Kohn Theorem Research Articles

Solution of the υ-representability problem on a one-dimensional torus

Abstract We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schrödinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn–Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg–Kohn theorem.

Wave functions consistent with experimental x-ray diffraction data: A hircocervus becomes reality

Since the early days of quantum physics, the possibility of obtaining wave functions consistent with experimental x-ray diffraction data has been envisioned. The idea is firmly grounded in the postulates of quantum mechanics and finds full support in the Hohenberg and Kohn theorem and Levy–Lieb search formulation of density functional theory. Within this framework, a rich history of research has unfolded over the years, introducing various strategies to obtain plausible one-electron reduced density matrices or wave functions that are compatible with x-ray structure factors. Approximately twenty-five years ago, all of this culminated with the development of the x-ray restrained wave function (XRW) approach. This method aims to determine wave functions that minimize the electronic energy of the examined systems while maximizing the statistical agreement between experimental and calculated x-ray diffraction data. Presently, the XRW technique stands as a well-established strategy, manifesting in various forms, and addressing numerous problems and challenges across chemistry, physics, and materials science. Moreover, there remains large room for improvement and extensions in the coming years. This paper will comprehensively review the current state of the x-ray restrained wave function approach, discussing its underlying foundations, historical background, theoretical details and extensions, practical applications, and forthcoming perspectives.

Extension of Bright Wilson's justification of the first Hohenberg Kohn theorem to non-nuclear maxima (NNM)

Extension of Bright Wilson's justification of the first Hohenberg Kohn theorem to non-nuclear maxima (NNM)

Probing many-body effects in harmonic traps with twisted light

We explore the potential of twisted light, a structured beam carrying orbital angular momentum, as a tool to unveil many-body effects in parabolically confined systems. According to the generalized Kohn theorem, the dipole response of such a multiparticle system to a spatially homogeneous probe is indistinguishable from the response of a system of noninteracting particles. Twisted light however can excite internal degrees of freedom, resulting in the appearance of new peaks in the multipole spectrum which are not present when the probe is a plane wave. We also demonstrate the ability of the proposed twisted light probe to capture the transition of interacting fermions into a strongly correlated regime in a one-dimensional harmonic trap. We report that, by suitable choice of the probe's parameters, the transition into a strongly correlated phase manifests itself as an approaching and ultimate superposition of peaks in the second-order quadrupole response. These features are observed in exact calculations for two electrons and well reproduced in adiabatic time-dependent density-functional theory simulations.

An exact one-particle theory of bosonic excitations: from a generalized Hohenberg–Kohn theorem to convexified N-representability

Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.

Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems

AbstractThe Hohenberg–Kohn (HK) theorem for interacting electrons is a cornerstone of modern electronic structure calculations. For a general quantum system, a HK‐type Hamiltonian in the form of can always be defined, by grouping those terms with fixed or preknown coefficients into an internal part of the Hamiltonian , and factorizing the remaining as the superposition of a set of Hermitian operators . It is asked whether the HK theorem can be extended to such a general setting, so that the ground‐state expectation values of as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. It is shown that the question can be addressed by the invertibility of generalized density correlation matrix (GDCM) defined with respect to the operators. This criterion is applied to several representative examples, including the quantum Ising dimer, frustration‐free systems, N‐level quantum systems and a fermionic Hubbard chain. It is suggested that for a finite‐size system, finding an invertible GDCM under one single configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space.

A remark on Kohn's theorem on sums of squares of complex vector fields

The plan of this paper is to give an alternate proof of Kohn's subelliptic estimate for systems of N smooth complex vector fields on an open set of Rn, and to improve it in extending the result to perturbations by a first-order term. A pseudodifferential generalization will also be given.

