Formulation Of Perturbation Theory Research Articles

Nambu-covariant many-body theory II: Self-consistent approximations

The theory of Self-Consistent Green’s Function (SCGF) is reformulated in an explicit Nambu-covariant fashion for applications to many-body systems at non-zero temperature in symmetry-broken phases. This is achieved by extending the Nambu-covariant formulation of perturbation theory, presented in the first part of this work, to non-perturbative schemes based on self-consistently dressed propagators and vertices. We work out in detail the self-consistent ladder approximation, motivated by a trade-off between numerical complexity and many-body phenomenology. Taking a complex general Hartree–Fock–Bogoliubov (HFB) propagator as a starting point, we also formulate and prove a sufficient condition on the stability of the HFB self-energy to ensure the convergence of the initial series of ladders at any energy. The self-consistent ladder approximation is written purely in terms of spectral functions and the resulting set of equations, when expressed in terms of Nambu tensors, are remarkably similar to those in the symmetry-conserving case. This puts the application of the self-consistent ladder approximation to symmetry-broken phases of infinite nuclear matter within reach.

The challenging equilibrium structure of HSSH: Another success of the rotational spectroscopy / quantum chemistry synergism

The simplest molecule with a disulfide bond, hydrogen disulfide (HSSH), represents an ideal test model for the determination of accurate gas-phase equilibrium structures for molecules containing third-row elements. First, pure theoretical composite schemes based on the coupled-cluster (CC) theory, which take into account the extrapolation to the complete basis set limit, core-valence correlation contributions, higher excitations in the CC expansion, and relativistic effects, allow for calculating accurate reference geometrical parameters. Second, using experimental vibrational ground-state rotational constants for a set of isotopologues, in conjunction with vibrational corrections based on second-order vibrational perturbation theory formulation and the recently developed Molecular Structure Refinement (MSR) software, we have determined the semi-experimental (SE) equilibrium structure of HSSH. The comparison of SE parameters with the computational best estimates shows an agreement within 0.001 Å for distances and 0.1° for angles, thus further validating the SE approach as cost-effective, provided that the required experimental data are available. Together with the intrinsic interest of HSSH, also in connection with astrochemistry, highly accurate structural properties of a prototypical disulfide bond can serve as references for future studies of larger molecules of biological interest containing this challenging moiety.

All-electron, real-space perturbation theory for homogeneous electric fields: theory, implementation, and application within DFT

Within density-functional theory, perturbation theory (PT) is the state-of-the-art formalism for assessing the response to homogeneous electric fields and the associated material properties, e.g., polarizabilities, dielectric constants, and Raman intensities. Here, we derive a real-space formulation of PT and present an implementation within the all-electron, numeric atom-centered orbitals electronic structure code FHI-aims that allows for massively parallel calculations. As demonstrated by extensive validation, we achieve a rapid computation of accurate response properties of molecules and solids. As an application showcase, we present harmonic and anharmonic Raman spectra, the latter obtained by combining hundreds of thousands of PT calculations with ab initio molecular dynamics. By using the PBE exchange-correlation functional with many-body van der Waals corrections, we obtain spectra in good agreement with experiment especially with respect to lineshapes for the isolated paracetamol molecule and two polymorphs of the paracetamol crystal.

Perturbation theory for arbitrary coupling strength?

We present a new formulation of perturbation theory for quantum systems, designated here as: “mean field perturbation theory” (MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since “renormalon”-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case.

The excitonic qubit coupled with a phonon bath on a star graph: anomalous decoherence and coherence revivals

Based on the operatorial formulation of perturbation theory, the dynamical properties of a Frenkel exciton coupled with a thermal phonon bath on a star graph are studied. Within this method, the dynamics is governed by an effective Hamiltonian which accounts for exciton---phonon entanglement. The exciton is dressed by a virtual phonon cloud, whereas the phonons are dressed by virtual excitonic transitions. Special attention is paid to the description of the coherence of a qubit state initially located on the central node of the graph. Within the nonadiabatic weak coupling limit, it is shown that several timescales govern the coherence dynamics. In the short time limit, the coherence behaves as if the exciton was insensitive to the phonon bath. Then, quantum decoherence takes place, this decoherence being enhanced by the size of the graph and by temperature. However, the coherence does not vanish in the long time limit. Instead, it exhibits incomplete revivals that occur periodically at specific revival times and it shows almost exact recurrences that take place at particular super-revival times, a singular behavior that has been corroborated by performing exact quantum calculations.

FROM ORDER TO CHAOS IN EARTH SATELLITE ORBITS

ABSTRACT We consider Earth satellite orbits in the range of semimajor axes where the perturbing effects of Earth’s oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the long-term orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angle-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances, which are of primary importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.

