QUANTUM PHASES OF MATTER

The course will provide a general introduction to the physics of strongly correlated systems at very low
temperature. The notion of quantum phase transition will be discussed. Specific examples will be
analysed. A first introduction to the modern theory of quantum phase of matter will be presented.

The course is made of three main parts:
I. Classical versus quantum critical phenomena. Dimensional crossover for Ising and Bose-Hubbard model
II. Scaling close to quantum phase transitions.
III. The quantum phase transition in the quantum Ising model. Cross-over at finite temperature.

Critical Phenomena.
Phase transitions, critical points, scaling, the role of dimensionality. The concepts of phase and
symmetry. Conformal invariance. Landau Theory: symmetry breaking.
Quantum Phase transitions.
The Ising model. Solidification transition. Transfer matrix formalism.
Correlation functions. The correspondence between statistical and quantum mechanics. The
notion of a quantum phase transition.Impact of quantum phase transitions.
Example of quantum phase transitions and their relevance for modern quantum material
science.Quantum technology.
The quantum Ising model.
Transverse Ising Model in one-dimension: ground state, quantum critical point, duality argument,
exact solution by Jordan-Wigner transformation.
Quantum phase transition in systems of strongly interacting bosons.
The Bose-Hubbard model.Phase diagram. Physical realizations:Josephson junctions arrays, Cold
atoms trapped in optical lattices.
The quantum critical regime.
The effects of quantum criticality at finite temperature. Thermal crossover and quantum critical
region. Thermal crossover in one dimensional Ising model in trasverse field.
Topological matter.
Beyond Landau-symmetry breaking. Topological order. Topological quantum phase transitions in
two spatial dimensions. The Kosterlitz-Thouless transition. Elements of lattice gauge theories.
Quantum phases of matter.

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

2. 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).

3. 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).

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

5. Lecture Notes