Probing many-body interactions in the cyclotron resonance of h -BN/bilayer graphene/ h -BN

Unlike a conventional two-dimensional electron gas system, which has parabolic band structure, the nonparabolic band dispersion of mono- to few-layer graphene violates Kohn's theorem. Thus, Landau levels (LLs) in graphene are sensitive to many-body interactions. This modifies the LL spacing, depending on the location of the Fermi energy (${E}_{\mathrm{F}}$). Such effects have been extensively studied in $h$-BN/monolayer graphene/$h$-BN through observation of inter-LL optical transitions known as cyclotron resonances (CRs). However, thus far, the influence of many-body interactions on the CR of bilayer graphene (BLG) has been rarely studied, even though BLG also possesses nonparabolic band dispersion. Here, we investigate CR in the $h$-BN/BLG/$h$-BN structure via magneto-photothermoelectric measurements under infrared laser irradiation. This method enables sensitive detection of cyclotron resonances while tuning ${E}_{\mathrm{F}}$ of BLG. The CR magnetic field value shifted significantly when ${E}_{\mathrm{F}}$ of BLG approached the charge-neutrality point (the Dirac point, DP). We attribute this to a change in the Fermi velocity of BLG near the DP, which occurs as a result of many-body interactions.

Polaritonic nonlocality in light–matter interaction

Sub-wavelength electromagnetic field localization has been central in photonic research in the last decade, allowing to enhance sensing capabilities as well as increasing the coupling between photons and material excitations. The ultrastrong light-matter coupling regime in the THz range with split-ring resonators coupled to magnetoplasmons has been widely investigated, achieving successive world-records for the largest light-matter coupling ever achieved. Ever shrinking resonators have allowed to approach the regime of few electrons strong coupling, in which single-dipole properties can be modified by the vacuum field. Here we demonstrate, theoretically and experimentally, the existence of a limit to the possibility of arbitrarily increasing electromagnetic confinement in polaritonic systems. Strongly sub-wavelength fields can excite a continuum of high-momenta propagative magnetoplasmons. This leads to peculiar nonlocal polaritonic effects, as certain polaritonic features disappear and the system enters in the regime of bound-to-continuum strong coupling. Emerging nonlinearities due to the local breaking of Kohn's theorem are also reported.

Non-Nuclear maxima and the universality of Bright Wilson’s justification of the first Hohenberg Kohn theorem revisited

The Kato cusp condition involves a spherically-averaged derivative of the electron density with respect to distance from a given nucleus weighted by the electron density at that nucleus. This “Kato ratio” is shown analytically to vanish at all points of the electron density (and of the underlying external potential) that do not exhibit a singularity including non-nuclear maxima/attractors (NNM/NNAs). Using prototypical test cases, the analytical result is confirmed numerically. This result, while quite simply demonstrated and possibly tacitly known, could not be located in the literature to the best of the authors’ efforts. Since the Kato ratio is zero at a NNM/NNA, this saves the DFT plausibility argument of E. Bright Wilson. The consideration of these ideas in excited electronic states is discussed.

THz intersubband absorption in n-type Si1−xGex parabolic quantum wells

High-quality n-type continuously graded Ge-rich Si1−xGex parabolic quantum wells with different doping levels were grown by using ultrahigh-vacuum chemical vapor deposition on Si(001) substrates. A thorough structural characterization study highlights an ideal parabolic compositional profile. THz intersubband absorption has been investigated in modulation-doped samples and samples directly doped in the wells. The comparison of experimental absorption data and theoretical calculations allowed us to quantify the impact of electron correlation effects on the absorption resonances in the different doping conditions and for electron sheet densities in the (1÷6)×1011 cm−2 range. A single optical resonance is present in modulation doped samples. Its peak energy and line shape are independent of temperature-induced variations of the electron distribution in the subbands up to 300 K, in agreement with the generalized Kohn theorem. This achievement represents a relevant step forward for the development of CMOS compatible optoelectronic devices in the THz spectral range, where thermal charge fluctuations play a key role.

Variational properties of auxiliary density functionals

The evolution of variational Coulomb fitting from a purely practical scheme to reduce computational burden to a formal variant of Hohenberg–Kohn–Sham density functional theory (auxiliary density functional theory, ADFT) is discussed. After a summary of the historical evolution, an analysis of its connection with the Hohenberg–Kohn theorem is given, some implications for the Euler equation and for time-dependent DFT are given and some implications for the deMon2k code delineated.

