## Simulations of collective quantum phenomena

The image shows the phonon spectrum of a harmonic chain (modeling for instance a charge-density wave) in an external random potential at zero temperature (Ph.D. thesis F. Krajewski). The springs are weak compared to the maximum curvature of the external potential, which explains the occurrence of a main phonon branch (yellow line). The existence of the second branch is non-trivial. It is due to a collective tunneling of the particles constituting the chain. At zero wavevector *q* , the probability of zero frequencies appears finite. This in turn implies that the chain can be moved with arbitrarily small external fields.