Large moir\'{e} superstructure of stacked incommensurate charge density waves

Recent advances in van der Waals heterostructures have opened the new frontier of moir\'{e} physics, whereby tuning the interlayer twist angle or adjusting lattice parameter mismatch have led to a plethora of exotic phenomena such as unconventional superconductivity and fractional quantum spin Hall effect. We extend the concept of moir\'{e} engineering to materials that host incommensurate orders, where we discovered a long-period, thermally-hysteretic moir\'{e} superlattice in a layered charge density wave (CDW) compound, EuTe$_\text{4}$. Using high-momentum-resolution X-ray diffraction performed on ultrathin flakes, we found two coexisting, incommensurate CDWs with slightly mismatched in-plane wavevectors. The interaction between these two CDWs leads to their joint commensuration with the high-symmetry lattice as well as a large moir\'{e} superstructure with an in-plane period of 13.6~nm. Due to different out-of-plane orders of the incommensurate CDWs, the moir\'{e} superstructure exhibits a clear thermal hysteresis, accounting for the large hysteresis observed in electrical resistivity and numerous metastable states induced by light or electrical pulses. Our findings pave the way for a new development in moir\'{e} engineering based on an incommensurate lattice. They further highlight the important role of interlayer ordering in determining the macroscopic properties of these stacked incommensurate structures.