Quantum-critical properties of the one- and two-dimensional random transverse-field Ising model from large-scale quantum Monte Carlo simulations

1- comparison of different finite-size-scaling techniques for disordered transverse-field Ising models
2- good level of detail in describing technical aspects
3- very transparent description of the difficulties of each technique, in particular concerning the temperature convergence
4- valuable for anyone who is interested in numerics for disordered systems or wants to start numerical research in this direction

1- with its strong focus on technical rigor, the physical message sometimes remains hidden

The authors apply the exact stochastic series expansion quantum Monte Carlo method to the random transverse field Ising model in one and two dimensions. The authors compare different techniques for finite-size scaling and illustrate their advantages/difficulties. The methods are tested for the 1D system, which is much better understood than the 2D case, and then applied to the 2D case to calculate critical couplings/exponents.

I find this numerical study very valuable, as it compares different finite-size-scaling techniques, discusses their strength/weaknesses, and shows what is necessary to arrive at consistent results. One can clearly see that it takes a lot of work to test all of these techniques and analyze the temperature convergence appropriately.

Given the strong technical aspect of this paper (which is very valuable for numerical studies), the physical insights are not always as clear to me. In particular, the authors state that they "resolve several inconsistencies in existing literature". Maybe it would be helpful to highlight these aspects again in the conclusions, as this is one of the parts which is most likely remembered by the reader. Then, it would be easier to find these parts again in the main text.

If I understand correctly, one of the main problems for the 2D case is that results from the sample-replication method are seemingly not consistent with the Binder ratios. Only if one takes the slow temperature and size convergence into account, these discrepancies get resolved. Is this correct?

All in all, I think this paper is of high quality and fulfils the publication criteria of SciPost Physics, once the authors have included my requested changes.

1- On page 4, between Eqs. (2) and (3), the authors write $h\ll \langle J \rangle$ or $h\gg \langle J \rangle$. The meaning of the expectation value has not been defined. Later $\langle \rangle$ has been defined as the thermal average for a given disorder configuration. Do the authors mean the disorder average here, which they denote by $[]$? The same notation also appears again on page 12.

2- On page 6, the definitions of the different parts of the Hamiltonian are slightly confusing. In Eqs. (8) and (9), the indices fulfill $i,j>1$ and $\mathcal{H}_{i,j}$ is only defined for $i\neq j$, because $\mathcal{H}_{i,i}$ includes a different part of the Hamiltonian. I'm also wondering what the constant $c$ in Eq. (7) is needed for, as all the necessary shifts are already included in $\mathcal{H}_{i,j}$. One should also mention at this point that $\mathcal{H}_{0,0}$ is not included in the Hamiltonian, as it is only needed for the fixed-length operator string (as mentioned in Sandvik's original paper).

3- Please check Eq. (19) again. From a quick dimensional analysis, it seems to me that the factor of $1/N^2$ should be $1/N$ and that the prefactor $1/\mathcal{L}$ should be something like $(\beta / \mathcal{L})^2$.

4- In Eq. (20), $\sigma^z_{i,p}$ is not defined.

5- Is Fig. 1 for a single disorder configuration? If yes, one could mention this in the caption.

6- If possible, it would be nice to include figures close to where they are first mentioned in the text. For example, one has to scroll quite a bit through the paper to find Figs. 1 or 3. I understand that this is not always possible, but it would improve readability.

7- I do not find the definition of Eq. (22) very clear, probably because I am not familiar with quasi-random numbers. What is $\lambda_s$?

8- Is there a reason why the periodic boundary conditions are chosen like this? For example, why is 7 connected vertically to 10 and not to 1? Maybe it is worth mentioning the advantage of this?

9- The expansion in Eq. (31) would be much clearer, if the authors referred to the scaling form of Eq. (23) .

10- In the caption of Fig. 10, the abbreviation RTFIC has not been defined, but is also never used anywhere else in the paper.

11- When first looking at Fig. 15, I was wondering why the critical values were not marked in the figure or mentioned in the caption. The caption and the text kind of implied to me that $d/z'=0$ marks the critical value. It took me some time to understand that there are significant finite-size/temperature effects, which are only discussed on the following pages. Maybe it is worth already adding a sentence here, which gives the reader a hint that this issue will be solved below.

12- I know that this is already mentioned in the main text, but I think it would be useful to add to the caption of Fig. 17 that the polynomial fits are of order 1 to 5. Do they appear in sequential order in the figure?

Ask for minor revision

A broad review of the current state of the problem

The manuscript presents extended computational results for the quantum Ising model, however it is not evident that they bring in the essential new physics or methods.

In the manuscript the random quantum Ising model has been examined using the stochastic series expansion quantum Monte Carlo method supplemented by the data analysis for finite-size systems. In the introduction the authors provided a rather broad review of the current state of the problem. It is clear that the study of the random quantum spin models is quite a hard problem with not many rigorous results, and where the results may strongly depend on the type of disorder.
The current study is focused on the accurate study of the critical properties of the model with two types of disorder: bond- and field-disorder, and bond-disorder only. The authors show their expertise in the study of the disordered systems using various methods to achieve the precise calculation of the critical points and exponents. From that point of view, the manuscript seems a bit technical.

