Density-density Response Function Research Articles

A mathematical analysis of the adiabatic Dyson equation from time-dependent density functional theory

In this article, we analyse the Dyson equation for the density–density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterise the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial DDRF and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation and the adiabatic local density approximation. In particular, these results show that neither of these approximations can shift the ionisation threshold of the Kohn–Sham system.

Plasmons in three-dimensional superconductors

We study the plasma branch of an homogeneous three-dimensional electron gas in an $s$-wave superconducting state. Although a sum rule guarantees that the departure of the plasma branch always coincides with the plasma frequency ${\ensuremath{\omega}}_{p}$, the dispersion and lifetime of plasmons is strongly affected by the presence of the pair condensate, especially when ${\ensuremath{\omega}}_{p}$ is close to the pair-breaking threshold $2\mathrm{\ensuremath{\Delta}}$. When ${\ensuremath{\omega}}_{p}$ is between $1.7\mathrm{\ensuremath{\Delta}}$ and $2\mathrm{\ensuremath{\Delta}}$, the level repulsion is strong enough to give the plasma branch an anomalous, downward dispersion and a dispersion minimum strictly lower than ${\ensuremath{\omega}}_{p}$. Then for ${\ensuremath{\omega}}_{p}>2\mathrm{\ensuremath{\Delta}}$, plasmons damp out in pair-breaking excitations, acquiring a small damping rate at zero temperature, which we compute in a non-perturbative way. Finally, the density-density response function displays a resonance near $2\mathrm{\ensuremath{\Delta}}$ (not to be confused with the amplitude mode), which can beat with the main plasma resonance, and subsists for ${\ensuremath{\omega}}_{p}$ large compared to $\mathrm{\ensuremath{\Delta}}$, thereby distinguishing charged from neutral condensates.

Diffusive density response of electrons in anisotropic multiband systems

We explicitly calculate the density-density response function with conserving vertex corrections for anisotropic multiband systems in the presence of impurities including long-range disorder. The direction dependence of the vertex corrections is correctly considered to obtain the diffusion constant which is given by a combination of componentwise transport relaxation times and velocities on the Fermi surface. We also investigate the diffusive density response of various anisotropic systems, propose some empirical rules for the corresponding diffusion constant, and demonstrate that it is crucial to consider the component dependence of the transport relaxation times to correctly interpret the transport properties of anisotropic systems, especially various topological materials with a different power-law dispersion in each direction.

Collective excitations and quantum incompressibility in electron-hole bilayers

We apply quantum continuum mechanics to the calculation of the excitation spectrum of a coupled electron-hole bilayer. The theory expresses excitation energies in terms of ground-state intra- and inter-layer pair correlation functions, which are available from Quantum Monte Carlo calculations. The final formulas for the collective modes deduced from this approach coincide with the formulas obtained in the "quasi-localized particle approximation" by Kalman et al., and likewise, the theory predicts the existence of gapped excitations in the charged channels, with the gap arising from electron-hole correlation. An immediate consequence of the gap is that the static density-density response function of the charged channel vanishes as $q^2$ for wave vector $q \to 0$, rather than linearly in $q$, as commonly expected. In this sense, the system is {\it incompressible}. This feature, which has no analogue in the classical electron-hole plasma, is consistent with the existence of an excitonic ground state and implies the existence of a discontinuity in the chemical potential of electrons and holes when the numbers of electrons and holes are equal. It should be experimentally observable by monitoring the densities of electrons and holes in response to potentials that attempt to change these densities in opposite directions.

Random phase approximation with exchange for an accurate description of crystalline polymorphism

We determine the correlation energy of BN, SiO$_2$ and ice polymorphs employing a recently developed RPAx (random phase approximation with exchange) approach. The RPAx provides larger and more accurate polarizabilities as compared to the RPA, and captures effects of anisotropy. In turn, the correlation energy, defined as an integral over the density-density response function, gives improved binding energies without the need for error cancellation. Here, we demonstrate that these features are crucial for predicting the relative energies between low- and high-pressure polymorphs of different coordination number as, e.g., between $\alpha$-quartz and stishovite in SiO$_2$, or layered and cubic BN. Furthermore, a reliable (H$_2$O)$_2$ potential energy surface is obtained, necessary for describing the various phases of ice. The RPAx gives results comparable to other high-level methods such as coupled cluster and quantum Monte Carlo, also in cases where the RPA breaks down. Although higher computational cost than RPA we observe a faster convergence with respect to the number of eigenvalues in the response function.

