Twisted van der Waals heterostructures represent a frontier in condensed matter physics, where the relative rotation between stacked two-dimensional layers introduces a moiré superlattice. This moiré pattern profoundly alters the electronic properties of the system, giving rise to correlated states such as superconductivity, Mott insulation, and quantum magnetism. The key parameter governing these phenomena is the twist angle between layers, which modulates the periodicity of the moiré potential and the resulting electronic band structure.

The formation of a moiré superlattice occurs when two atomically thin layers with a slight lattice mismatch or rotational misalignment are stacked. For instance, in twisted bilayer graphene (TBG), rotating one graphene sheet relative to another by a small angle θ generates a long-wavelength periodic potential. At specific "magic angles," typically around 1.1 degrees for TBG, the Fermi velocity of electrons vanishes, leading to nearly flat bands. These flat bands enhance electronic correlations, as the kinetic energy of electrons is quenched, making interactions dominate.

The continuum model provides a theoretical framework for describing the low-energy electronic structure of twisted heterostructures. It treats the interlayer coupling as a perturbation to the Dirac cones of the individual layers, incorporating the moiré potential as a spatially varying term. This model successfully predicts the emergence of flat bands and the renormalization of the Fermi velocity near magic angles. Numerical simulations based on this approach have been validated by experimental observations, confirming the critical role of twist angle in tuning electronic behavior.

Experimental techniques for controlling twist angles with high precision have advanced significantly. Methods such as tear-and-stack assembly allow for manual rotation of layers, followed by alignment using optical microscopy or atomic force microscopy. More recently, scanning probe-assisted manipulation has enabled sub-0.1-degree accuracy in angle control. In situ annealing further refines the alignment by minimizing strain-induced disorder. These techniques are essential for reproducibly accessing correlated states, as deviations of even 0.1 degrees from the magic angle can drastically alter the electronic properties.

At magic angles, twisted graphene systems exhibit a rich phase diagram as a function of carrier density and temperature. At half-filling of the flat bands, strong Coulomb interactions can induce a Mott insulating state, where electrons localize due to repulsion. Slightly away from half-filling, unconventional superconductivity emerges, with critical temperatures up to a few Kelvin. The pairing mechanism is believed to involve non-phonon-mediated interactions, possibly arising from spin fluctuations or other exotic correlations. Similar phenomena have been observed in other twisted systems, such as transition metal dichalcogenide (TMD) heterobilayers, where moiré potentials also lead to flat bands and correlated insulators.

Spectroscopic tools like scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) have been instrumental in probing these states. STM reveals the real-space charge distribution modulated by the moiré pattern, while ARPES maps the momentum-space band structure, directly visualizing flat bands. Transport measurements further characterize the correlated phases, showing signatures such as insulating gaps, superconducting domes, and anomalous Hall effects.

Theoretical efforts extend beyond the continuum model to include many-body effects. Hartree-Fock and dynamical mean-field theory (DMFT) calculations capture the competition between kinetic energy and interactions, predicting the stability of various ordered phases. Recent work has also explored the role of lattice relaxation, where the layers deform slightly to minimize interlayer energy, further modifying the moiré potential.

Beyond graphene, twisted TMD heterostructures offer additional tunability through their strong spin-orbit coupling and valley degrees of freedom. For example, in twisted MoSe2/WSe2 bilayers, the moiré potential traps excitons, leading to spatially ordered quantum emitters. The interplay between spin, valley, and charge in these systems opens new avenues for engineering topological states and quantum simulators.

The study of twisted van der Waals heterostructures is still in its early stages, with many open questions remaining. The precise mechanism of superconductivity, the nature of competing orders, and the role of disorder are active areas of research. Future directions include exploring higher-order moiré systems with three or more layers, which could host even richer phase diagrams. Advances in fabrication and theoretical modeling will continue to drive discoveries in this field, offering insights into fundamental quantum phenomena and potential applications in quantum materials engineering.

In summary, twisted van der Waals heterostructures provide a versatile platform for investigating strongly correlated electrons in tunable, low-dimensional systems. The interplay between twist angle, moiré potential, and electronic interactions leads to emergent states that challenge existing theoretical paradigms. Experimental progress in angle control and characterization techniques has enabled systematic exploration of these phenomena, while theoretical frameworks like the continuum model offer a foundation for understanding their origins. As research progresses, these systems may yield new principles for designing quantum materials with tailored electronic properties.