Perturbation Theory, Response Functions, and Green’s Functions

The critical dynamics of charge-density waves (CDW's) just above the threshold for steady-state sliding motion are studied via a renormalization-group expansion in 4-\ensuremath{\epsilon} dimensions. This necessitates the simultaneous treatment of the long-wavelength low-frequency dynamics as well as the singular behavior associated with the jumps of local regions during which the local velocities become large. Scaling forms are obtained for response and correlation functions in the critical region, and exponent identities are derived. To leading order in \ensuremath{\epsilon}, the exponent \ensuremath{\zeta} of the CDW current ${\mathit{I}}_{\mathrm{CDW}}$\ensuremath{\sim}(V-${\mathit{V}}_{\mathit{T}}$${)}^{\mathrm{\ensuremath{\zeta}}}$ just above the threshold voltage ${\mathit{V}}_{\mathit{T}}$ is found to be \ensuremath{\zeta}=1-\ensuremath{\epsilon}/6+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), in reasonable agreement with recent numerical simulations in two and three dimensions. The spatial correlations in the local velocities of the CDW decay exponentially with a correlation length, which diverges with an exponent \ensuremath{\nu} exactly equal to 1/2. Nevertheless, finite-size scaling of many quantities will be determined by a different exponent ${\ensuremath{\nu}}_{\mathrm{FS}}$=1/2+\ensuremath{\epsilon}/8+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$)=2/d+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). The other exponents are related to \ensuremath{\zeta}, \ensuremath{\nu}, and ${\ensuremath{\nu}}_{\mathrm{FS}}$. Simple forms of long-wavelength low-frequency correlation and response functions in the moving phase are obtained to all orders in perturbation theory.

We present a methodology to calculate frequency and momentum dependent all-electron response functions determined within Kohn-Sham density functional theory. It overcomes the main obstacle in calculating response functions in practice, which is the slow convergence with respect to the number of unoccupied states and the basis-set size. In this approach, the usual sum-over-states expression of perturbation theory is complemented by the response of the orbital basis functions, explicitly constructed by radial integrations of frequency-dependent Sternheimer equations. To an essential extent an infinite number of unoccupied states are included in this way. Furthermore, the response of the core electrons is treated virtually exactly, which is out of reach otherwise. The method is an extension of the recently introduced incomplete-basis-set correction (IBC) [Betzinger et al., Phys. Rev. B85, 245124 (2012); 88, 075130 (2013)] to the frequency and momentum domain. We have implemented the generalized IBC within the all-electron full-potential linearized augmented-plane-wave method and demonstrate for rocksalt BaO the improved convergence of the dynamical Kohn-Sham polarizability. We apply this technique to compute (a) quasiparticle energies employing the COHSEX approximation for the self-energy of many-body perturbation theory and (b) all-electron RPA correlation energies. It is shown that the favorable convergence of the polarizability is passed over to the COHSEX and RPA calculation.

The rotational g factors of the hydrogen halides, HX (X=F,Cl,Br,I), and noble gas hydride cations, XH+ (X=Ne,Ar,Kr,Xe), have been calculated at the level of the random phase approximation (RPA) as relativistic four-component linear response functions as well as nonrelativistic linear response functions. In addition, using perturbation theory with the mass-velocity and Darwin operators as perturbations, the relativistic corrections have been estimated as quadratic response functions. It was found that the four-component relativistic calculations give in general a more negative electronic contribution to the rotational g factor than the nonrelativistic calculations with relativistic corrections ranging from 0.2% for HF and NeH+ to 2.9% for XeH+ and 3.5% for HI. The estimates of the relativistic corrections obtained by perturbation theory with the mass-velocity and Darwin operators are in good agreement with the four-component results for HF, HCl, NeH+, and ArH+, whereas for HI, KrH+, and XeH+ they have the wrong sign.

In this article we report the first calculations of second harmonic generation second hyperpolarizability of liquid water using coupled cluster/molecular mechanics (CC/MM) methods or coupled cluster/dielectric continuum (CC/DC) methods. The latter approach treats the solvent as an isotropic homogeneous fluid while the former accounts for the discrete nature of the solvent molecules. The CC/MM approach may include or exclude polarization effects explicitly. Alternatively, polarization effects may be included using perturbation theory. The CC descriptions implemented are the coupled cluster second-order approximate singles and doubles (CC2) and coupled cluster singles and doubles models. The second harmonic generation second hyperpolarizabilities are, depending on the model, obtained using either an analytical implementation of the cubic response function or using an analytical implementation of the quadratic response function combined with the finite field technique. The CC/MM results for the second harmonic generation second hyperpolarizability compare excellently with experimental data while a significant overestimation is found when using the CC/DC model. Particular, the cavity radius in the CC/DC calculations have an enormous effects on this fourth-order property.

The pion in electromagnetic and weak neutral-current nuclear response functions

The response function, which is directly related to the cross section for inelastic scattering of electrons from nuclear matter, is calculated with the help of a time-dependent perturbation theory using a separable potential. All diagrams through second order in the interaction are computed and their contributions compared. The self-energy problem is examined, and it is concluded that a Hartree-Fock approximation is preferable to one which attempts to takeK-matrix insertions into account. Results for the response function are compared with those of previous workers.

Frequency-dependent second hyperpolarizabilities using coupled cluster cubic response theory

The response function of a finite nucleus is calculated with the help of perturbation theory. Realistic hard core nucleon-nucleon potentials are used. For theT-matrix elements a Bethe-Goldstone like equation is solved using the method of Kallio and Day. Numerical calculations of the response function are carried out for oxygen.

