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Pivot Hamiltonians as generators of symmetry and entanglement
by Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath and Ruben Verresen
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Submission summary
As Contributors:  Nathanan Tantivasadakarn · Ruben Verresen 
Preprint link:  scipost_202204_00017v1 
Date submitted:  20220411 21:29 
Submitted by:  Tantivasadakarn, Nathanan 
Submitted to:  SciPost Physics Core 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
It is wellknown that symmetryprotected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finitedepth unitary operator $U$. Here, we consider obtaining the entangler from a local `pivot' Hamiltonian $H_{piv}$ such that $U = e^{i\pi H_{piv}}$. This perspective of Hamiltonians pivoting between the trivial and SPT phase opens up two new directions: (i) Since SPT Hamiltonians and entanglers are now on the same footing, can we iterate this process to create other interesting states? (ii) Since entanglers are known to arise as discrete symmetries at SPT transitions, under what conditions can this be enhanced to $U(1)$ pivot symmetry generated by $H_{piv}$? In this work we explore both of these questions. With regard to the first, we give examples of a rich web of dualities obtained by iteratively using an SPT model as a pivot to generate the next one. For the second question, we derive a simple criterion for when the direct interpolation between the trivial and SPT Hamiltonian has a $U(1)$ pivot symmetry. We illustrate this in a variety of examples, assuming various forms for $H_{piv}$, including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a $U(1)$ pivot symmetry is that it shares a mutual anomaly with the symmetry protecting the nearby SPT phase. We discuss how such anomalous and nononsite $U(1)$ symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixedpoint toric code state.
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Anonymous Report 2 on 2022630 (Invited Report)
Report
The manuscript introduces a general kind of duality that may lead to various symmetry protected topological phases. The duality is established by means of a pivot Hamiltonian that realizes a oneparameter family of unitary transformations, only at the "end points" of which the symmetry in question is preserved. Specifically, the pivot Hamiltonian itself is chosen to break that symmetry. This allows for the possibility that said end points correspond to different SPTs, and the authors often find this to be the case. Moreover, a linear interpolation between these end points must then feature a phase transition between different SPTs, and there is a good chance for this to occur at the midpoint of this linear interpolation. Related to that, there may be an additional symmetry emerging at this midpoint, generated by the pivot Hamiltonian (which, again, does not commute with the generators of the original symmetry, leading to possible "anomalies"). A useful general criterion for the emergence of this additional symmetry at the midpoint is derived. The convex hull of the three Hamiltonians given by some starting point H0, its dual under the pivot operation, and the pivot itself then span a twodimensional phase diagram, much of which is under control or at least informed by the methods derived in this paper. The pivotoperation can be repeated indefinitely, using the dual of H0 as the new pivot. This leads to a "web of dualities". There is much interest recently in dualities of this kind, and the resulting web is studied in detail in one dimension, using a trivial H0 and and Ising chain Hamiltonian as the pivot. The method is then explored in higher dimensions. First, in the context of a threebody Ising Hamiltonian on the triangular lattice as the pivot, and then by pivoting with toric code Hamiltonians.
In all, I view this contribution as a powerful construction to explore the phase diagrams of SPTs. The presentation is lucid and clear with some exceptions. To me, section 4 was essentially unintelligible, perhaps largely due to me lack of familiarity with Ref. [52], but if that's the case, this deviation from selfcontainedness came without warning. The idea of creating a 2D pivot from a 1D pivot still eludes me based on what is written. There is a H(1)_pivot, which is really "zero dimensional" if I understand correctly (or a sum is missing in its introduction), and its (j,k) dependence in the first line of (21) is suppressed. The latter defines H(2)_pivot, and the relation to H_pivot in subsequent subsections, e.g. Eq. (26), is not clear. Moreover, the procedure usually starts by defining an H0, see above, but this is left unclear in 4.1 and 4.2, until H0 is suddenly referenced in 4.3, without definition. I am hopeful that the presentation of this section can be improved. Similarly, in Section 5.4, is would be great if the authors could say clearly if the scenario they develop there involves conjecture, is based on details that they prefer to leave to future work, or should be selfevident from the present context (which, however, then largely escapes me). Other than that, the paper is well written and certainly represents a contribution worthy of publication in SciPost.
Note in passing: There were some typos: "as as" appears twice on p. 17, p. 18 speaks of "a suitable chose".
Anonymous Report 1 on 2022528 (Invited Report)
Report
The authors introduce a technique for generating models exhibiting symmetry protected topological (SPT) order by using a "pivot Hamiltonian" as the generator of a unitary transformation that performs a $\pi$ rotation of a model with trivial order to a model with SPT order. They showcase this technique to generate a range of models and study the corresponding phase diagrams. Interestingly, they provide conditions for models to be invariant under continuous rotations generated by these pivot Hamiltonians and argue that the resulting $U(1)$ symmetry has to be anomalous.
The paper is well written and offers an interesting perspective on SPT phases and the connections between models exhibiting them. The method is based on an original idea and has broad applicability with examples in various dimensions. I think this paper is a interesting contribution to the field and is sufficiently relevant to warrant publication in SciPost Physics Core, upon clarifying the issues below.
Requested changes
Specific comments:
1. It is stated above Eq. (3) that $H_{pivot}$ could have a smaller symmetry group than $G$, implying that it is possible that it also has the symmetry G. It is my understanding, it is crucial that it does not have the symmetry $G$ that protects the resulting SPT, as otherwise $H(\theta)$ would be a path of $G$symmetric Hamiltonians, along which the SPT order can not change.
2. Below equation 6, it is stated that $N$ is the "smallest integer" such that $e^{2\pi i H_{pivot}} = 1$. If this condition holds for some $N$ it will hold for $N=1$, so I believe this should say "largest integer" instead. (Additionally, in the paragraph right below, there is a broken reference to Appendix 2.2, which does not exist.)
3. (optional) Figure 2 and 6 might benefit from having axis like Figure 3, although this is not strictly necessary for their interpretation.
General comment:
It is mentioned that the SPT entanglers are finite depth unitaries. There has been previous work on dualities relating SPTs to trivial phases, such as the KennedyTasaki transformation for the Haldane phase and its generalizations. These dualities all have the feature that a local order operator is mapped to a nonlocal string order operator, which is not something that can be achieved with a finite depth unitary. I understand that by breaking the symmetry the SPT entanglers can map between different SPT phases, but a comment (or speculation) on the distinction between these two seemingly different kinds of duality mappings might be helpful. Additionally, a generic pivot Hamiltonian will not generate a constant depth unitary, so some discussion on the assumptions required for this to be the case would be useful.
Author: Nathanan Tantivasadakarn on 20221015 [id 2923]
(in reply to Report 1 on 20220528)
We thank the referee for their comments.

