The quantum dynamics of interacting many-body systems has become a uniquevenue for the realization of novel states of matter. Here we unveil a new classof nonequilibrium states that are eigenstates of an emergent local Hamiltonian.The latter is explicitly time dependent and, even though it does not commutewith the physical Hamiltonian, it behaves as a conserved quantity of thetime-evolving system. We discuss two examples in which the emergent eigenstatesolution can be applied for an extensive (in system size) time: transport inone-dimensional lattices with initial particle (or spin) imbalance, and suddenexpansion of quantum gases in optical lattices. We focus on noninteractingspinless fermions, hard-core bosons, and the Heisenberg model. We show thatcurrent-carrying states can be ground states of emergent local Hamiltonians,and that they can exhibit a quasimomentum distribution function that is peakedat nonzero (and tunable) quasimomentum. We also show that time-evolving statescan be highly-excited eigenstates of emergent local Hamiltonians, with anentanglement entropy that does not exhibit volume-law scaling.
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