Standard Perturbation Expansion Research Articles

Semi-analytic theory of multiphonon effects on the static structure factors of warm solids.

A semi-analytic formula for the temperature-dependent static structure factor S(k) of polycrystalline and amorphous solids applicable to the entire wavenumber (k) range is derived. The formula describes thermal diffuse scattering due to multiphonon processes entirely by a single kernel function without resorting to the standard perturbation expansion. It is analytically proven that S(k → 0) is determined from the one-phonon term, whereas the asymptotic limit S(k → ∞) = 1 can be reproduced through a Gaussian integral of the multiphonon term. The formula also reveals that an enhancement of the one-phonon scattering intensity at each Bragg point is expressed as a logarithmic singularity. Numerical examples for a face-centred cubic polycrystal near the melting point are shown. The present formula is computationally more efficient than other theoretical models, requiring only a one-dimensional integration to obtain S(k) once the elastic part of the structure factor and the Debye-Waller factor are given.

Stabilization and pattern formation in chemotaxis models with acceleration and logistic source.

We consider the following chemotaxis-growth system with an acceleration assumption, \begin{align*} \begin{cases} u_t= \Delta u -\nabla \cdot\left(u \bw \right)+\gamma\xkh{u-u^\alpha}, & x\in\Omega,\ t>0,\\ v_t=\Delta v- v+u, & x\in\Omega,\ t>0,\\ \bw_t= \Delta \bw -\bw +\chi\nabla v, & x\in\Omega,\ t>0, \end{cases} \end{align*} under the homogeneous Neumann boundary condition for $u,v$ and the homogeneous Dirichlet boundary condition for $\bw$ in a smooth bounded domain $\Omega\subset\R^{n}$ ($n\geq1$) with given parameters $\chi>0$, $\gamma\geq0$ and $\alpha>1$. It is proved that for reasonable initial data with either $n\leq3$, $\gamma\geq0$, $\alpha>1$ or $n\geq4,\ \gamma>0,\ \alpha>\frac12+\frac n4$, the system admits global bounded solutions, which significantly differs from the classical chemotaxis model that may have blow-up solutions in two and three dimensions. For given $\gamma$ and $\alpha$, the obtained global bounded solutions are shown to convergence exponentially to the spatially homogeneous steady state $(m,m,\mathbf 0$) in the large time limit for appropriately small $\chi$, where $m=\frac1{|\Omega|}\jfo u_0(x)$ if $\gamma=0$ and $m=1$ if $\gamma>0$. Outside the stable parameter regime, we conduct linear analysis to specify possible patterning regimes. In weakly nonlinear parameter regimes, with a standard perturbation expansion approach, we show that the above asymmetric model can generate pitchfork bifurcations which occur generically in symmetric systems. Moreover, our numerical simulations demonstrate that the model can generate rich aggregation patterns, including stationary, single merging aggregation, merging and emerging chaotic, and spatially inhomogeneous time-periodic. Some open questions for further research are discussed.

Renormalization group optimized λϕ4 pressure at next-to-next-to-leading order

We investigate the renormalization group optimized perturbation theory (RGOPT) at the next-to-next-to-leading order (NNLO) for the thermal scalar field theory. From comparing three thus available successive RGOPT orders, we illustrate the efficient resummation and very good apparent convergence properties of the method. In particular, the remnant renormalization scale dependence of thermodynamical quantities is drastically improved as compared to both standard perturbative expansions and other related resummation methods, such as the screened perturbation theory. Our present results thus constitute a useful first NNLO illustration in view of NNLO applications of this approach to the more involved thermal QCD.