Carrier concentration independent plasmons in biased twisted bilayer graphene

We calculate the density-density response function of biased twisted bilayer graphene (BTBG) and study its plasmon dispersion within the random phase approximation (RPA). At long wavelengths ($q\ensuremath{\rightarrow}0$), plasmon dispersion shows local classical behavior $\ensuremath{\omega}={\ensuremath{\omega}}_{0}\sqrt{q}$. Unlike the situation in conventional two-dimensional electron gas (2DEG), where the density dependence of the plasmon energy is of the form ${\ensuremath{\omega}}_{0}\ensuremath{\propto}\sqrt{n}$ ($n$ is the carrier concentration), the plasmon energy ${\ensuremath{\omega}}_{0}$ is independent of the carrier concentration ($n$) in biased twisted bilayer graphene. Furthermore, the plasmon energy (${\ensuremath{\omega}}_{0}$) is also independent of the Fermi energy ($\ensuremath{\mu}$) which is decided by the carrier concentration ($n$).

Dynamical polarization function and plasmons in monolayer XSe ( X=In,Ga )

By using the single-band and multiband effective Hamiltonian for monolayer $X\mathrm{Se}$ ($X=\mathrm{In},\mathrm{Ga}$), we calculate the density-density response function of monolayer $X\mathrm{Se}$ ($X=\mathrm{In},\mathrm{Ga}$) and study its plasmon dispersion within the random-phase approximation. At long wavelengths ($q\ensuremath{\rightarrow}0$), plasmon dispersion shows the local classical behavior $\ensuremath{\omega}={\ensuremath{\omega}}_{0}\sqrt{q}$. Although, monolayer $X\mathrm{Se}$ ($X=\mathrm{In},\mathrm{Ga}$) has a nonparabolic ``Mexican hat'' topmost valence band dispersion which is so different from the parabolic band structure in two-dimensional electron gas, the corresponding density dependence of the plasmon energy is still of the form ${\ensuremath{\omega}}_{0}\ensuremath{\propto}\sqrt{n}$ ($n$ is the carrier concentration) as in a conventional two-dimensional electron gas (2DEGs). However, the Fermi energy ($|\ensuremath{\mu}|$) dependence of the plasmon energy is of the form ${\ensuremath{\omega}}_{0}\ensuremath{\propto}{|\ensuremath{\mu}|}^{1/4}$, which is different from the conventional 2DEGs (${\ensuremath{\omega}}_{0}\ensuremath{\propto}\sqrt{|\ensuremath{\mu}|}$).

Plasmons in anisotropic Dirac systems

We consider the plasmon excitations in anisotropic two-dimensional Dirac systems, be it either anisotropic graphene or surfaces of topological insulators. Generalizing the exact density-density response function one finds a plasmon dispersion that is anisotropic already at the lowest frequencies. Asymptotic expressions are obtained for the dispersion in this regime. We show that the plasmon properties of the complete material class of anisotropic Dirac systems are characterized by just two dimensionless material parameters. The strong anisotropy can be used to guide the plasmon modes, introducing new functionalities to the field of Dirac plasmonics.

Floquet metal-to-insulator phase transitions in semiconductor nanowires.

We study steady states of semiconductor nanowires subjected to strong resonant time-periodic drives. The steady states arise from the balance between electron-phonon scattering, electron-hole recombination via photoemission, and Auger scattering processes. We show that tuning the strength of the driving field drives a transition between an electron-hole metal (EHM) phase and a Floquet insulator (FI) phase. We study the critical point controlling this transition. The EHM-to-FI transition can be observed by monitoring the presence of peaks in the density-density response function, which are associated with the Fermi momentum of the EHM phase and are absent in the FI phase. Our results may help guide future studies toward inducing exotic nonequilibrium phases of matter by periodic driving.

ON A RESPONSE FUNCTION AND TWO-BODY DENSITY MATRIX OF NUCLEAR MATTER

We present a summary of on-going calculations that address the static and dynamic structure of nuclear matter. Specific projects include (i) evaluation of the density-density response function and corresponding dynamic structure factor, based on the correlated random-phase approximation (CRPA_I) and generalizations of this method, and (ii) low-order variational calculation of the reduced two-body density matrix and corresponding generalized momentum distribution. The numerical applications involve the model interaction V2.

