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We investigate emergent quantum dynamics of the tilted Ising chain in theregime of a weak transverse field. Within the leading order perturbationtheory, the Hilbert space is fragmented into exponentially many decoupledsectors. We find that the sector made of isolated magnons is integrable withdynamics being governed by a constrained version of the XXZ spin Hamiltonian.As a consequence, when initiated in this sector, the Ising chain exhibitsballistic transport on unexpectedly long times scales. We quantitativelydescribe its rich phenomenology employing exact integrable techniques such asGeneralized Hydrodynamics. Finally, we initiate studies ofintegrability-breaking magnon clusters whose leading-order transport isactivated by scattering with surrounding isolated magnons.

Fragmentation and emergent integrable transport in the weakly tilted Ising chain