Microscopic derivation of density functional theory for superfluid systems based on effective action formalism

Abstract A density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and non-local pairing densities and demonstrate that the Hohenberg–Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from $0$ to $1$, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn–Sham potential generalized to include the pairing potentials. The resultant Kohn–Sham potential has a nice feature in that it expresses the microscopic formulae of the external, Hartree, pairing and exchange–correlation terms separately. It is shown that our Kohn–Sham potential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglecting the correlations. An advantage of our exact formalism lies in the fact that it provides ways to improve the correlation part systematically.

Long-wave Absorption of Few-Hole Gas in Prolate Ellipsoidal Ge/Si Quantum Dot: Implementation of Analytically Solvable Moshinsky Model.

In this paper, the behavior of a heavy hole gas in a strongly prolate ellipsoidal Ge/Si quantum dot has been investigated. Due to the specific geometry of the quantum dot, the interaction between holes is considered one-dimensional. Based on the adiabatic approximation, it is shown that in the z-direction, hole gas is localized in a one-dimensional parabolic well. By modeling the potential of pair interaction between holes in the framework of oscillatory law, the problem is reduced to a one-dimensional, analytically solvable Moshinsky model. The exact energy spectrum of the few-hole gas has been calculated. A detailed analysis of the energy spectrum is presented. The character of long-wave transitions between the center-of-mass levels of the system has been obtained when Kohn theorem is realized.

Realization of Harmonic Oscillator Arrays with Graded Semiconductor Quantum Wells.

The harmonic oscillator is a foundational concept in both theoretical and experimental quantum mechanics. Here, we demonstrate harmonic oscillators in a semiconductor platform by faithfully implementing continuously graded alloy semiconductor quantum wells. Unlike current technology, this technique avoids interfaces that can hamper the system and allows for the production of multiwell stacks several micrometers thick. The experimentally measured system oscillations are at 3THz for two structures containing 18 and 54 parabolic quantum wells. Absorption at room temperature is achieved: this is as expected from a parabolic potential and is unlike square quantum wells that require cryogenic operation. Linewidths below 11% of the central frequency are obtained up to 150K, with a 5.6% linewidth obtained at 10K. Furthermore, we show that the system correctly displays an absence of nonlinearity despite electron-electron interactions-analogous to the Kohn theorem. These high-quality structures already open up several new experimental vistas.

Many-body effects in nodal-line semimetals: Correction to the optical conductivity

Coulomb interaction might have important effects on the physical observables in topological semimetals with vanishing density of states at the band touching due to the weak screening. In this work, we show that Kohn's theorem is not fulfilled in nodal-line semimetals (NLSMs), which implies non-vanishing interaction corrections to the conductivity. Using renormalized perturbation theory, we determine the first-order optical conductivity in a clean NLSM to be $\sigma_{\perp \perp}(\Omega) = 2 \sigma_{\parallel \parallel}(\Omega) = \sigma_0 [1 + C_2 \alpha_R(\Omega)]$, where $\perp$ and $\parallel$ denote the perpendicular and parallel components with respect to the nodal loop, $\sigma_0 = (2 \pi k_0) e^2/(16h)$ is the conductivity in the noninteracting limit, $2 \pi k_0$ is the nodal loop perimeter, $C_2 = (19-6\pi)/12 \simeq 0.013$ is a numerical constant and $\alpha_R(\Omega)$ is the renormalized fine structure constant in the NLSM. The analogies between NLSMs and 2D Dirac fermions are reflected in the universal character of the correction $C_2 \alpha_R(\Omega)$, which is exactly parallel to that of graphene. Finally, we analyze some experiments that have determined the optical conductivity in NLSMs, discussing the possibility of experimentally measuring our result.

