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Yang-Mills observables: from $\mathrm{KMOC}$ to eikonal through $\mathrm{EFT}$
We obtain a conservative Hamiltonian describing the interactions of two charged bodies in Yang-Mills through $\mathcal{O}$ (α 2 ) and to all orders in velocity. Our calculation extends a recently-introduced framework based on scattering amplitudes and effective field theory (EFT) to consider color-charged objects. These results are checked against the direct integration of the observables in the Kosower-Maybee-O’Connell (KMOC) formalism. At the order we consider we find that the linear and color impulses in a scattering event can be concisely described in terms of the eikonal phase, thus extending the domain of applicability of a formula originally proposed in the context of spinning particles.
Color-Dual Fates of F 3 , R 3 , and N = 4 Supergravity
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Kinetics of hydrodynamic pions in chiral perturbation theory
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Analytic MHD Theory for Earth's Bow Shock at Low Mach Numbers
A previous MHD theory for the density jump at the Earth's bow shock, which assumed the Alfven M(A) and sonic M(s) Mach numbers are both much greater than 1, is reanalyzed and generalized. It is shown that the MHD jump equation can be analytically solved much more directly using perturbation theory, with the ordering determined by M(A) and M(s), and that the first-order perturbation solution is identical to the solution found in the earlier theory. The second-order perturbation solution is calculated, whereas the earlier approach cannot be used to obtain it. The second-order terms generally are important over most of the range of M(A) and M(s) in the solar wind when the angle theta between the normal to the bow shock and magnetic field is not close to 0 deg or 180 deg (the solutions are symmetric about 90 deg). This new perturbation solution is generally accurate under most solar wind conditions at 1 AU, with the exception of low Mach numbers when theta is close to 90 deg. In this exceptional case the new solution does not improve on the first-order solutions obtained earlier, and the predicted density ratio can vary by 10-20% from the exact numerical MHD solutions. For theta approx. = 90 deg another perturbation solution is derived that predicts the density ratio much more accurately. This second solution is typically accurate for quasi-perpendicular conditions. Taken together, these two analytical solutions are generally accurate for the Earth's bow shock, except in the rare circumstance that M(A) is less than or = 2. MHD and gasdynamic simulations have produced empirical models in which the shock's standoff distance a(s) is linearly related to the density jump ratio X at the subsolar point. Using an empirical relationship between a(s) and X obtained from MHD simulations, a(s) values predicted using the MHD solutions for X are compared with the predictions of phenomenological models commonly used for modeling observational data, and with the predictions of a modified phenomenological model proposed recently. The similarities and differences between these results are illustrated using plots of X and a(s) predicted for the Earth's bow shock. The plots show that the new analytic solutions agree very well with the exact numerical MHD solutions and that these MHD solutions should replace the corresponding phenomenological relations in comparisons with data. Furthermore, significant differences exist between the standoff distances predicted at low M(A) using the MHD models versus those predicted by the new modified phenomenological model. These differences should be amenable to observational testing.
Adiabatic invariants in stellar dynamics. 2: Gravitational shocking
A new theory of gravitational shocking based on time-dependent perturbation theory shows that the changes in energy and angular momentum due to a slowly varying disturbance are not exponentially small for stellar dynamical systems in general. It predicts significant shock heating by slowly varying perturbations previously thought to be negligible according to the adiabatic criterion. The theory extends the scenarios traditionally computed only with the impulse approximation and is applicable to a wide class of disturbances. The approach is applied specifically to the problem of disk shocking of star clusters.
Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory
We develop expressions for electron density defined through the linear response for general density functional approximations, demonstrating results for orbital functionals and for many-body perturbation theory, and explore the connections to developments in DFT.
