Higher-order Perturbation Theory Research Articles

Investigations of the electron paramagnetic resonance parameters and defect structures for Cu2+ ions in BeO crystal with trigonally distorted tetrahedral symmetry

Abstract The electron paramagnetic resonance(EPR) parameters (g factors gi and hyperfine structure constants Ai, where i = x, z) for Cu2+ ions embedded in beryllium oxide (BeO) crystal at Be2+ sites with C3v local symmetry have been calculated by the complete diagonalization method (CDM) and high-order perturbation theory method (PTM). The calculated results from the two methods are in reasonable agreement with the observed data available. The defect structural data of the trigonal Cu2+ center in beryllium oxide are determined from the calculations and analyses. The calculated results show that the Be2+ ions are replaced by Cu2+ ions in BeO crystal and the ligands undergo outward movement.

Exploring high multiplicity amplitudes in quantum mechanics

Calculations of $1\to N$ amplitudes in scalar field theories at very high multiplicities exhibit an extremely rapid growth with the number $N$ of final state particles. This either indicates an end of perturbative behaviour, or possibly even a breakdown of the theory itself. It has recently been proposed that in the Standard Model this could even lead to a solution of the hierarchy problem in the form of a "Higgsplosion". To shed light on this question we consider the quantum mechanical analogue of the scattering amplitude for $N$ particle production in $\phi^4$ scalar quantum field theory, which corresponds to transitions $\langle N \lvert \hat{x} \rvert 0 \rangle$ in the anharmonic oscillator with quartic coupling $\lambda$. We use recursion relations to calculate the $\langle N \lvert \hat{x} \rvert 0 \rangle$ amplitudes to high order in perturbation theory. Using this we provide evidence that the amplitude can be written as $\langle N \lvert \hat{x} \rvert 0 \rangle \sim \exp(F(\lambda N)/\lambda)$ in the limit of large $N$ and $\lambda N$ fixed. We go beyond the leading order and provide a systematic expansion in powers of $1/N$. We then resum the perturbative results and investigate the behaviour of the amplitude in the region where tree-level perturbation theory violates unitarity constraints. The resummed amplitudes are in line with unitarity as well as stronger constraints derived by Bachas. We generalize our result to arbitrary states and powers of local operators $\langle N \lvert \hat{x}^q \rvert M \rangle$ and confirm that, to exponential accuracy, amplitudes in the large $N$ limit are independent of the explicit form of the local operator, i.e. in our case $q$.

Interplay between Electronic Correlation and Metal-Ligand Delocalization in the Spectroscopy of Transition Metal Compounds: Case Study on a Series of Planar Cu2+ Complexes.

We present a comprehensive theoretical study of the physical phenomena that determine the relative energies of three of the lowest electronic states of each of the square-planar copper complexes [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+ and present a detailed analysis of the extent to which truncated configuration interaction (CI) and coupled cluster (CC) theories succeed in predicing the excitation energies. We find that ligand-metal charge transfer (CT) single excitations play a crucial role in the correct determination of the properties of these systems, even though the first impact of these CT on the energetics of these systems appears at fourth-order in perturbation theory. We provide a minimal selected CI space for describing these systems with multireference theories and use a high-order perturbation theory analysis within this space to derive a simple and general physical picture for the LMCT process. We find that coupled cluster singles and doubles (CCSD) energy differences agree very well with near full CI values even though the D1 diagnostics are large, which casts doubt on the usefulness of single-amplitude-based multireference diagnostics. Configuration interaction singles and doubles (CISD) severely underestimates the excitation energies, and the failure is a direct consequence of the size-inconsisency errors in CISD. Finally, we present reference values for the energy differences computed using explicitly correlated CCSD(T) and BCCD(T) theory.

