Quantum spin spiral ground state of the ferrimagnetic sawtooth chain

1- Numerics appears to be quite robust
2- Good introduction and explains nicely the problem that is addressed
3- Includes relevant technical details

1- the advantages of SU(2) in the calculation stated in the abstract and section IIA are manifestly incorrect
2- the author's missed some straightforward followup calculations for some of the symmetry groups (eg the S^2 value for the U(1) symmetric calculations)

The authors study a saw-tooth chain using a combination of finite and infinite MPS methods, using DMRG and VUMPS. This is an interesting model that has a ferrimagnetic state, previously supposed to be commensurate but the authors provide compelling evidence that it is incommensurate.

The use of $SU(2)$ symmetry in the calculations however is misrepresented. Specifically the way the author's calculate the effective bond dimension when using $SU(2)$ is invalid. It is widely known since the start of non-abelian symmetries in DMRG that for non-zero magnetization the advantages of using $SU(2)$ symmetry are lessened. For total spin zero states the correspondence between $SU(2)$ symmetry and $U(1)$ (or no symmetry) is straightforward because there is an exact correspondence that each basis state of spin $\vert j \rangle$ corresponds to a multiplet of $2j+1$ states in a $U(1)$ basis. However no such direct relationship applies to states of non-zero total spin. The author's don't state exactly how they calculate $\chi_\text{eff}$, however they appear to have done something similar to multiplying the maximum number of states kept in the basis by the maximum dimension of the $SU(2)$ representation. This is not a valid way to calculate $\chi_\text{eff}$, as demonstrated by the example of a fully polarized ferromagnet. For a concrete example, suppose the system size is $L=400$, and therefore the fully polarized ferromagnet is solvable with $\chi=1$ $SU(2)$ state, that has total spin $j=200$ at the edge of the system. The author's calculation would determine that this is equivalent to $\chi_\text{eff}=2j+1 = 401$ states using a $U(1)$ basis, however a fully polarized ferromagnetic state of all spins up (for example) is a simple product state that only requires bond dimension $\chi=1$.

I don't know how to to calculate an exact correspondence for $\chi_\text{eff}$ (indeed, there probably isn't an exact correspondence), but to see why the advantages of $SU(2)$ are so restricted for polarized states, one can consider the dimensions of the Hilbert space, visualized on a Bratteli diagram [1]. The dimension of the Hilbert space is the number of ways of getting from one side of the diagram (with spin $s=0$) to the other side (with spin $s=j$). With $U(1)$ symmetry the z-spin can be both positive and negative, so there are many more paths (and hence higher dimensional Hilbert space) than using an $SU(2)$ basis where the spin $s$ is strictly non-negative. However when the magnetization of the state is large, there are clearly very few paths that can venture into negative values of z-spin, and hence the difference in Hilbert space dimension between $U(1)$ and $SU(2)$ basis sets becomes smaller, until finally at full polarization both basis sets have the same dimension.

One way of making a concrete estimate of the advantage of $SU(2)$ symmetry is to compare the relative sizes of the Hilbert spaces, which can be calculated using combinatorial techniques. Eg for L=100 and spin j=27, dimension of the $SU(2)$ Hilbert space in this sector is 17,533,203,421,077,004,122,000, compared with the $U(1)$ sector of $s^z=27$ of 24,865,270,306,254,660,391,200. The ratio is only 1.41818..., so the two basis sets are exploring Hilbert spaces which are very similar in size. Indeed, this is surely an upper bound on the possible value of $\chi_\text{eff}$. So I believe that there is essentially no advantage to using $SU(2)$ symmetry in this calculation. This is also consistent with my own very small scale test - for size $L=100$ magnetization $s=27$ with $\chi=100$ the variational energy using $SU(2)$ and $U(1)$ is barely distinguishable (matching to 8 significant figures). This is also consistent with the author's own results, which indicate that the iMPS results using VUMPS using $U(1)$ symmetry or no symmetry are at least as accurate (if not more so) than the finite size calculations using $SU(2)$, as shown eg in Fig 2.