On the connection between perturbation theory and new semiclassical expansion in quantum mechanics
Here, it is shown that for the one-dimensional anharmonic oscillator with potential V(x) = ax 2 + bgx 3 + ... = $\frac{1}{g^2}$ $\hat{V}$ (gx), as well as for the radial oscillator V(r) = $\frac{1}{g^2}$ $\hat{V}$ (gr) and for the perturbed Coulomb problem V(r) = $\frac{a}{r}$ + βgr + ... = g $\tilde{V}$ (gr), the Perturbation Theory in powers of the coupling constant g (weak coupling regime) and the semiclassical expansion in powers of $\hbar$ 1/2 for the energies coincide. This is related to the fact that the dynamics developed in two spaces: x (r)-space and gx (gr)-space, lead to the same energy spectra. The equations which govern dynamics in these two spaces, the Riccati-Bloch equation and the Generalized Bloch equation, respectively, are presented. It is shown that the perturbation theory for the logarithmic derivative of the wavefunction in - space leads to (true) semiclassical expansion in powers of $\hbar$ 1/2 ; for the one-dimensional case this corresponds to the flucton calculus for the density matrix in the path integral formalism in Euclidean (imaginary) time proposed by one of the authors. Matching the perturbation theory in powers of g and the semiclassical expansion in powers of $\hbar$ 1/2 for the wavefunction leads to a highly accurate local approximation in the entire coordinate space, its expectation value for the Hamiltonian provides a prescription for the summation of the perturbative (trans)-series.
Robust root clustering for linear uncertain systems using generalized Lyapunov theory
Consideration is given to the problem of matrix root clustering in subregions of a complex plane for linear state space models with real parameter uncertainty. The nominal matrix root clustering theory of Gutman & Jury (1981) using the generalized Liapunov equation is extended to the perturbed matrix case, and bounds are derived on the perturbation to maintain root clustering inside a given region. The theory makes it possible to obtain an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space.
Ladder symmetries and Love numbers of Reissner-Nordström black holes
It is well known that asymptotically flat black holes in general relativity have vanishing tidal Love numbers. In the case of Schwarzschild and Kerr black holes, this property has been shown to be a consequence of a hidden structure of ladder symmetries for the perturbations. In this work, we extend the ladder symmetries to non-rotating charged black holes in general relativity. As opposed to previous works in this context, we adopt a more general definition of Love numbers, including quadratic operators that mix gravitational and electromagnetic perturbations in the point-particle effective field theory. We show that the calculation of a subset of those couplings in full general relativity is affected by an ambiguity in the split between source and response, which we resolve through an analytic continuation. As a result, we derive a novel master equation that unifies scalar, electromagnetic and gravitational perturbations around Reissner-Nordström black holes. The equation is hypergeometric and can be obtained from previous formulations via nontrivial field redefinitions, which allow to systematically remove some of the singularities and make the presence of the ladder symmetries more manifest.
Consequences of using nonlinear particle trajectories to compute spatial diffusion coefficients
The propagation of charged particles through interstellar and interplanetary space has often been described as a random process in which the particles are scattered by ambient electromagnetic turbulence. In general, this changes both the magnitude and direction of the particles' momentum. Some situations for which scattering in direction (pitch angle) is of primary interest were studied. A perturbed orbit, resonant scattering theory for pitch-angle diffusion in magnetostatic turbulence was slightly generalized and then utilized to compute the diffusion coefficient for spatial propagation parallel to the mean magnetic field, Kappa. All divergences inherent in the quasilinear formalism when the power spectrum of the fluctuation field falls off as K to the minus Q power (Q less than 2) were removed. Various methods of computing Kappa were compared and limits on the validity of the theory discussed. For Q less than 1 or 2, the various methods give roughly comparable values of Kappa, but use of perturbed orbits systematically results in a somewhat smaller Kappa than can be obtained from quasilinear theory.
Topological Superconductivity Based on Antisymmetric Spin–Orbit Coupling
Topological superconductivity (TSC) has drawn much attention for its fundamental interest and application in quantum computation. An outstanding challenge is the lack of intrinsic TSC materials with a p-wave pairing gap, which has led to the development of an effective p-wave theory of coupling s-wave gap with Rashba spin–orbit coupling (RSOC). However, the RSOC-strict mechanism and materials pose still both fundamental and practical limitations. Here, we generalize this theory to antisymmetric SOC (ASOC). Using k·p perturbation theory, we demonstrate that 2D crystals, with point groups of C 2 , C 4 , C 6 , C 2v , C 4v , C 6v , D 2 , D 4 , D 6 , S 4 , or D 2d , can all facilitate the desired ASOC. Remarkably, this enables us to discover 314 TSC candidates by screening 2D material databases, which are further confirmed by first-principles calculations of Majorana boundary modes and the topological invariant of the superconducting gap. As a result, our work fundamentally enriches TSC theory and greatly expands the classes of TSC materials for experimental exploration.
