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Conformal Field Theories (CFTs) have been used extensively to understand thephysics of critical lattice models at equilibrium. However, the applicabilityof CFT calculations to the behavior of the lattice systems in theout-of-equilibrium setting is not entirely understood. In this work, we comparethe CFT results of the evolution of the entanglement spectrum after a quantumquench with numerical calculations of the entanglement spectrum of thethree-state Potts model using matrix product state simulations. Our resultslead us to conjecture that CFT does not describe the entanglement spectrum ofthe three-state Potts model at long times, contrary to what happens in theIsing model. We thus numerically simulate the out-of-equilibrium behaviour ofthe Potts model according to the CFT protocol - i.e. by taking a particularproduct state and "cooling" it, then quenching to the critical point and findthat, in this case, the entanglement spectrum is indeed described by the CFT atlong times.

On quenches to the critical point of the three states Potts model -- Matrix Product State simulations and CFT