Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model

1. Interesting combination of quantum group symmetry with recent ideas on generalised symmetry and topological phases
2. Clear and precise exposition, self-contained

1. The model under consideration is very specific, as is the case with most quantum group symmetric models beyond the spin 1/2 XXZ chain, making it unclear how generally applicable these ideas are
2. The results in this paper are of a rather technical nature, and several relevant questions on physical interpretation as well as how this work fits with other more general approaches are left to future work.

The authors consider a quantum group deformation of the well known Affleck-Kennedy-Lieb-Tasaki (AKLT) model, a prototypical example of a symmetry protected topological (SPT) phase with non-trivial edge modes. In the standard AKLT model, there exists a non-local transformation by Kennedy and Tasaki (KT) that maps the different SPT ground states of the AKLT model to a dual Hamiltonian with the same spectrum that exhibits conventional symmetry breaking and ferromagnetic order. In this paper, the authors argue that in the quantum AKLT (qAKLT) model this transformation needs to be generalized in order to retain the same physical implications. Technically, this is done by first mapping the qAKLT Hamiltonian to the ordinary AKLT Hamiltonian via a locality preserving mapping between frustration-free Hamiltonians known as Witten's conjugation. Then, they employ the standard KT transformation to map to a symmetry breaking Hamiltonian, after which the inverse Witten conjugation is applied to obtain a quantum deformation of the symmetry breaking Hamiltonian.

I think this is an interesting paper that stands out among much of the recent work on generalized symmetries. The application of Witten's conjugation adds a new ingredient to the study of duality transformations in spin systems, and looks like it might have a more widespread application. At the same time however, it is unclear what the more general implications of this work are due to the extremely specific nature of the model in question. In particular, unlike the quantum-group symmetric XXZ model, the spin 1 qAKLT model (eq. 14) appears to be highly finetuned and it is difficult to imagine it could serve as a model for a physical system; I would have appreciated some discussion on this aspect. If the authors are able to make a case that these results shed light on systems beyond the specific model under consideration, or that these techniques can be applied in a more general setting, I would recommend this article for publication. If not, I think this paper is better suited to a more specialized journal.

In addition, I have the following comments/questions:

- At the top of page 8, it is stated that "... the non-locality means that any attempt at a correspondence fails to fit into systematic approaches to constructing dualities ...". This paragraph is a bit misleading. In the general approaches cited in this paragraph, it is made very clear that dualities are defined with respect to a certain symmetry, and that only symmetric operators remain local under the duality transformation. The usual Kennedy-Tasaki transformation is defined with respect to an ordinary Z2 x Z2 symmetry, and as the authors point out, the q-deformed AKLT Hamiltonian breaks the Z2 x Z2 symmetry.

- As pointed out in the conclusion, the correct way to apply these general frameworks would be to properly identify the symmetry, which in this case is the quantum group symmetry, and dualize with respect to that symmetry. In fact, in reference [], the authors consider a duality based on the quantum group symmetry of the XXZ model, and show one can understand the well known IRF/Vertex correspondence in terms of their general categorical framework. It would be interesting to consider whether the KT transformation can be generalized by considering the SO(3)_q symmetry from this general framework.

- Can the operation q -> 1/q be realised as a linear operator on the Hilbert space?

- In (12), it would be a bit clearer to just write out |+> and |->, which I am assuming are defined as the eigenvectors of the Pauli X matrix, to avoid any ambiguity.

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The authors address the connection between string order and hidden symmetry breaking in the $q$AKLT model with $SO_q$(3) quantum group symmetry. Unlike the undeformed case, this model breaks all the symmetries that are known to protect the spin-1 Haldane phase, a prototypical example of a symmetry-protected topological phase. For instance, it breaks the $\mathbb{Z}_2 \times \mathbb{Z}_2$ subgroup of $\pi$-rotations about the principal axes. Nevertheless, the previous work (Ref. [30]) by one of the authors argued that the Hamiltonian of the model is invariant under these transformations if they are accompanied by a change of the parameter $q \to q^{-1}$, which is dubbed the duality group $\mathbb{Z}_2 \times \mathbb{Z}^{(q)}_2$. Despite this difference, the undeformed and $q$AKLT models share many common features; for example, the Kennedy-Tasaki (KT) transformation maps non-local string order present in both models to local order, as argued in Ref. [29]. However, the catch here is that the KT transformation sends the $q$AKLT Hamiltonian to another Hamiltonian that is no longer local. To fix this issue, the authors construct a modified duality transformation that is implemented by Witten's conjugation, a tool to deform frustration-free models systematically. Equipped with this, the authors demonstrate that the dual Hamiltonian obtained via the duality transformation is local. Since the transformation is no longer a similarity transformation, it is not immediately clear whether there is still a spectral gap above the ground state of the dual Hamiltonian. However, the authors provide strong evidence that this is the case.

