High-order Perturbation Expansion Research Articles

An adaptive divided-difference perturbation method for solving stochastic problems

Most stochastic problems are based on the independent and symmetrically distributed random variables. Using this principle, in this study, we propose an efficient calculation scheme to obtain the statistical moments based on high-order stochastic perturbation (SP) expansion. This solves the long-existing problem of the SP method, namely, the low calculation efficiency of statistical moments using high-order perturbation expansion, and the complexity of the calculation format. The scheme involves adopting the divided-difference method to approximate the partial derivative items of the SP method. The proposed method, which is similar to the collocation point method, only requires the deterministic calculation result for each node in the divided-difference method. In addition, as the adoption of high-order perturbation expansion is limited in multidimensional random problems due to the curse of dimensionality, an adaptive partial derivative items selection method is proposed in this paper to efficiently and reasonably ignore the expansion items that have an insignificant influence on statistical moments. This considerably improves the calculation efficiency of the statistical moments. Finally, the adaptive divided-difference perturbation (ADDP) method proposed in this paper is compared with the quasi-Monte Carlo method and the adaptive sparse grid methods using three numerical examples. The results prove that the ADDP method not only has high efficiency but also leads to significant improvements in the accuracy of statistical moments.

Learning Feynman Diagrams with Tensor Trains

We use tensor network techniques to obtain high-order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm. It yields the full time evolution of physical quantities in the presence of any arbitrary time-dependent interaction. Our benchmarks on the Anderson quantum impurity problem, within the real-time nonequilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates 1/N2 or faster, where N is the number of function evaluations. The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in quantum Monte Carlo calculations. Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two-dimensional lattice.11 MoreReceived 13 July 2022Revised 5 September 2022Accepted 8 September 2022DOI:https://doi.org/10.1103/PhysRevX.12.041018Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsQuantum dotsTechniquesDiagrammatic methodsGreen's function methodsMachine learningQuantum Monte CarloSingle impurity modelTensor network methodsCondensed Matter, Materials & Applied Physics

VarRCWA: An Adaptive High-Order Rigorous Coupled Wave Analysis Method.

Semianalytical methods, such as rigorous coupled wave analysis, have been pivotal in the numerical analysis of photonic structures. In comparison to other numerical methods, they have a much lower computational cost, especially for structures with constant cross-sectional shapes (such as metasurface units). However, when the cross-sectional shape varies even mildly (such as a taper), existing semianalytical methods suffer from high computational costs. We show that the existing methods can be viewed as a zeroth-order approximation with respect to the structure's cross-sectional variation. We derive a high-order perturbative expansion with respect to the cross-sectional variation. Based on this expansion, we propose a new semianalytical method that is fast to compute even in the presence of large cross-sectional shape variation. Furthermore, we design an algorithm that automatically discretizes the structure in a way that achieves a user-specified accuracy level while at the same time reducing the computational cost.

Generalized master equation for charge transport in a molecular junction: Exact memory kernels and their high order expansion.

We derive a set of generalized master equations (GMEs) to study charge transport dynamics in molecular junctions using the Nakajima-Zwanzig-Mori projection operator approach. In the new GME, time derivatives of population on each quantum state of the molecule, as well as the tunneling current, are calculated as the convolution of time non-local memory kernels with populations on all system states. The non-Markovian memory kernels are obtained by combining the hierarchical equations of motion (HEOM) method and a previous derived Dyson relation for the exact kernel. A perturbative expansion of these memory kernels is then calculated using the extended HEOM developed in our previous work [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. By using the resonant level model and the Anderson impurity model, we study properties of the exact memory kernels and analyze convergence properties of their perturbative expansions with respect to the system-bath coupling strength and the electron-electron repulsive energy. It is found that exact memory kernels calculated from HEOM exhibit short memory times and decay faster than the population and current dynamics. The high order perturbation expansion of the memory kernels can give converged results in certain parameter regimes. The Padé and Landau-Zener resummation schemes are also found to give improved results over low order perturbation theory.

