1
Path Integral and its application in quantum mechanics, statistical mechanics, quantum field theory.


2
Classical phase transitions. Singularities and order of the transition. Symmetry, symmetry breaking and order paramer. Ginzburg Landau Theory.



3
Dimensional scaling. Relation among critical exponents. Wilson Renormalization Group and determination of critical exponents. Epsilon expansion. Connection with the renormalization of Quantum Field Theory



4
Mermin-Wagner theorem and no ferromagnetic ordered phase in two dimensions. Topological Kosterlitz-Thouless phase transition.



5
Quantum Phase Transitions. Relation between d quantum, and d+1 classical phase transitions



6
Examples of quantum-classical dimensional crossover: one and two dimensional Ising model. Transfer matrix formalism. Quantum Rotor model.



7
Examples of Quantum Phase Transitions. The Bose-Hubbard model and physical realizations.



8
Transverse Ising Model in one-dimension: ground state, quantum critical point, duality argument, exact solution by Jordan-Wigner transformation.



9
Effects of quantum criticality at finite temperature. Thermal crossover and quantum critical region. Thermal crossover in one dimensional Ising model.



10
Quantum fluids of matter and light. Superradiance and subradiance.



11
Goldstone theorem and the Anderson-Higgs mechanism

R. Feynmann, "Statistical Mechanics: A Set Of Lectures", (Frontiers in Physics) CRC press, 1972.


S. Sachdev, “Quantum Phase Transitions” (Cambridge University press 2011).
X.G. Wen, “Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons”, (Oxford University press 2007).


X.G. Wen, “Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light
and Electrons”, (Oxford University press 2007).


G. Mussardo, "Il modello di Ising. Introduzione alla teoria dei campi e delle transizioni di fase", Boringheri 2010


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