Efficient representation of the linear density-density response function.

We present a thorough derivation of the mathematical foundations of the representation of the molecular linear electronic density-density response function in terms of a computationally highly efficient moment expansion. Our new representation avoids the necessities of computing and storing numerous eigenfunctions of the response kernel by means of a considerable dimensionality reduction about from 103 to 101 . As the scheme is applicable to any compact, self-adjoint, and positive definite linear operator, we present a general formulation, which can be transferred to other applications with little effort. We also present an explicit application, which illustrates the actual procedure for applying the moment expansion of the linear density-density response function to a water molecule that is subject to a varying external perturbation potential. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

On the response function of a degenerate Bose gas

The space-time density-density response function for degenerate ideal Bose gas is considered. On this basis, it is shown that upon exposure to a weak external field, a space-time wave of average non-uniform density fluctuation is formed in the ideal Bose gas, within the limit of strong degeneracy, which decays only in time in an exponential manner.

Interaction-disorder-driven characteristic momentum in graphene, approach of multi-body distribution functions

Multi-point probability measures along with the dielectric function of Dirac Fermions in mono-layer graphene containing particle-particle and white-noise (out-plane) disorder interactions on an equal footing in the Thomas-Fermi-Dirac approximation is investigated. By calculating the one-body carrier density probability measure of the graphene sheet, we show that the density fluctuation (ζ−1) is related to the disorder strength (ni), the interaction parameter (rs) and the average density (bar{{boldsymbol{n}}}) via the relation {{boldsymbol{zeta }}}^{-{bf{1}}}{boldsymbol{propto }}{{boldsymbol{r}}}_{{boldsymbol{s}}}{{boldsymbol{n}}}_{{boldsymbol{i}}}^{{bf{2}}}{bar{{boldsymbol{n}}}}^{-{bf{1}}} for which bar{{boldsymbol{n}}}to {bf{0}} leads to strong density inhomogeneities, i.e. electron-hole puddles (EHPs), in agreement with the previous works. The general equation governing the two-body distribution probability is obtained and analyzed. We present the analytical solution for some limits which is used for calculating density-density response function. We show that the resulting function shows power-law behaviors in terms of ζ with fractional exponents which are reported. The disorder-averaged polarization operator is shown to be a decreasing function of momentum like ordinary 2D parabolic band systems. It is seen that a disorder-driven momentum qch emerges in the system which controls the behaviors of the screened potential. We show that in small densities an instability occurs in which imaginary part of the dielectric function becomes negative and the screened potential changes sign. Corresponding to this instability, some oscillations in charge density along with a screening-anti-screening transition are observed. These effects become dominant in very low densities, strong disorders and strong interactions, the state in which EHPs appear. The total charge probability measure is another quantity which has been investigated in this paper. The resulting equation is analytically solved for large carrier densities, which admits the calculation of arbitrary-point correlation function.

Iterative approach for the moment representation of the density-density response function

The linear density-density response function χ(r,r′) can be transformed from its Eigensystem representation into a computationally more efficient moment expansion representation using a suitable unitary transformation. Here, we propose an iterative approach for the direct calculation of this moment representation without resorting to either the direct-space or the conventional Eigensystem representation.

Adiabatic Connection without Coupling Constant Integration.

Using a second-order approximation to Random Phase Approximation renormalized (RPAr) many-body perturbation theory for the interacting density-density response function, we have developed a so-called higher-order terms (HOT) approximation for the correlation energy. In combination with the first-order RPAr correction, our new method faithfully captures the infinite-order correlation for a given exchange-correlation kernel, yielding errors of the total correlation energy on the order of 1% or less for most systems. For exchange-like kernels, our new method has the further benefit that the coupling-strength integration can be completely eliminated resulting in a modest reduction in computational cost compared to the traditional approach. When the correlation energy is accurately reproduced by the HOT approximation, structural properties and energy differences are also accurately reproduced, as we demonstrate for several periodic solids and some molecular systems. Energy differences involving fragmentation are challenging for the HOT method, however, due to errors that may not cancel between a composite system and its constituent pieces.

