Spatial symmetry constraint of charge-ordered kagome superconductor CsV$_3$Sb$_5$

Elucidating the symmetry of intertwined orders in exotic superconductors is at the quantum frontier. Recent surface sensitive studies of the topological kagome superconductor CsV$_3$Sb$_5$ discovered a cascade 4a$_0$ superlattice below the charge density wave (CDW) ordering temperature, which can be related to the pair density modulations in the superconducting state. If the 4a$_0$ phase is a bulk and intrinsic property of the kagome lattice, this would form a striking analogy to the stripe order and pair density wave discovered in the cuprate high-temperature superconductors, and the cascade ordering found in twisted bilayer graphene. High-resolution X-ray diffraction has recently been established as an ultra-sensitive probe for bulk translational symmetry-breaking orders, even for short-range orders at the diffusive limit. Here, combining high-resolution X-ray diffraction, scanning tunneling microscopy and scanning transmission electron microscopy, we demonstrate that the 4a$_0$ superstructure emerges uniquely on the surface and hence exclude the 4a$_0$ phase as the origin of any bulk transport or spectroscopic anomaly. Crucially, we show that our detected 2$\times$2$\times$2 CDW order breaks the bulk rotational symmetry to C2, which can be the driver for the bulk nematic orders and nematic surface superlattices including the 4a$_0$ phase. Our high-resolution data impose decisive spatial symmetry constraints on emergent electronic orders in the kagome superconductor CsV$_3$Sb$_5$.