Nonperturbative Approach Research Articles

A novel pattern in a class of fractal models with the non-perturbative approach

This paper provides evidence on how the fractal vibration system can be converted into a damping system in a continuous space. The fractal Duffing vibration is considered an example. It is shown that the inertia force in the fractal space can be transformed into a combination of the inertia force and the damping force in a continuum space employing the fractional parameter. In addition, the velocity force can be converted to both the damping force and the stiffness force in the smooth space. The nonlinear system is solved using the non-perturbation method. The calculations showed that the approximate solution agrees with the corresponding numerical one when the fraction parameter becomes the unity value. The new approach is represented as the best one because it highlights the damping influence of fraction derivatives. This approach can help a large number of researchers interested in studying nonlinear fractal oscillations.

Spinor GW /Bethe-Salpeter calculations in BerkeleyGW: Implementation, symmetries, benchmarking, and performance

Computing the $GW$ quasiparticle bandstructure and Bethe-Salpeter Equation (BSE) absorption spectra for materials with spin-orbit coupling has commonly been done by treating $GW$ corrections and spin-orbit coupling as separate perturbations to density-functional theory. However, accurate treatment of materials with strong spin-orbit coupling often requires a fully relativistic approach using spinor wavefunctions in the Kohn-Sham equation and $GW$/BSE. Such calculations have only recently become available, in particular for the BSE. We have implemented this approach in the plane-wave pseudopotential $GW$/BSE code BerkeleyGW, which is highly parallelized and widely used in the electronic-structure community. We present reference results for quasiparticle bandstructures and optical absorption spectra of solids with different strengths of spin-orbit coupling, including Si, Ge, GaAs, GaSb, CdSe, Au, and Bi$_2$Se$_3$. The calculated quasiparticle band gaps of these systems are found to agree with experiment to within a few tens of meV. The absorption spectrum of GaSb calculated with the fully-relativistic $GW$-BSE captures the large spin-orbit splitting of peaks in the spectrum. For Bi$_2$Se$_3$, we find a drastic change in the low-energy bandstructure compared to that of DFT, with the fully-relativistic treatment of the $GW$ approximation correctly capturing the parabolic nature of the valence and conduction bands after including off-diagonal self-energy matrix elements. We present the detailed methodology, approach to spatial symmetries for spinors, comparison against other codes, and performance compared to spinless $GW$/BSE calculations and perturbative approaches to SOC. This work aims to spur further development of spinor $GW$/BSE methodology in excited-state research software.

Scattering of plane-wave and twisted photons by helical media

By using quantum electrodynamics in a dispersive medium, we describe scattering of plane-wave and twisted photons by a slab made of a helical medium, the helix axis being normal to the slab plane and the medium being not translation invariant in this plane, in general. In the particular cases, the permittivity tensor of a helical medium corresponds to cholesteric liquid crystals, C*-smectics, biaxial chiral nematics and smectics, Q-plates, chiral sculptured thin films, and helical dislocations. Both perturbative and nonperturbative approaches are considered. The explicit expressions for scattering amplitudes, probabilities, and Stokes parameters of photons are found taking into account the form of the photon wave packet. The selection rules are established showing that the helical medium transfers the momentum and the angular momentum to scattered photons. This property can be employed for production of twisted photons with large projection of the total angular momentum. We describe the device for shifting the projection of the total angular momentum of a photon and the principal scheme for signal coding in terms of twisted photons.

Renormalization of Transverse-Momentum-Dependent Parton Distribution on the Lattice.

To calculate the transverse-momentum-dependent parton distribution functions (TMDPDFs) from lattice QCD, an important goal yet to be realized, it is crucial to establish a viable nonperturbative renormalization approach for linear divergences in the corresponding Euclidean quasi-TMDPDF correlators in large-momentum effective theory. We perform a first systematic study of the renormalization property of the quasi-TMDPDFs by calculating the relevant matrix elements in a pion state at five lattice spacings ranging from 0.03fm to 0.12fm. We demonstrate that the square root of the Wilson loop combined with the short distance hadron matrix element provides a successful method to remove all ultraviolet divergences of the quasi-TMD operator, and thus provides the necessary justification to perform a continuum limit calculation of TMDPDFs. In contrast, the popular regularization independent momentum subtraction renormalization (RI/MOM) scheme fails to eliminate all linear divergences.

