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Topological insulators are materials that behave as insulators in their interior but have conductive states on their edges or surfaces that are topologically protected. They are significant for their potential applications in quantum computing and spintronics.

The Chern number is an integer that characterizes the topological order of certain quantum states and can be associated with quantized Hall conductance. It plays a crucial role in classifying the phases of matter that emerge from topological properties.

The Fractional Quantum Hall Effect is a phenomenon observed when electrons form a new type of quantum fluid characterized by fractionalized charge and anyonic statistics. It is significant for revealing interactions that lead to complex quantum states.

The Quantum Hall Effect is a quantum phenomenon observed in 2D electron systems under low temperatures and strong magnetic fields, where the Hall conductance exhibits quantized plateaus. It is significant as it led to the discovery of topologically protected quantum states.

The Quantum Spin Hall Effect is a state in which a material has a quantized spin Hall conductance, leading to edge states that carry spin-polarized currents without dissipation. This is significant for potential applications in spintronic devices.