The XVIII Lisbon Autumn School brought together physicists from different areas, ranging from QCD to condensed matter. This subject will be of ever-growing importance in the coming years. The topics covered are: Anomalies, Physical Charges, Chiral Symmetry, Vortices (Superconductivity, Solitons, Kosterlitz-Thouless Transitions), Non-trivial Topology on the Lattice, Confinement (Wilson Loops and Strings, Instantons, Abelian Higgs Model, Dual QCD).

The XVIII Lisbon Autumn School brought together physicists from different areas, ranging from QCD to condensed matter. This subject will be of ever-growing importance in the coming years. The topics covered are: Anomalies, Physical Charges, Chiral Symmetry, Vortices (Superconductivity, Solitons, Kosterlitz-Thouless Transitions), Non-trivial Topology on the Lattice, Confinement (Wilson Loops and Strings, Instantons, Abelian Higgs Model, Dual QCD).

For the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1 = SO(7)1/(G2)1 × (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.

In this thesis, I study a class of exotic quantum matter named Symmetry-Protected Topological (SPT) phases. These are short-range-entangled quantum phases hosting non-trivial states on their boundaries. In the free-fermion limit, they are famously known as Topological Insulators (TI). Huge progress has been made recently in understanding SPT phases beyond free fermions. Here I will discuss three aspects of SPT phases in interacting systems, mostly in three dimensions: (1) Novel SPT phases could emerge in strongly correlated systems, with no non-interacting counterpart. In particular, I will discuss interaction-enabled electron topological insulators, including their classification, construction, characterization and realization. (2) When strong interactions are present, the surface of many SPT phases (including the familiar free fermion topological insulator) can be gapped without breaking any symmetry, at the expense of having intrinsic topological order on the surface. (3) Some topological phases that are non-trivial in the free fermion theory become trivial once strong interactions are introduced. The material of this thesis closely parallels that of Refs. [1, 2, 3, 4, 5, 6].

For many years, the physics of strongly correlated systems was considered a theorists' playground, right at the border with pure mathematics, where physicists from the `real world' did not venture. The time has come, however, when healthy physics cannot exist without these techniques and results. Lectures on selected topics in the theory of strongly correlated systems are here presented by the leading experts in the field. Topics covered include a use of the form factor approach in low-dimensional systems, applications of quantum field theory to disorder, and dynamical mean field theory. The main divisions of the book deal with: I) Quantum Critical Points; (II) Strongly Correlated One-Dimensional Systems; (III) Strong Correlations and Disorder; and (IV) Dynamical Mean Field Theory.

"Topological Systems: Energy and Correlation" is a book that explores the intricate interplay between topology, energy, and correlation in various physical systems, particularly in the realm of quantum matter. The book delves into the fascinating phenomena arising from topological structures and the strong correlations among their constituent particles, such as zero-energy modes, non-trivial topology, and emergent excitations. It discusses the theoretical framework and experimental observations of topological phases of matter, such as topological insulators, superconductors, and semimetals, that exhibit robust and quantized electronic properties that are protected by symmetries and topology. The book is aimed at graduate students and researchers in condensed matter physics, materials science, and related fields who are interested in understanding the fundamental principles and exciting applications of topological systems.

In this book the author extends the concepts introduced in his Quantum Field Theory in Condensed Matter Physics to situations in which the strong electronic correlations are crucial for the understanding of the observed phenomena. Starting from a model field theory to illustrate the basic ideas, more complex systems are analyzed in turn. A special chapter is devoted to the description of antiferromagnets, doped Mott insulators, and quantum Hall liquids from the point of view of gauge theory.