Integrable Floquet QFT: Elasticity and factorization under periodic driving

1- Axiomatic definition of integrability for Floquet systems
2-Manuscript well-written and easy to follow

1- The examples are mostly non-interacting and not especially exciting from a physics perspective.

In this paper, the author introduces a definition of integrability for periodically driven (Floquet) systems. Whereas the idea of Floquet integrability was proposed before, this is the first concrete and complete definition of what integrability could mean for such systems. The author follows an axiomatic approach that could be useful to identify and analyze integrable Floquet systems. Examples and counter examples of integrability and quasi-integrability are given, mostly for non-interacting theories. Overall I think this paper is interesting and it is certainly satisfying to have a concrete framework for Floquet integrability. My main concern with this work is that it remains unclear whether there exists any interacting, non-trivial example of Floquet integrable system. This is an important issue that should be discussed in the paper. The only interacting example uses the boost operator and already appeared in a previous work by Gritsev and Polkovnikov (and clearly such protocols are quite artificial). On the other hand, it remains very unclear to me that a non-trivial example like eq 53 is integrable in any meaningful sense. Even if it is hard to show integrability in the way introduced by the author, I think some arguments to explain why this model could be integrable are warranted. Could the author exhibit at least one or two local conserved quantities for this model? A thorough discussion of interacting examples would make this framework much more exciting.

1- Provide a concrete path to find interacting examples not based on the boost operator
2- Extend the discussion of interacting models, especially the XXX chain since this is the most interesting example.

1-Axiomatic construction of integrable quantum field theory under periodic driving based on general principles in the style of the S-matrix bootstrap program.

2-Nice presentation, including clear introduction to the standard S-matrix bootstrap program and detailed calculations.

1-Definition of Floquet integrability based on infinite set of local conserved charges may be too restricting for practical uses.

2-Examples are mostly trivial, approximate and not related to many-body interactions.

In this work the author discusses aspects of integrability in periodically-driven quantum field theories. Following the reasoning of standard integrable QFT, the author proposes a definition of Floquet integrability associated with existence of an infinite set of local charges that commute with the Floquet Hamiltonian (the effective Hamiltonian describing the time-evolution of the system over each driving period). In analogy to the standard scattering theory arguments on integrability and the S-matrix bootstrap program, the author then argues that Floquet integrability guarantees elasticity and factorisation of particle collisions under stroboscopic observations of the system which imply certain constraints on the stroboscopic time evolution. The rest of this work focuses on examples of driving protocols: it is first shown that the simple choice of periodic mass quenches in free bosonic or fermionic field theories does not satisfy the conditions of Floquet integrability and later on some trivial cases of approximate Floquet integrability are discussed.

Overall this work introduces an extension of concepts and ideas of integrable QFT to the case of periodic driving that are useful for an axiomatic construction of stroboscopically integrable QFT. Motivation for the study of periodic driving dynamics comes from quantum or condensed matter physics and experimental applications for which quantum field theories are useful as effective descriptions at least at equilibrium or close to it. However the relativistic invariance of QFT, which is crucial in the S-matrix bootstrap program, is only approximate in condensed matter or experimental applications and so the dynamics under periodic driving would diverge from the QFT predictions rather quickly. This fact restricts somehow the potential use of these results in this direction. Moreover the definition of Floquet integrability introduced here, even though directly analogous to the standard one, may be too strong to describe any example of practical interest. For example even the simple protocol of free field theory periodic driving that is exactly solvable due to the closeness of the algebra of quadratic operators and which does not lead to infinite heating up, is excluded from this definition of Floquet integrability. Almost all examples discussed later are rather trivial cases of approximate Floquet integrability that would work even for a small quantum system or are based on very exceptional fine-tuning that reduces the dynamics again to a quantum mechanical rather than statistical physics problem (exact revivals or fully degenerate energy spectrum). Perhaps the only non-trivial case is the boost-operator protocol of sec. 6.3.2.

Therefore one conclusion of this work could have been that defining Floquet integrability in QFT by requiring existence of an infinite set of local conserved charges, in direct analogy to the standard case, is too constraining, which is certainly an interesting observation.

In section 4, the author introduces a function F that describes the time evolution of a single-particle state over one driving period and determines the analytical properties that it must satisfy. This is essentially the equivalent of the travelling phase factor. Question: shouldn’t one allow also for an independent S-matrix describing the elastic collisions during one driving period? In sec.4 collisions seem to be allowed only during the part of the driving steps when time-evolution follows $H_{int}$ and so the corresponding S-matrix is the standard one. How would the axioms be modified if one took into account this possibility?

p.12 explain better how the requirement of rapidity set conservation after n periods implies the same after each one period separately

p.12 “If we consider a large enough number of periods, n, the particles in the final state (12), can be considered to be well separated from each other. This assumption that the final state can be represented in terms of well separated asymptotic particle states is sensible in translationally invariant systems”
I think translational invariance is not sufficient to guarantee well-separation of outgoing particles. Locality of the Floquet Hamiltonian seems to be necessary. It should anyway be clarified if the Floquet Hamiltonian is assumed to be local or if the infinite set of conserved charges is assumed to be local without the Floquet Hamiltonian being such.


Minor changes:

- I think this paper doesn't really fit in the subject area "Condensed Matter Physics - Theory" but rather in "high energy".

- eq. (12): sum over i must be a typo

- it would be good to cite P. Dorey’s “exact S-matrices” or refs. therein as a reference on the standard scattering theory arguments relating presence of local conserved charges to factorisability, elasticity etc. that have been used in this paper.

- Shouldn’t F(\theta) be unitary?

- p.22: “Suppose for example, that we choose the initial state to be a one-particle eigenstate…” I think the requirement that the Bogolyubov coefficient d should vanish needs at least a two-particle state to demonstrate (because d is the coefficient of the annihilation operator which annihilates the vacuum).

- “For a general system a full revival is expected when the system is placed in a finite volume, however, the time when a revival is expected grows exponentially with system size…” It’s not accurate to say that exact revivals are possible in general systems but correspond to exponentially large revival times. For a general system exact revivals are just impossible (exact revivals require either all energy levels being multiples of a fundamental frequency as in CFT or fine-tuning). In general systems the dynamics is quasi-periodic meaning that revivals are only approximate (in the sense of the quantum recurrence theorem mentioned in 6.2.1).

- p.27: “Bogoliubov transformation describing each step of the driving, as d(2),p = 0, for all momenta, p, and integer n.”: erase “integer n” since there isn’t here.

-(47) and (48) can be simplified further: e.g. $E_{1,p}^2-E_{2,p}^2$ is just $m_1^2-m_2^2$.

- p.31: isn’t this protocol the same as the Onsager algebra protocol of [13] discussed later? so that the Floquet Hamiltonian can be found from the BCH expansion as linear combination of the generators (of course this doesn’t mean it has a trivial form)

- p.33: notation (3^n) and Z[0] not so clear in the unnumbered eq. defining U,V. The meaning of parity is also unclear (in terms of the fermionic fields?)