Conformal mappings in perturbative QCD

We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on the Borel-Laplace integral regulated with the principal value prescription and the expansion of the Borel transform in powers of the variable which performs the conformal mapping of the cut Borel plane onto the unit disk. We write down the expression of the conformal mapping for the most general location of the singularities of the Borel transform and review the properties of the corresponding expansions of the correlators. Unlike the standard perturbative expansions, which are divergent, the modified expansions have a tamed behaviour at large orders and may even converge under some conditions. On the other hand, the expansion functions exhibit nonperturbative features similar to those of the expanded function. Using these properties, it was suggested recently that the expansions based on the conformal mapping of the Borel plane may provide an alternative to the standard OPE. We briefly review the arguments in favour of this conjecture and discuss the application of the method to the Adler function for massless quarks and the static quark self-energy calculated in lattice QCD.

On the molecular electronic flux: Role of nonadiabaticity and violation of conservation

Analysis of electron flux within and in between molecules is crucial in the study of real-time dynamics of molecular electron wavepacket evolution such as those in attosecond laser chemistry and ultrafast chemical reaction dynamics. We here address two mutually correlated issues on the conservation law of molecular electronic flux, which serves as a key consistency condition for electron dynamics. The first one is about a close relation between "weak" nonadiabaticity and the electron dynamics in low-energy chemical reactions. We show that the electronic flux in adiabatic reactions can be consistently reproduced by taking account of nonadiabaticity. Such nonadiabaticity is usually weak in the sense that it does not have a major effect on nuclear dynamics, whereas it plays an important role in electronic dynamics. Our discussion is based on a nonadiabatic extension of the electronic wavefunction similar in idea to the complete adiabatic formalism developed by Nafie [J. Chem. Phys. 79, 4950 (1983)], which has also recently been reformulated by Patchkovskii [J. Chem. Phys. 137, 084109 (2012)]. We give straightforward proof of the theoretical assertion presented by Nafie using a time-dependent mixed quantum-classical framework and a standard perturbation expansion. Explicitly taking account of the flux conservation, we show that the nonadiabatically induced flux realizes the adiabatic time evolution of the electronic density. In other words, the divergence of the nonadiabatic flux equals the time derivative of the electronic density along an adiabatic time evolution of the target molecule. The second issue is about the accurate computationability of the flux. The calculation of flux needs an accurate representation of the (relative) quantum phase, in addition to the amplitude factor, of a total wavefunction and demands special attention for practical calculations. This paper is the first one to approach this issue directly and show how the difficulties arise explicitly. In doing so, we reveal that a number of widely accepted truncation techniques for static property calculations are potential sources of numerical flux non-conservation. We also theoretically propose alternative strategies to realize better flux conservation.

Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré

We present an extension of a classical result of Poincare (1892) about continuation of periodic orbits and breaking of completely resonant tori in a class of nearly integrable Hamiltonian systems, which covers most Hamiltonian Lattice models. The result is based on the fixed point method of the period map and exploits a standard perturbation expansion of the solution with respect to a small parameter. Two different statements are given, about existence and linear stability: a first one, in the so called non-degenerate case, and a second one, in the completely degenerate case. A pair of examples inspired to the existence of localized solutions in the discrete NLS lattice is provided.

Reconstructing Nonequilibrium Regimes of Quantum Many-Body Systems from the Analytical Structure of Perturbative Expansions

We propose a systematic approach to the non-equilibrium dynamics of strongly interacting many-body quantum systems, building upon the standard perturbative expansion in the Coulomb interaction. High order series are derived from the Keldysh version of determinantal diagrammatic Quantum Monte Carlo, and the reconstruction beyond the weak coupling regime of physical quantities is obtained by considering them as analytic functions of a complex-valued interaction $U$. Our advances rely on two crucial ingredients: i) a conformal change of variable, based on the approximate location of the singularities of these functions in the complex $U$-plane; ii) a Bayesian inference technique, that takes into account additional known non-perturbative relations, in order to control the amplification of noise occurring at large $U$. This general methodology is applied to the strongly correlated Anderson quantum impurity model, and is thoroughly tested both in- and out-of-equilibrium. In the situation of a finite voltage bias, our method is able to extend previous studies, by bridging with the regime of unitary conductance, and by dealing with energy offsets from particle-hole symmetry. We also confirm the existence of a voltage splitting of the impurity density of states, and find that it is tied to a non-trivial behavior of the non-equilibrium distribution function. Beyond impurity problems, our approach could be directly applied to Hubbard-like models, as well as other types of expansions.