Quantum field perturbation theory revisited

Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in $D$ dimension with exponential interactions, such as $\mu^D\exp(\alpha\phi)$. In particular, we use the relation $$ \exp\big(\alpha{\delta\over \delta J(x)}\big)\exp(-Z_0[J])=\exp(-Z_0[J+\alpha_x]) $$ with $J$ the external source, and $\alpha_x(y)=\alpha\delta(y-x)$. Such a shift is strictly related to the normal ordering of $\exp(\alpha\phi)$ and to a scaling relation which follows by renormalizing $\mu$. Next, we derive a new formulation of perturbation theory for the potentials $V(\phi)={\lambda\over n!}:\phi^n:$, using the generating functional associated to $:\exp(\alpha\phi):$. The $\Delta(0)$-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to $J$ to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.

A time-dependent formulation of multi-reference perturbation theory.

We discuss the time-dependent formulation of perturbation theory in the context of the interacting zeroth-order Hamiltonians that appear in multi-reference situations. As an example, we present a time-dependent formulation and implementation of second-order n-electron valence perturbation theory. The resulting time-dependent n-electron valence second-order perturbation theory (t-NEVPT2) method yields the fully uncontracted n-electron valence perturbation wavefunction and energy, but has a lower computational scaling than the usual contracted variants, and also avoids the construction of high-order density matrices and the diagonalization of metrics. We present results of t-NEVPT2 for the water, nitrogen, carbon, and chromium molecules and outline directions for the future.

Equation of state and liquid–vapour equilibrium in a triangle-well fluid

The second-order thermodynamic perturbation theory formulation of Barker and Henderson is used to derive the equation of state of the triangle-well fluid. This is combined with the rational function approximation to the radial distribution function of the hard-sphere fluid. Results are obtained for the critical parameters and the liquid–vapour coexistence curve for various values of the range of the potential. A comparison with available simulation data is presented.

Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations

Various kinetic balances for constructing appropriate basis sets in four-component relativistic calculations are examined in great detail. These include the well-known restricted (RKB) and unrestricted (UKB) kinetic balances, the less-known dual kinetic balance (DKB) as well as the unknown inverse kinetic balance (IKB). The RKB and IKB are complementary to each other: The former is good for positive-energy states, whereas the latter good for negative-energy states. The DKB combines the good of both RKB and IKB and even provides full variational safety. However, such an advantage is largely offset by its complicated nature. The UKB does not offer any particular advantages as well. Overall, the RKB is the simplest ansatz. Although the negative-energy states by a finite RKB basis are in error of O(c0), there is no objection to using them as intermediates for a sum-over-states formulation of perturbation theory, provided that the magnetic balance is also incorporated in the case of magnetic properties. In particular, the RKB is also an essential ingredient for formulating two-component relativistic theories, while all the others are simply incompatible. As such, the RKB should be regarded as the cornerstone of relativistic electronic structure calculations.

Spin Component Scaling in Multiconfiguration Perturbation Theory

We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decomposed into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are determined from the stationary condition of the total energy calculated up to order 3. The decomposition schemes are tested numerically on the example of bond dissociation profiles and energy differences. We conclude that Grimme's parameters determined for single-reference Møller-Plesset theory may give a modest error reduction along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with molecular geometry, at the same time they are more efficient in reducing the error when following a bond dissociation process.

Modified perturbation theory for the Yukawa model

A new formulation of perturbation theory for a description of the Dirac and scalar fields (the Yukawa model) is suggested. For a main approximation, the self-consistent field model that considers to a certain degree forces caused by interaction of fields is chosen. This approximation leads to a normally ordered form of the interaction Hamiltonian. Generation of fermion mass in the process of interaction with exchange by a scalar boson is investigated. It is demonstrated that for zero bare mass, the fermion can acquire mass only if its coupling constant exceeds the critical value determined by the boson mass. In this regard, the problem of the neutrino mass is discussed.

Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions.

Perturbation theory formulation of Maxwell's equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electromagnetic systems. It is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fails when applied to the problem of shifted material boundaries. In this paper we developed coupled mode and perturbation theory formulations for treating generic perturbations of a waveguide cross section based on Hamiltonian formulation of Maxwell equations in curvilinear coordinates. We show that our formulation is accurate beyond the first order and rapidly converges to an exact result when used in a coupled mode theory framework even for the high-index-contrast discontinuous dielectric profiles. Among others, our formulation allows for an efficient numerical evaluation of such quantities as deterministic PMD and change in the GV and GVD of a mode due to generic profile distortion in a waveguide of arbitrary cross section. To our knowledge, this is the first time perturbation and coupled mode theories are developed to deal with arbitrary profile variations in high-index-contrast waveguides.

