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Integrable Floquet spin chains are known to host strong zero and $\pi$ modeswhich are boundary operators that respectively commute and anticommute with theFloquet unitary generating stroboscopic time-evolution, in addition toanticommuting with a discrete symmetry of the Floquet unitary. Thus theexistence of strong modes imply a characteristic pairing structure of the fullspectrum. Weak interactions modify the strong modes to almost strong modes thatalmost commute or anticommute with the Floquet unitary. Manifestations ofstrong and almost strong modes are presented in two different Krylov subspaces.One is a Krylov subspace obtained from a Lanczos iteration that maps thetime-evolution generated by the Floquet Hamiltonian onto dynamics of a singleparticle on a fictitious chain with nearest neighbor hopping. The second is aKrylov subspace obtained from the Arnoldi iteration that maps thetime-evolution generated directly by the Floquet unitary onto dynamics of asingle particle on a fictitious chain with longer range hopping. While theformer Krylov subspace is sensitive to the branch of the logarithm of theFloquet unitary, the latter obtained from the Arnoldi scheme is not. Theeffective single particle models in the Krylov subspace are discussed, and thetopological properties of the Krylov chain that ensure stable $0$ and $\pi$modes at the boundaries are highlighted. The role of interactions is discussed.Expressions for the lifetime of the almost strong modes are derived in terms ofthe parameters of the Krylov subspace, and are compared with exactdiagonalization.

Strong and almost strong modes of Floquet spin chains in Krylov subspaces