Scale Invariant Resummed Thermal Perturbative Expansions

We illustrate how our recently developed renormalization group optimized perturbation (RGOPT) efficiently resums perturbative expansions in thermal field theories. The residual renormalization scale dependence of optimized thermodynamical quantities is drastically improved as compared to either standard perturbative expansions, or related methods such as the screened perturbation or (resummed) hard-thermal-loop perturbation. Our approach is illustrated briefly for the nonlinear sigma model, as a toy model for thermal QCD. Finally, preliminary applications of RGOPT to hard thermal loop resummation for the QCD pressure are sketched.

Convergent series for lattice models with polynomial interactions

The standard perturbative weak-coupling expansions in lattice models are asymptotic. The reason for this is hidden in the incorrect interchange of the summation and integration. However, substituting the Gaussian initial approximation of the perturbative expansions by a certain interacting model or regularizing original lattice integrals, one can construct desired convergent series. In this paper we develop methods, which are based on the joint and separate utilization of the regularization and new initial approximation. We prove, that the convergent series exist and can be expressed as re-summed standard perturbation theory for any model on the finite lattice with the polynomial interaction of even degree. We discuss properties of such series and study their applicability to practical computations on the example of the lattice ϕ4-model. We calculate 〈ϕn2〉 expectation value using the convergent series, the comparison of the results with the Borel re-summation and Monte Carlo simulations shows a good agreement between all these methods.

Wick polynomials and time-evolution of cumulants

We show how Wick polynomials of random variables can be defined combinatorially as the unique choice, which removes all “internal contractions” from the related cumulant expansions, also in a non-Gaussian case. We discuss how an expansion in terms of the Wick polynomials can be used for derivation of a hierarchy of equations for the time-evolution of cumulants. These methods are then applied to simplify the formal derivation of the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete nonlinear Schödinger equation (DNLS) with suitable random initial data. We also present a reformulation of the standard perturbation expansion using cumulants, which could simplify the problem of a rigorous derivation of the Boltzmann-Peierls equation by separating the analysis of the solutions to the Boltzmann-Peierls equation from the analysis of the corrections. This latter scheme is general and not tied to the DNLS evolution equations.

Dynamical Landau theory of the glass crossover

I introduce a dynamical field theory to describe the glassy behavior in supercooled liquids. The mean-field approximation of the theory predicts a dynamical arrest transition, as in ideal Mode-Coupling-Theory and mean-field discontinuous Spin-Glass Models. Instead {\it beyond} the mean-field approximation the theory predicts that the transition is avoided and transformed into a crossover, as observed in experiments and simulations. To go beyond mean-field a standard perturbative loop expansion is performed at first. Approaching the ideal critical point this expansion is divergent at all orders and I show that the leading divergent term at any given order is the same of a dynamical stochastic equation, called Stochastic-Beta-Relaxation (SBR) in {\it EPL 106, 56003 (2014)}. At variance with the original theory SBR can be studied beyond mean-field directly, without the need to resort to a perturbative expansion. Thus it provides a qualitative and quantitative description of the dynamical crossover. For consistency reasons it is important to establish the connection between the dynamical field theory and SBR beyond perturbation theory. This can be done with the help of a stronger result: the dynamical field theory is {\it exactly} equivalent to a theory with quenched disorder. Qualitatively the non-perturbative mechanism leading to the crossover is therefore the same of SBR. Quantitatively SBR corresponds to make the mean-field approximation once the quenched disorder has been generated.

Scale-Invariant Resummed Perturbation at Finite Temperatures.