Perturbation theory for bound states and resonances where potentials and propagators have arbitrary energy dependence

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a nonlinear function of energy. Such derivations cannot be used, for example, in the context of relativistic quantum field theory. Here we solve this problem by providing a new, general formulation of perturbation theory for calculating the changes in the energy spectrum and wave function of bound states and resonances induced by perturbations to the Hamiltonian. Although our derivation is valid for energy-dependent potentials and is not restricted to inverse free Green functions that are linear in the energy, the expressions obtained for the energy and wave function corrections are compact, practical, and maximally similar to the ones of quantum mechanics. For the case of relativistic quantum field theory, our approach provides a direct covariant way of obtaining corrections to wave functions that are not in the center of mass frame.

Cosmological Perturbation Theory Using the Schrödinger Equation

We introduce the theory of nonlinear cosmological perturbations using the correspondence limit of the Schrodinger equation. The resulting formalism is equivalent to using the collisionless Boltzmann (or Vlasov) equations, which remain valid during the whole evolution, even after shell crossing. Other formulations of perturbation theory explicitly break down at shell crossing, e.g., Eulerean perturbation theory, which describes gravitational collapse in the fluid limit. This Letter lays the groundwork by introducing the new formalism, calculating the perturbation theory kernels that form the basis of all subsequent calculations. We also establish the connection with conventional perturbation theories, by showing that third-order tree-level results, such as bispectrum, skewness, cumulant correlators, and three-point function, are exactly reproduced in the appropriate expansion of our results. We explicitly show that cumulants up to N = 5 predicted by Eulerian perturbation theory for the dark matter field δ are exactly recovered in the corresponding limit. A logarithmic mapping of the field naturally arises in the Schrodinger context, which means that tree-level perturbation theory translates into (possibly incomplete) loop corrections for the conventional perturbation theory. We show that the first loop correction for the variance is σ2 = σ + (-1.14 - n)σ for a field with spectral index n. This yields 1.86 and 0.86 for n = -3 and -2, respectively, to be compared with the exact loop order corrections 1.82 and 0.88. Thus, our tree-level theory recovers the dominant part of first-order loop corrections of the conventional theory, while including (partial) loop corrections to infinite order in terms of δ.

Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarization-mode dispersion and group-velocity dispersion

We develop a novel perturbation theory formulation to evaluate polarization-mode dispersion (PMD) for a general class of scaling perturbations of a waveguide profile based on generalized Hermitian Hamiltonian formulation of Maxwell’s equations. Such perturbations include elipticity and uniform scaling of a fiber cross section, as well as changes in the horizontal or vertical sizes of a planar waveguide. Our theory is valid even for discontinuous high-index contrast variations of the refractive index across a waveguide cross section. We establish that, if at some frequencies a particular mode behaves like pure TE or TM polarized mode (polarization is judged by the relative amounts of the electric and magnetic longitudinal energies in the waveguide cross section), then at such frequencies for fibers under elliptical deformation its PMD as defined by an intermode dispersion parameter t becomes proportional to group-velocity dispersion D such that t 5 lduDu, where d is a measure of the fiber elipticity and l is a wavelength of operation. As an example, we investigate a relation between PMD and group-velocity dispersion of a multiple-core step-index fiber as a function of the core‐clad index contrast. We establish that in this case the positions of the maximum PMD and maximum absolute value of group-velocity dispersion are strongly correlated, with the ratio of PMD to group-velocity dispersion being proportional to the core‐clad dielectric contrast. © 2002 Optical Society of America OCIS codes: 060.2310, 060.2400, 060.2280, 060.0060.

Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates

Perturbation theory formulation of Maxwell's equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electro-magnetic systems. It is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fail when applied to the problem of shifted material boundaries. In this paper we developed a novel coupled mode and perturbation theory formulations for treating generic non-uniform (varying along the direction of propagation) perturbations of a waveguide cross-section based on Hamiltonian formulation of Maxwell equations in curvilinear coordinates. We show that our formulation is accurate and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index-contrast discontinuous dielectric profiles. Among others, our formulation allows for an efficient numerical evaluation of induced PMD due to a generic distortion of a waveguide profile, analysis of mode filters, mode converters and other optical elements such as strong Bragg gratings, tapers, bends etc., and arbitrary combinations of thereof. To our knowledge, this is the first time perturbation and coupled mode theories are developed to deal with arbitrary non-uniform profile variations in high index-contrast waveguides.

