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Markus Penz

Publications and source records attributed to Markus Penz.

At least 37 records · Page 2Linked to original sources
arXiv2024

Exchange energies with forces in density-functional theory

We propose exchanging the energy functionals in ground-state DFT with physically equivalent exact force expressions as a new promising route towards approximations to the exchange-correlation potential and energy. In analogy to the usual energy-based procedure, we split the force difference between the interacting and auxiliary Kohn-Sham system into a Hartree, an exchange, and a correlation force. The corresponding scalar potential is obtained by solving a Poisson equation, while an additional transverse part of the force yields a vector potential. These vector potentials obey an exact constraint between the exchange and correlation contribution and can further be related to the atomic-shell structure. Numerically, the force-based local-exchange potential and the corresponding exchange energy compare well with the numerically more involved optimized effective-potential method. Overall, the force-based method has several benefits when compared to the usual energy-based approach and opens a route towards numerically inexpensive non-local and (in the time-dependent case) non-adiabatic approximations.

physics.chem-ph↗
arXiv2023

The structure of the density-potential mapping. Part II: Including magnetic fields

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. In this Part~II of a series of two articles, we aim at clarifying the status of this theorem within different extensions of DFT including magnetic fields. We will in particular discuss current-density-functional theory (CDFT) and review the different formulations known in the literature, including the conventional paramagnetic CDFT and some non-standard alternatives. For the former, it is known that the Hohenberg-Kohn theorem is no longer valid due to counterexamples. Nonetheless, paramagnetic CDFT has the mathematical framework closest to standard DFT and, just like in standard DFT, non-differentiability of the density functional can be mitigated through Moreau-Yosida regularization. Interesting insights can be drawn from both Maxwell-Schrödinger DFT and quantum-electrodynamical DFT, which are also discussed here.

physics.chem-ph↗
arXiv2023

Density-potential inversion from Moreau-Yosida regularization

For a quantum-mechanical many-electron system, given a density, the Zhao-Morrison-Parr method allows to compute the effective potential that yields precisely that density. In this work, we demonstrate how this and similar inversion procedures mathematically relate to the Moreau-Yosida regularization of density functionals on Banach spaces. It is shown that these inversion procedures can in fact be understood as a limit process as the regularization parameter approaches zero. This sheds new insight on the role of Moreau-Yosida regularization in density-functional theory and allows to systematically improve density-potential inversion. Our results apply to the Kohn-Sham setting with fractional occupation that determines an effective one-body potential that in turn reproduces an interacting density.

physics.chem-ph↗
arXiv2023

The structure of the density-potential mapping. Part I: Standard density-functional theory

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg-Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg-Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.

physics.chem-ph↗
arXiv2022

Polaritonic Hofstadter Butterfly and Cavity-Control of the Quantized Hall Conductance

In a previous work [Phys. Rev. Lett. 123, 047202 (2019)] a translationally invariant framework called quantum-electrodynamical Bloch (QED-Bloch) theory was introduced for the description of periodic materials in homogeneous magnetic fields and strongly coupled to the quantized photon field in the optical limit. For such systems, we show that QED-Bloch theory predicts the existence of fractal polaritonic spectra as a function of the cavity coupling strength. In addition, for the energy spectrum as a function of the relative magnetic flux we find that a terahertz cavity can modify the standard Hofstadter butterfly. In the limit of no quantized photon field, QED-Bloch theory captures the well-known fractal spectrum of the Hofstadter butterfly and can be used for the description of 2D materials in strong magnetic fields, which are of great experimental interest. As a further application, we consider Landau levels under cavity confinement and show that the cavity alters the quantized Hall conductance and that the Hall plateaus are modified as $σ_{xy}=e^2ν/h(1+η^2)$ by the light-matter coupling $η$. Most of the aforementioned phenomena should be experimentally accessible and corresponding implications are discussed.

cond-mat.mes-hall↗
arXiv2022

Fully self-consistent optimized effective potentials from a convex minimization problem

The optimized effective potential method is formulated as a convex minimization problem. This formulation does not require assumptions about $v$-representability nor functional differentiability. The formulation provides a natural framework for fully self-consistent calculations where both a Kohn--Sham system with a non-local potential and an additional system with a local potential are jointly optimized. The formulation is also well suited for extensions to other flavors of density-functional theory, e.g. current-density functional theory, where there are additional potentials besides the ordinary electrostatic potential.

physics.chem-ph↗
arXiv2022

Density-Functional Theory on Graphs

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.

quant-ph↗
arXiv2021

Making ab initio QED functional(s): Non-perturbative and photon-free effective frameworks for strong light-matter coupling

