Probing many-body localization in a disordered quantum dimer model on the honeycomb lattice

1- a disordered quantum dimer model on a honey comb lattice is studied, which is an open problem
2- state-of-the-art exact diagonalization is used to investigate the problem

1- the presented data does not sufficiently support the claim that an MBL transition exists in this system --- different observables show different behavior and finite size effects are strong.

The authors study a disordered quantum dimer model on the honey comb lattice using exact diagonalization approaches (full diagonalization, shift and invert techniques, and Lanczos time evolution). The goal is to identify whether the system possesses a many-body localization (MBL) transition at some critical value of the disorder. This is a relevant question, because it is a subject of debate whether such a transition can exist in dimensions greater than one. The authors provide a comprehensive numerical study of the problem, by both looking at spectral statistics and non-equilibrium dynamics.

When studying the data, it seems to me that there is no clear evidence for an MBL transition. Here are a couple of examples from which one would rather expect the absence of a transition in the thermodynamic limit:
1- Fig. 1: the level spacing statistics shows significant finite size effects in the crossing point of different system sizes
2- Fig. 8: the peak in the variance of the entanglement entropy is significantly shifting to larger values of the disorder strength when the system size is increased
3- Fig 9: The dynamics of the imbalance is still strongly finite size dependent with a trend that actually at larger system sizes there is stronger decay (even though times are not as large as in panel a)

I think that the authors need to critically discuss these issues when revising the manuscript. Also another challenge (which is presented somewhat differently in the manuscript) is indeed the slow growth of the Hilbert space with system size N, because in that case the argument for destabilization of many-body localization provided in Phys. Rev. B 95, 155129 (2017), will be even weaker. Due to these considerations it seems to be very challenging to affirmatively answer whether there exists an MBL transition in this 2D quantum dimer model. It would be great if the authors could discuss this point too.

1- critically discuss possible discrepancies between the different investigated observables, in particular, also focussing on finite size effects.
2- Pg. 5, 2nd paragraph: rewrite the terms localized phase and ETH phase.
3- Fig. 3: Maybe it makes sense to show the KL divergence divided by its system size
4- Fig. 10 is misleading. It is not justified to take finite time date (on top of that at different times) and make an attempt to extrapolate them to infinite system size --- in particular given the residual flow of the imbalance for the larger system at strong disorder. Please remove this figure.
5- On page 11 top and the appendix the description of the RK point is misleading as one would think that it is attained at V=0 (it is characterized as model 1 in the absence of disorder). Also what is meant with the ket of ones (all possible dimer coverings in the zero winding sector, I suppose?) should be clarified.
6- The authors state that the RK point has area law scaling of the entanglement entropy. This statement is clear when considering all winding sectors. When projecting onto winding zero it is not obvious to me because of the entropy which results from fixing the winding constraint. Please elaborate on that point.
7- Sometimes in the text the figures are referred to as top and bottom panel instead of left and right panel. I suggest to introduce a) and b) labels for clarity.

1-The authors consider a model that (to my knowledge) has not been considered before in the context of many-body localization (MBL), though bearing similarities to the previous study Ref. [48].
2-The authors use many different measures as a probe for MBL.
3-Relatively few studies into 2-dimensional MBL have been performed so far, the work is timely and relevant.
4-The findings are relevant to state-of-the-art experiments.

1-The authors do not clearly demonstrate that MBL properties survive in the thermodynamic limit - although it is not necessarily expected that this would be the case.

This work addresses the problem of many-body localization (MBL), which pertains to the interplay between disorder and interactions in many-body systems. At this point many numerical works have been devoted to MBL in one-dimensional systems, particularly in the archetypal model: the XXZ Heisenberg chain. There are strong indications that in this system there is a transition in the thermodynamic limit between an ergodic and a localized phase at a certain finite (model-dependent) strength of the disorder.

Of major interest is what happens in the case of two geometric dimensions (2D). Some theoretical approaches and recent experiments indicate that a "true" transition remains in this case, whereas some theoretical methods and our own recent numerics (Ref. [39]) suggest instead that in 2D the "transition" is more like a crossover, with an effective critical disorder that grows with system size.

The challenge for numerical approaches in 2D is that in even the simplest unconstrained many-body systems the Hilbert space growth exponentially in the number of sites $N$ as $2^N$. Since exact numerics ("unbiased" in the words of the authors) can handle up to 20-ish sites, this leaves the available system sizes prohibitively small. A way to circumvent this problem is by introducing constraints in the Hamiltonian, which reduces the scaling of the size of the Hilbert space with the number of sites.

The present manuscript considers such an approach in a constrained dimer model on a honeycomb lattice. The authors then proceed to apply a large distinct number of measures of the MBL transition to the model, including both static properties such as level statistics and entanglement, as well as dynamics. They argue that their results indicate an MBL transition in this model at a disorder value of $V \approx 20$-$25$.

This work represents a nice addition to the existing literature on MBL, in particular adding to the results in two dimensions where relatively few results are available, providing high-quality numerical results. In my view there is still a lot of potential for understanding MBL through the pathway of similar constrained models. I would recommend publication in SciPost Physics as the manuscript successfully addresses Expectation criteria 1 and 3.

I have the following comments/questions:

1-I am confused concerning the discussion around Fig. 5. How are the black dashed lines computed here? What is their precise relation to the basis states?

2-I assume that the way the system is "sliced" for the bipartite entropy is not crucial for the results. Did the authors explicitly verify this?

3-A key question concerns the extrapolation of the results to large systems. The curves for the imbalance dynamics in Fig. 9 show quite a strong dependence on system size, seemingly much stronger than similar curves for the imbalance in 1D. I wonder if a crossover-type scenario along the lines of [Phys. Rev. B 99, 134305] might be a better explanation of the authors' results for this reason - although one should always be careful extrapolating from small systems of course. The authors fairly acknowledge such a possibility in the Conclusion, but perhaps it would be useful to explore this scenario in more detail.

1-The axis labels are often tiny and hard to read. Please update the figures and make them more readable, using fewer columns if needed. Also, if there is a left and right panel, explain the left panel first (Fig. 2). If a figure has many colours, I recommend using a gradient (e.g. equally spaced colours from a "viridis" colourmap).

2-I'm not a big fan of this "biased/unbiased"-nomenclature, but if you must use it, please define clearly what is meant by this (exact vs. approximate approaches).