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## "Radical chiral Floquet phases in a periodically driven Kitaev model and beyond", Hoi Chun Po et al., 2017

*Reviewed December 29, 2023*

*Citation:* Po, Hoi Chun, et al. "Radical chiral Floquet phases in a periodically driven Kitaev model and beyond." Physical Review B 96.24 (2017): 245116.

*Web:* https://arxiv.org/abs/1701.01440

*Tags:* Physical, Floquet-theory, Topological-insulators

This paper discusses a family of non-equilibrium fractional
topological phases. In these phases, a time-periodic Hamiltonian
produces excitations with fractional statistics, and produces chiral
quantum channels that propagate a quantized fractional number of qubits along the sample edge
during each driving period. They describe some of the properties as "sharply distinct" standard topological phases.

This paper is part of the more general Floquet theory of topological phases of matter.
Here, Floquet is a catch-all term for periodically driven systems. It is interesting to take a history of the word.
It originated in the theory of differential equations, when Gaston Floquet studied linear ODEs with
periodic coefficients:

> Floquet, Gaston. "Sur les équations différentielles linéaires à coefficients périodiques." Annales scientifiques de l'École normale supérieure. Vol. 12. 1883.

His main contribution was Floquet's theorem, which states that
the fundamental matrix solution of a periodic ODE can be decomposed into
a periodic part and an exponential part. This theory is obviously useful
for the study of physical systems in which the Hamiltonian is periodically driven:

> Oka, Takashi, and Sota Kitamura. "Floquet engineering of quantum materials." Annual Review of Condensed Matter Physics 10 (2019): 387-408.

Amazingly, the key features of Floquet's theorem allow for new sorts of topological order to emerge,
and have led to fantastic developments:

> Cayssol, Jérôme, et al. "Floquet topological insulators." physica status solidi (RRL)-Rapid Research Letters 7.1-2 (2013): 101-108.

> Ye, Bingtian, Francisco Machado, and Norman Y. Yao. "Floquet phases of matter via classical prethermalization." Physical Review Letters 127.14 (2021): 140603.

Now, periodically driven Hamiltonians are also being used at the core of some error correcting codes
such as the Hastings-Haah Floquet code:

> Hastings, Matthew B., and Jeongwan Haah. "Dynamically generated logical qubits." Quantum 5 (2021): 564.