In this work, we give a comprehensive derivation of an exact and numerically feasible method to perform ab initio calculations of quantum particles interacting with a quantized electromagnetic field. We present a hierarchy of density-functional-type theories that describe the interaction of charged particles with photons and introduce the appropriate Kohn-Sham schemes. We show how the evolution of a system described by quantum electrodynamics in Coulomb gauge is uniquely determined by its initial state and two reduced quantities. These two fundamental observables, the polarization of the Dirac field and the vector potential of the photon field, can be calculated by solving two coupled, nonlinear evolution equations without the need to explicitly determine the (numerically infeasible) many-body wave function of the coupled quantum system. To find reliable approximations to the implicit functionals, we present the appropriate Kohn-Sham construction. In the nonrelativistic limit, this density-functional-type theory of quantum electrodynamics reduces to the densityfunctional reformulation of the Pauli-Fierz Hamiltonian, which is based on the current density of the electrons and the vector potential of the photon field. By making further approximations, e.g., restricting the allowed modes of the photon field, we derive further density-functional-type theories of coupled matter-photon systems for the corresponding approximate Hamiltonians. In the limit of only two sites and one mode we deduce the appropriate effective theory for the two-site Hubbard model coupled to one photonic mode. This model system is used to illustrate the basic ideas of a density-functional reformulation in great detail and we present the exact Kohn-Sham potentials for our coupled matter-photon model system.

We propose an orbital exchange-correlation functional for applying time-dependent density functional theory to many-electron systems coupled to cavity photons. The time nonlocal equation for the electronphoton optimized effective potential (OEP) is derived. In the static limit our OEP energy functional reduces to the Lamb shift of the ground state energy. We test the new approximation in the Rabi model. It is shown that the OEP (i) reproduces quantitatively the exact ground-state energy from the weak to the deep strong coupling regime and (ii) accurately captures the dynamics entering the ultrastrong coupling regime. The present formalism opens the path to a first-principles description of correlated electron-photon systems, bridging the gap between electronic structure methods and quantum optics for real material applications. [7,8]. The description of realistic coupled matterphoton systems requires combining electronic structure methods from materials science with quantum optical models. Recently, a time-dependent density-functional theory (TDDFT) for QED has been developed [9][10][11][12], allowing for such a first-principles treatment. However, any application of this theory requires approximations to the electron-photon exchange-correlation (xc) functional, which are currently not available.In this Letter we construct the first approximation to the xc functional of QED-(TD)DFT. To achieve this goal, we extend the widely used optimized effective potential (OEP) approach in electronic structure methods [13][14][15][16][17][18] to the photon-mediated electron-electron coupling. The new functional is tested from low to high coupling regime in the Rabi model [19][20][21], through comparison with the exact and classical solutions. We also address the functional dependence on the initial many-body state, assumed to be either a fully interacting or a factorizable state. In both cases, the electron-photon OEP for the model performs well, providing a promising path to the ab initio description of strongly coupled matter-photon systems.Consider a system with an arbitrarily large number N of electrons at coordinates fr i g N i¼1 , e.g., an atom, an ion, or a molecule, interacting with M quantized electromagnetic modes of a microcavity with frequencies ω α . We denote bŷ H 0 ¼T þV ee þV ext the Hamiltonian of the electronic system with kinetic energyT, Coulomb interactionV ee , and (time-dependent) external potentialV ext ¼ P N i¼1 v ext ðr i tÞ, due to the nuclei and any classical field applied to the system. In the dipole approximation [22] the length-gauge Hamiltonian [9,23,24] of the total electron-photon system can be represented as follows:where the second term corresponds to the energy 1=8π R ðB 2 þÊ 2 Þdr of the transverse radiation field. The magnetic fieldB α ¼ ffiffiffiffiffi ffi 4π pp α in the α mode is proportional to the photon canonical momentump α , while the electric fieldÊ α ¼ ffiffiffiffiffi ffi 4π p ðω αqα − λ αR Þ is related to the canonical coordinateq α . The latter is defined via the displacemen...

We propose a practical alternative to Eliashberg equations for the ab initio calculation of superconducting transition temperatures and gap functions. Within the recent density functional theory for superconductors, we develop an exchange-correlation functional that retains the accuracy of Migdal's approximation to the many-body electron-phonon self-energy, while having a simple analytic form. Our functional is based on a parametrization of the Eliashberg self-energy for a superconductor with a single Einstein frequency, and enables density functional calculations of experimental excitation gaps. By merging electronic structure methods and Eliashberg theory, the present approach sets a new standard in quality and computational feasibility for the prediction of superconducting properties.

Designing materials with advanced functionalities is the main focus of contemporary solid-state physics and chemistry. Research efforts worldwide are funneled into a few high-end goals, one of the oldest, and most fascinating of which is the search for an ambient temperature superconductor (A-SC). The reason is clear: superconductivity at ambient conditions implies being able to handle, measure and access a single, coherent, macroscopic quantum mechanical state without the limitations associated with cryogenics and pressurization. This would not only open exciting avenues for fundamental research, but also pave the road for a wide range of technological applications, affecting strategic areas such as energy conservation and climate change. In this roadmap we have collected contributions from many of the main actors working on superconductivity, and asked them to share their personal viewpoint on the field. The hope is that this article will serve not only as an instantaneous picture of the status of research, but also as a true roadmap defining the main long-term theoretical and experimental challenges that lie ahead. Interestingly, although the current research in superconductor design is dominated by conventional (phonon-mediated) superconductors, there seems to be a widespread consensus that achieving A-SC may require different pairing mechanisms. In memoriam, to Neil Ashcroft, who inspired us all.