Moiré systems, formed by stacking atomically thin layers with a relative twist or lattice mismatch, have emerged as a versatile platform for exploring strongly correlated electron physics. The periodic potential arising from the moiré superlattice dramatically modifies the electronic band structure, leading to flat bands with enhanced Coulomb interactions relative to kinetic energy. This regime enables the stabilization of exotic quantum phases, including Wigner crystals, stripe phases, and quantum critical points, which can be probed through transport measurements and scanning probe microscopy.

Wigner crystallization is a hallmark of strong correlation physics, where electrons localize into an ordered lattice due to Coulomb repulsion, overcoming their kinetic energy. In moiré systems, the large effective electron mass and reduced screening in flat bands lower the critical density for Wigner crystallization. Experimental signatures include a sharp drop in conductivity at low carrier densities, accompanied by a threshold electric field for depinning of the electron lattice. Scanning tunneling microscopy (STM) reveals periodic charge modulations with a characteristic wavevector corresponding to the expected electron density. For example, in twisted bilayer graphene, STM measurements have identified hexagonal charge order at densities below one electron per moiré unit cell, consistent with a Wigner crystal phase. Transport measurements show insulating behavior with an activation gap, typically in the range of 1 to 10 meV, depending on the twist angle and dielectric environment.

Stripe phases, another correlated state, arise from the interplay between Coulomb repulsion and kinetic energy, leading to unidirectional charge density waves. These phases are often observed near half-filling of the moiré band, where the system balances between localization and delocalization. In twisted transition metal dichalcogenides, such as WSe2/WS2 heterobilayers, transport measurements exhibit anisotropic resistance, reflecting the broken rotational symmetry of stripe order. STM and atomic force microscopy (AFM) have visualized stripe patterns with periods of several nanometers, modulated by the moiré superlattice. The stripe orientation can be tuned by external electric fields, demonstrating the delicate balance of interactions in these systems.

Quantum criticality in moiré systems occurs at the boundary between competing phases, such as between a Wigner crystal and a Fermi liquid or between a stripe phase and a uniform state. The proximity to a quantum critical point leads to non-Fermi liquid behavior, characterized by power-law divergences in thermodynamic and transport properties. Resistivity measurements often show a linear temperature dependence over a wide range, indicative of strong quantum fluctuations. In magic-angle twisted bilayer graphene, quantum criticality has been inferred from the scaling of the resistivity near half-filling, with a crossover temperature scale of around 10 K. Scanning probes have further revealed fluctuating charge patterns near critical points, suggesting the presence of emergent symmetries.

Transport experiments provide key insights into these correlated phases. The Hall coefficient, for instance, shows deviations from classical behavior in the Wigner crystal regime, reflecting the contribution of pinned electron lattices to magnetotransport. In stripe phases, the Hall effect becomes anisotropic, with different responses parallel and perpendicular to the stripe direction. Nonlinear transport measurements, such as differential conductance, reveal depinning transitions and sliding dynamics of charge-ordered states. For example, in twisted bilayer graphene, threshold voltages of a few millivolts have been observed, corresponding to the depinning of the Wigner crystal.

Scanning probe techniques offer complementary real-space information. STM can directly image the charge order with sub-nanometer resolution, while AFM measures the associated mechanical distortions. Spectroscopy modes, such as scanning tunneling spectroscopy (STS), reveal the electronic density of states, showing gaps or pseudogaps in the Wigner crystal and stripe phases. In some cases, moiré systems exhibit coexisting orders, such as Wigner crystals superimposed on charge density waves, which can be disentangled through spatial mapping at different energies.

The role of the dielectric environment and substrate interactions is crucial in stabilizing these phases. For instance, encapsulating moiré systems in hexagonal boron nitride (hBN) reduces disorder and enhances correlation effects. The dielectric constant of the surrounding materials influences the Coulomb interaction strength, with lower dielectric constants favoring Wigner crystallization. Experimental studies have systematically varied the dielectric environment to tune the phase diagram, observing shifts in the critical densities for Wigner crystals and stripe phases.

Temperature and magnetic field are additional tuning parameters. At finite temperatures, thermal fluctuations can melt the Wigner crystal or disrupt stripe order, leading to crossovers into disordered phases. Magnetic fields can induce transitions between different correlated states, as seen in the quantum Hall regime of moiré systems. For example, in twisted bilayer graphene, quantized Hall plateaus appear at high fields, coexisting with or replacing the correlated insulating states observed at zero field.

Theoretical models of moiré systems often employ extended Hubbard models or continuum descriptions incorporating the moiré potential. These models predict phase diagrams with Wigner crystals, stripe phases, and quantum critical points, in qualitative agreement with experiments. However, quantitative discrepancies remain, particularly in the exact critical parameters and the nature of intermediate phases. Experimental advances in sample homogeneity and measurement techniques are expected to refine these comparisons.

Future directions include exploring the interplay between correlation effects and superconductivity in moiré systems, as well as the potential for topological phases in the presence of strong interactions. The development of high-resolution scanning probes with improved energy resolution will enable more detailed studies of quantum critical fluctuations and emergent symmetries. Additionally, the integration of moiré systems with other quantum materials could lead to hybrid phases with novel properties.

In summary, moiré systems provide a rich playground for studying Wigner crystallization, stripe phases, and quantum criticality. Transport and scanning probe measurements have revealed a wealth of experimental signatures, from insulating behavior and charge order to non-Fermi liquid transport. These findings deepen our understanding of strongly correlated electron physics in tunable, two-dimensional platforms.