Charged black hole bombs in a Minkowski cavity

Press and Teukolsky famously introduced the concept of a black hole bomb system: a scalar field scattering a Kerr black hole confined inside a mirror undergoes superradiant amplification that keeps repeating due to the reflecting boundary conditions at the mirror. A similar charged black hole bomb system exists if we have a charged scalar field propagating in a Reissner–Nordström black hole confined inside a box. We point out that scalar fields propagating in such a background are unstable not only to superradiance but also to a mechanism known as the near-horizon scalar condensation instability. The two instabilities are typically entangled but we identify regimes in the phase space where one of them is suppressed but the other is present, and vice-versa (we do this explicitly for the charged but non-rotating black hole bomb). These ‘corners’ in the phase space, together with a numerical study of the instabilities allow us to identify accurately the onset of the instabilities. Our results should thus be useful to make educated choices of initial data for the Cauchy problem that follows the time evolution and endpoint of the instabilities. Finally, we use a simple thermodynamic model (that makes no use of the equations of motion) to find the leading order thermodynamic properties of hairy black holes and solitons that should exist as a consequence (and that should be the endpoint) of these instabilities. To find the properties of these hairy solutions at higher order in perturbation theory, the Einstein–Maxwell-scalar equations of motion would have to be solved.

Determination of quark masses from nf=4 lattice QCD and the RI-SMOM intermediate scheme

We determine the charm and strange quark masses in the $\overline{\text{MS}}$ scheme, using $n_f=2+1+1$ lattice QCD calculations with highly improved staggered quarks (HISQ) and the RI-SMOM intermediate scheme to connect the bare lattice quark masses to continuum renormalisation schemes. Our study covers analysis of systematic uncertainties from this method, including nonperturbative artefacts and the impact of the non-zero physical sea quark masses. We find $m_c^{\overline{\text{MS}}}(3 \text{GeV}) = 0.9896(61)$ GeV and $m_s^{\overline{\text{MS}}}(3 \text{GeV}) = 0.08536(85)$ GeV, where the uncertainties are dominated by the tuning of the bare lattice quark masses. These results are consistent with, and of similar accuracy to, those using the current-current correlator approach coupled to high-order continuum QCD perturbation theory, implemented in the same quark formalism and on the same gauge field configurations. This provides a strong test of the consistency of methods for determining the quark masses to high precision from lattice QCD. We also give updated lattice QCD world averages for $c$ and $s$ quark masses.

A first step towards effectively nonperturbative scattering amplitudes in the perturbative regime

We propose an effectively nonperturbative approach to calculating scattering amplitudes in the perturbative regime. We do this in a discretized momentum space by using the QSE method to calculate all the contributions (to all orders in perturbation theory) to the scattering eigenstates that are above a precision cutoff. We then calculate the scattering amplitude by directly taking the inner product between these eigenstates. In the current work we have analyzed this procedure for a λϕ4 theory in one spatial dimension and compared our results with perturbation theory obtaining favorable results suggestive that further research in this direction might be worthwhile. In particular, we show that the efficiency of our method scales much better than second- and higher-order perturbation theory as the momentum lattice spacing decreases and as the eigenstate energy increases.

Aspects of perturbation theory in quantum mechanics: The BenderWuMathematica® package

We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The latter may form a prototype for regularized quantum field theory. We first generalize the method of Bender–Wu,and derive exact recursion relations which allow the determination of the perturbative wave-function and energy corrections to an arbitrary order, at least in principle. For 1D systems, we implement these equations in an easy to use Mathematica® package we call BenderWu. Our package enables quick home-computer computation of high orders of perturbation theory (about 100 orders in 10–30 s, and 250 orders in 1–2 h) and enables practical study of a large class of problems in Quantum Mechanics. We have two hopes concerning the BenderWu package. One is that due to resurgence, large amount of non-perturbative information, such as non-perturbative energies and wave-functions (e.g. WKB wave functions), can in principle be extracted from the perturbative data. We also hope that the package may be used as a teaching tool, providing an effective bridge between perturbation theory and non-perturbative physics in textbooks. Finally, we show that for the multi-variable case, the recursion relation acquires a geometric character, and has a structure which allows parallelization to computer clusters. Program summaryProgram Title: BenderWuProgram Files doi:http://dx.doi.org/10.17632/vpg2zsbryc.1Licensing provisions: CC By 4.0Programming language: Wolfram MathematicaNature of problem: In 1D quantum mechanics, a perturbative expansions are known to generically be divergent. An analysis of such problems was so far limited to a case-by-case basis. The Mathematica package presented here allows a quick computation and analysis of all such 1D quantum mechanical problems.Solution method: The program uses a general recursive relation, inspired by the works of Bender and Wu [1], which allows quick computation of the perturbative data.