I am surprised by the author's statement that they don't know of a practical way to calculate the total spin per unit length using $U(1)$ symmetry. MPO techniques to calculate moments of operators, for example $S^2$ (which has a simple MPO form) have been known for a long time [2,3], which is independent of the symmetry group, and it is essentially the same calculation that the author's perform themselves for the correlation function in section IIC using very similar transfer matrix techniques [manuscript ref 47] - the total spin is essentially the integral of the correlation function. But for $U(1)$ a calculation isn't necessary as it can be calculated analytically. The total spin in the thermodynamic limit is
$$
\langle S^2 \rangle = \left \langle \sum_{i,j} \left( S^z_i S^z_j + \frac{1}{2} \left( S^+_i S^-_j + S^-_i S^+_j \right) \right) \right\rangle
$$

For an injective iMPS, the connected part of the 2-point correlation function is necessarily zero at long distances; the only contribution in the limit of $|j-i| \rightarrow \infty$ is from the disconnected part $\langle S^z_i \rangle \langle S^z_j \rangle$ (and likewise for $S^+$ and $S^-$). Now a state with $U(1)$ symmetry is an eigenstate of $S^z$, call it magnetization $s = \langle S^z_i \rangle$ per site. Since the $U(1)$ symmetry is enforced, the local expectation values $\langle S^+_i \rangle$ and $\langle S^-_i \rangle$ are identically zero, hence the leading contribution to $\langle S^2 \rangle$ is $L^2 s^2$, coming from the local $S^z$ contributions. All other contributions are sub-leading ($O(L)$), hence the total magnetization per site $\frac{\sqrt{\langle S^2 \rangle}}{L}$ is identically equal to $s$, being the $U(1)$ quantum number per site. An interesting question is what is the behavior of the sub-leading term, since this is the variance per site of the (square of the) total spin. There is clearly a contribution equal to $sL$ coming from the purely on-site contributions (i.e. the terms with $i=j$ in the summation above), and if there are no other contributions then it is an eigenstate of $S^2$ with eigenvalue $sL(sL+1)$, but this need not necessarily be the case. The remaining term is essentially the integral of the spin-spin correlation function so this should be possible to check.

It is good that the author's calculate the energy variance in the DMRG calculations. This is a much more reliable estimate of the error than the truncation error. However it is unclear whether they use the variance scaling to extrapolate their observables to the limit of zero variance. Also, it is unclear why the authors didn't also calculate the energy variance for the VUMPS calculations, as this is a widely used procedure [1, 4, manuscript ref 47].

[1] https://en.wikipedia.org/wiki/Bratteli_diagram
[2] https://doi.org/10.48550/arXiv.1008.4667
[3] https://doi.org/10.1103/PhysRevB.100.235140
[4] https://doi.org/10.1103/PhysRevB.97.045125

1- the authors should remove/revise the grossly misleading characterization of the efficiency of the finite-size $SU(2)$ calculations.
2- the authors should consider the other comments, but I understand they will not significantly contribute to the overall conclusions of the paper and likely take time time to implement, so this is not required for publication.

1- state-of-the-art numerical simulations
2- complementary use of finite-size DMRG and infinite-size VUMPS formalism
3- efficient DMRG by exploiting the SU(2) spin symmetry
4- detailed analysis about obtained data
5- possible explanation about differences from previous studies
6- interesting results providing new physics

1- Only a figure for an optimal case may be shown in Figure 2.

The ferrimagnetic state of the S = 1/2 AFM-AFM sawtooth chain was studied using DMRG technique and VUMPS formalism. Finite-size and infinite-size calculations were performed in a complementary style, and eventually convincing results were provided. Particularly, it is nice idea to directly target the multiplet states by exploiting the SU(2) spin symmetry.

Their findings are very interesting and give a deeper insight in to the field of low-dimensional frustrated magnets: (i) The total spin takes irrational values 0.25<Stot/L<0.28 and reaches 1/4 only in the limit of J’=infinity. (ii) The ground state is characterized by an incommensurate quantum spin spiral in the apex-apex correlations, and on the other hand, the basal-basal correlation is still commensurate.

