## Admission Requirements

Bachelor in Physics and knowledge of basic statistical mechanics.

## Description

What do a magnet, a Bose-Einstein condensate and a flock of birds have in common? All these systems exhibit a collective behavior and have large-scale physical properties that cannot be understood in terms of a simple extrapolation of the properties of a few particles. Conversely, systems comprising many interacting subunits often present entirely new properties, that scientists refer to as emergent.

Statistical Physics A, is the first part of a two-part introductory course on emergent phenomena in equilibrium and non-equilibrium systems. The course provides an introduction to phase transitions and critical phenomena at equilibrium. The second (elective) part of this course is given in Statistical Physics B and is focused on emergent phenomena in non-equilibrium systems (i.e. collective motion in animals and other biological systems etc.)

The course consists of 10 lectures and 5 tutorials (exercise classes), during which the students will practice the concepts and the techniques learned during the lectures by solving problems.

Topics

• Introduction to phase transitions and critical phenomena in statistical mechanics.

• The one-dimensional Ising model: exact solution via transfer matrix method.

• The two-dimensional Ising model: domain walls and Peierls’ argument.

• Mean field theory.

• Fluctuations theory and field-theoretical approach to critical phenomena.

• Universality, scaling and critical dimensions.

• Real-space renormalization group.

• Momentum-shell renormalization group and ε-expansion.

## Course Objectives

The aim of the course is to develop a strong foundation in advanced statistical mechanics with an emphasis on emergent phenomena. Furthermore, the course aims to provide the students with a toolbox of mathematical techniques that can be readily used in theoretical and experimental research projects.

Specifically, at the end of the course, successful students will have learned how to:

• Solve 1-dimensional spin models and calculate quantities of physical interests such as the average magnetization, magnetic susceptibility, energy, entropy etc.

• Set up a mean-field theory for the Ising model and analogous models and use it to calculate the critical exponents.

• Recognize the scale-invariance and how it connects with second order phase transitions.

• Construct a field theory for magnetism by perfuming the continuum limit of the mean-field Ising model and use it to calculate the spin-spin correlation function and other relevant quantities.

• Apply Wilson’s renormalization group, in real and Fourier space, to go beyond mean-field and accurately describe the critical behavior of ferromagnets.

## Generic skills (soft skills)

At the end of the course, students will have been trained how to:

• Decompose a complex concept into simpler steps, each of which can be rationalized at an intuitive level.

• Think across individual academic disciplines and use analogies with other fields (e.g. computer graphics) to build up physical intuition.

• Not to stop after the first hints of success, but aim at greater goals.

## Timetable

## Mode of instruction

Lectures and tutorials.

## Assessment method

Final exam with open questions and homework assignments for bonus points.

## Blackboard

To have access to Blackboard you need a ULCN-account.Blackboard UL

## Reading list

Nigel Goldenfeld, Lectures of phase transitions and the renormalization group (Perseus Books, 1992).

## Contact

Lecturer: Dr. Luca Giomi