Construction of Hamiltonians by supervised learning of energy and entanglement spectra

Correlated many-body problems ubiquitously appear in various fields of physics such as condensed matter physics, nuclear physics, and statistical physics. However, due to the interplay of the large number of degrees of freedom, it is generically impossible to treat these problems from first principles. Thus the construction of a proper model, namely effective Hamiltonian, is essential. Here, we propose a simple scheme of constructing Hamiltonians from given energy or entanglement spectra with machine learning. We apply the proposed scheme to the Hubbard model at the half-filling, and compare the obtained effective low-energy spin-1/2 model with several analytic results based on the high order perturbation theory which have been inconsistent with each other. We also show that our approach can be used to construct the entanglement Hamiltonian of a quantum many-body state from its entanglement spectrum as well. We exemplify this using the ground states of the $S=1/2$ two-leg Heisenberg ladders. We observe a qualitative difference between the entanglement Hamiltonians of the two phases (the Haldane phase and the Rung Singlet phase) of the model due to the different origin of the entanglement. In the Haldane phase, we find that the entanglement Hamiltonian is non-local by nature, and the locality can be restored by introducing the anisotropy and turning the system into the large-$D$ phase. Possible applications to the study of strongly-correlated systems and the model construction from experimental data are discussed.

Directed Percolation: Calculation of Feynman Diagrams in the Three-Loop Approximation

The directed bond percolation process is an important model in statistical physics. By now its universal properties are known only up to the second-order of the perturbation theory. Here, our aim is to put forward a numerical technique with anomalous dimensions of directed percolation to higher orders of perturbation theory and is focused on the most complicated Feynman diagrams with problems in calculation. The anomalous dimensions are computed up to three-loop order in ε = 4 − d.

Quantification of control rod worth uncertainties propagated from nuclear data via a hybrid high-order perturbation and efficient sampling method

Based on the high order perturbation theory and few-group neutron diffusion equation, the formula to evaluate control rod worth is derived for various core state, which is equivalent to establish a function between CRW and eigenvalue, few-group homogenized cross sections, flux and adjoint flux. Then, an effective hybrid high order perturbation and efficient sampling (HOPES) method to quantify the uncertainty of control rod worth propagated from uncertainties in input parameters is proposed. To verify the validity of the HOPES method, a three dimensional mini core model with typical AP1000 fuel assemblies in a 3 × 3 checkerboard pattern is built and uncertainty in nuclear data is chosen as the main uncertainty source. Numerical results indicate that the HOPES method is much more effective to directly propagate uncertainties in input parameters to the final control rod worth and can quantify the uncertainty of either differential or integral control rod worth more accurately. Finally, the uncertainty of differential and integral CRW propagated from nuclear cross sections was quantified for the first time by using the HOPES method.

High order perturbation theory for difference equations and Borel summability of quantum mirror curves

We adapt the Bender-Wu algorithm to solve perturbatively but very efficiently the eigenvalue problem of "relativistic" quantum mechanical problems whose Hamiltonians are difference operators of the exponential-polynomial type. We implement the algorithm in the function BWDifference in the updated Mathematica package BenderWu. With the help of BWDifference, we survey quantum mirror curves of toric fano Calabi-Yau threefolds, and find strong evidence that not only are the perturbative eigenenergies of the associated 1d quantum mechanical problems Borel summable, but also that the Borel sums are exact.