Quantum quasi Monte Carlo algorithm for out-of-equilibrium Green functions at long times

We extend the recently developed quantum quasi Monte Carlo (QQMC) approach to obtain the full frequency dependence of Green functions in a single calculation. QQMC is a general approach for calculating high-order perturbative expansions in power of the electron-electron interaction strength. In contrast to conventional Markov chain Monte Carlo sampling, QQMC uses low-discrepancy sequences for a more uniform sampling of the multidimensional integrals involved and can potentially outperform Monte Carlo by several orders of magnitude. A core concept of QQMC is the a priori construction of a ``model function'' that approximates the integrand and is used to optimize the sampling distribution. In this paper, we show that the model function concept extends to a kernel approach for the computation of Green functions. We illustrate the approach on the Anderson impurity model and show that the scaling of the error with the number of integrand evaluations $N$ is $\ensuremath{\sim}1/{N}^{0.86}$ in the best cases and comparable to Monte Carlo scaling $\ensuremath{\sim}1/{N}^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the fork tensor product state (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Pad\'e approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.

Lagrangian Cosmological Perturbation Theory at Shell Crossing.

We consider the growth of primordial dark matter halos seeded by three crossed initial sine waves of various amplitudes. Using a Lagrangian treatment of cosmological gravitational dynamics, we examine the convergence properties of a high-order perturbative expansion in the vicinity of shell crossing by comparing the analytical results with state-of-the-art high resolution Vlasov-Poisson simulations. Based on a quantitative exploration of parameter space, we study explicitly for the first time the convergence speed of the perturbative series and find, in agreement with intuition, that it slows down when going from quasi-one-dimensional initial conditions (one sine wave dominating) to quasitriaxial symmetry (three sine waves with same amplitude). In most cases, the system structure at collapse time is, as expected, very similar to what is obtained with simple one-dimensional dynamics, except in the quasitriaxial regime, where the phase-space sheet presents a velocity spike. In all cases, the perturbative series exhibits a generic convergence behavior as fast as an exponential of a power law of the order of the expansion, allowing one to numerically extrapolate it to infinite order. The results of such an extrapolation agree remarkably well with the simulations, even at shell crossing.

Higher Order Algorithm for Solving Lambert’s Problem

This work presents a high-order perturbation expansion method for solving Lambert’s problem. The necessary condition for the problem is defined by a fourth-order Taylor expansion of the terminal error vector. The Taylor expansion partial derivative models are generated by Computational Differentiation (CD) tools. A novel derivative enhanced numerical integration algorithm is presented for computing nonlinear state transition tensors, where only the equation of motion is coded. A high-order successive approximation algorithm is presented for inverting the problems nonlinear necessary condition. Closed-form expressions are obtained for the first, second,third, and fourth order perturbation expansion coefficients. Numerical results are presented that compare the convergence rate and accuracy of first-through fourth-order expansions. The initial p-iteration starting guess is used as the Lambert’s algorithm initial condition. Numerical experiments demonstrate that accelerated convergence is achieved for the second-, third-, and fourth-order expansions, when compared to a classical first-order Newton method.

Exact generator and its high order expansions in time-convolutionless generalized master equation: Applications to spin-boson model and excitation energy transfer

The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.

On high-order perturbation expansion for the study of long–short wave interactions

In high-order analysis and simulation of long–short surface wave interaction using mode decomposition, ‘divergent’ terms of the form $k_{S}a_{L}=O(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D716})\gg 1$ appear in the high-order expansions, where $k_{L,S}$, $a_{L,S}$ are respectively the long, short modal wavenumbers and amplitudes, with $\unicode[STIX]{x1D6FE}\equiv k_{S}/k_{L}\gg 1$ and $k_{L}a_{L}\sim k_{S}a_{S}=O(\unicode[STIX]{x1D716})$ finite. We address the effect of these terms on the numerical scheme, showing numerical cancellation at all orders $m$; but increasing ill-conditioning of the numerics with $\unicode[STIX]{x1D6FE}$ and $m$, which we quantify. In the context of mode decomposition, we show theoretical exact cancellation of the divergent terms up to $m=3$, extending the existing result of Brueckner & West (J. Fluid Mech., vol. 196, 1988, pp. 585–592) and supporting the conjecture that this is obtained for all orders $m$. We show the latter by developing a theoretical proof for any $m$ using a Dirichlet–Neumann operator and mathematical induction. The implication of the theoretical proof on the numerical simulation of long–short wave interaction is discussed.