Finite-temperature dynamics of the Mott insulating Hubbard chain

We study the dynamical response of the half-filled one-dimensional(1d) Hubbard model for a range of interaction strengths $U$ and temperatures $T$ by a combination of numerical and analytical techniques. Using time-dependent density matrix renormalization group (tDMRG) computations we find that the single-particle spectral function undergoes a crossover to a spin-incoherent Luttinger liquid regime at temperatures $T \sim J=4t^2/U$ for sufficiently large $U > 4t$. At smaller values of $U$ and elevated temperatures the spectral function is found to exhibit two thermally broadened bands of excitations, reminiscent of what is found in the Hubbard-I approximation. The dynamical density-density response function is shown to exhibit a finite temperature resonance at low frequencies inside the Mott gap, with a physical origin similar to the Villain mode in gapped quantum spin chains. We complement our numerical computations by developing an analytic strong-coupling approach to the low-temperature dynamics in the spin-incoherent regime.

Lippmann-Schwinger theory for two-dimensional plasmon scattering

Long-lived and ultra-confined plasmons in two-dimensional (2D) electron systems may provide a sub-wavelength diagnostic tool to investigate localized dielectric, electromagnetic, and pseudo-electromagnetic perturbations. In this Article, we present a general theoretical framework to study the scattering of 2D plasmons against such perturbations in the non-retarded limit. We discuss both parabolic-band and massless Dirac fermion 2D electron systems. Our theory starts from a Lippmann-Schwinger equation for the screened potential in an inhomogeneous 2D electron system and utilizes as inputs analytical long-wavelength expressions for the density-density response function, going beyond the local approximation. We present illustrative results for the scattering of 2D plasmons against a point-like charged impurity and a one-dimensional electrostatic barrier due to a line of charges. Exact numerical results obtained from the solution of the Lippmann-Schwinger equation are compared with approximate results based on the Born and eikonal approximations. The importance of nonlocal effects is finally emphasized.

Anisotropic plasmons, Friedel oscillations, and screening in8−Pmmnborophene

$8-Pmmn$ borophene is a polymorph of borophene which hosts an anisotropic tilted Dirac cone. Using a low energy effective Hamiltonian for borophene, we analytically calculate the density-density response function of borophene and study its anisotropic plasmon dispersion and screening properties. We find that the anisotropic plasmon mode in $8-Pmmn$ borophene, remains undamped for higher energies along the mirror symmetry direction. The friedel oscillation in borophene is found to decay as $r^{-3}$ in the large $r$ limit, with both the amplitude as well as the periodicity of the Friedel oscillations being anisotropic.

Exchange, correlation, and scattering effects on surface plasmons in arm-chair graphene nanoribbons.

Using Maxwell's equations for the incoming and outgoing electromagnetic field, in interaction with a metallic arm-chair graphene nanoribbon (AGNR), and the relationship between the density-density response function and the conductivity, we study surface plasmons (SPs) in a AGNR following the Lindhard, random-phase approximation (RPA), and Hubbard approaches. For transverse magnetic (TM) modes we obtain analytical dispersion relations (DRs) valid for q ≤ kF and assess their width dependence. In all approaches we include screening. In the long-wavelength limit q → 0 there is a small but noticeable difference between the DRs of the three approaches. In this limit the respective, scattering-free conductivities differ drastically from those obtained when scattering by impurities is included. We demonstrate that the SP field is proportional to the square of the quality factor Q. The reflection amplitude shows that metallic AGNRs do not support Brewster angles. In addition, AGNRs do not support transverse electric (TE) SPs.

Optical properties of periodic systems within the current-current response framework: Pitfalls and remedies

We compare the optical absorption of extended systems using the density-density and current-current linear response functions calculated within many-body perturbation theory. The two approaches are formally equivalent for a finite momentum $\mathbf{q}$ of the external perturbation. At $\mathbf{q}=\mathbf{0}$, however, the equivalence is maintained only if a small $q$ expansion of the density-density response function is used. Moreover, in practical calculations, this equivalence can be lost if one naively extends the strategies usually employed in the density-based approach to the current-based approach. Specifically, we discuss the use of a smearing parameter or of the quasiparticle lifetimes to describe the finite width of the spectral peaks and the inclusion of electron-hole interaction. In those instances, we show that the incorrect definition of the velocity operator and the violation of the conductivity sum rule introduce unphysical features in the optical absorption spectra of three paradigmatic systems: silicon (semiconductor), copper (metal) and lithium fluoride (insulator). We then demonstrate how to correctly introduce lifetime effects and electron-hole interactions within the current-based approach.