of NQS and should be tested at least with the other methods in Figure 9

1- Figure 9: Discuss and demonstrate the stability of the result with respect to changing learning rate, initial parameters distribution, standard deviation and alpha. Add results for RBM with phase and amplitude encoded by separated networks and preferably also something even simpler than RBM, e.g., Jastrow. Also show results for symmetric RBM with at least as many parameters as for cRBM.

2- Reevaluate or explain the validity of the estimation of the critical field as well as the significance of the hysteresis.

3- Show at least some tests for transfer learning where the direction of the learning was reversed at finite fields.

4- For further suggestions see the report.

Accept in alternative Journal (see Report)

1- clear presentation, 2- proposes a specific NQS architecture change which performs better in studied task, 3- studies a relevant physical system

1- Does not demonstrate that earlier findings about phase transitions of the X-cube model and Haah’s code under uniform magnetic fields can be replicated with NQS, which would show that the analysis done with NQS is trustworthy

This work studies the potential of neural quantum states (NQS) in representing 3D lattice systems with long-range entanglement, specifically the checkerboard, X-cube, and Haah's code models. The authors show that unspecialized NQS architectures, such as multilayer perceptrons and restricted Boltzmann machines, do not perform well in this task. They propose a more specialized architecture, the correlation-enhanced RBM, to address this issue. The proposed approach is tested by studying a phase transition in the checkerboard model.

I believe this work is relevant and substantial enough for publication in SciPost Physics. However, it could be improved by including an analysis of the X-cube and Haah’s code models and demonstrating that previous results obtained with other methods can be replicated using NQS. This comparison would be very interesting and would more clearly indicate the advantages or disadvantages of NQS, strengthening the claim that NQS methods can serve as a new "workhorse" in studying such systems.

Publish (meets expectations and criteria for this Journal)