Translation Groups for arbitrary Gauge Fields in Synthetic Crystals with real hopping amplitudes

This paper studies tight-binding models defined on graphs with vertices corresponding to elements of a discrete group $G$ (Cayley graphs), and uniform hopping given by left multiplication in $G$. The main result of the paper is that such "Cayley crystals" realize generalized magnetic translation groups for a broad class of discrete gauge groups, with the commutator subgroup $[G,G]$ playing the role of the gauge group. $[G,G]$ is assumed to be finite.

To show this, the Cayley graph is decomposed into "pillars" labeled by elements of the abelianized group $G/[G,G]!\cong!\mathbb Z^n$ (ignoring torsion), with sites within pillars (later forming internal degrees of freedom) related by elements of the commutator subgroup $C!=![G,G]$. By decomposing the Hilbert space into unitary irreps of $C$, it is shown that a tight-binding Hamiltonian on the Cayley crystal decomposes into representation sectors in which inter-pillar hopping is dressed by matrices taking values in a particular irrep of $C$, and that these behave analogously to background gauge fields.

I believe the paper is interesting and merits publication. However, in its current form, the presentation is overly technical and assumes a level of familiarity with group theory that is not yet standard within the condensed matter community. At times, explicit examples and physical explanations are eschewed in favor of formal constructions or definitions. It would greatly improve readability if the formal discussions in core sections 3-8 were accompanied by at least one or two explicit examples, instead of relegating the latter to the end of the paper in section 9. Also, it would be very helpful if a short opening paragraph was added to most (sub)sections outlining the key goals/results of that (sub)section.

One question I have is in what sense this framework in terms of Cayley-Schreier lattices goes beyond standard magnetic translation groups for abelian $[G,G]$, since all the examples considered in this paper involve abelian commutator subgroups.

Some specific comments and suggestions:

1. page 3, second paragraph, "...generalizations of MTGs have mostly focused on fundamental physics..." -- What is meant by "fundamental physics" here?

2. page 3, fourth paragraph -- better phrasing would be "hyperbolic lattices, characterized by infinite-order translation groups that are non-amenable and Fuchsian". It is also not clear what "asymptotics of finite quotients" means there.

3. section 3.1 -- the first two paragraphs of this section really need to be expanded with more explanations, and will be bewildering to most condensed matter physicists who are not immersed in the theory of group extensions and cohomology of groups. The author should at least provide enough detail so that a reader can pick up the essential facts by consulting references. For instance, the author should elaborate that the essential problem is that of a group extension

   $$1 \to [G,G] \to G \to G/[G,G] \to 1,$$
   where $[G,G]$ is the commutator subgroup of $G$ defining the Cayley crystal. Then $G/[G,G] = \mathbb{Z}^n !\times! X_\tilde{n}$ in general and the author is considering extension problems in which there is no torsion so that $X_\tilde{n}$ is trivial. The Schreier transversal then seems to be a choice of section in this extension.

4. page 7 -- eq. (4) is bit opaque. Why define the link this way? Is it the correction i.e. the gauge transformation in $C$ picked up when we compose translations? If so, doesn't it follow from the theory of group extensions? A simple example would go a long way to help understand why these definitions of links and cables make sense. That the link variable defined coincides with a 2-cocycle in a group extension deserves at least a few sentences of explanation, rather than just an assertion.

5. page 8, eq. (6) -- It seems there are a few unstated facts used here? The left regular rep of $C$ decomposes into finite-dimensional unitary irreps: $T_C^L = \oplus_\xi \sigma_\xi^{\oplus d_\xi}$. Next, a rep of $G$ is induced on both sides and the fact that $\mathrm{Ind}$ is linear on direct sums is used? Since $T_\xi = \mathrm{Ind}^G_C(\sigma_\xi)$ and $T^L = \mathrm{Ind}^G_C(T_C^L)$, combining everything, we get eq. (6)? Also a minor comment around eq. (7): the Peter-Weyl theorem specifically applies to compact topological groups. the analogous results for finite groups mostly just follow from Schur orthogonality, but the intended meaning of the author is clear.

6. page 9, paragraph below eq. (9) -- "integrals of $C$" have not been defined before this point. Only after reading further is it clear that this means a group whose commutator subgroup is $C$. In the same paragraph, it isn't clear what "dynamics subspaces" means. I presume this means the hoppings become complex-valued only after $T^C_L$ is decomposed into unitary irreps of $C$, like in eq.(8)?

Ask for minor revision

In the paper entitled "Translation Groups for arbitrary Gauge Fields in Synthetic Crystals with real hopping amplitudes", the author has introduced and studied the Cayley-Schreier (CS) lattices, in particular, by showing that these systems naturally realize the generalization of the magnetic translation groups to arbitrary discrete gauge groups G. By focusing on the Abelian cases, the author has shown the Bloch decomposition and corresponding Bloch Hamiltonian eigenstates (related to the generalized magnetic translations) for some concrete lattice systems.

I find the paper very interesting with several original results although the style of the paper is very technical and not of easy access for condensed matter physicists who are not very familiar with advanced mathematical formalism of the paper such as advanced group theory, Peter-Weil theorem and Cayley graphs.

Honestly, I would have expected a deeper and more detailed discussion about generalized Bloch theorem for the CS lattices, somehow a more explicit presentation, similar to what has been done in some works on hyperbolic lattices, see for instance Refs [21,22] cited by the author. I believe it would be beneficial for the readers if this point is presented and discussed in a more physics-friendly way instead of the very formal presentation by the author concerning Bloch decomposition.

More importantly, I have two questions for the author:

1) For the examples discussed in section 9, the author explicitly discusses the spectra which result to be gapless as shown in Fig.6. On the other hand, it is well known that magnetic translations are key ingredient that ensure the topological stability and gapped nature of the Quantum Hall states for instance. The situation changes when we consider different types of crystalline solids, especially those with more complex symmetries. In these materials, magnetic translations (or generalizations of them) can conspire with other crystal symmetries to protect gapless points in the electronic band structure. How is the situation in the case of CS lattices? Is there any way to understand if a given model, related to a group G, supports or not a well-defined gap before doing brute-force calculations? What is the role of discrete symmetries in this context?

2) How the topology of the Cayley graphs influence the physical features of the CS lattice models such as their generalized magnetic translations? It is well known that Cayley graphs can possess non-trivial topology, see for instance the following work:

Mark Brittenham, Susan Hermiller, Derek Holt, Algorithms and topology of Cayley graphs for groups, Journal of Algebra 415, 112 (2014).

Thus, it would be interesting and relevant if the author could comment the role of graph topology and its possible physical implications for the CS lattices (although it may be highly non-trivial to get a complete picture at this stage).

Overall, I support the publication of this work on SciPost once the author addresses the points I raised above.

Publish (meets expectations and criteria for this Journal)