Tunable Competing Electronic Orders in Double Quantum Spin Hall Superlattices

Competing superconducting (SC) and density-wave orders are of key importance in generating unconventional superconductivity and emergent electronic responses. Quasi-one-dimensional models provide insight into these competing orders and suggest higher-dimensional realizations through coupled-wire constructions, but analysis of such systems remains limited. Recent studies suggest that double helical edge states (DHESs) in double quantum spin Hall insulators (DQSHIs) form a two-channel Luttinger liquid that exhibits SC and spin density wave (SDW) phases and their $\pi$-junction analogs. Here, we analyze weakly coupled DHESs from the surface of a periodically stacked layered structure consisting of DQSHIs and dielectrics, where inter-edge interactions approximately develop a tunable helical sliding Luttinger liquid (HSLL) order. Using a renormalization-group analysis, we construct phase diagrams and identify a regime of HSLL parameters that favor competing two-dimensional $\pi$-SC and $\pi$-SDW orders. We identify parameter regimes where the competing orders could be realized experimentally in nanoscale devices. Our study suggests a promising materials platform for exploring tunable $\pi$-SC and $\pi$-SDW orders in double quantum spin Hall superlattices.