The effects of higher orders of perturbation theory on the correlation energy of atoms and bonds in molecules

AbstractWe examine, for the first time, the effects of higher orders of Møller–Plesset perturbation theory on the individual atoms within a molecule and the bonds between them, via the topological energy partitioning method of interacting quantum atoms. In real terms (i.e., not by absolute value) MP3 decreases the correlation energy of a bond, and MP4SDQ also decreases the energy of the atoms at either end of the bond. In addition, we investigated long‐range through‐space dispersive effects on a H2 oligomer. Overall, MP3 is the largest correction to the correlation energy, and most of that energy is allocated to chemical bonds, reducing their values in actual terms. The MP4SDQ bond correlation correction, despite being relatively small, tends to have two effects: (i) for small or negative correlation energies MP4SDQ tends to decrease the bond correlation values even more, and (ii) for large (positive) bond correlation energies MP4SDQ tends to restore the bond correlation energies from the MP3 back toward the MP2 values. Furthermore, each individual part of a molecule or complex (atom or bond) has a specific convergence pattern for the MPn series: through‐space interactions converge at MP2 but bonds converge at MP3 level. The atomic correlation energy appears to head toward convergence at the MP4 level.

BRST renormalization of the first order Yang–Mills theory

We examine the renormalization of the first order formulation of the Yang–Mills theory, by using the BRST identities. These preserve the gauge invariance of the theory and enable a recursive proof of renormalizability to higher orders in perturbation theory. The renormalization involves non-linear mixings as well as re-scalings of the fields and sources, which lead to a renormalized action at all orders.

The analysis of harmonic generation coefficients in the ablative Rayleigh-Taylor instability

In this research, we use the numerical simulation method to investigate the generation coefficients of the first three harmonics and the zeroth harmonic in the Ablative Rayleigh–Taylor Instability. It is shown that the interface shifts to the low temperature side during the ablation process. In consideration of the third-order perturbation theory, the first three harmonic amplitudes of the weakly nonlinear regime are calculated and then the harmonic generation coefficients are obtained by curve fitting. The simulation results show that the harmonic generation coefficients changed with time and wavelength. Using the higher-order perturbation theory, we find that more and more harmonics are generated in the later weakly nonlinear stage, which is caused by the negative feedback of the later higher harmonics. Furthermore, extending the third-order theory to the fifth-order theory, we find that the second and the third harmonics coefficients linearly depend on the wavelength, while the feedback coefficients are almost constant. Further analysis also shows that when the fifth-order theory is considered, the normalized effective amplitudes of second and third harmonics can reach about 25%–40%, which are only 15%–25% in the frame of the previous third-order theory. Therefore, the third order perturbation theory is needed to be modified by the higher-order theory when ηL reaches about 20% of the perturbation wavelength.

Classical gluon and graviton radiation from the bi-adjoint scalar double copy

We find double copy relations between classical radiating solutions in Yang-Mills theory coupled to dynamical color charges and their counterparts in a cubic bi-adjoint scalar field theory which interacts linearly with particles carrying bi-adjoint charge. The particular color-to-kinematics replacements we employ are motivated by the BCJ double copy correspondence for on-shell amplitudes in gauge and gravity theories. They are identical to those recently used to establish relations between classical radiating solutions in gauge theory and in dilaton gravity. Our explicit bi-adjoint solutions are constructed to second order in a perturbative expansion, and map under the double copy onto gauge theory solutions which involve at most cubic gluon self-interactions. If the correspondence is found to persist to higher orders in perturbation theory, our results suggest the possibility of calculating gravitational radiation from colliding compact objects, directly from a scalar field with vastly simpler (purely cubic) Feynman vertices.

