For a closed quantum system evolving out of equilibrium, thermalization means that local expectation values attain some For integrable systems, for which local conserved quantities restrict stationary values, independent of the details of the initial state.the relaxation process, convergence to a non-thermal state has beenpredicted. For non-integrable models, instead, thermalization is ingenerally expected. But recent experimental results showing nothermalization even away from integrability have triggered a lot oftheoretical interest to the problem.

Studying non-integrable quantum systems is challenging, due to thelack of analytical tools and the limited applicability of numericaltechniques. In particular, the applicability of Matrix Product States(MPS) based algorithms for the simulation of time evolution is limitedto short times due to the fast growth of entanglement in the evolvedstate. We have devised a new algorithms which allows the simulation oflonger times than previous methods. This represents a powerful toolfor the study of out-of-equilibrium dynamics.

Using this new numerical algorithm, we have studied the time evolutionof an infinite spin chain under a non-integrable Hamiltonian.Surprisingly, we have found that different regimes of thermalizationare possible, depending on the initial state of the system. Somestates thermalize "strongly", meaning that all local expectationvalues converge to the thermal values. Other states, instead, exhibit"weak" thermalization, a situation in which local expectation valuesdo not relax, and the thermal values are only recovered after takingtime average. Finally, there is a third group of states for which wedo not observe any sort of thermalization, to the longest times we maysimulate.