The electronic and magnetic properties of topological materials are strongly influenced by temperature and external magnetic fields. Topological materials are a class of materials that exhibit unique electronic properties due to their topological order, which arises from the non-trivial arrangement of their electronic states. Some well-known examples of topological materials include topological insulators, Weyl semimetals, and Dirac semimetals.1. Temperature dependence:The electronic properties of topological materials are generally robust against temperature variations, as they are protected by topological invariants. However, at higher temperatures, several factors can influence their properties:a. Thermal fluctuations: As temperature increases, thermal fluctuations can cause the electronic states to become more disordered, which may affect the topological properties of the material.b. Phase transitions: Topological materials can undergo phase transitions at certain critical temperatures, where their topological properties may change. For example, a topological insulator may transition into a trivial insulator or a metal at high temperatures.c. Electron-phonon coupling: At higher temperatures, the interaction between electrons and lattice vibrations phonons becomes more significant. This can lead to changes in the electronic band structure and affect the topological properties of the material.2. External magnetic field dependence:The magnetic properties of topological materials are strongly influenced by external magnetic fields. The application of an external magnetic field can lead to several effects:a. Quantum Hall effect: In two-dimensional topological materials, an external magnetic field can induce the quantum Hall effect, where the electronic states become quantized into Landau levels. This can lead to the emergence of new topological phases and edge states.b. Time-reversal symmetry breaking: An external magnetic field can break time-reversal symmetry, which is an essential ingredient for the existence of certain topological phases, such as topological insulators. This can lead to a transition from a topological phase to a trivial phase or a different topological phase.c. Magnetic order: In some topological materials, an external magnetic field can induce magnetic order, such as ferromagnetism or antiferromagnetism. This can affect the topological properties of the material, as well as its electronic and transport properties.d. Tuning of Weyl and Dirac points: In Weyl and Dirac semimetals, an external magnetic field can be used to tune the position and number of Weyl and Dirac points in the electronic band structure. This can lead to the emergence of new topological phases and properties, such as chiral anomaly and anomalous Hall effect.In summary, the electronic and magnetic properties of topological materials can be significantly influenced by temperature and external magnetic fields. Understanding these dependencies is crucial for the development of novel applications and devices based on topological materials, such as topological quantum computing, spintronics, and thermoelectric devices.