Theoretical explanation of spin-Hamiltonian parameters and local structure for the orthorhombic MnO2 −4 clusters in K2CrO4 : Mn6 + crystal

ABSTRACTIn this paper, the spin-Hamiltonian parameters (g factors gx, gy, gz and hyperfine structure constants A Ax, Ay, Az) and the absorption spectrum of K2CrO4 : Mn6 + crystal are theoretically explained by using the high-order perturbation theory, the double-spin-orbit-coupling model theory and the double-mechanism theory (the crystal field mechanism and the charge-transfer (CT) mechanism). The calculation results show that the contribution of the CT mechanism cannot be neglected for Mn6 + ions in orthorhombic clusters with the ground state .

A critical evaluation of perturbation theories by Monte Carlo simulation of the first four perturbation terms in a Helmholtz energy expansion for the Lennard-Jones fluid.

The Helmholtz energy of a fluid interacting by a Lennard-Jones pair potential is expanded in a perturbation series. Both the methods of Barker-Henderson (BH) and of Weeks-Chandler-Andersen (WCA) are evaluated for the division of the intermolecular potential into reference and perturbation parts. The first four perturbation terms are evaluated for various densities and temperatures (in the ranges ρ*=0-1.5 and T*=0.5-12) using Monte Carlo simulations in the canonical ensemble. The simulation results are used to test several approximate theoretical methods for describing perturbation terms or for developing an approximate infinite order perturbation series. Additionally, the simulations serve as a basis for developing fully analytical third order BH and WCA perturbation theories. The development of analytical theories allows (1) a careful comparison between the BH and WCA formalisms, and (2) a systematic examination of the effect of higher-order perturbation terms on calculated thermodynamic properties of fluids. Properties included in the comparison are supercritical thermodynamic properties (pressure, internal energy, and chemical potential), vapor-liquid phase equilibria, second virial coefficients, and heat capacities. For all properties studied, we find a systematically improved description upon using a higher-order perturbation theory. A result of particular relevance is that a third order perturbation theory is capable of providing a quantitative description of second virial coefficients to temperatures as low as the triple-point of the Lennard-Jones fluid. We find no reason to prefer the WCA formalism over the BH formalism.

Merging of photons in the field of a multielectron atom: Higher orders of perturbation theory

The contribution of higher orders of relativistic quantum perturbation theory to the differential cross section for the merging of three X-ray photons into one photon in the field of a multielectron atom has been studied in the Tamm–Dankov approximation. It has been shown that higher orders do not change the physical results obtained in the leading (second) order of perturbation theory.

Double hard scattering without double counting

Double parton scattering in proton-proton collisions includes kinematic regions in which two partons inside a proton originate from the perturbative splitting of a single parton. This leads to a double counting problem between single and double hard scattering. We present a solution to this problem, which allows for the definition of double parton distributions as operator matrix elements in a proton, and which can be used at higher orders in perturbation theory. We show how the evaluation of double hard scattering in this framework can provide a rough estimate for the size of the higher-order contributions to single hard scattering that are affected by double counting. In a numeric study, we identify situations in which these higher-order contributions must be explicitly calculated and included if one wants to attain an accuracy at which double hard scattering becomes relevant, and other situations where such contributions may be neglected.

Non-linear perturbation of black branes at large D

The Einstein equations describing the black-brane dynamics both in Minkowski and AdS background were recently recast in the form of coupled diffusion equations in the large-D(imension) limit. Using such results in the literature, we formulate a higher-order perturbation theory of black branes in time domain and present the general form of solutions for arbitrary initial conditions. For illustrative purposes, the solutions up to the first or second order are explicitly written down for several kind of initial conditions, such as a Gaussian wave packet, shock wave, and rather general superposed sinusoidal waves. These could be the first examples describing the non-trivial evolution of black-brane horizons in time domain. In particular, we learn some interesting aspects of black-brane dynamics such as the Gregory-Laflamme (GL) instability and non-equilibrium steady state (NESS). The formalism presented here would be applicable to the analysis of various black branes and their holographically dual field theories.