Strong light-matter coupling provides a promising path for the control of quantum matter where the latter is routinely described from first-principles. However, combining the quantized nature of light with this ab initio tool set is challenging and merely developing, as the coupled light-matter Hilbert space is conceptually different and computational cost quickly becomes overwhelming. In this work, we provide a non-perturbative photon-free formulation of quantum electrodynamics (QED) in the long-wavelength limit, which is formulated solely on the matter Hilbert space and can serve as an accurate starting point for such ab initio methods. The present formulation is an extension of quantum mechanics that recovers the exact results of QED for the zero- and infinite-coupling limit, the infinite-frequency as well as the homogeneous limit and we can constructively increase its accuracy. We show how this formulation can be used to devise approximations for quantum-electrodynamical density-functional theory (QEDFT), which in turn also allows to extend the ansatz to the full minimal-coupling problem and to non-adiabatic situations. Finally, we provide a simple local-density-type functional that takes the strong coupling to the transverse photon-degrees of freedom into account and includes the correct frequency and polarization dependence. This is the first QEDFT functional that accounts for the quantized nature of light while remaining computationally simple enough to allow its application to a large range of systems. All approximations allow the seamless application to periodic systems.

quant-ph↗
arXiv2021

Revisiting density-functional theory of the total current density

Density-functional theory requires an extra variable besides the electron density in order to properly incorporate magnetic-field effects. In a time-dependent setting, the gauge-invariant, total current density takes that role. A peculiar feature of the static ground-state setting is, however, that the gauge-dependent paramagnetic current density appears as the additional variable instead. An alternative, exact reformulation in terms of the total current density has long been sought but to date a work by Diener is the only available candidate. In that work, an unorthodox variational principle was used to establish a ground-state density-functional theory of the total current density as well as an accompanying Hohenberg-Kohn-like result. We here reinterpret and clarify Diener's formulation based on a maximin variational principle. Using simple facts about convexity implied by the resulting variational expressions, we prove that Diener's formulation is unfortunately not capable of reproducing the correct ground-state energy and, furthermore, that the suggested construction of a Hohenberg-Kohn map contains an irreparable mistake.

physics.chem-ph↗
arXiv2020

Virial relations for electrons coupled to quantum field modes

In this work we present a set of virial relations for many electron systems coupled to field modes, described by the Pauli--Fierz Hamiltonian in dipole approximation and using length gauge. Currently, there is growing interest in solutions of this Hamiltonian due to its relevance for describing molecular systems strongly coupled to photonic modes in cavities, and in the possible modification of chemical properties of such systems compared to the ones in free space. The relevance of such virial relations is demonstrated by showing a connection to mass renormalization and by providing an exact way to obtain total energies from potentials in the framework of Quantum Electrodynamical Density Functional Theory.

quant-ph↗
arXiv2020

On a combinatorial problem in the Secret Santa ritual

The Secret Santa ritual, where in a group of people every member presents a gift to a randomly assigned partner, poses a combinatorial problem when considering the probabilities involved in the formation of pairs, where two persons exchange gifts mutually. We give different possible derivations for such probabilities by counting fixed-point-free permutations with certain numbers of 2-cycles.

math.HO↗
arXiv2020

Unique Continuation for the Magnetic Schrödinger Equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.

math-ph↗
arXiv2019

Force Balance Approach for Advanced Approximations in Density Functional Theories

We propose a systematic and constructive way to determine the exchange-correlation potentials of density-functional theories including vector potentials. The approach does not rely on energy or action functionals. Instead it is based on equations of motion of current quantities (force balance equations) and is feasible both in the ground-state and the time-dependent setting. This avoids, besides differentiability and causality issues, the optimized-effective-potential procedure of orbital-dependent functionals. We provide straightforward exchange-type approximations for different density functional theories that for a homogeneous system and no external vector potential reduce to the exchange-only local-density and Slater X$α$ approximations.

physics.chem-ph↗
arXiv2019

Quantum Electrodynamical Bloch Theory with Homogeneous Magnetic Fields

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.

cond-mat.mes-hall↗
arXiv2019

Kohn-Sham theory with paramagnetic currents: compatibility and functional differentiability

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-defined Kohn-Sham iteration scheme with a partial convergence result. To this end, we rely on a formulation of Moreau-Yosida regularization for reflexive and strictly convex function spaces. The optimal $L^p$-characterization of the paramagnetic current density $L^1\cap L^{3/2}$ is derived from the $N$-representability conditions. A crucial prerequisite for the convex formulation of paramagnetic current-density-functional theory, termed compatibility between function spaces for the particle density and the current density, is pointed out and analyzed. Several results about compatible function spaces are given, including their recursive construction. The regularized, exact functionals are calculated numerically for a Kohn-Sham iteration on a quantum ring, illustrating their performance for different regularization parameters.

physics.chem-ph↗
arXiv2018

Regularity for evolution equations with non-autonomous perturbations in Banach spaces

We provide regularity of solutions to a large class of evolution equations on Banach spaces where the generator is composed of a static principal part plus a non-autonomous perturbation. Regularity is examined with respect to the graph norm of the iterations of the principal part. The results are applied to the Schrödinger equation and conditions on a time-dependent scalar potential for regularity of the solution in higher Sobolev spaces are derived.

math-ph↗
arXiv2018

Generalized Kohn-Sham iteration on Banach spaces

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.

physics.chem-ph↗