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Integrable Floquet QFT: Elasticity and factorization under periodic driving
by Axel Cortes Cubero
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Axel Cortes Cubero |
| Submission information | |
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| Preprint Link: | http://arxiv.org/abs/1804.07728v1 (pdf) |
| Date submitted: | April 26, 2018, 2 a.m. |
| Submitted by: | Axel Cortes Cubero |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
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.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2018-6-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1804.07728v1, delivered 2018-06-15, doi: 10.21468/SciPost.Report.502
Strengths
1- Axiomatic definition of integrability for Floquet systems 2-Manuscript well-written and easy to follow
Weaknesses
1- The examples are mostly non-interacting and not especially exciting from a physics perspective.
Report
Requested changes
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.
Report #1 by Anonymous (Referee 1) on 2018-6-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1804.07728v1, delivered 2018-06-15, doi: 10.21468/SciPost.Report.501
Strengths
2-Nice presentation, including clear introduction to the standard S-matrix bootstrap program and detailed calculations.
Weaknesses
2-Examples are mostly trivial, approximate and not related to many-body interactions.
Report
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.
Requested changes
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:
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I think this paper doesn't really fit in the subject area "Condensed Matter Physics - Theory" but rather in "high energy".
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eq. (12): sum over i must be a typo
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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.
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Shouldn’t F(\theta) be unitary?
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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).
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“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).
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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 $m_1^2-m_2^2$.
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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)
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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?)