# Ten-week online school

Perimeter Institute hosted a 10-week online school from May 24 to July 28, 2022. Participants learned about cutting-edge topics in modern theoretical physics through four courses, group projects on specialized topics, as well as other special events such as presentations from researchers and panel discussions. The school involved 4.5 hours of synchronous content and one-to-two hours of independent work or study each week.

Courses for the 2022 online school:

- Path integrals
- Symmetries
- Topics in quantum science
- Numerical methods

In order to accommodate students from different time zones, online school content is offered on Tuesdays and Thursdays at two different times each day: 9 am to 11:15 am, and 3:15 pm to 5:30 pm (**Eastern Time**). You can check these times in your own time zone here and here. Students will specify their time preferences in the online application.

## Sample course outlines

Below are some sample course outlines from the 2020 online program.

Note: These course outlines are for previous versions of the courses and do not reflect the topics that were covered in the 2022 PSI Start program.

#### Course description

This course has two main goals: (1) to introduce some key models from condensed matter physics; and (2) to introduce some numerical approaches to studying these (and other) models. As a precursor to these objectives, we will carefully understand many-body states and operators from the perspective of condensed matter theory. (However, I will cover only spin models. We will not discuss or use second quantization.)

Once this background is established, we will study the method of exact diagonalization and write simple python programs to find ground states, correlation functions, energy gaps, and other properties of the transverse-ﬁeld Ising model and XXZ model. We will also discuss the computational limitations of exact diagonalization. Finally, I will introduce the concept of matrix product states, and we will see how these can be used with algorithms such as the density matrix renormalization group (DMRG) to study ground state properties for much larger systems than can be studied with exact diagonalization.

Each session will include substantial programming exercises in addition to lecture. Prior programming experience is not expected or required, but I would like everyone to have python (version 3) installed on their computer prior to the first class, including Jupyter notebooks; see “Resources” below, and please email me about any difficulty with the installation.

#### Instructor information

#### Learning outcomes

By the end of this course students should:

- Understand many-body Hilbert space and the operators that act on it, as well as the assumptions typically made in the setting of condensed matter physics
- Be familiar with some of the basic models of condensed matter, namely Ising and XXZ
- Be able to write simple python programs to study these models by exact diagonalization, and understand the limitations of this method
- Understand the concept of matrix product states, as well as algorithms like iTEBD and DMRG that are based on matrix product states

#### Resources

We will use Python 3 with Jupyter notebooks. If you have not used these before, I recommend installing them using Anaconda. I will distribute partially completed programs in the form of Jupyter notebooks, and your exercises will consist primarily of filling in the missing pieces.

#### Tentative course schedule

Lecture 1 | Introduction to many-particle states and operators, and to Ising and XXZ models; programming basics; finding expectation values |

Lecture 2 | Exact diagonalization part 1: representing models; finding eigenstates, energy gaps, and phase transitions |

Lecture 3 | Exact diagonalization part 2: limitations of the method; using symmetries; (if time permits) dynamics |

Lecture 4 | Matrix product states part 1: entanglement and the singular value decomposition, what is a matrix product state and why is it useful? |

Lecture 5 | Matrix product states part 2: algorithms for finding ground states using matrix product states -- iTEBD and DMRG |

#### Course description

The goal of this course is to introduce the path integral formulation of quantum mechanics and a few of its applications. We will begin by motivating the path integral formulation and explaining its connections to other formulations of quantum mechanics and its relation to classical mechanics. We will then explore some applications of path integrals.

Each session will include roughly equal amounts of lecture time and activities. The activities are designed to enhance your learning experience and allow you to assess your own level of understanding.

#### Instructor information

#### Learning outcomes

By the end of this course students should be able to:

- Explain the connection between quantum mechanics and statistical physics
- Use the saddle-point approximation to evaluate path integrals
- Determine tunneling rates using the instanton method
- Explain the connection between particle statistics and spacetime dimension

#### Resources

For a part of each class you will collaboratively work on problems in groups of 3 or 4 in Zoom breakout rooms. Before the course begins you should determine a method to share mathematical expressions over Zoom. You will be provided with a document that gives several suggestions.

#### Tentative course schedule

Lecture 1 | Introduction to path integrals and the semi-classical limit |

Lecture 2 | Propagator in real and imaginary time |

Lecture 3 | Perturbation theory |

Lecture 4 | Non-perturbative physics and quantum tunneling |

Lecture 5 | Topology and path integrals |

#### Course description

The aim of this course is to understand the thermodynamics of quantum systems and in the process to learn some fundamental tools in Quantum Information. We will focus on the topics of foundations of quantum statistical mechanics, resource theories, entanglement, fluctuation theorems, and quantum machines.

Each session will include roughly equal amounts of lecture time and tutorials, in which you will work in groups to solve problems related to the lecture content.

#### Instructor information

Alioscia Hamma

#### Learning outcomes

By the end of this course students should be able to:

- Understanding how statistical mechanics emerges from the unitary evolution of a quantum system.
- Understanding the concepts of work and thermal machines in a quantum setting.
- Understanding the quantum thermodynamic demons, Landauer's principle and the like.
- Becoming ready to study the literature in the ﬁeld and to start a research project.

#### Tentative course schedule

Lecture 1 | Foundations of Quantum Statistical Mechanics - Entanglement |

Lecture 2 | Resource Theories and Quantum Information |

Lecture 3 | Quantum Thermal Operations |

Lecture 4 | Fluctuation Theorems and Quantum Information |

Lecture 5 | Quantum Thermal Machines |

#### Course description

The aim of this course is to explore some of the many ways in which symmetries play a role in physics. We’ll start with an overview of the concept of symmetries and their description in the language of group theory. We will then discuss continuous symmetries and infinitesimal symmetries, their fundamental role in Noether's theorem, and their formalisation in terms of Lie groups and Lie algebras. In the last part of the course we will focus on symmetries in quantum theory and introduce representations of (Lie) groups and Lie algebras.

Each session will include roughly equal amounts of lecture time and activities. The activities are designed to enhance your learning experience and allow you to assess your own level of understanding.

#### Instructor information

#### Learning outcomes

By the end of this course students should be able to:

- Evaluate the symmetries of a Lagrangian or action functional and construct the associated Noether's charges/currents
- Construct the Lie algebras for the classical Lie groups and specify their structure constants
- Classify the unitary irreducible representations of some simple Lie groups such as U(1), SO(3), and SU(2)
- Justify the appearance of spin in quantum mechanics from the point of view of representation theory

#### Resources

These are some of the resources that we will use during the lectures/activities:

- Socrative and Slido. These are online apps that we will use for activities. They don’t need to be installed (they can run from a browser) and don’t require an account.
- GeoGebra. Willed be used this to show you interactive simulations or visualisations during the lectures, and then share the applets that are created with you. You can either use the online version or install it, which is recommended since it’s a very useful piece of (open-source) software.

#### Tentative course schedule

Lecture 1 | Overview/definition of symmetry, elements of group theory, examples of applications of symmetries in physical problems |

Lecture 2 | Continuous and discrete symmetries, infinitesimal symmetries, Noether's theorem |

Lecture 3 | Lie groups and Lie algebras |

Lecture 4 | Symmetries in quantum mechanics |

Lecture 5 | Representation theory |

## Main Navigation

*The PSI Start program is supported by Michael Serbinis and Laura Adams. *