Schedule:

• Monday, July 6th, from 10AM to 12PM and from 2PM to 4PM
• Tuesday, July 7th, from 10AM to 12PM.

Lecture 1: The renormalisation group and functional methods for quantum many-body systems This first lecture will introduce the overall concept of the renormalisation group, including a brief history of its development. Then, we will revise the path-integral formalism to describe dilute quantum gases and the Matsubara approach to account for finite temperatures. Specifically, we will examine this formalism for describing Bose gases, which will enable us to extract the known mean-field and Gaussian solutions. After understanding these functional methods, we will be able to define the effective action, which will become one of the central physical objects of these lectures. Finally, we will introduce the functional renormalisation group (FRG) approach, including its flow equation.

Lecture 2: Functional renormalisation group for Bose gases Having introduced the FRG, this lecture will focus on its application to Bose gases. Firstly, this lecture will focus on Bose gases around the superfluid phase transition, which can be described by a classical O(2)-model by means of a dimensionality reduction. This will enable us to derive simple, explicit FRG flow equations for the couplings of the model. We will examine how the couplings flow at different scales and determine the phase transition properties. After examining this classical model, we will study the full quantum Bose gas. We will derive the full set of flow equations and perform numerical calculations to extract physical observables. The obtained observables will be compared with known solutions obtained with other approaches.

Lecture 3: Functional renormalisation group for Fermi gases and other applications After examining bosonic systems, we will extend the FRG approach to fermionic degrees of freedom. To do this, we will introduce Grassmann fields with anticommutative properties. This will enable us to study a spin-1/2 Fermi gas and its strongly correlated BCS-BEC crossover, including the Unitary limit. We will compare the FRG results with Quantum Monte Carlo simulations, which will show the high predictive power of the FRG, despite its modest computational cost. To finish these lectures, we will then briefly summarise the application of the FRG to related systems, including lattice configurations, nuclear matter, and few-body physics. About the speaker: I am a theoretical physicist specialised in quantum many-body calculations of ultracold atomic systems. I did my PhD at the University of Manchester, where I specialised in functional methods for quantum gases. Since then, I have held postdoctoral positions at the University of Barcelona, the University of Glasgow, and the Catholic University of Chile, where I have worked on a variety of problems, including quantum mixtures, polarons, and atomtronic circuits. The overall aim of my research is to combine state-of-the-art theoretical and computational techniques to address many-body systems across different configurations, such as different geometries, statistics, among others. During July, I will do a research stay in Barcelona as a visiting researcher funded by the Catalonia Quantum Academy.