The paper contains a lot of material, and I think that its organization could be improved:

1) In Sec.2 the random transverse-field Ising model is introduced, but an accurate description of the averaging over disordered realization is missing here. Therefore, e.g. the definition of the magnetization in Eq.(3) is ambiguous. The right-hand side does not contain thermodynamic or/and random averaging.
Then, in Sec.3.3 the authors introduced Sobol sequences for the disorder average and gave the approximate Eq.(21), while the definition of the disorder averaging was not provided before.

2) In Sec.4.2 the magnetization $m^2(h,L)$ is not defined. Therefore, it is not clear if it is just thermally averaged and also randomly averaged quantity.

Ask for minor revision

1- Careful analysis of a highly complex problem where there are not many results.

2- Benchmarks of the techniques used have been provided for a simpler 1D version, which is fairly well understood.

3- Improved data analysis methods are implemented, such as more efficient sampling of the disorder phase space, and different extrapolations to the infinite size limit

1- Due to the large amount of content covered in this manuscript, the reader often finds themselves having to go back to remind themselves of the quantities being considered and the techniques. (See requested changes)

The authors have studied the random transverse field Ising model (RTFIM) using an unbiased quantum Monte Carlo method. The technique is know to be robust and the results are expected to be trustworthy. The analysis of this disorder problem has been carried out by different members of the community over the last few years and conflicting outcomes have been reported. The authors suggest that this may be the result of extreme sensitivity to temperature for this model and have introduced a method for careful data analysis in the low temperature regime. Although the primary target is to understand the 2D RTFIM, the data analysis techniques introduced are benchmarked using the 1D version, which is well understood. The main contribution of their work is to show that temperature effects need to be carefully treated, and that this treatment leads to results consistent with previous works.

1- As there exist conflicting values for the universal exponents from previous studies, it is worth mentioning in detail in the conclusion how this is resolved by quoting the values from previous references and showing how this work makes them consistent.

2- At the end of Sec 4, a small recap would be very useful for the reader. This just needs to mention which quantities are going to be extracted using which method. Same for Sec 5, where this would include a discussion of different exponents, and show the consistency across methods.

3- typo on page 22 : "hardy visible" instead of "hardly visible"

Ask for minor revision

see report

see report

The authors consider the random transverse field Ising model in one and two space dimensions and study its critical properties using stochastic series expansion (SSE) quantum Monte Carlo (QMC) and finite size scaling. For 1d the known results for the critical point as well as the critical exponents are reproduced within statistical and extrapolation error bars, in addition new insights into convergence and finite size properties are revealed. In 2d the authors determine a new critical field value (for the box distribution), which is higher than previous numerical estimates and critical exponent values that agree within the error margins with previous values extracted from QMC simulations and SDRG calculations. Again, due to the enormous numerical effort made and the new methods applied in the data analysis, novel insights into convergence and finite size properties are revealed, paving the way for future numerical studies of disordered systems governed by an infinite randomness fixed point (IRFP).

The paper is very well written and discusses thoroughly all potential problems that occur in QMC studies of critical random systems with activated dynamics, meaning with exponentially diverging time scales at the critical point). This is an excellent work and in my view it presents a breakthrough in analysis of an extremely hard problem, namely the computational determination of the critical properties of higher-dimensional disordered systems that are supposedly governed by an infinite randomness fixed point, a previously-identified and long-standing research stumbling block in Sci. Post. expectation terms. It also fulfills *all* general acceptance criteria. Therefore, I recommend its publication in Sci. Post. in its present form after the following minor points have been considered:

1) To my knowledge, there is a zero-temperature version of SSE (T=0 SSE) which can be applied to the TFIM (Section 1.4.2 in Stochastic series expansion quantum Monte Carlo by Roger G. Melko). As discussed thoroughly by the authors, an extrapolation to T=0 (especially for the RTFIM) requires one to approach very low temperature, so it would be advantageous if such a T=0 SSE can capture T=0 properties without extrapolation. It would be useful if the authors could discuss briefly whether T=0 SSE captures the T=0-properties of the random TFIM or not, or whether the T=0 SSE is hard or impossible to be applied in the present case.

2) The SDRG for the two-dimensional RTFIM (ref. [35] in the paper) predicts that critical exponents of h-fix and h-box are equal, but ref. [40] concludes, also numerically with QMC, that they are different. Although the authors focus on the h-box distribution, exclusively, it would be useful if they found indications pointing in the same direction as in [40] or not.

3) p.4: “… rare regions with finite clusters with very special disorder configurations …”
I wonder why the authors denote these configurations as “very special”: the commn understanding is that these configurations comprise strongly coupled clusters, i.e. clusters (compact or fractal) that are locally in the ordered phase.

4) p.5: “The critical exponent of the dynamical scaling is given by psi=1/2”
Since usually the dynamic exponent is denoted as z I would more precisely write “… of the activated dynamic scaling …”

5) p.9: “causing different disorder configurations to converge in temperature at different β values”: I wonder whether similar convergence variations appear in the number of necessary MCS, N_MC, for equilibration. The authors state that they use for all samples N_MC<100 sweeps (p.10) – did they observe that this is sufficient for ALL samples to *equilibrate*?

6) Section 3.3 and Appendix A: The authors explain that the advantage of the Sobol sequences they use for disorder realizations is to sample the disorder space more evenly, but is this legitimate: Fig. 2 demonstrates impressively that the usual pseud-random numbers tend to cluster (and leave holes), but isn’t that exactly what leads to a higher probability for strongly coupled clusters (stronger Griffiths singularities), which is suppressed by Sobol sequences?

see report

Publish (easily meets expectations and criteria for this Journal; among top 50%)