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As Contributors: | Axel Cortes Cubero |

Arxiv Link: | https://arxiv.org/abs/1804.07728v2 |

Date submitted: | 2018-08-01 |

Submitted by: | Cortes Cubero, Axel |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | High-Energy Physics - Theory |

In (1+1)-dimensional quantum field theory, integrability is typically defined as the existence of an infinite number of local charges of different Lorentz spin, which commute with the Hamiltonian. A well known consequence of integrability is that scattering of particles is elastic and factorizable. These properties are the basis for the bootstrap program, which leads to the exact computation of S-matrices and form factors. We consider periodically-driven field theories, whose stroboscopic time-evolution is described by a Floquet Hamiltonian. It was recently proposed by Gritsev and Polkovnikov that it is possible for some form of integrability to be preserved even in driven systems. If a driving protocol exists such that the Floquet Hamiltonian is integrable (such that there is an infinite number of local and independent charges, a subset of which are parity-even, that commute with it), we show that there are strong conditions on the stroboscopic time evolution of particle trajectories, analogous to S-matrix elasticity and factorization. We propose a new set of axioms for the time evolution of particles which outline a new bootstrap program, which can be used to identify and classify integrable Floquet protocols. We present some simple examples of driving protocols where Floquet integrability is manifest; in particular, we also show that under certain conditions, some integrable protocols proposed by Gritsev and Polkovnikov are solutions of our new bootstrap equations.

Editor-in-charge assigned

I first want to thank the referees for their time and useful comments. I have incorporated their suggestions into this new version of the manuscript, which I hope they will find more suitable. I list below the changes made to the manuscript, in reply to the requests from the referees.

Referee report 1:

I first remark, that it is true that at present, the examples of Floquet integrability discussed are very simple, and it would be much more interesting in the future to see if some more intricate and truly interacting integrable protocols can be found. Nevertheless, there does not seem to be any particular “no-go theorem” prohibiting the discovery of more interesting examples, and I list throughout the manuscript several promising future directions to explore. I therefore don’t really agree with the suggested conclusion that integrability is “too constraining”, since I do think it will soon be possible to find other interesting examples.

-I clarified the discussion on the possibility two particles scattering within the “shaded region” governed by $H’(t)$. In general, this would be described by some combined phase, $F(\theta_1,\theta_2)$. I argue, that as a consequence of stroboscopic factorizability, it must be possible to express this function in terms of the one-particle phase, $F(\theta)$, and the standard two-particle S-matrix. I extended the discussion of the Floquet Yang-Baxter equation, to include this. This is similar to the case of three-particle scattering in standard equilibrium integrable QFT, where the general three-particle S-matrix can be reduced in terms of products of 2-particle S-matrices.

-The assumption that the set of rapidities are conserved after each period, is just the simplest solution to the requirement that rapidities are conserved after $n$-periods. The beginning of Section 4 has been slightly rewritten to make this more clear. This is similar to the standard equilibrium case, where one can only directly derive the fact that the set of rapidities is conserved between the asymptotic in and out states, yet the simplest assumption is that the set of particle rapidities is conserved at intermediate times as well.

-We are not requiring locality of the Floquet Hamiltonian, however, we require that it commutes with some local conserved charges. The discussion regarding “well separated particles” has been modified and made clearer in the text. The only requirements are that the basis of asymptotic particle states is complete and spans the Hilbert space, such that the state after $n$ periods can be expressed in terms of particle states, and then we require that the conserved charges be local. From these requirements, Eq. (13) follows, and it is not necessary that the Floquet Hamiltonian be local.

-I have now changed the main subject to High Energy Theory, and kept Condensed Matter as a secondary subject. I think the subject of the manuscript lies somewhere in between, and it is hard to classify, and perhaps may be of interest to both communities. I would leave the final decision up to the editor in charge in any case.

-Indeed the sum over $i$ in (12) was a mistake and has been corrected now.

-I have now included a reference to P. Dorey at the beginning of Section, I agree it is a relevant review.

-The condition of unitarity on $F(\theta)$ is mainly what I call the Floquet Annihilation axiom. This is the statement that $F(\theta)$ can be cancelled against the opposite phase, arising from an antiparticle, which is the manifestation of unitarity in the $F(\theta)$ function.

-p.22: “Suppose for example, that we choose the initial state to be a one-particle eigenstate…” I agree it is necessary to make the argument with a multi particle state, that the Bogoliubov coefficient d should vanish, and this was a mistake in the previous version. It has now been corrected with a multi particle state.

-I clarified the issue of revivals in generic systems, it is made clear that in general one can only expect approximate revivals, and I have made sure to point this out where relevant in the manuscript.

-p.27: “Bogoliubov transformation describing each step of the driving, as d(2),p = 0, for all momenta, p, and integer n.” I have corrected this typo, now it reads $d_{(2n),p}=0$

-Equations (47) and (48) (now 48 and 49 in this version) have been simplified, inserting the dispersion relations.

-The protocol described in 6.2.3 is not exactly the same as the Onsager protocol described in [13]. The Onsager protocol consists of alternating between the Hamiltonians $A_0$ and $A_1$ from Eq. (53). The protocol in 6.2.3 consists of a two-step driving process where one of the Hamiltonians is either $A_0$ or $A_1$, but the second Hamiltonian can be a full transverse field Ising chain, involving both terms, So it is in this sense more general than the Onsager protocol. The Protocol of 6.2.3 is initially expected to be integrable only when tuned to the appropriate revival time. The Onsager protocol of [13], being simpler, was proposed to be integrable for any driving period, though we find that it doesn’t fulfill all our requirements of integrability except at the fine-tuned revival times.

-The notation (3^n) and Z[0], was taken from the original reference [27], but I have now made more transparent in the text what this means. The references to parity were also modified and made clearer in this section. Conserved charges built out of only the U operators are invariant under spatial inversion, $x\to-x$.

Referee report 2:

-I have expanded the discussion in the Conclusions section about how other less trivial integrable Floquet systems may be found. We point out that recently a protocol very similar to the “quantum dilatation clock” proposed in the conclusions was studied, and the results seem to be compatible with our definition of integrability, but this has to be studied further. I also extended the discussion on the study of Floquet critical phenomena, and outlined clearly the steps that would need to be followed to show any connection between critical phenomena and integrability in 2d driven systems.

-I have added a paragraph at the end of 6.3.1, discussing the XXX protocol proposed in [13]. While we are not able at present to study the properties of this protocol as we did with those related to free systems, there is reason to be hopeful that it may agree with our definition of integrability. As we discuss, the first few terms in the BCH expansion of the Floquet Hamiltonian do seem to commute with local, and parity-even conserved charges, so the protocol seems to be integrable at least at this perturbative level. This is not true for the generic Onsager protocol also discussed in this section, which does not commute with parity-even charges.

In general, as I replied to the first referee, there does not seem to be any “no-go theorem” preventing more interesting interacting integrable protocols from being discovered in the future, and there are possible open pathways to discover new protocols. So the fact that at this point we can only discuss very simple protocols in detail should not mean that only these protocols are possible.

Resubmission 1804.07728v2 (1 August 2018)

Submission 1804.07728v1 (26 April 2018)