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. Elements of lattice gauge theories. Topological order in one spatial dimension.

• Quantum phases of matter.

Entanglement in many-body systems. Short Vs long range entanglement in extended systems. Modern classification of quantum phases of matter.

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).

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