In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schrödinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.
Abstract. We derive some rigorous results concerning the backflow operator introduced by Bracken and Melloy. We show that it is linear bounded, self adjoint, and not compact. Thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator. From the position representation of the backflow operator we obtain a more efficient method to determine the backflow constant. Finally, detailed position probability flow properties of a numerical approximation to the (perhaps improper) wave function of maximal backflow are displayed.
In a recent letter [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed point problem. This idea was used to generalize the existence and uniqueness theorems underlying timedependent density functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case as it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that in the most physically relevant cases the fixed point procedure converges. This is further demonstrated with an example.
A detailed account of the Kohn-Sham algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows to rigorously introduce, in contrast to the common unregularized approach, a well-defined Kohn-Sham iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.
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.
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