Fermi two-atom problem: Nonperturbative approach via relativistic quantum information and algebraic quantum field theory

In this work we revisit the famous Fermi two-atom problem, which concerns how relativistic causality impacts atomic transition probabilities, using the tools from relativistic quantum information and algebraic quantum field theory. The problem has sparked different analyses from many directions and angles since the proposed solution by Buchholz and Yngvason [Phys. Rev. Lett. 73, 613 (1994)]. Some of these analyses employ various approximations, heuristics, and perturbative methods, which tends to render some of the otherwise useful insights somewhat obscured. It is also noted that they are all studied in flat spacetime. We show that current tools in relativistic quantum information, combined with an algebraic approach to quantum field theory, are now powerful enough to provide fuller and cleaner analysis of the Fermi two-atom problem for arbitrary curved spacetimes in a completely nonperturbative manner. Our result gives the original solution of Buchholz and Yngvason a very operational reinterpretation in terms of qubits interacting with a quantum field and allows for various natural generalizations and inclusion of detector-based local measurement for the quantum field [Phys. Rev. D 105, 065003 (2022)].

Vacuum polarization energy decline and spontaneous positron emission in QED under Coulomb supercriticality

The properties of QED-vacuum polarization, caused by the supercritical external Coulomb source with charge $Z$ and size $R$, are explored in an essentially nonperturbative approach with emphasis on the vacuum energy ${\mathcal{E}}_{\mathrm{VP}}$ (where VP stands for vacuum polarization.) It is shown that in the supercritical region ${\mathcal{E}}_{\mathrm{VP}}$ turns out to be a decreasing function of the Coulomb charge, resulting in decay into the negative range as $\ensuremath{\sim}\ensuremath{-}{Z}^{4}/R$. Moreover, it is indeed the decline of ${\mathcal{E}}_{\mathrm{VP}}$, which provides the required energy for (presumably possible) positron emission. Found this way, properties of the resonance decay are in agreement with those achieved by other methods. The additional problems of spontaneous emission, caused by lepton number, are also discussed.

Generalized spin-boson models with non-normalizable form factors

Generalized spin-boson (GSB) models describe the interaction between a quantum mechanical system and a structured boson environment, mediated by a family of coupling functions known as form factors. We propose an extension of the class of GSB models, which can accommodate non-normalizable form factors provided that they satisfy a weaker growth constraint, thus accounting for a rigorous description of a wider range of physical scenarios; we also show that such “singular” GSB models can be rigorously approximated by GSB models with normalizable form factors. Furthermore, we discuss in greater detail the structure of the spin-boson model with a rotating wave approximation: for this model, the result is improved via a nonperturbative approach that enables us to further extend the class of admissible form factors as well as to compute its resolvent and characterize its self-adjointness domain.

Bound States in QED from a light-front approach


 We calculate the binding energy and structure of the positronium system in a light-front nonperturbative approach. The input interaction is from Quantum Electrodynamics first principles and one dynamical photon is explicitly included in the basis. We obtain the low-lying mass spectrum, which is found to be consistent with that from nonrelativistic quantum mechanics. Various excitation modes of the low-lying states can be identified from the associated wave functions. The photon distribution in the longitudinal direction is calculated for the low-lying positronium states.

Fundamental length scale and the bending of light in a gravitational field

The canonical approach to quantizing quantum gravity is understood to suffer from pathological non-renomalizability. Nevertheless in the context of effective field theory, a viable perturbative approach to calculating elementary processes is possible. Some non-perturbative approaches, most notably loop quantum gravity and combinatorial quantum gravity imply the existence of a minimal length. To circumvent the seeming contradiction between the existence of a minimum length and the principle of special relativity, Double Special Relativity introduces modified dispersion relationships that reconcile the conflict. In this work, we combine these dispersion relationships with an effective field theory approach to compute the first post Newtonian correction to the bending of light by a massive object. The calculation offers the prospect of a directly measurable effect that rests upon both the existence of a quantized gravitational field and a minimal length. Experimental verification would provide evidence of the existence of a quantum theory of gravity, and the fundamental quantization of spacetime with a bound on the minimal distance.

Nonperturbative ab initio approach for calculating the electrical conductivity of a liquid metal

We propose a nonperturbative ab initio approach to calculate the electrical conductivity of a liquid metal. Our approach is based on the Kubo formula and the theory of electron-phonon coupling (EPC) and, unlike the conventional empirical approach based on the Kubo-Greenwood formula, fully takes into account the effect of coupling between electrons and moving ions. We show that the electrical conductivity at high temperature is determined by an EPC parameter ${\ensuremath{\lambda}}_{\mathrm{tr}}$, which can be inferred, nonperturbatively, from the correlation of electron scattering matrices induced by ions. The latter can be evaluated in a molecular dynamics simulation. Based on the density-functional theory and pseudopotential methods, we implement the approach in an ab initio manner. We apply it to liquid sodium and obtain results in good agreement with experiments. This approach is efficient and based on a rigorous theory suitable for applying to general metallic liquid systems.