The paper is overall well-written, with a broad introduction and pedagogical review of technical aspects, making the manuscript accessible to those who are unfamiliar with quantum groups. The results are certainly interesting and warrant further investigation. Thus, I recommend the paper for publication after the authors address the following questions/comments:

1. Possibility of a symmetry-protected trivial phase
It is briefly mentioned in the Conclusion that the exchange of $q$ and $q^{-1}$ corresponds to the action of inversion transformation on the $q$AKLT Hamiltonian. An immediate consequence is that this Hamiltonian is invariant under the combination of $\mathbb{Z}_2 \times \mathbb{Z}_2$ and inversion. This reminds me of several references discussing trivial phases protected by point group symmetries: Y. Fuji, F. Pollmann, and M. Oshikawa, Phys. Rev. Lett. \textbf{114}, 177204 (2015); A. Kshetrimayum, H-H. Tu, and R. Orus, Phys. Rev. B \textbf{93}, 245112 (2016). I wonder if it is possible to think of the ground state phase of the $q$AKLT as a symmetry-protected trivial phase.

2. Hamiltonians with $\mathbb{Z}_2 \times \mathbb{Z}^{(q)}_2$ symmetry
This question is somewhat related to the first: What is the most general form of a Hamiltonian that is invariant under $\mathbb{Z}_2 \times \mathbb{Z}^{(q)}_2$? I know one of the authors discussed the $q$-deformed bilinear-biquadratic Hamiltonian in Ref. [31], which includes the $q$AKLT model as a limiting case. But does that Hamiltonian represent the most general form?

3. The parameter range of $q$
In Appendix A, it is stated that a generic $q$ refers to $q>0$ and finite. However, in the main text, the parameter range of $q$ does not seem to be mentioned explicitly. This could be problematic, as the term $C_{j,j+1}$ in Eq. (35) and its counterpart in Eq. (46) may not be positive semi-definite for general $q \in \mathbb{C}$. For instance, it is positive-semidefinite but not positive definite at $q=\pm i$. I suggest the authors explicitly specify the range of $q$ in the main text.

4. Explicit form of ${\cal N}_{q{\rm KT}}$
Is it easy to derive the action of ${\cal N}_{q{\rm KT}}=M U_{\rm KT}M^{-1}$ on the standard basis states $|m_1, m_2, ..., m_L\rangle$? It seems that the answer is almost provided in Eq. (55). However, it would still be beneficial if the authors could elaborate on this further.

5. The existence of a spectral gap in the dual Hamiltonian $H_{q{\rm SSB}}$
The authors argued that Knabe's method is not applicable to $H_{q{\rm SSB}}$ due to its lack of translation invariance. Is this assertion really true? As discussed by Lemm and Mozgunov [J. Math. Phys. \textbf{60}, 051901 (2019)], and also in Appendix B of Ref. [32], Knabe's method has been generalized to cover systems with open boundaries. While they assume translation invariance in the bulk for the argument, this is not strictly necessary; the local interactions may depend on positions as long as the resulting Hamiltonian remains frustration-free. The situation is particularly simple when considering a subsystem of three sites to get a lower bound on a gap, in which case the subsystem Hamiltonian is defined as $h_{3,j} = H^{q{\rm SSB}}_{j,j+1}+H^{q{\rm SSB}}_{j+2,j+3}$. I am not sure whether one can diagonalize $h_{3,j}$ analytically. But if one can and can get a lower bound uniform in the system size $L$, one can establish the existence of a spectral gap in $H_{q{\rm SSB}}$, at least for certain values of $q$.

6. Minor comment
I suggest the authors define $A^\pm_j$ and $A^0_j$ precisely around Eq. (2).

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