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Integrable deformations of superintegrable quantum circuits

by Tamás Gombor, Balázs Pozsgay

Submission summary

Authors (as registered SciPost users): Tamás Gombor · Balázs Pozsgay
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-10-04 10:17
Submitted by: Pozsgay, Balázs
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


Superintegrable models are very special dynamical systems: they possess more conservation laws than what is necessary for complete integrability. This severely constrains their dynamical processes, and it often leads to their exact solvability, even in non-equilibrium situations. In this paper we consider special Hamiltonian deformations of superintegrable quantum circuits. The deformations break superintegrability, but they preserve integrability. We focus on a selection of concrete models and show that for each model there is an (at least) one parameter family of integrable deformations. Our most interesting example is the so-called Rule54 model. We show that the model is compatible with a one parameter family of Yang-Baxter integrable spin chains with six-site interaction. Therefore, the Rule54 model does not have a unique integrability structure, instead it lies at the intersection of a family of quantum integrable models.

Current status:
In refereeing

Submission & Refereeing History

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Submission 2205.02038v4 on 4 October 2023

Reports on this Submission

Anonymous Report 1 on 2024-1-22 (Invited Report)


1- Solves an outstanding problem of the integrability structure behind "Rule 54".
2- Clearly written and accessible.


The manuscript discusses integrable deformations of "superintegrable" quantum circuits: They superimpose an integrable Hamiltonian to a superintegrable circuit so that the resulting model no-longer exhibits an exponential number of conserved quantities. They show that the integrable Hamiltonian (and all its higher conserved charges) commute with the original model, which implies that the new model is also integrable in the usual sense. They consider three examples, a circuit made of swap gates, a circuit of swap gates with phases, and the so-called "Rule 54" circuit. The latter example is the most interesting (and nontrivial) of the three.

The result on the integrability of Rule 54 represents a fundamental breakthrough: the model has been observed to be "solvable" in many different contexts, but until now the algebraic reasons for its integrability had not been understood. This paper answers the question by exhibiting a family of integrable models, whose transfer matrices commute with the unitary operator, and thus positions the model at an intersection of a (at least one) family of Yang-Baxter integrable models. Moreover, the contents of the paper are clearly and sensibly structured, which makes the manuscript easy to follow and accessible. Therefore I recommend it for publication in SciPost Physics.

Requested changes

I have a few comments/questions for the authors.

1- Paragraph before (4.13): "therefore its action makes the two sublattices highly entangled, both in equilibrium, and non-equilibrium situations."

I find this sentence a bit confusing - what do "equilibrium and non-equilibrium situations" refer to? The way I understand it, if one starts with a state that is a product state with respect to the two sublattices, and applies the operator $\mathcal{D}$ on it, the resulting state is entangled between the two sublattices. But this is a general statement for states, I don't see how this relates to either equilibrium or non-equilibrium.

2- Section 4:
How do we know that all the higher charges also commute with $\mathcal{V}$? From the discussion it is clear that $H(\Delta)$ commutes with $\mathcal{V}$, and that it admits a family of commuting charges. However, it is not obvious to me that all these additional charges commute with $\mathcal{V}$.

3- Last few paragraphs of Sec. 5.2:
Why are there so many exclamation marks? I can understand the excitement of the authors, but in my personal opinion some of the exclamation marks could be replaced by fullstops.

4- A common point of all the three examples is the fact that "evolution in space" can be formulated as a valid local dynamical map. Do the authors expect that this plays any role? A related question is, whether the deterministic map of Ref. [17] could be treated in an analogous way. Could the treatment be extended to that case? I would naively expect its integrable structure to be somehow related. I understand that this might not be necessarily a straightforward question to answer.

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

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