The one-dimensional Holstein model and Holstein-Hubbard model have no nontrivial local conserved quantities other than the Hamiltonian and total fermion number.
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Quantum quenches in the Ising chain exhibit qualitatively distinct out-of-equilibrium dynamics when crossing continuous versus first-order quantum transitions depending on the transverse field strength.
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Proof of the absence of local conserved quantities in the Holstein model
The one-dimensional Holstein model and Holstein-Hubbard model have no nontrivial local conserved quantities other than the Hamiltonian and total fermion number.
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Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models
Quantum quenches in the Ising chain exhibit qualitatively distinct out-of-equilibrium dynamics when crossing continuous versus first-order quantum transitions depending on the transverse field strength.