We agree that the symmetry of the pivot is strictly smaller than $G$ if the symmetry is unitary. However, there is is a subtlety when $G$ contains an antiunitary symmetry. For example, the Ising Hamiltonian Eq.1 as a pivot commutes with the full symmetry $G=\mathbb Z_2 \times \mathbb Z_2^T$ in Eq. 12. However, the rotated Hamiltonian Eq.3 only commutes with timereversal at $\theta=0,\pi$. We have added a footnote to clarify this point.

We thank the referee for pointing this out. This has been corrected. The link now correctly refers to Appendix B.

We have added a clarification to the captions of Figure 2 and 6 that they are plotted using barycentric coordinates. We thank the referee for this suggestion.
Reply to general comment:
It is our understanding that the KennedyTasaki transformation maps between a symmetry broken phase to an SPT phase, which explains why it cannot be written as a finite depth circuit. However, it should be possible to realize the KennedyTasaki transformation by augmenting the SPT entanglers with KramersWannier dualities. The pivot Hamiltonians in our paper are defined to to be local, and therefore it will always generate a constant depth unitary. We are unaware of nonlocal versions of our pivot Hamiltonians.
Author: Nathanan Tantivasadakarn on 20221015 [id 2924]
(in reply to Report 2 on 20220630)We thank the referee for their careful reading and for pointing out where we can improve the work. Firstly, section 4.1 has been improved, clarifying the distinction of the local pivot Hamiltonian vs the total pivot Hamiltonian. Secondly, the presentation of Sec. 5.4 is derived based on field theory arguments. Verifying this numerically in a lattice model is left to future work. In the updated version, we have added a description of a lattice model that is likely to realize this $O(2)/\mathbb Z_2$ transition, which future work can numerically explore.