We use the scalar model with quartic interaction to illustrate how a nonperturbative variational technique combined with renormalization group (RG) properties efficiently resums perturbative expansions in thermal field theories. The resulting convergence and scale dependence of optimized thermodynamical quantities, here illustrated up to two-loop order, are drastically improved as compared to standard perturbative expansions, as well as to other related methods such as the screened perturbation or (resummed) hard-thermal-loop perturbation, that miss RG invariance (as we explain). Being very general and easy to implement, our method is a potential analytical alternative to dealing with the phase transitions of field theories such as thermal QCD.

New branches of massive gravity

The basic building block for Lorentz invariant and ghost free massive gravity is the square root of the combination $g^{-1}\eta\,$, where $g^{-1}$ is the inverse of the physical metric and $\eta$ is a reference metric. Since the square root of a matrix is not uniquely defined, it is possible to have physically inequivalent potentials corresponding to different branches. We show that around Minkowski background the only perturbatively well defined branch is the potential proposed by de Rham, Gabadadze and Tolley. On the other hand, if Lorentz symmetry is broken spontaneously, other potentials exist with a standard perturbative expansion. We show this explicitly building new Lorentz invariant, ghost-free massive gravity potentials for theories that in the background preserve rotational invariance, but break Lorentz boosts.

Spin Glass Field Theory with Replica Fourier Transforms

We develop a field theory for spin glasses using Replica Fourier Transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas n and the number of replica symmetry breaking steps R generic integers. We show how the RFT applied to the two-replica fields allows to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows to express any i-replica vertex in the new RFT basis and hence enables to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi's solution for the free-energy of an Ising spin glass.

Spin glass field theory with replica Fourier transforms

We develop a field theory for spin glasses using replica Fourier transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas n and the number of replica symmetry breaking steps R generic integers. We show how the RFT applied to the two-replica fields allows one to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows one to express any i-replica vertex in the new RFT basis and hence enables one to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass.

Accuracy of analytical models of the large-scale matter distribution

We investigate the possible accuracy that can be reached by analytical models for the matter density power spectrum and correlation function. Using a realistic description of the power spectrum that combines perturbation theory with a halo model, we study the convergence rate of several perturbative expansion schemes and the impact of nonperturbative effects, as well as the sensitivity to phenomenological halo parameters. We check that the simple reorganization of the standard perturbative expansion, with a Gaussian damping prefactor, provides a well-ordered convergence and a finite correlation function that yields a percent accuracy at the baryon acoustic oscillation peak (as soon as one goes to second order). Lagrangian-space expansions are somewhat more efficient, when truncated at low orders, but may diverge at high orders. We find that whereas the uncertainty on the halo-profile mass-concentration relation is not a strong limitation, the uncertainty on the halo mass function can severely limit the accuracy of theoretical predictions for $P(k)$ (this also applies to the power spectra measured in numerical simulations). The real-space correlation function provides a better separation between perturbative and nonperturbative effects, which are restricted to $x \lesssim 10 h^{-1}$Mpc at all redshifts.

Taming Landau singularities in QCD perturbation theory: The analytic approach 2.0

The aim of this topical article is to outline the fundamental ideas underlying the recently developed Fractional Analytic Perturbation Theory (FAPT) of QCD and present its main calculational tools together with key applications. For this, it is first necessary to review previous methods to apply QCD perturbation theory at low spacelike momentum scales, where the influence of the Landau singularities becomes inevitable. Several concepts are considered and their limitations are pointed out. The usefulness of FAPT is discussed in terms of two characteristic hadronic quantities: the perturbatively calculable part of the pion's electromagnetic form factor in the spacelike region and the Higgs-boson decay into a $b\bar b$ pair in the timelike region. In the first case, the focus is on the optimization of the prediction with respect to the choice of the renormalization scheme and the dependence on the renormalization and the factorization scales. The second case serves to show that the application of FAPT to this reaction reaches already at the four-loop level an accuracy of the order of 1%, avoiding difficulties inherent in the standard perturbative expansion. The obtained results are compared with estimates from fixed-order and contour-improved QCD perturbation theory. Using the brand-new Higgs mass value of about 125 GeV, measured at the Large Hadron Collider (CERN), a prediction for $\Gamma_{H\to b\bar{b}}=2.4 \pm 0.15 {\rm MeV}$ is extracted.

