QUANTUM PHASES OF MATTER
Academic Year 2020/2021 - 1° Year - Curriculum CONDENSED MATTER PHYSICS and Curriculum THEORETICAL PHYSICSCredit Value: 6
Scientific field: FIS/02 - Theoretical physics, mathematical models and methods
Taught classes: 42 hours
Term / Semester: 2°
Learning Objectives
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.
Course Structure
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.
Detailed Course Content
- 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.
Entanglement in many-body systems. Short Vs long range entanglement in extended systems. Modern classification of quantum phases of matter.
Textbook Information
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