A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets

A continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

More on the three-gluon vertex in SU(2) Yang-Mills theory in three and four dimensions

The three-gluon vertex has been found to be a vital ingredient in non-perturbative functional approaches. We present an updated lattice calculation of it in various kinematical configurations for all tensor structures and multiple lattice parameters in three dimensions, and in a subset of those in four dimensions, for SU(2) Yang-Mills theory in minimal Landau gauge. In three dimensions an unambiguous zero crossing for the tree-level form-factor is established, and consistency for all investigated form factors with a power-like divergence towards the infrared is observed. Using very coarse lattices this is even seen towards momenta as low as about 15 MeV. The results in four dimensions are consistent with such a behavior, but do not yet reach deep enough into the infrared to establish it.

Flowing bosonization in the nonperturbative functional renormalization-group approach

Bosonization allows one to describe the low-energy physics of one-dimensional quantum fluids within a bosonic effective field theory formulated in terms of two fields: the ``density’’ field \varphiφ and its conjugate partner, the phase \varthetaϑ of the superfluid order parameter. We discuss the implementation of the nonperturbative functional renormalization group in this formalism, considering a Luttinger liquid in a periodic potential as an example. We show that in order for \varthetaϑ and \varphiφ to remain conjugate variables at all energy scales, one must dynamically redefine the field \varthetaϑ along the renormalization-group flow. We derive explicit flow equations using a derivative expansion of the scale-dependent effective action to second order and show that they reproduce the flow equations of the sine-Gordon model (obtained by integrating out the field \varthetaϑ from the outset) derived within the same approximation. Only with the scale-dependent (flowing) reparametrization of the phase field \varthetaϑ do we obtain the standard phenomenology of the Luttinger liquid (when the periodic potential is sufficiently weak so as to avoid the Mott-insulating phase) characterized by two low-energy parameters, the velocity of the sound mode and the renormalized Luttinger parameter.

Robust Nonequilibrium Edge Currents with and without Band Topology.

We study two-dimensional bosonic and fermionic lattice systems under nonequilibrium conditions corresponding to a sharp gradient of temperature imposed by two thermal baths. In particular, we consider a lattice model with broken time-reversal symmetry that exhibits both topologically trivial and nontrivial phases. Using a nonperturbative Green function approach, we characterize the nonequilibrium current distribution in different parameter regimes. For both bosonic and fermionic systems, we find chiral edge currents that are robust against coupling to reservoirs and to the presence of defects on the boundary or in the bulk. This robustness not only originates from topological effects at zero temperature but, remarkably, also persists as a result of dissipative symmetries in regimes where band topology plays no role. Chirality of the edge currents implies that energy locally flows against the temperature gradient without any external work input. In the fermionic case, there is also a regime with topologically protected boundary currents, which nonetheless do not circulate around all system edges.

Critical Phenomena and Phase Transitions in Large Lattices within Monte-Carlo Based Non-perturbative Approaches

Critical phenomena and Goldstone mode effects in spin models with the O(n) rotational symmetry are considered. Starting with Goldstone mode singularities in the XY and O(4) models, we briefly review various theoretical concepts, as well as state-of-the-art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for lattices with linear sizes up to L = 1536, which are very large as compared to L ≤ 128 usually used in the finite-size scaling analysis. These results are obtained, using a parallel OpenMP implementation of the Wolff single-cluster algorithm. The finite-size scaling analysis of the critical exponent η, assuming the usually accepted correction-to-scaling exponent ω ≈ 0.8, shows that η is likely to be somewhat larger than the value 0.0335 ± 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: η = ω =1/8 and ν = 2/3, found within the GFD theory.

Production of massive bosons from the decay of a massless particle beam

Taking a two interacting scalar toy model with interaction term $g\ensuremath{\phi}{\ensuremath{\chi}}^{2}$, we study the production of $\ensuremath{\chi}$ particles coming from the decay of an asymptotic and highly occupied beam of $\ensuremath{\phi}$ particles. We perform a nonperturbative analysis coming from parametric resonant instabilities and investigate the possibility that massive $\ensuremath{\chi}$ particles are produced from decays of massless $\ensuremath{\phi}$ particles from the beam. Although this process is not present in a perturbative analysis, our nonperturbative approach allows it to happen under certain conditions. For a momentum $p$ of the beam particles and a mass ${m}_{\ensuremath{\chi}}$ of the produced ones, we find that the decay is allowed if the energy density of the beam exceeds the instability threshold ${p}^{2}{m}_{\ensuremath{\chi}}^{4}/(2{g}^{2})$. We also provide an analytical expression for the spontaneous decay rate at the earliest time.

