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Condensed Matter Systems in Cavity Quantum Electrodynamics

Condensed matter physics and quantum electrodynamics (QED) have been long considered as distinct disciplines. This situation is changing by the progress in cavity QED materials. Motivated by these advances we aim to bridge these fields by merging fundamental concepts coming from both sides. In the first part of the thesis we present how non-relativistic QED can be constructed and we discuss the light-matter interaction in different gauges and that neglecting particular quadratic terms can lead to instabilities. In the second part, we revisit the Sommerfeld model of the free electron gas in cavity QED and provide the analytic solution for this paradigmatic system coupled to the cavity. We show that the cavity field modifies the optical conductivity of the electron gas and suppresses its Drude peak. Further, by constructing an effective field theory in the continuum of photon modes we show how the photon field leads to a many-body renormalization of the electron mass, which modifies the fermionic quasiparticle excitations of the Fermi liquid. In the last part, we show that translational symmetry for periodic materials in homogeneous magnetic fields can be restored by embedding the problem into QED. This leads to a generalization of Bloch's theory for electron-photon systems, that we named as QED-Bloch theory, which can be applied for the description of periodic materials in homogeneous magnetic fields and strongly coupled to the quantized cavity field. As a first application we consider Landau levels coupled to a cavity and we show that quasiparticle excitations between Landau levels and photons appear, called Landau polaritons. Further, for periodic materials in such setups, QED-Bloch theory predicts the emergence of novel fractal polaritonic energy spectra, which we name as fractal polaritons. The fractal polaritons are a polaritonic, QED analogue of the Hofstadter butterfly.