Evolution from BardeenCooper-Schrieffer to Bose-Einstein Condensation in Two Dimensions: Crossovers and Topological Quantum Phase Transitions

We review aspects of the evolution from Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) in two dimensions, which have now become relevant in systems with low densities, such as gated superconductors LixZrNCl, magic-angle twisted trilayer graphene, FeSe, FeSe1 and ultracold Fermi superfluids. We emphasize the important role played by chemical potentials in determining crossovers or topological quantum phase transitions during the BCS-BEC evolution in one-band and two-band superfluids and superconductors. We highlight that crossovers from BCS to BEC occur for pairing in nonnodal s-wave channels, whereas topological quantum phase transitions, in which the order parameter symmetry does not change, arise for pairing in any nodal higher angular momentum channels, such as d-wave. We conclude by discussing a few open questions regarding the BCS-to-BEC evolution in 2D, including modulus fluctuations of the order parameter, tighter upper bounds on critical temperatures, and the exploration of lattice effects in two-band superconductors and superfluids.