Spontaneous non-Hermiticity in the ( 2+1 )-dimensional Gross-Neveu model

Using a nonperturbative approach based on the Cornwall-Jackiw-Tomboulis (CJT) effective action $\mathrm{\ensuremath{\Gamma}}(S)$ for composite operators ($S$ is the full fermion propagator), the phase structure of the simplest massless ($2+1$)-dimensional Gross-Neveu model is investigated. We have calculated $\mathrm{\ensuremath{\Gamma}}(S)$ and its stationary (or Dyson-Schwinger) equation in the first order of the bare coupling constant $G$ and have shown that there exist a well-defined dependence of $G\ensuremath{\equiv}G(\mathrm{\ensuremath{\Lambda}})$ on the cutoff parameter $\mathrm{\ensuremath{\Lambda}}$, such that the Dyson-Schwinger equation is renormalized. It has three different solutions for fermion propagator $S$ corresponding to possible dynamical appearance of three different mass terms in the model. One is a Hermitian, but two others are non-Hermitian and $\mathcal{P}\mathcal{T}$ even or odd. It means that two phases with spontaneous non-Hermiticity can be emerged in the system. Moreover, mass spectrum of quasiparticles is real in these non-Hermitian and $\mathcal{P}\mathcal{T}$ even/odd phases.

The damping Helmholtz–Rayleigh–Duffing oscillator with the non-perturbative approach

The present study suggests a very simple, effective new method to study the damping quadratic–cubic nonlinear oscillation in physical phenomena such as fluid, solid-state physics, optics, plasma physics, dispersion, and convection systems with no perturbation. The damping nonlinear oscillation is converted to its equivalent linear oscillation. The exact solution of the corresponding linear oscillation is verified by the first-order homotopy perturbation method which shows an exact agreement. Stability conditions are imposed from the frequency formula. The accuracy of the proposed approach has shown that the approximate analytical solutions are in excellent agreement with the corresponding exact solutions.

The traditional approximation of rotation for rapidly rotating stars and planets

Context. We examine the dynamics of low-frequency gravito-inertial waves (GIWs) in differentially rotating deformed radiation zones in stars and planets by generalising the traditional approximation of rotation (TAR). The TAR treatment was built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating. However, it has been generalised in our previous work by including the effects of the centrifugal deformation using a non-perturbative approach. In the meantime, TAR has been generalised in spherical geometry to take the differential rotation into account. Aims. We aim to carry out a new generalisation of the TAR treatment to account for the differential rotation and the strong centrifugal deformation simultaneously. Methods. We generalise our previous work by taking into account the differential rotation in the derivation of our complete analytical formalism that allows the study of the dynamics of GIWs in differentially and rapidly rotating stars. Results. We derived the complete set of equations that generalises the TAR, simultaneously taking the full centrifugal acceleration and the differential rotation into account. Within the validity domain of the TAR, we derived a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods of the GIWs, which can be used to probe the structure and dynamics of differentially rotating deformed stars with asteroseismology. Conclusions. A new generalisation of the TAR, which simultaneously takes into account the differential rotation and the centrifugal acceleration in a non-perturbative way, was derived. This generalisation allowed us to study the detectability and the signature of the differential rotation on GIWs in rapidly rotating deformed stars and planets. We found that the effects of the differential rotation in early-type deformed stars on GIWs is theoretically largely detectable in modern space photometry using observations from Kepler and TESS.

Composite operator approach to dynamical mass generation in the (2 + 1)-dimensional Gross–Neveu model

Using a nonperturbative approach based on the Cornwall–Jackiw–Tomboulis (CJT) effective action [Formula: see text] for composite operators, the phase structure of the simplest massless [Formula: see text]-dimensional Gross–Neveu model is investigated. We have calculated [Formula: see text] in the first-order of the bare coupling constant [Formula: see text] and have shown that there exist three different specific dependences of [Formula: see text] on the cutoff parameter [Formula: see text], and in each case, the effective action and its stationarity equations have been obtained. The solutions of these equations correspond to the fact that three different masses of fermions can arise dynamically and, respectively, three different nontrivial phases can be observed in the model.