Information‐theoretic approach in density functional theory and its recent applications to chemical problems

AbstractUsing simple functionals of the electron density to appreciate and quantify molecular structure and chemical reactivity properties is a recent endeavor in density functional theory (DFT) toward the development of a new chemical reactivity theory. According to the first Hohenberg–Kohn theorem in DFT, the electron density alone should be able to determine any property in the ground state. Exchange and correlation energies are such properties, so are molecular structure and chemical reactivity, and hence they all should accurately be determined by electron density functionals. Quantities such as Shannon entropy, Fisher information, Ghosh–Berkowitz–Parr entropy, information gain, Onicescu information energy, etc., from the information‐theoretic approach are simple electron density functionals, whose analytical forms are exactly known. In this article, we demonstrate their usefulness and validity to quantify regioselectivity, stereoselectivity, and other structure and reactivity properties. We will outline the current understanding of its theoretical framework at first, and then highlight recent applications to chemical problems including isomeric and conformational stability, electrophilicity and nucleophilicity, strong covalent and weak noncovalent interactions, acidity and basicity, aromaticity and antiaromaticity, and numerous other properties. The effort of employing electron density functionals to quantify chemical concepts should open up a new door for us to ultimately develop a chemical reactivity theory using the DFT language.This article is characterized under:Structure and Mechanism > Reaction Mechanisms and CatalysisElectronic Structure Theory > Density Functional TheoryStructure and Mechanism > Molecular StructuresMolecular and Statistical Mechanics > Molecular Interactions

Collective modes of a photon Bose–Einstein condensate with thermo-optic interaction

Although for photon Bose–Einstein condensates the main mechanism of the observed photon–photon interaction has already been identified to be of a thermo-optic nature, its influence on the condensate dynamics is still unknown. Here a mean-field description of this effect is derived, which consists of an open-dissipative Schrödinger equation for the condensate wave function coupled to a diffusion equation for the temperature of the dye solution. With this system at hand, the lowest-lying collective modes of a harmonically trapped photon Bose–Einstein condensate are calculated analytically via a linear stability analysis. As a result, the collective frequencies and, thus, the strength of the effective photon–photon interaction turn out to strongly depend on the thermal diffusion in the cavity mirrors. In particular, a breakdown of the Kohn theorem is predicted, i.e. the frequency of the centre-of-mass oscillation is reduced due to the thermo-optic photon–photon interaction.

Inverting the external‐potential‐to‐electron density map in systems of electrons by minimization of a least‐squares fit functional for analytic potentials

AbstractIn a system of electrons, there is a map connecting any external potential v with its electron density ρ v. In this work, we describe a procedure for inverting this potential‐to‐density map, so that potentials (if any) corresponding to a target density ρt can be obtained. We give the trial external potential v α, an analytic expression depending on a number of parameters α = (α1, …) and then minimize the least‐squares integral ∫(ρ α − ρt)2 dr by the conjugate gradient method, where ρ α is the density corresponding to v α. The implementation takes advantage of the analytic nature of v α. The procedure can be applied to any system and quantum chemistry model, and works both for ground and excited states, as well as for ensembles of states. The method is tested with some excited states of the particle‐in‐a‐box model, confirming the lack of a Hohenberg–Kohn theorem for excited states. It is also applied to the first singlet excited state of the helium atom, where, apart from the nucleus–electron attraction potential, some generalized external potentials are found.

Realization of the Kohn's Theorem in Ge/Si Quantum Dots with Hole Gas: Theory and Experiment.

This article discusses specific quantum transitions in a few-particle hole gas, localized in a strongly oblate lens-shaped quantum dot. Based on the adiabatic method, the possibility of realizing the generalized Kohn theorem in such a system is shown. The criteria for the implementation of this theorem in a lens-shaped quantum dot, fulfilled in the experiment, is presented. An analytical expression is obtained for the frequencies of resonant absorption of far-infrared radiation by a gas of heavy holes, which depends on the geometric parameters of the quantum dot. The results of experiments on far-infrared absorption in the arrays of p-doped Ge/Si quantum dots grown by molecular beam epitaxy (MBE) with gradually increasing average number of holes in dot are presented. Experimental results show that the Coulomb interaction between the holes does not affect the resonant frequency of the transitions. A good agreement between the theoretical and experimental results is shown.