The manuscript considers a spin-1/2 model with 3-spin interaction and a magnetic field on a diamond chain, that has a quantum phase transition. The combination of the interaction and geometry of the chain ensure an extensive number of local symmetries and respective immobile excitations. The authors work out a bijective mapping to transverse field Ising model, allowing (semi-)analytical solution for periodic and open boundary conditions. This allows to work out the details of the immobile excitations: their energies and behaviour upon approaching the transition.

The results are convincing and well presented: the manuscript can be published essentially as is, with only minor issues requiring fixes -- see below.

Questions

• I wonder whether the properties of the model are related to the geometry of the diamond chain and local symmetry of flipping individual rombii? This was explored by Peter Schmelcher, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.035161, in the context of compact localised states of flatbands (single particle physics).
• Consequently, I suspect there might be local unitary transformations that transform the Hamiltonian (1) into 2 decoupled Hamiltonians, one of these corresponding to the immobile modes.
• I wonder if the results of the manuscript, especially the immobile excitations and splitting the model into isolated segments by walls, are related to results in these publications:
• Lee et al: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.109.245137
• Kwan et al: https://arxiv.org/abs/2304.02669:
• Naik et al: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.110.L220303 I would be grateful for any comments.

Remarks

• Above Eq. 3: perhaps a typo "... we summaries its symmetries"?
• Sec. 4, p.11: "Further, one immobile mode energetically located between these two modes that could overlap at certain crystal moments (see Fig. 2)" Does this mode overlap has any physical consequences?
• Eqs. 65, 67: there seem to be an extra indentation after the equations

Publish (meets expectations and criteria for this Journal)

1. The novel exactly solvable quantum spin chain model is proposed.
2. The model is an example of the models with local conservation laws which also possess a global topological order.
3. The model has topologically protected localized modes.

1. p.2, Beginning of Sec.2, The authors referred to their model as a "quasi-one-dimensional", though it is exactly one-dimensional.

2. p. 3, Fig.1, The illustration of the structure of the chain is a bit strange. It looks like the site $C$ doesn't liked directly to the sites $A$ and $B$, and the three-links junction between them actually do not contain any spin. I understand the idea behind that, the authors want to emphasize the structure of the model after a duality transformation leading to just a chain with alternating bond $a$ and $b$. But, in my opinion the conventional picture of a diamond chain, a corner charing squares, is more relevant here. The emergent quantum Ising chain structure can be depicted as an additional dashed horizontal middle line with bonds $a$ and $b$.

3. The system has the symmetry with respect to permutation of $A$ and $B$ spins at each dimer part. So, the topological global symmetry operator given in Eq. (4) can be also constructed replacing $\sigma_{i, A}$ with $\sigma_{i, B}$. Doesn't it responsible for additional symmetry ?

4. p.6, The periodic case corresponding to symmetric sector after diagonalization (Bololiubov transformation) lead to only two bands, while the matrix $M$ is 8 by 8. This means that each eigenvalue is fourfold degenerate. If there any physical symmetry behind this degeneracy ?

5. Appendix B, The authors consider the system with different magnetic fields $h_1$ and $h_2$ on the single and on dimer sites. How does i affect the conserved quantities ?

6. Although, the authors declared the direct link between their model and fractons in higher dimensional models, they did not demonstrate it explicitly or by any formal evidence.

Publish (meets expectations and criteria for this Journal)

1- Although the manuscript is motivated by fracton physics, the immobile excitations do not exhibit some defining properties of fractons. It would improve clarity of the manuscript if the authors explicitly state in what precise sense the excitations are fracton-like and in what sense they are not genuine fractons.
2- The mapped model corresponds to a transverse-field Ising chain with regularly alternating couplings. There exists relevant earlier literature on such models as for instance O. Derzhko et al., Physical Review B 66, 144401 (2002). The authors should cite and briefly discuss this and related works on the bond alternating transverse Ising chains to better place their results in the context of existing literature.
3- The observed quantum phase transition is found to belong to 2D Ising universality class, while the studied diamond spin chain is one-dimensional and maps to a transverse-field Ising chain. A short clarification that this refers to the 2D classical Ising universality class via the quantum-to-classical mapping would avoid confusion.
4- To bring a deeper insight, it would be valuable to add to the manuscript typical field dependencies of the magnetization and triplet correlation function as conjugated quantities to the field term and three-spin coupling, which would provide a further confirmation of the nature of quantum phase transition.
5- The immobile excitations are defined in terms of symmetry sectors. It would strengthen the physical interpretation if the authors briefly discuss, which local operator(s) create such an excitation and how such excitations could be detected experimentally.
6- There seem to be a few typos in equations as for instance missing brackets after summation symbols in Eqs. (8)-(9), (21) at the effective field terms, in the second and third rows in Eq. (13), and in Eq. (26).

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