Quantum phase transitions occur at zero temperature when a non-thermal parameter, such as pressure or magnetic field, is varied. These transitions are driven by quantum fluctuations, which arise from Heisenberg's uncertainty principle. This lecture notes provide a pedagogical introduction to quantum phase transitions, discussing their importance, relation to classical transitions, and experimental relevance. The Ising model in a transverse field is used as a primary example, along with quantum phase transitions in itinerant electron systems and their connection to non-Fermi liquid behavior. Phase transitions involve changes in the properties of a system as it moves between different states. First-order transitions involve latent heat, while second-order transitions, like the magnetic transition of iron, do not. The critical point, where the properties of two phases become indistinguishable, is a key concept in phase transitions. Quantum phase transitions occur at zero temperature and are driven by quantum fluctuations rather than thermal ones. They are distinct from classical phase transitions and have significant experimental relevance, such as in quantum Hall effects, localization, non-Fermi liquid behavior, and high-temperature superconductivity. The Ising model in a transverse field is a paradigmatic example of a quantum phase transition. It demonstrates how quantum fluctuations can destroy long-range order. The model is also used to illustrate the quantum-to-classical mapping, where a quantum spin in a transverse field can be mapped to a classical Ising chain. This mapping helps in understanding the critical behavior of quantum phase transitions. Quantum phase transitions are also important in understanding non-Fermi liquid behavior in itinerant electron systems. For example, in heavy-fermion systems like CeCu₆₋ₓAuₓ, the specific heat coefficient shows a logarithmic temperature dependence near the quantum critical point, indicating deviations from the Fermi liquid theory. These deviations are due to the scattering of electrons off diverging magnetic fluctuations at the critical point. Quantum phase transitions are a fascinating area of condensed matter physics, offering new insights into complex systems where conventional methods fail. They are crucial for understanding a wide range of phenomena, from superconductivity to magnetic materials. Future research in this area promises to uncover more about the behavior of quantum systems and their applications.Quantum phase transitions occur at zero temperature when a non-thermal parameter, such as pressure or magnetic field, is varied. These transitions are driven by quantum fluctuations, which arise from Heisenberg's uncertainty principle. This lecture notes provide a pedagogical introduction to quantum phase transitions, discussing their importance, relation to classical transitions, and experimental relevance. The Ising model in a transverse field is used as a primary example, along with quantum phase transitions in itinerant electron systems and their connection to non-Fermi liquid behavior. Phase transitions involve changes in the properties of a system as it moves between different states. First-order transitions involve latent heat, while second-order transitions, like the magnetic transition of iron, do not. The critical point, where the properties of two phases become indistinguishable, is a key concept in phase transitions. Quantum phase transitions occur at zero temperature and are driven by quantum fluctuations rather than thermal ones. They are distinct from classical phase transitions and have significant experimental relevance, such as in quantum Hall effects, localization, non-Fermi liquid behavior, and high-temperature superconductivity. The Ising model in a transverse field is a paradigmatic example of a quantum phase transition. It demonstrates how quantum fluctuations can destroy long-range order. The model is also used to illustrate the quantum-to-classical mapping, where a quantum spin in a transverse field can be mapped to a classical Ising chain. This mapping helps in understanding the critical behavior of quantum phase transitions. Quantum phase transitions are also important in understanding non-Fermi liquid behavior in itinerant electron systems. For example, in heavy-fermion systems like CeCu₆₋ₓAuₓ, the specific heat coefficient shows a logarithmic temperature dependence near the quantum critical point, indicating deviations from the Fermi liquid theory. These deviations are due to the scattering of electrons off diverging magnetic fluctuations at the critical point. Quantum phase transitions are a fascinating area of condensed matter physics, offering new insights into complex systems where conventional methods fail. They are crucial for understanding a wide range of phenomena, from superconductivity to magnetic materials. Future research in this area promises to uncover more about the behavior of quantum systems and their applications.