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In this thesis, we investigated the evolution of exotic quantum systems through the lenses of their dynamics. Dynamics unveil properties of systems that in absence of motion remain obscure. Not only the nature of physical systems is manifested during their evolution, but also their intimate relation with time can be elucidated.

Such theories describe electronic properties of one-dimensional systems (as very thin wires) or two-dimensional ones (like sheets). Their main peculiarity is the fact that components are forced, by dimensional constraints, to interact very strongly with their neighbours. Hence, individual motion of constituents is always accompanied by collective one; in one dimension the situation is so dramatic that individual motion is totally devoted to the collective, while in two dimensions the two phenomena coexist.

The emergence of new, collective, modalities of transport of quantities as charge or heat makes these models (which are also realised in laboratories) appealing for future applications. It also begs for different mathematical methods and the standard intuition breaks down in front of them. In this work, we predict novel properties of these materials and we reconcile them with the standard knowledge of the field.