Most theoretical studies for correlated light-matter systems are performed within the longwavelength limit, i.e., the electromagnetic field is assumed to be spatially uniform. In this limit the so-called length-gauge transformation for a fully quantized light-matter system gives rise to a dipole self-energy term in the Hamiltonian, i.e., a harmonic potential of the total dipole matter moment. In practice this term is often discarded as it is assumed to be subsumed in the kinetic energy term. In this work we show the necessity of the dipole self-energy term. First and foremost, without it the light-matter system in the long-wavelength limit does not have a ground-state, i.e., the combined light-matter system is unstable. Further, the mixing of matter and photon degrees of freedom due to the length-gauge transformation, which also changes the representation of the translation operator for matter, gives rise to the Maxwell equations in matter and the omittance of the dipole self-energy leads to a violation of these equations. Specifically we show that without the dipole self-energy the so-called "depolarization shift" is not properly described. Finally we show that this term also arises if we perform the semi-classical limit after the length-gauge transformation. In contrast to the standard approach where the semi-classical limit is performed before the length-gauge transformation, the resulting Hamiltonian is bounded from below and thus supports ground states. This is very important for practical calculations and for density-functional variational implementations of the non-relativistic QED formalism. For example, the existence of a combined light-matter ground state allows to calculate the Stark shift non-perturbatively. *

Experiments at the interface of quantum-optics and chemistry have revealed that strong coupling between light and matter can substantially modify chemical and physical properties of molecules and solids. While the theoretical description of such situations is usually based on non-relativistic quantum electrodynamics, which contains quadratic light-matter coupling terms, it is commonplace to disregard these terms and restrict to purely bilinear couplings. In this work we clarify the physical origin and the substantial impact of the most common quadratic terms, the diamagnetic and self-polarization terms, and highlight why neglecting them can lead to rather unphysical results. Specifically we demonstrate its relevance by showing that neglecting it leads to the loss of gauge invariance, basis-set dependence, disintegration (loss of bound states) of any system in the basis set-limit, unphysical radiation of the ground state and an artificial dependence on the static dipole. Besides providing important guidance for modeling strongly coupled light-matter systems, the presented results do also indicate under which conditions those effects might become accessible. * Electronic address: christian.schaefer@mpsd.mpg.de †

We propose a solution to the problem of Bloch electrons in a homogeneous magnetic field by including the quantum fluctuations of the photon field. A generalized quantum electrodynamical (QED) Bloch theory from first principles is presented. In the limit of vanishing quantum fluctuations we recover the standard results of solid-state physics, for instance, the fractal spectrum of the Hofstadter butterfly. As a further application we show how the well known Landau physics is modified by the photon field and that Landau polaritons emerge. This shows that our QED-Bloch theory does not only allow to capture the physics of solid-state systems in homogeneous magnetic fields, but also novel features that appear at the interface of condensed matter physics and quantum optics.Cavity QED materials is a growing research field bridging quantum optics [1, 2], polaritonic chemistry [3-7], and materials science, such as light-induced new states of matter achieved with classical laser fields [8,9]. Photonmatter interactions have recently been suggested to modify electronic properties of solids, such as superconductivity and electron-phonon coupling [10][11][12][13][14]. On the other hand, materials in classical magnetic fields are known to give rise to a variety of novel phenomena such as the Landau levels [15], the integer [16,17] and the fractional quantum Hall effect [18], and the quantum fractal of the Hofstadter butterfly [19] which can be now accessed experimentally with high resolution [20][21][22]. One of the open questions in this field is whether Bloch theory is applicable for solids in the presence of a homogeneous magnetic field. The homogeneous magnetic field breaks explicitly translational symmetry. This issue was solved to some extent by introducing the magnetic translation group. However, the magnetic translation group puts fundamental limitations on the possible directions and values of the strength of the magnetic field [17,23,24].In this Letter, by combining QED with solid-state physics, we provide a consistent and comprehensive theory for solids interacting with homogeneous electromagnetic fields, both classical and quantum. Our main findings are as follows: (i) The quantum fluctuations of the electromagnetic field allow us to restore translational symmetry that is broken due to an external homogeneous magnetic field (see Fig. 1). (ii) We generalize Bloch theory and provide a Bloch central equation for electrons in a solid in the presence of a homogeneous magnetic field and its quantum fluctuations. (iii) Applying our framework to the case of a 2D solid in a perpendicular homogeneous magnetic field, in the limit of no quantum fluctuations, we recover the fractal spectrum of the Hofstadter butterfly (see Fig. 2). (iv) In the case of a 2D * vasil.rokaj@mpsd.mpg.de † angel.rubio@mpsd.mpg.de A A A A A u y x v w = 1 A = 2 A = 3 A = 4 A = 5 A = 1 = 2 = 3 = 4 = 5 a y √ 2 ωc ay A t o t = c o n s t ext tot tot tot tot tot ext ext ext ext FIG.