Spin Liquid versus Spin Orbit Coupling on the Triangular Lattice

We would like to thank the referee for their review of our manuscript and for recommending its publication. However, we believe that they have likely misunderstood key ideas of our method. The main point of emphasis in our method is that, instead of working directly with the modified wave function, we use the variational Monte Carlo to directly calculate the perturbative corrections to operator expectation values at a given order in $\alpha$. Critically, after expanding Eq.(15), these corrections can be found by calculating only the connected contributions to the expectation value. Note, that this is just a numerical application of the well known `Linked Cluster Theorem', which states that all contributions to the quantum correlation function can be found by summing only the connected Feynman diagrams. We use Eq.(16) as an intermediate step to obtain the numerical form of these connected contributions, which we describe in Eqs.(17) and (18).

Referee: "Any expansion is not size extensive. "

If one were to naively apply our Eq.(16), one would find such a sub-extensive correction after optimizing over $\alpha$. However, then one formally retains all powers of $n$ in the perturbative expansion and groups together powers of $\alpha$, the linked cluster theorem guarantees an exact cancelation of 'disconnected' terms in the numerator with the normalization factor in the denominator. Each term in the perturbative expansion is therefore given only by the connected part of the operator expansion, which scales linearly with system size, giving an extensive correction to all correlation functions. The linked cluster theorem is proven rigorously in standard many-body textbooks, and clearly results in an expansion of the partition function which is size extensive.

Referee: "In fact, Eq.(16) is EXACTLY what is done when people perform on Lanczos step: $|Psi_{1LS}\rangle = (1+ \alpha H)|Psi_{VMC} \rangle. $"

First of all, Eq.(16) is different from the result one gets using one Lanczos step. In that case, the last term in the numerator appearing at order $\alpha^2$, denoted as $\mathrm{Re} [ \langle \sf{H}^2 \mathcal{O} \rangle_0 ]$, would not be present. As has long been appreciated in perturbative QFT, it is critical to properly count the number of diagrams contributing at any order in a perturbative expansion in order to ensure an exact cancelation of vacuum diagrams. Additionally, we never actually apply Eq.(16) directly in our numerical simulation, but only use it as a starting point for further derivations. When calculating the correction to an observable, we expand both the numerator and the denominator in Eq.(16) and truncate our results at a finite order in $\alpha$. In fact, we could safely add terms of order $\alpha^3$ and further to Eq.(16) without changing any of our results.

Referee: " Therefore, their approach IS equivalent to one Lanczos step. Naively, Eq.(16) may seem size extensive, but by considering the variation with respect to alpha, in order to minimize the energy, one can easily realize that alpha is not O(1/L) and therefore the energy gain is not size extensive."

We reiterate, that we do not apply Eq.(16) directly, but expand this equation further to arrive at our final expressions. The resulting expressions for the corrections to general observables are given in Eq.(17) and to our specific choice of operators in Eq.(18). Notice how all these expressions scale with system size as $O(N)$, due to an exact cancelation of "vacuum diagrams".

Referee: "Anyway, Eq.(16) is completely equivalent to performing one Lanczos step, as described in the references cited."

As we have explained, Eq.(16) is both different from the result one finds by performing one Lanczos step and is only used as a starting point for further derivations in our numerical method. Furthermore, we do not work directly with the modified wave function, as is done when applying a single Lanczos step, but instead calculate the corrections to operator expectation values, using the connected correlation functions. To clarify this point, we have added the term ``$+O(\alpha^3)$'' to the numerator and denominator of Eq.(16), emphasizing that this equation is only used as an intermediate step in our derivation.

Referee: "Minor points: 1) In the introduction the expression ``one performs Monte Carlo sampling of the quantum wave function in real space" is sloppy. I suppose that this means that the sampling is performed in a (many-body) basis set where electrons/spins are localized on each site. "Real space" is reminiscent of a one-body sampling. "

We have reworded the corresponding statement for clarity.

Referee: "2) I suppose that both $C_2$ and $S_6$ in Eq.(2) also affect the spin operators. This should be mentioned. Moreover, a picture for the symmetry generators would help the reader. Is the $S_6$ symmetry a combination of a 120 rotation plus an inversion?"

We have added one more equation [Eq.(3)] where we describe how each symmetry operation transforms the spin components. We have also included a sentence above Eq.(2) where we clarify that the sixfold roto-reflection $\mathcal{S}_6$ is indeed ``a combination of a $120^\circ$ rotation and an inversion''.

"3) I do not find the definition of the vector $n_{ij}$ in Eq.(3)."

We have included an appropriate definition of $n_{ij}$ below Eq.~(4) [which used to be Eq.~(3)].

4) Just to clarify: on page 5, the number 18 (different mean-field ansatze) is referred to the Z-2 classification, correct?

Indeed, this is correct. To clarify this even further in the text, we now explicitly state that ``there are at least 18 different $\mathbb{Z}_2$ mean-field ansatze.

Referee: "At page 6, what do you mean by ``all spin-dependent quadratic terms are prohibited"?"

We mean that the PSG equations do not allow any hopping terms in the mean-field Hamiltonian which depend on the orientation of the spin [and therefore break SU(2) invariance]. To make this even more transparent, we have slightly reworded the corresponding sentence.

Referee: "6) I suggest to introduce the extended model with $J_2$ and $J_3$ at the beginning. At present, $J_2$ is mentioned at page 8 without any reference to the actual Hamiltonian."

We have added a line directing readers to the definition of the next-nearest neighbor term $J_2$.

Referee: "7) In the caption of Fig.3, what is the red region? I suppose the ordered 120-degree state, but this is not written. I am also disappointed by the layout of the figure: why not putting the label on the x axis? Why showing the long arrow? It would be better to have 5 panels with labels a), b), c)..., and no arrow. (The same for Fig.5).}"

We do not understand the referee's first question; we explicitly state it in the caption of Fig.~3 that the red region corresponds to the $120^\circ$ order. In accordance with the referee's further suggestions, we have added a label to the $J_{\pm\pm}$ axes of Figs.~3 and 5. For compactness, however, we have not separated each plot into different panels.

Referee: "8) I suppose that the value of the local moment reported at page 9 are for the simple Heisenberg model on the triangular lattice, but this is not written."

For clarity, we now state it explicitly that the corresponding DMRG results are ``for the triangular-lattice Heisenberg model''.

Referee: " 9) Finally, figures are too small and must be enlarged."

We have increased the sizes of our figures (where possible).

Referee: "In summary, the paper contains a wrong statement, but nevertheless it contains relevant numerical calculations on an interesting model. Therefore, I ask for a revision; after that the paper can be published."

As we have thoroughly addressed, the referenced statement is in fact correct. Since we have also addressed all of the referee's minor points by making minor revisions in the manuscript, we believe that it can be published in its current form.