Recently, the concept of quantum simulators has built unexpected bridges between various fields of modern physics. By exploiting the interaction between atomic gases and external fields, theorists and experimentalists have developed and designed remarkable setups that reproduce large families of quantum systems [1].

Remarkably, such setups allow the tuning of fundamental features - such as the geometry and the dimensionality of the system, the interaction between the particles - with an exquisite precision. In this context, specific configurations of the external electromagnetic fields lead to non-trivial Berry’s phases [2], which mimic the presence of synthetic gauge fields in the cold atoms dynamics [3, 4]. Such gauge fields are extremely malleable and could lead to various fascinating effects stemming from condensed-matter and even high-energy physics [5].

In this talk, I will show how synthetic gauge fields can be dressed in optical lattice setups in order to realize topological states of matter. These quantum phases encode non-trivial topological order [6, 7] and lead to fascinating effects, such as the quantum Hall [8] and the quantum spin Hall effects [9]. Topological phases are extremely robust against perturbations caused by the experimental environments, and therefore, constitute potential candidates for quantum computing devices [9]. Generating topologically protected phases with cold atoms is an extremely attractive and realistic goal [10, 11], as it would offer an ideal playground to investigate their fundamental properties, such as their robustness against interactions and disorder.

I will discuss how topological phases can be engineered in an optical lattice and how one can trigger and detect phase transitions in these setups. Besides, I will discuss the interplay between topology, quantum transport and the pseudorelativistic regimes which can be reached in these systems.