The field dependence of the magnetic susceptibility and collective mode frequencies of superfluid $^{3}\mathit{B}$ are calculated perturbatively in (\ensuremath{\gamma}H/\ensuremath{\Delta}) using the quasiclassical theory. With the aid of this perturbation theory, we interpret the gap distortion, nonlinear susceptibility, and collective excitations of the order parameter in terms of additional correlations between Cooper pairs that are induced by a magnetic field. We also calculate the dispersion splittings of the real squashing modes in a magnetic field. In addition to the nonlinear Zeeman shifts of these modes, which occur in strong magnetic fields (\ensuremath{\gamma}H\ensuremath{\sim}0.10\ensuremath{\Delta}), there are low-field (\ensuremath{\gamma}H\ensuremath{\sim}${q}^{2}$${v}_{f}^{2}$/\ensuremath{\Delta}) nonlinearities of the collective mode frequencies resulting from the field dependence of the quantization axis. Our results for all of these response properties of $^{3}\mathit{B}$ depend upon a small number of material parameters; thus measurements of these properties can provide detailed information on the quasiparticle interactions.

A new formalism was recently proposed to improve random phase approximation (RPA) correlation energies by including approximate exchange effects [B. Mussard et al., J. Chem. Theory Comput. 12, 2191 (2016)]. Within this framework, by keeping only the electron-hole contributions to the exchange kernel, two approximations can be obtained: An adiabatic connection analog of the second order screened exchange (AC-SOSEX) and an approximate electron-hole time-dependent Hartree-Fock (eh-TDHF). Here we show how this formalism is suitable for an efficient implementation within the plane-wave basis set. The response functions involved in the AC-SOSEX and eh-TDHF equations can indeed be compactly represented by an auxiliary basis set obtained from the diagonalization of an approximate dielectric matrix. Additionally, the explicit calculation of unoccupied states can be avoided by using density functional perturbation theory techniques and the matrix elements of dynamical response functions can be efficiently computed by applying the Lanczos algorithm. As shown by several applications to reaction energies and weakly bound dimers, the inclusion of the electron-hole kernel significantly improves the accuracy of ground-state correlation energies with respect to RPA and semi-local functionals.

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.

The Kohn-Sham system is the prototypical example of an auxiliary system that targets, in principle exactly, an observable like the electronic density without the need to calculate the complicated many-body wavefunction. Although the Kohn-Sham system does not describe excited-state properties directly, it also represents a very successful strategy guideline for many spectroscopy applications. Here we propose a generalization of the Kohn-Sham idea. In many situations one is interested only in limited answers to specific questions, whereas in state-of-the-art approaches a lot of information is generally calculated that is not needed for the interpretation of experimental spectra. For example, when the target is a spectrum S(ω) like the optical absorption of a solid, within time-dependent density-functional theory (TDDFT) one calculates the whole response function χ(r,r',ω). Analogously, within many-body perturbation theory (MBPT) one calculates the whole one-particle Green's function G(r,r',ω), while only the total spectral function A(ω) is needed for angle-integrated photoemission spectra. In this contribution, we advocate the possibility of designing auxiliary systems with effective potentials or kernels that target only the specific spectral properties of interest and are simpler than the self-energy of MBPT or the exchange-correlation kernel of TDDFT. In particular, we discuss the fundamentals and prototypical applications of simplified effective kernels for optical absorption and spectral potentials for photoemission, and we discuss how to express these potentials or kernels as functionals of the density.

Most material properties of great physical interest are directly related to nuclear dynamics, e.g. the ionic thermal conductivity, Raman/IR vibrational spectra, inelastic X-ray, and Neutron scattering. A theory able to compute from first principles these properties, accounting for the anharmonicity and quantum fluctuations in the nuclear energy landscape that can be implemented in systems with hundreds of atoms is missing. Here, we derive an approximate theory for the quantum time evolution of lattice vibrations at finite temperature. This theory introduces the time dynamics in the Self-Consistent Harmonic Approximation (SCHA) and shares with the static case the same computational cost. It is nonempirical, as pure states evolve according to the Dirac least action principle and the dynamics of the thermal ensemble conserves both energy and entropy. The static SCHA is recovered as a stationary solution of the dynamical equations. We apply perturbation theory around the static SCHA solution and derive an algorithm to compute efficiently quantum dynamical response functions. Thanks to this new algorithm, we have access to the response function of any general external time-dependent perturbation, enabling the simulation of phonon spectra without following any perturbative expansion of the nuclear potential or empirical methods. We benchmark the algorithm on the IR and Raman spectroscopy of high-pressure hydrogen phase III, with a simulation cell of 96 atoms. Our work also explores the nonlinear regime of the dynamical nuclear motion, providing a paradigm to simulate the interaction with intense or multiple probes, as in pump-probe spectroscopy, or chemical reactions involving light atoms, as the proton transfer in biomolecules

Renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

The non-Fermi-liquid fixed point of the overscreened Kondo problem is studied in the limit of a large number of conduction electron channels, k\ensuremath{\gg}1, using perturbation theory in 1/k. By expanding the beta function and self-energy to sub-leading-order in 1/k, we examine the scaling, thermodynamics, and dynamic response functions in detail. Our results compliment and extend previous work based on conformal field theory. We present the dynamical spin susceptibility for the first time and show that the impurity spin is marginally quenched, with an inelastic linewidth of order T.