Interaction and coherence in two-dimensional bilayers

Two-dimensional electron gas (2DEG) bilayers provide suitable platforms for electronic phases and transitions that are fundamental to both theoretical physics and practical applications in device technology. In bilayer systems, the additional pseudospin, representing the layer degree of freedom, enables the emergence of interlayer coherence, which is a direct consequence of the interlayer Coulomb interaction. This study presents a comprehensive Hartree-Fock (HF) mean-field investigation of the interlayer coherence in 2D bilayers, uncovering ground-state behaviors and temperature-dependent phase transitions that are distinct from single-layer 2DEG. This interlayer coherence signals a spontaneous breaking of the U(1) symmetry in layer pseudospin. We explore the zero-temperature phase diagrams as a function of the electron density and interlayer separation within the HF formalism. We also calculate the critical temperature (${T}_{c}$) of the interlayer coherence onset by self-consistently solving the HF gaplike equation. We contrast this interlayer coherent phase in electron-electron (e-e) bilayers with the closely related excitonic superfluid phase in electron-hole (e-h) bilayers. Although both e-e and e-h bilayers spontaneously break the pseudospin U(1) symmetry, e-h bilayers produce Bardeen-Cooper-Schrieffer--Bose-Einstein condensates crossover intrinsic to the excitons acting as effective bosons or Cooper pairs, whereas the symmetry-broken phase in e-e bilayers is akin to the XY or easy-plane pseudospin ferromagnetism. Using the same system parameters and a similar theoretical framework, we find that ${T}_{c}$ of the interlayer coherent phase in e-e bilayers is about one-third of that in exciton condensates, suggesting a weaker interlayer coherence in e-e bilayers. In addition, we examine the effect of a weak interlayer tunneling on the interlayer coherence order parameter, drawing parallels with the influence of an effective in-plane magnetic field on the XY pseudospin ferromagnetism. Our findings provide a comparative theoretical framework that bridges the gap between the interlayer coherence physics in e-e and e-h bilayers, contributing to a unified understanding of phase transitions in low-dimensional electron-hole systems and establishing in particular the same universality class for interlayer phase coherence in both e-e and e-h bilayers.