  We study the quantum dynamics of a simple translation invariant,
center-of-mass (CoM) preserving model of interacting fermions in one dimension
(1D), which arises in multiple experimentally realizable contexts. We show that
this model naturally displays the phenomenology associated with fractonic
systems, wherein single charges can only move by emitting dipoles. This allows
us to demonstrate the rich Krylov fractured structure of this model, whose
Hilbert space shatters into exponentially many dynamically disconnected
subspaces. Focusing on exponentially large Krylov subspaces, we show that these
can be either be integrable or non-integrable, thereby establishing the notion
of Krylov-restricted thermalization. We analytically find a tower of integrable
Krylov subspaces of this Hamiltonian, all of which map onto spin-1/2 XX models
of various system sizes. We also discuss the physics of the non-integrable
subspaces, where we show evidence for weak Eigenstate Thermalization Hypothesis
(ETH) restricted to each non-integrable Krylov subspace. Further, we show that
constraints in some of the thermal Krylov subspaces cause the long-time
expectation values of local operators to deviate from behavior typically
expected from translation-invariant systems. Finally, we show using a
Schrieffer-Wolff transformation that such models naturally appear as effective
Hamiltonians in the large electric field limit of the interacting Wannier-Stark
problem, and comment on connections of our work with the phenomenon of Bloch
many-body localization.