High-order perturbative expansions in massless gauge theories with NSPT

We investigate the possibility of using numerical stochastic perturbation theory (NSPT) to probe high orders in the perturbative expansion of lattice gauge theories with massless Wilson fermions. Twisted boundary conditions are used to regularise the gauge zero-mode; the extension of these boundary conditions to include fermions in the fundamental representation requires to introduce a smell degree of freedom. Moreover, the mass of Wilson fermions is affected by an additive renormalisation: we study how to determine the mass counterterms consistently in finite volume. The knowledge of the critical masses will enable high-order perturbative computations in massless QCD, e.g. (as a first application) for the plaquette.

Switches in a genetic regulatory system under multiplicative non-Gaussian noise

The non-Gaussian noise is multiplicatively introduced to model the universal fluctuation in the gene regulation of the bacteriophage λ. To investigate the key effect of non-Gaussian noise on the genetic on/off switch dynamics from the viewpoint of quantitative analysis, we employ the high-order perturbation expansion to deduce the stationary probability density of repressor concentration and the mean first passage time from low concentration to high concentration and vice versa. The occupation probability of different concentration states can be estimated from the height and shape of the peaks of the stationary probability density, which could be used to determine the overall expression level. A further concern is the mean first passage time, also referred to as the mean switching time, which can be adopted as an important measure to characterize the adaptability of gene expression to the environmental variation. Through our investigation, it is observed that the non-Gaussian heavy-tailed noise can better induce the switches between distinct genetic expression states and additionally, it accelerates the switching process more evidently compared to the Gaussian noise and the bounded noise.

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.

Investigation of new methods for numerical stochastic perturbation theory in φ4 theory

Numerical stochastic perturbation theory is a powerful tool for estimating high-order perturbative expansions in lattice field theory. The standard algorithms based on the Langevin equation, however, suffer from several limitations which in practice restrict the potential of this technique. In this work we investigate some alternative methods which could in principle improve on the standard approach. In particular, we present a study of the recently proposed Instantaneous Stochastic Perturbation Theory, as well as a formulation of numerical stochastic perturbation theory based on Generalized Hybrid Molecular Dynamics algorithms. The viability of these methods is investigated in $\varphi^4$ theory.

Convergence of high order perturbative expansions in open system quantum dynamics.

We propose a new method to directly calculate high order perturbative expansion terms in open system quantum dynamics. They are first written explicitly in path integral expressions. A set of differential equations are then derived by extending the hierarchical equation of motion (HEOM) approach. As two typical examples for the bosonic and fermionic baths, specific forms of the extended HEOM are obtained for the spin-boson model and the Anderson impurity model. Numerical results are then presented for these two models. General trends of the high order perturbation terms as well as the necessary orders for the perturbative expansions to converge are analyzed.

Properties of the one-dimensional Bose–Hubbard model from a high-order perturbative expansion

We employ a high-order perturbative expansion to characterize the ground state of the Mott phase of the one-dimensional Bose–Hubbard model. We compute for different integer filling factors the energy per lattice site, the two-point and density–density correlations, and expectation values of powers of the on-site number operator determining the local atom number fluctuations (variance, skewness, kurtosis). We compare these expansions to numerical simulations of the infinite-size system to determine their range of applicability. We also discuss a new sum rule for the density–density correlations that can be used in both equilibrium and non-equilibrium systems.