Full nonlinear growing and decaying modes of superhorizon curvature perturbations

We clarify the behavior of curvature perturbations in a nonlinear theory in case the inflaton temporarily stops during inflation. We focus on the evolution of curvature perturbation on superhorizon scales by adopting the spatial gradient expansion and show the nonlinear theory, called the beyond $\ensuremath{\delta}N$ formalism, for a general single scalar field as the next-leading order in the expansion. Both the leading order in the expansion ($\ensuremath{\delta}N$ formalism) and our nonlinear theory include the solutions of full-nonlinear orders in the standard perturbative expansion. Additionally, in our formalism, we can deal with the time evolution in contrast to $\ensuremath{\delta}N$ formalism, where curvature perturbations remain just constant, and show decaying modes do not couple with growing modes as similar to the case with the linear theory. We can conclude that, although the decaying mode diverges when $\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}$ vanishes, there appears no trouble for both the linear and nonlinear theory since these modes will vanish at late times.

Exact mathematical treatment of the modifications of finite‐dimensional quantum systems

AbstractLow rank modification (LRM) is a new mathematical formalism by which one can express eigenvalues and eigenstates of the modified system B in terms of the eigenvalues and eigenstates of the original system A. In this respect, LRM is similar to a standard perturbation expansion, which also expresses eigenvalues and eigenstates of the perturbed system B in terms of the eigenvalues and eigenstates of the unperturbed system A. However, unlike perturbation expansion, LRM produces correct results however strong the “perturbation” of the original system A. LRM is here applied to finite n‐dimensional systems A and B that are described by generalized n × n eigenvalue equations. In the LRM approach, modified system B is described by a ρ × ρ matrix equation, where ρ is the dimension of the space affected by the “modification” of the original system A. In mathematical terms, ρ is the rank of the operators that describe this modification. In many important cases, ρ << n, which results in the substantial numerical efficiency of the LRM approach. The method is illustrated with two examples. In the first example, LRM is applied to the vibrational isotope effect within the harmonic approximation. In particular, out‐of‐plane vibrations of benzene (H,D)‐isotopomers are analyzed in this way. In the second example, LRM is applied to some problems in a solid state physics. LRM provides a new mathematical treatment of all those features that break translational symmetry of a solid, such as point defects, the surface of a solid, etc. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Completing Irving and Kirkwood’s Molecular Theory of Transport Processes: Nonequilibrium Entropy Conservation

Following Irving and Kirkwood’s (J. Chem. Phys. 1950, 18, 817) classical approach to the statistical mechanics of transport processes and the conservation equations for mass, momentum, and energy, we introduce a particular dynamical variable for entropy and derive the general nonequilibrium entropy conservation equation. This particular formalism is shown to encompass both Boltzmann’s and Gibbs’ entropy definitions as special cases. Entropy generation is shown to follow from phase-space dimensionality loss and truncations or approximations in higher-order space, the latter of which is consistent with the thesis of Jaynes (Am. J. Phys. 1965, 33, 391). The general approach to entropy conservation given here not only completes Irving and Kirkwood’s treatment of the transport equations but also allows for a consistent analysis of all transport equations for any given system. Following standard perturbation expansion methods about local equilibrium states, we derive the closed form of the entropy conservation equation for isolated systems, which is shown to be in agreement with well-known phenomenological results and the principles of irreversible thermodynamics. In addition, the generalized nonequilibrium entropy developed here is fully consistent with its equilibrium counterpart. As an example, our formalism allows the analysis of entropy changes in dense gases and liquids through the introduction of a nonequilibrium Green’s entropy. This study provides a firm molecular basis of entropy conservation by consistent methods across the transport equations, allowing ready extensions to complex systems. Such foundations are of contemporary importance in designing energy-efficient or minimum entropy generating engineering systems.