Large scale simulations of topological defects in pair density waves

Pair density wave (PDW) has been in the central stage of high-Tc cuprate since its direct experimental observation. A PDW is a unique superconducting state in which the electron pairing function varies periodically as a function of position. Experimentally, density-wave states of cuprates have been shown to be highly inhomogeneous, indicating the importance of both quenched disorder and topological defects. To investigate the effects of spatial inhomogeneity, we combine the kernel polynomial method (KPM), which offers linear-scaling electronic structure calculation, with the renormalized mean-field theory (RMFT) for $t$-$J$ model. This allows us to perform unprecedented large-scale simulations, with number of lattice sites up to $N\sim 10^5$, of PDW states. In particular, we are trying to investigate the solutions with fractionalized half-vortices with a dislocation of the charge stripes. We also compute the local electronic density of states of these novel inhomogeneous PDW states.