Topologically Protected States
Topologically protected states are a fascinating and fundamental concept in the fields of condensed matter physics and quantum computing. These states are characterized by their resilience to local perturbations due to the global properties of the system they inhabit, which are often described using the language of topology.
Fundamental Concepts
Topological Insulators
A topological insulator is a material with insulating properties in its interior but conducting states on its surface. These surface states are topologically protected, meaning they are robust against disturbances such as impurities or defects. This protection arises because the surface states are a result of the non-trivial topological order of the system. The distinction between an insulator and a topological insulator can be understood through the concept of band structure, where the presence of an energy gap in the bulk allows for conduction only at the surface.
Symmetry-Protected Topological Order
In addition to intrinsic topological order, certain systems exhibit symmetry-protected topological (SPT) order. These are states that, unlike intrinsic topological orders, rely on symmetries for their protection. An example of an SPT phase is the topological insulator of non-interacting fermions. SPT states are characterized by short-range entanglement and are protected by symmetries such as time-reversal symmetry and particle-hole symmetry.
Topological Order
Topological order is a property of certain quantum states of matter that goes beyond the traditional Landau symmetry-breaking theory. These states exhibit long-range entanglement and are characterized by the emergence of anyonic excitations, which do not obey the conventional Fermi-Dirac or Bose-Einstein statistics.
Edge States
In many topologically non-trivial systems, edge states emerge at the boundaries. These are electronic states confined to the boundary of a material, such as a topological insulator, and are protected by the bulk properties of the system. The existence of edge states is a hallmark of the bulk-boundary correspondence principle.
Applications
Topological Quantum Computing
One of the most promising applications of topologically protected states is in quantum computing. Topological quantum computers exploit the properties of anyons, particularly Majorana fermions, to perform computations that are inherently protected from local sources of decoherence. This protection is due to the non-local nature of the information stored in topological qubits, making topological quantum computers less susceptible to errors compared to traditional quantum computers.
Photonic Systems
Recent advances in topological photonics have allowed the realization of high-coherence quantum interference on photonic chips by utilizing topologically protected states. These systems leverage the robustness of topological edge states to create devices with reduced sensitivity to imperfections in fabrication.