Unified Approach to Polarons and Phonon-Induced Band Structure Renormalization
Ab initio calculations of the phonon-induced band structure renormalization are currently based on the perturbative Allen-Heine theory and its many-body generalizations. These approaches are unsuitable to describe materials where electrons form localized polarons. Here, we develop a self-consistent, many-body Green’s function theory of band structure renormalization that incorporates localization and self-trapping. Here, we show that the present approach reduces to the Allen-Heine theory in the weak-coupling limit, and to total energy calculations of self-trapped polarons in the strong-coupling limit. To demonstrate this methodology, we reproduce the path-integral results of Feynman and diagrammatic Monte Carlo calculations for the Fröhlich model at all couplings, and we calculate the zero point renormalization of the band gap of an ionic insulator including polaronic effects.
Machine-assisted discovery of integrable symplectic mappings
Integrable systems possess a hidden symmetry associated with the existence of conserved quantities known as integrals of motion. These systems play an important role in understanding general dynamics in accelerators and have potential for future designs. This work will cover two automated methods for finding integrable symplectic maps of the plane. The first algorithm is based on the observation that the evolution of an integrable system in phase space is confined to a lower-dimensional submanifold of a specific type. The second algorithm relies on an analysis of dynamical variables. Both methods rediscover some of the famous McMillan-Suris integrable mappings and ultra-discrete Painlev\'e equations. Over 100 new integrable families are presented and analyzed, some of which are isolated in the space of parameters, while others are families with one parameter (or the ratio of parameters) being either continuous or discrete. In addition, the newly discovered maps are related to a general 2D symplectic map through the use of discrete perturbation theory. A method is proposed for constructing smooth near-integrable dynamical systems based on mappings with polygon invariants.
Machine-Assisted Discovery of Integrable Symplectic Mappings
Integrable systems possess a hidden symmetry associated with the existence of conserved quantities known as integrals of motion. These systems play an important role in understanding general dynamics in accelerators and have potential for future designs. This work will cover two automated methods for finding integrable symplectic maps of the plane. The first algorithm is based on the observation that the evolution of an integrable system in phase space is confined to a lower-dimensional submanifold of a specific type. The second algorithm relies on an analysis of dynamical variables. Both methods rediscover some of the famous McMillan-Suris integrable mappings and ultra-discrete Painlev\'e equations. Over 100 new integrable families are presented and analyzed, some of which are isolated in the space of parameters, while others are families with one parameter (or the ratio of parameters) being either continuous or discrete. In addition, the newly discovered maps are related to a general 2D symplectic map through the use of discrete perturbation theory. A method is proposed for constructing smooth near-integrable dynamical systems based on mappings with polygon invariants.
Ground-state energies of simple metals
A structural expansion for the static ground-state energy of a simple metal is derived. Two methods are presented, one an approach based on single-particle band structure which treats the electron gas as a nonlinear dielectric, the other a more general many-particle analysis using finite-temperature perturbation theory. The two methods are compared, and it is shown in detail how band-structure effects, Fermi-surface distortions, and chemical-potential shifts affect the total energy. These are of special interest in corrections to the total energy beyond third order in the electron-ion interaction and hence to systems where differences in energies for various crystal structures are exceptionally small. Preliminary calculations using these methods for the zero-temperature thermodynamic functions of atomic hydrogen are reported.
Analytic gradients for coupled-cluster energies that include noniterative connected triple excitations - Application to cis- and trans-HONO
An efficient formulation of the analytic energy gradient for the single and double excitation coupled-cluster method that includes a perturbational estimate of the effects of connected triple excitations is presented. The formulation has a small computational cost, and the algebraic manipulations may be applied generally to the analytic gradient of Moller-Plesset perturbation theory energies. The new formulation has been implemented in an efficient set of programs that utilize highly vectorized algorithms and has been used to investigate the equilibrium structures, harmonic vibrational frequencies, IR intensities, and energy separation of cis- and trans-HONO.
Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory
Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). In conclusion, our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.