  Two-dimensional (2D) bilayers, twisted to particular angles to display
electronic flat bands, are being extensively explored for physics of strongly
correlated 2D systems. However, the similar rich physics of one-dimensional
(1D) strongly correlated systems remains elusive as it is largely inaccessible
by twists. Here, a distinctive way to create 1D flat bands is proposed, by
either stamping or growing a 2D monolayer on a non-Euclidean
topography-patterned surface. Using boron nitride (hBN) as an example, our
analysis employing elastic plate theory, density-functional and coarse-grained
tight-binding method reveals that hBN's bi-periodic sinusoidal deformation
creates pseudo-electric and magnetic fields with unexpected spatial dependence.
A combination of these fields leads to anisotropic confinement and 1D flat
bands. Moreover, changing the periodic undulations can tune the bandwidth, to
drive the system to different strongly correlated regimes such as density
waves, Luttinger liquid, and Mott insulator. The 1D nature of these states
differs from those obtained in twisted materials and can be exploited to study
the exciting physics of 1D quantum systems.