SU(3) inDdecays: From 30% symmetry breaking to10−4precision

Flavor SU(3) symmetry, including $30\%$ first order SU(3) breaking, has been shown to describe adequately a vast amount of data for charmed meson decays to two pseudoscalar mesons and to a vector and a pseudoscalar meson. We review a recent dramatic progress achieved by applying a high order perturbation expansion in flavor SU(3) breaking and treating carefully isospin breaking. We identify a class of U-spin related $D^0$ decays to pairs involving charged pseudoscalar or vector mesons, for which high-precision nonlinear amplitude relations are predicted. Symmetry breaking terms affecting these relations are fourth order U-spin breaking, and terms which are first order in isospin breaking and second order in U-spin breaking. The predicted relations are shown to hold experimentally at a precision varying between $10^{-3}$ and $10^{-4}$, in agreement with estimates of high order terms. We also discuss amplitude relations for $D^0$ decays to pairs of neutral pseudoscalar mesons, and relations for rate asymmetries between decays involving $K^0_S$ and $K^0_L$ which hold up to second order U-spin breaking.

Relevance of the axial anomaly at the finite-temperature chiral transition in QCD

We investigate the nature of the finite-temperature chiral transition in QCD with two light flavors, in the case of an effective suppression of the the U(1)_A symmetry breaking induced by the axial anomaly, which implies the symmetry breaking U(2)_L X U(2)_R -> U(2)_V, instead of SU(2)_L X SU(2)_R -> SU(2)_V. For this purpose, we perform a high-order field-theoretical perturbative study of the renormalization-group (RG) flow of the corresponding three-dimensional multiparameter Landau-Ginzburg-Wilson Phi4 theory with the same symmetry-breaking pattern. We confirm the existence of a stable fixed point (FP), and determine its attraction domain in the space of the bare quartic parameters. Therefore, the chiral QCD transition might be continuous also if the U(1)_A symmetry is effectively restored at Tc. However, the corresponding universality class differs from the O(4) vector universality class which would describe a continuous transition in the presence of a substantial U(1)_A symmetry breaking at Tc. We estimate the critical exponents of the U(2)_L X U(2)_R -> U(2)_V universality class by computing and analyzing their high-order perturbative expansions. These results are important to discriminate among the different scenarios for the scaling behavior of QCD with two light flavors close to the chiral transition.

Marginal dimensions for multicritical phase transitions

The field-theoretical model describing multicritical phenomena with two coupled order parameters with n_{||} and n_{\perp} components and of O(n_{||}) \oplus O(n_{\perp}) symmetry is considered. Conditions for realization of different types of multicritical behaviour are studied within the field-theoretical renormalization group approach. Surfaces separating stability regions for certain types of multicritical behaviour in parametric space of order parameter dimensions and space dimension d are calculated using the two-loop renormalization group functions. Series for the order parameter marginal dimensions that control the crossover between different universality classes are extracted up to the fourth order in \varepsilon=4-d and to the fifth order in a pseudo-\varepsilon parameter using the known high-order perturbative expansions for isotropic and cubic models. Special attention is paid to a particular case of O(1) \oplus O(2) symmetric model relevant for description of anisotropic antiferromagnets in an external magnetic field.

Compelling Evidence of Renormalons in QCD from High Order Perturbative Expansions

We compute the static self-energy of SU(3) gauge theory in four spacetime dimensions to order α20 in the strong coupling constant α. We employ lattice regularization to enable a numerical simulation within the framework of stochastic perturbation theory. We find perfect agreement with the factorial growth of high order coefficients predicted by the conjectured renormalon picture based on the operator product expansion.

Kitaev model and dimer coverings on the honeycomb lattice

We consider an extension of the Kitaev honeycomb model based on arbitrary dimercoverings satisfying matching rules. We focus on three different dimer coverings having thesmallest unit cells for which we calculate the ground-state phase diagram. We also studyone- and two-vortex properties for these coverings in the Abelian phases and show thatvortex–vortex interactions can be attractive or repulsive. These qualitative differences areconfirmed analytically by high-order perturbative expansions around the isolated-dimerlimit. Similarities and differences with the original Kitaev honeycomb model are discussed.