Motivated by recent progress in the superconductivity nonreciprocal phenomena, we study the general theory of Josephson diodes. The central ingredient for Josephson diodes is the asymmetric proximity process inside the tunneling barrier. From the symmetry breaking point of view, there are two types of Josephson diodes: inversion breaking and time-reversal breaking. For the inversion breaking case, applying voltage bias could effectively tune the proximity process like the voltage-dependent Rashba coupling or electric polarization giving rise to $I_c(V)\ne I_c(-V)$ and $I_{r+}\ne I_{r-}$. For the time-reversal breaking case, the current flow could adjust the internal time-reversal breaking field like magnetism or time-reversal breaking electron-electron pairing, which leads to $I_{c+}\ne I_{c-}$. All these results provide a complete understanding and the general principles of realizing Josephson diodes, especially the recently found ${\mathrm{NbSe}}_2/{\mathrm{Nb}}_3{\mathrm{Br}}_8/{\mathrm{NbSe}}_2$ Josephson diodes.

Motivated by the recently observed insulating states in twisted bilayer graphene, we study the nature of the correlated insulating phases of the twisted bilayer graphene at commensurate filling fractions. We use the continuum model and project the Coulomb interaction onto the flat bands to study the ground states by using a Hartree-Fock approximation. In the absence of the hexagonal boron nitride substrate, the ground states are the intervalley coherence states at charge neutrality (filling $\nu =0$, or four electrons per moiré cell) and at $\nu =-1/4$ and $-1/2$ (three and two electrons per cell, respectively) and the $C_2\mathcal{T}$ symmetry-broken state at $\nu =-3/4$ (one electron per cell). The hexagonal boron nitride substrate drives the ground states at all $\nu$ into $C_2\mathcal{T}$ symmetry broken-states. Our results provide good reference points for further study of the rich correlated physics in the twisted bilayer graphene.

Motivated by recent experiments on ${\mathrm{AV}}_3{\mathrm{Sb}}_5$ (A = K, Rb, Cs), the chiral flux phase has been proposed to explain time-reversal symmetry breaking. To fully understand the physics behind the chiral flux phase, we construct a low-energy effective theory based on the van Hove points around the Fermi surface. The possible symmetry-breaking states and their classifications of the low-energy effective theory are completely studied, especially the flux phases on the kagome lattice. In addition, we discuss the relations between the low-energy symmetry breaking orders, the chiral flux, and charge bond orders. We find all possible 183 flux phases on the kagome lattice within a 2*2 unit cell by brute-force approach and classify them by point-group symmetry. Among the 183 phases, we find 3 classes in a 1*1 unit cell, 8 classes in a 1*2 unit cell, and 18 classes in a 2*2 unit cell, respectively. These results provide a full picture of the time-reversal symmetry breaking in kagome lattices.