Topological matter and its exploration with quantum gases

This lecture explores the concept of topological matter in two-dimensional systems, focusing on the absence of long-range order at non-zero temperature. The Peierls argument and the Mermin-Wagner-Hohenberg theorem are discussed, highlighting the unconventional phase transitions in 2D systems. The role of vortices and the Kosterlitz-Thouless mechanism are explained, along with the implications for superconductors and graphene. The lecture delves into the 2D ideal Bose gas, Gross-Pitaevskii approach, and the Bogoliubov spectrum, emphasizing the quasi-long range order and sound propagation. Experimental observations and theoretical models in atomic and photonic fluids are presented, shedding light on the critical points and superfluid behavior.