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Pivot Hamiltonians as generators of symmetry and entanglement

by Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath and Ruben Verresen

Submission summary

As Contributors: Nathanan Tantivasadakarn · Ruben Verresen
Preprint link: scipost_202204_00017v1
Date submitted: 2022-04-11 21:29
Submitted by: Tantivasadakarn, Nathanan
Submitted to: SciPost Physics Core
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical

Abstract

It is well-known that symmetry-protected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finite-depth 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 non-onsite $U(1)$ symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixed-point toric code state.

Current status:
Editor-in-charge assigned


Submission & Refereeing History


Reports on this Submission

Anonymous Report 2 on 2022-6-30 (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 one-parameter 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 two-dimensional phase diagram, much of which is under control or at least informed by the methods derived in this paper. The pivot-operation 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 three-body 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 self-containedness 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 sub-sections, 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 self-evident 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".

  • validity: high
  • significance: top
  • originality: top
  • clarity: good
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2022-5-28 (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 Kennedy-Tasaki transformation for the Haldane phase and its generalizations. These dualities all have the feature that a local order operator is mapped to a non-local 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.

  • validity: high
  • significance: high
  • originality: top
  • clarity: good
  • formatting: good
  • grammar: excellent

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