Quantum information theory and strongly correlated electron systems share a common theme of massive quantum entanglement. In both topological error correction codes and theories of quantum materials (spin liquid, heavy fermion and high-Tc systems), this entanglement is implemented by means of an emergent gauge symmetry. Inspired by these connections, I will first review the emergence of dynamical gauge fields in solid state theory, concentrating on lattice models with constrained Hilbert spaces. Subsequently, the main focus of the lecture is to introduce the simplest kind of topological order, which appears in Z2 gauge theories and, equivalently, in Kitaev's toric code. More general topologically ordered states will be discussed in passing.
Electrons interweaving with topological order will be introduced by means of a simple model of fermions coupled to Kitaev's toric code. This model has the benefit of an exactly solvable ground state and subtleties in Luttinger's theorem can be explicitly exemplified. The arguably simplest non-Fermi liquid, the orthogonal metal, is introduced in this context and experimental relevance in the context of cuprate and heavy-fermion materials will be discussed. If time permits, I will conclude the talks with recent work on the interplay of fermions with emergent U(1) gauge fields and with a conjecture of the notion of topological order in impurity models.