Quantum Monte Carlo simulations in the trimer basis: first-order transitions and thermal critical points in frustrated trilayer magnets

1. Numerical simulations are very clean and well-made

2. Useful results to illustrate why some compounds/models could show two lines of maxima of specific heat, while some others (e.g. water) have only. Connection to recent experiments on frustrated magnets (Ref. 32).

1. Unclear if the rotated Ising model discussion actually explains the scaling of the specific heat. This should be clarified.

This paper describes quantum Monte Carlo simulations of a frustrated spin system, a trilateral magnet, which have for specificities to have no sign problem in a 3-site (trimer) basis.

At the technical level, this is an extension of previous work (including by the authors) in Refs 23-27 which were mostly concerned with bilayer systems, using a 2-sites basis. There is no major breakthrough here.

Concerning physics, there is a first-order quantum phase transition (driven by the competition between the couplings within or between trimers) similar to the one in the bilayer system, further continued at finite temperature by a line of first-order transitions ending a critical point (of Ising type), similar to the water phase diagram. The main point of the paper is that this frustrated model has an extra tuning parameter (here one of the three couplings inside the trimer) which allows to change the slope of the first-order line, as well as to create/enhance a second maximum in the specific heat (beyond the one existing that continues the first-order line). This second line of maxima is not present in the water phase diagram. This is perhaps not a major physical result, but it is an interesting point that is made, specially in light of recent experiments in SrCu2(BO3)2 (Ref. 32) where the similarity between specific heat data in this frustrated magnet and water was made. This manuscript allows to rationalise (and tune) the different behaviours between models which admit one line of maxima of specific heat, and those which admit two.

The paper is very clear and otherwise well written. I think it satisfies all the criteria for publication in SciPost Physics. Below, I have only one main concern and the rest is mostly simple comments / suggestions.

- My main concern is the discussion on 4.3 using a “rotation” of the critical Ising model to explain the scaling behaviours of specific heat. From my understanding of this discussion, it seems that the specific heat should *always* diverge as L^{gamma / nu} for all models which admit a 2d Ising critical point in an extended phase diagram, except when the Z_2 symmetry is not explicitly broken (phi=0). This sounds a bit surprising, and I wonder whether finite-size scaling of the rotated Ising model (of the kind shown in Figure 7) actually confirm this. My concern is mostly that this perhaps true, but only very close to the critical point. I am not sure this actually explains the scaling reported in Fig 8c, where none of the data scale as expected L^\gamma/nu (neither the quartet susceptibility nor the specific heat). Can the authors report the values of the (quartet) correlation length corresponding to the data in Fig. 8 ? (I assume these must have been obtained). This would allow to see if the data presented there are actually inside the critical region. If this is not the case, I am not sure the full rotated Ising model argument is worth adding to the paper then.

- I would suggest to move the full section 3 into an Appendix, as this is mostly a technical/formal discussion. I don’t think this is a major point of the paper and I also think that quantum Monte Carlo experts (to which perhaps this discussion is aimed at) would not be too much surprised by the abstract directed loop.

- As a minor point, the value of c in Appendix A should be quoted (even if not universal).