Second-harmonic generation of I-III-VI2chalcopyrite semiconductors: Effects of chemical substitutions

We report first-principles calculations of the second-order optical response coefficients in the I-III-${\mathrm{VI}}_{2}$ $(\mathrm{I}=\mathrm{A}\mathrm{g},\mathrm{C}\mathrm{u};$ $\mathrm{I}\mathrm{I}\mathrm{I}=\mathrm{G}\mathrm{a},\mathrm{I}\mathrm{n};$ $\mathrm{V}\mathrm{I}=\mathrm{S},\mathrm{S}\mathrm{e},\mathrm{T}\mathrm{e})$ chalcopyrite semiconductors. The computational approach uses the length-gauge formulation of perturbation theory which explicitly separates pure interband from mixed intraband-interband contributions. The expressions for static and frequency dependent second-harmonic generation coefficients are evaluated from band structures based on the local density approximation but including semiempirical gap corrections. The linear muffin-tin orbital method is used to calculate the required band structures and matrix elements. The results are in good agreement with experiment for the compounds for which data are available and provide predictions in the other cases. The trends show that the dominating factor determining ${\ensuremath{\chi}}^{(2)}$ is the anion rather than the group I or group III cation. The ${\ensuremath{\chi}}^{(2)}$ values clearly fall into separated groups with increasing value going from S to Se to Te. While this correlates approximately inversely with the band gap, several exceptions are notable: (1) Cu compounds have smaller gaps than corresponding Ag compounds and nevertheless have slightly lower ${\ensuremath{\chi}}^{(2)};$ (2) ${\mathrm{AgGaTe}}_{2}$ has a higher gap than ${\mathrm{AgInSe}}_{2}$ but nevertheless has a much higher ${\ensuremath{\chi}}^{(2)}.$ An analysis of the various contributions to the frequency dependent imaginary part of the response functions, $\mathrm{Im}{{\ensuremath{\chi}}^{(2)}(\ensuremath{-}2\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})},$ is presented in an attempt to correlate the ${\ensuremath{\chi}}^{(2)}$ values with band structure features. The main findings of this analysis are that (1) there is a large compensation between intra/inter- and interband contributions frequency by frequency as well as in the static values; (2) the static ${\ensuremath{\chi}}^{(2)}$ value is strongly affected by the sign of the low frequency parts of these separate contributions; (3) these low frequency parts correspond to only a few valence and conduction bands and only to so-called $2\ensuremath{\omega}$ resonances; (4) the general shape of the $\mathrm{Im}{{\ensuremath{\chi}}^{(2)}(\ensuremath{-}2\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})}$ response functions is determined by the band structures alone while the intensity, which ultimately explains the difference between tellurides and selenides, arises from the magnitude of the matrix elements. Starting from ${\mathrm{AgGaSe}}_{2},$ the smaller effect on the ${\ensuremath{\chi}}^{(2)}$ due to In subsitution for Ga than to Te substitution for Se can be explained by the fact that the Ga to In substitution changes the gap only in a small region near the center of the Brillouin zone, while the Se to Te substitution changes the gap throughout the Brillouin zone. This shows that contributions from other parts of the Brillouin zone than the center dominate the behavior. The difference between Cu and Ag based compounds can be explained on the basis of a different degree of compensation of inter- and intra/interband contributions.

Eigenvalue perturbation and generalized Krylov subspace method

In this paper, we study the computational aspect of eigenvalue perturbation theory. In previous research, high order perturbation terms were often derived from Taylor series expansion. Computations based on such an approach can be both unstable and highly complicated. We present here an approach based on the differential formulation of perturbation theory where the high order perturbation can be naturally obtained. The high order perturbation can be interpreted as a generalized Krylov subspace approximation and its convergence rate can be analyzed accordingly. This approach provides a simple and stable method to compute a few eigenvalues of a slightly modified system.

Magnetic Scaling in Superconductors

The Ginzburg-Landau-Wilson superconductor in a magnetic field $B$ is considered in the approximation that magnetic-field fluctuations are neglected. A formulation of perturbation theory is presented in which multiloop calculations fully retaining all Landau levels are tractable. A 2-loop calculation shows that, near the zero-field critical point, the singular part of the free energy scales as ${F}_{\mathrm{sing}}\ensuremath{\approx}\phantom{\rule{0ex}{0ex}}|t{|}^{2\ensuremath{-}\ensuremath{\alpha}}F(B|t{|}^{\ensuremath{-}2\ensuremath{\nu}})$, where $\ensuremath{\nu}$ is the coherence-length exponent---a result which has hitherto been assumed on purely dimensional grounds.