69699 - Theory Of Many Body Systems s Theory

Learning outcomes

At the end of the course, the student is able to study strongly interacting quantum many body systems. More specifically, he/she is able to use both exact results and approximation techniques for the analysis of phase transitions.

Course contents

PART 1 (Prof.ssa Ercolessi)

We will present analytical techniques for the study of quantum many body systems (bosonic, fermionic, magnetic) and the description of their phases. More specifically exact solution techniques (transfer matrix and Bethe ansatz) for integrable systems and perturbative techniques (mean field and variational approach).

These techniques will be seen in action by analyzing a series of important models for applications in physics.

Classical models:   Ising model,  Ice-type models

Quantum models: Spin models (XX, XY, Heisenberg, XXZ)      

  

PART 2 (Prof. Degli Esposti Boschi)

Qubit, quantum computation and simulation.

Algorithmic complexity, intrinsic parallelism and exponential speed-up.

Physical implementations.

Di Vincenzo criteria.

Notes on decoherence.

Deutsch-Josza algorithm.

Hilbert spaces for many qubits.

Separable States.

Computational and Bell bases.

Reduced density matrices and Schmidt decomposition.

Einstein-Podolsky-Rosen paradox and Bell inequality.

Cirel'son's limit and generalized inequalities for entangles states of two qubits.  

Bloch sphere and convexity properties of the space of density matrices.

Fidelity.

Teleportation and channels.

No-cloning theorem.

Shannon e von Neumann entropy as entanglement measure for pure states.

Distillation and formation Entanglement.

Wootters concurrence and spin correlations.

Monogamy properties and multipartite entanglement.

Entanglement and quantum statistical thermodynamics.

Entanglement witnesses in magnetic systems at finite temperature.

Comments on the area law for von Neumann entropy.

Readings/Bibliography

[1] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press. 

[2] M. Takahashi, Thermodynamics of One-dimensional Solvable Models.

[3] M.A. Nielsen, I.L. Chuang, Quantum Information and Quantum Computattion.

Teaching methods

Topics are fully explained in class by the teacher.

Assessment methods

The exam is oral and it consists of a presentation of a topic that further develops one of the subjects presented in class, to be  arranged with the teacher in advance.

Office hours

See the website of Elisa Ercolessi

See the website of Cristian Degli Esposti Boschi