Topological crystalline insulators represent a distinct class of quantum materials where the topological protection of surface states arises from crystalline symmetries rather than time-reversal symmetry alone. Unlike conventional topological insulators, which rely on strong spin-orbit coupling and time-reversal symmetry to protect conducting surface states, TCIs derive their robustness from point-group symmetries of the crystal lattice, such as mirror or rotational symmetry. This fundamental difference leads to unique electronic properties and expands the range of potential material platforms for topological phenomena.

The defining feature of TCIs is the presence of symmetry-protected surface states that remain gapless as long as the relevant crystalline symmetry remains unbroken. In materials like SnTe and Pb1-xSnxSe, mirror symmetry plays a crucial role in protecting the Dirac-like surface states. These states emerge on specific crystal surfaces that preserve the mirror symmetry, while they may be gapped on surfaces where the symmetry is broken. For instance, SnTe, a prototypical TCI with a rocksalt structure, exhibits Dirac cones on the (001), (011), and (111) surfaces due to mirror symmetry protection, while the (110) surface does not host protected states because mirror symmetry is absent.

The electronic structure of TCIs can be understood through band inversion mechanisms similar to conventional topological insulators but with additional constraints imposed by crystalline symmetry. In SnTe, the band inversion occurs between the Te p and Sn p orbitals at multiple points in the Brillouin zone, leading to multiple Dirac cones on certain surfaces. The number and location of these Dirac cones are dictated by the crystal symmetry, making the surface state configuration highly anisotropic compared to conventional topological insulators. Pb1-xSnxSe alloys exhibit a topological phase transition as a function of composition and temperature, where the material transitions from a trivial insulator to a TCI as the band inversion occurs.

One key distinction between TCIs and conventional topological insulators lies in their response to symmetry-breaking perturbations. While time-reversal-invariant topological insulators are robust against non-magnetic disorder, their surface states can be gapped by magnetic impurities or fields. In contrast, TCIs are sensitive to perturbations that break the protecting crystalline symmetry. For example, applying uniaxial strain or introducing surface reconstructions that lower the mirror symmetry can destroy the topological protection in TCIs. This sensitivity provides a knob to control the surface states through external stimuli, offering potential advantages for device applications.

Material realizations of TCIs extend beyond SnTe and Pb1-xSnxSe. The family of IV-VI semiconductors, including related compounds like PbS and PbSe, have been predicted and confirmed to exhibit TCI phases under certain conditions. More recently, oxide perovskites and antiperovskites have emerged as candidates for TCI behavior, expanding the chemical and structural diversity of these materials. The exploration of ternary and quaternary compounds has further enriched the phase space, with materials like (Pb,Sn)(Se,Te) alloys allowing fine-tuning of topological properties through composition.

The unique properties of TCIs open several avenues for device applications. One promising direction is in low-power electronics, where the spin-momentum locked surface states could enable efficient charge transport with reduced scattering. The anisotropy of surface states in TCIs may be exploited for directional transport devices or sensors that respond selectively to certain crystallographic orientations. Another potential application lies in thermoelectric devices, where the combination of bulk insulating behavior and metallic surface states could help achieve high thermoelectric figures of merit by independently optimizing bulk and surface contributions.

In the realm of quantum computing, TCIs offer an alternative platform for realizing Majorana zero modes when combined with superconductivity. The symmetry-protected surface states may provide additional control knobs for manipulating these exotic quasiparticles compared to conventional topological insulators. The possibility of tuning the topological phase transition in materials like Pb1-xSnxSe through temperature or composition could lead to switchable topological devices with reconfigurable functionality.

Challenges remain in the practical implementation of TCI-based devices. Controlling surface disorder and maintaining crystalline symmetry at interfaces are critical for preserving the topological surface states. The relatively small bulk band gaps in many TCI materials, typically on the order of a few hundred meV, limit operating temperatures for some applications. Progress in materials growth and interface engineering will be essential to overcome these limitations and harness the full potential of TCIs.

Recent advances in angle-resolved photoemission spectroscopy and scanning tunneling microscopy have enabled detailed characterization of the symmetry-protected surface states in TCIs, confirming theoretical predictions and revealing new phenomena. Transport measurements have demonstrated the existence of surface-dominated conduction in thin films of SnTe and related compounds, providing evidence for the topological protection of surface states. The development of novel characterization techniques tailored to probe symmetry-protected states continues to push the understanding of these materials forward.

The study of TCIs has also stimulated theoretical developments in topological classification schemes beyond the standard tenfold way. The concept of symmetry-indicated topology has expanded to include various crystalline symmetries, leading to the prediction of new topological phases with different types of protected boundary states. This broader framework has implications beyond TCIs, influencing the search for topological materials across different crystal systems and symmetry classes.

As research progresses, the interplay between different types of topology—such as the combination of crystalline topology with time-reversal topology—presents rich opportunities for discovering new quantum phenomena. The exploration of heterostructures combining TCIs with other quantum materials may lead to emergent properties not present in either constituent alone. The field continues to evolve rapidly, with ongoing efforts to identify new TCI materials, understand their fundamental properties, and develop practical applications that leverage their unique characteristics.

The distinct nature of TCIs compared to conventional topological insulators offers complementary advantages for both fundamental studies and technological applications. While conventional topological insulators may be more robust against certain types of disorder, TCIs provide additional control through crystalline symmetry and exhibit surface state configurations that are intrinsically tied to the crystal structure. This connection between topology and crystallography makes TCIs particularly interesting for materials engineering approaches aimed at tailoring topological properties for specific applications. The continued investigation of these materials promises to yield both scientific insights and practical advances in quantum materials science.