Nonsymmorphic spin-space cubic groups and SU(2)$_1$ conformal invariance in one-dimensional spin-1/2 models

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Emergent symmetries in many body systems are a timely and central theme of modern research. It was already known that a SU(2) symmetry emerges in the so called Kitaev-Gamma spin chain. Here the authors show that SU(2) symmetry emerges as an extended gapless phase in a much more general way in spin 1/2 chain with cubic symmetry in the Hamiltonian. It is shown that all the five nonsymmorphic cubic groups can stabilize SU(2) phases, whereas nonsymmorphic planar groups cannot. The authors also built minimal models for the corresponding nonsymmorphic cubic groups, and provided numerical evidence of emergent SU(2) conformal invariance.This is definitively an important addition to what was know about emergent symmetry in lattice spin models and deserves publication.

1-This manuscript can be considered as a key reference for anyone interested to learn about the emergence of SU(2) symmetry in discrete Kitaev-like spin chains.

2-The results of the paper are very convincing given a very careful analysis, and given that it combines complementary approaches: group theory, field theory, and numerical density matrix renormalization group.

3-It is very exhaustive, studying symmetry breaking terms, resulting symmetry groups, and resulting effective theories.

4-It is very pedagogical and at the same time delivers new results.

1-The key ideas of this manuscript have been worked out in multiple previous papers by the authors (Refs.24,34 and many more). But this does not take away from the key role of this paper which is (i) to explore in details general symmetry groups and (ii) provide a full detail exposition.

2-An experimental relevance of the model is not particularly clear.

Previous works have demonstrated an emergent SU(2) symmetry in the Kitaev-Gamma spin chain. The present work presents a breakthrough on these previously-identified research directions, by extending the prediction of emergent SU(2) symmetry to other discrete cubic point groups. In addition to these new results, the paper is particularly well written, containing many useful details, citing relevant references, and hence can also serve as a indispensable reference for a new-comer to the field. I strongly recommend it publication. Few minor comments remarks appear below.

---In the caption of Fig. 1 please add: "This figure is taken from Ref. [34].".
---Below Eq.4 please define $T_a$ (translation by one lattice site) - in addition to the definition of $T_{na}$.
---Below Eq. 23 (or elsewhere) please address the question: is there any numerical evidence for different velocities for the left and right moving sectors?
---Eq.27 typo - second equation should be $\lambda^{-1} \tau - i x$.
---Above Eq.29 "and similar for..."-->"and similarly for...".
---Eq.36 the $\ell$ is probably a typo.
---In the paragraph below Eq.38 "an extend region" -->"an extended region"
---Eqs.41 and 42 and , one of the L subscript should be R.
---Below Eq.103 typo: "are shown are shown"