The manuscript is well written with containing clear comparisons with previous studies as well as careful analyses of obtained data. Therefore, I think that this manuscript meets the acceptance criteria for publication in SciPost Physics. Nevertheless, I just have some questions for the finishing touch.
- Figure 2 looks fine but the finite-size effects seem to accidentally small at J=-1 and J’=1. Is it possible to give a similar figure for the other parameter (even without L=252 and L=400 curves) in Appendix. It would be useful to understand the difficulty of studying this system.
- It is unclear if a specific Stot is targeted in exact diagonalization calculations including static spin structure factor. Maybe related to this, some values in Figure 3 and 4 look different, for example, Stot/L for L=36 and J’/|J|=1.3. Furthermore, in Figure 4 a sudden change of the Stot/L behaviour for smaller systems between J’/|J|=1.3 and J’/|J|=1.3 is a bit surprising. This seems to be inconsistent with FIG.3 of Ref.5. Do you have any suggestions about it?
- f(1/L)=0.25+1/L in Figure 4 should be a straight line because the x-axis is just 1/L. But it seems to be not. What is the reason?

1- Please check the consistency between Figure 3 and 4.
2- If possible, please add a figure like Figure 2 for another parameter in Appendix.
3- Please make it clear whether a specific Stot is targeted in exact diagonalization calculations.
4- Please check the validity of scaling function in Figure 4.

1- State-of-the-art numerical calculations
2- exhaustive introduction also accessible to non-experts and students, including a review of previous results on the analysed and related models
3- descriptive and well prepared figures
4- exhaustive description of the used numerical methods as well as reasoning on the choice of selected methods and their limitations
5- analyses of the numerical errors
6- clear description of the physical phenomena detected in the model
7- explanation as to why the presented results are different from previous ones on the same model

The authors study a sawtooth chain with FM J interactions between A and B spins and AFM J' interactions on the B chain. The model has already been studied and the existence of a ferromagnetic to ferrimagnetic transition is known. However, the ferrimagnetic phase was believe to be commensurate, while this work claims the ground state is characterised by incommensurate spin spiral correlations between the apical A spins. This results was achieved by exploiting the SU(2) symmetry in the numerical calculations, allowing for a very high value of the effective bond dimension. Moreover, the authors compare their finite-DMRG results with VUMPS calculations for infinite system while also discussing how the latter are limited by the available exploitable symmetries.
They go on to discuss the case with J=-1 and J'=1, which allows them to also underline why their results are different from previous ones thanks to the exploited SU(2) symmetry which allows them to target all possible $S_{tot}$ sectors and compare them to define the energy minimum as well as access fairly large rings. In turn, it also allows them to calculate the correct $S_{tot}/L$ in the ferrimagnetic phase. This shows a different scaling behavior to what previously established. This different behavior was determined thanks to the access to far larger systems, once more reminding the community that large system sizes are often important in determining the physics of strongly correlated systems.
Lastly, the authors discuss the static spin structure factor which allows them to claim incommensurate spin spiral correlations between A spins. Again, they are able to access the incommensurability thanks to large system size. Moreover, they underline how the spiral wavelength is very large for certain J'/|J| values, once more explaining why previous studies had not detected it.

Overall, I believe this work meets the criteria for acceptance in SciPost Physics, in particular it opens a new pathway in an existing research direction, with clear potential for multipronged follow-up work, for example in magnetic molecules such as Mn wheels.
My only comment is to include data and scripts for reproducing the figures presented in the manuscript, especially in the main text.

1- In the general criteria for acceptance in SciPost Physics, it is stated that the paper mus "provide (directly in appendices, or via links to external repositories) all reproducibility-enabling resources: explicit details of experimental protocols, datasets and processing methods, processed data and code snippets used to produce figures, etc.". I urge the authors to provide such repository for reproducing figures, it does not have to contain their DMRG code, but at least the data presented in the main text and, if possible, the script to transform the data to figures.