Nonperturbative Scheme Research Articles

Nambu-covariant many-body theory II: Self-consistent approximations

The theory of Self-Consistent Green’s Function (SCGF) is reformulated in an explicit Nambu-covariant fashion for applications to many-body systems at non-zero temperature in symmetry-broken phases. This is achieved by extending the Nambu-covariant formulation of perturbation theory, presented in the first part of this work, to non-perturbative schemes based on self-consistently dressed propagators and vertices. We work out in detail the self-consistent ladder approximation, motivated by a trade-off between numerical complexity and many-body phenomenology. Taking a complex general Hartree–Fock–Bogoliubov (HFB) propagator as a starting point, we also formulate and prove a sufficient condition on the stability of the HFB self-energy to ensure the convergence of the initial series of ladders at any energy. The self-consistent ladder approximation is written purely in terms of spectral functions and the resulting set of equations, when expressed in terms of Nambu tensors, are remarkably similar to those in the symmetry-conserving case. This puts the application of the self-consistent ladder approximation to symmetry-broken phases of infinite nuclear matter within reach.

Dark matter vorticity and velocity dispersion from truncated Dyson-Schwinger equations

Large-scale structure formation is studied in a kinetic theory approach, extending the standard perfect pressureless fluid description for dark matter by including the velocity dispersion tensor as a dynamical degree of freedom. The evolution of power spectra for density, velocity and velocity dispersion degrees of freedom is investigated in a non-perturbative approximation scheme based on the Dyson-Schwinger equations. In particular, the generation of vorticity and velocity dispersion is studied and predictions for the corresponding power spectra are made, which qualitatively agree well with results obtained from N-body simulations. It is found that velocity dispersion grows strongly due to non-linear effects and at late times its mean value seems to be largely independent of the initial conditions. By taking this into account, a rather realistic picture of non-linear large-scale structure formation can be obtained, albeit the numerical treatment remains challenging, especially for very cold dark matter models.

Relativistic calculation of the orbital hyperfine splitting in complex microscopic structures

Theoretical spectroscopy based on double perturbation theory is typically challenged by systems with large orbital hyperfine splitting. Therefore, we here derive a rigorous, non-perturbative scheme starting from Dirac’s equation which allows to calculate the contribution of the orbital HFI for complex structures including heavy atoms with strong spin-orbit coupling (SOC). Using the PAW formalism, the method has been implemented in the software package Quantum ESPRESSO. We show that the ‘orbital part’ actually scales with SOC strength if orbital quenching is hindered by low local symmetry, i.e. in case of dimers or atoms at surfaces. This holds true in particular when the unpaired electron is localized in quasi-atomic p-like orbitals. Here, the orbital part is by far not negligible, but becomes dominant by surpassing the dipolar contribution by a factor of five.

Renormalised spectral flows

We derive renormalised finite functional flow equations for quantum field theories in real and imaginary time that incorporate scale transformations of the renormalisation conditions, hence implementing a flowing renormalisation. The flows are manifestly finite in general non-perturbative truncation schemes also for regularisation schemes that do not implement an infrared suppression of the loops in the flow. Specifically, this formulation includes finite functional flows for the effective action with a spectral Callan-Symanzik cutoff, and therefore gives access to Lorentz invariant spectral flows. The functional setup is fully non-perturbative and allows for the spectral treatment of general theories. In particular, this includes theories that do not admit a perturbative renormalisation such as asymptotically safe theories. Finally, the application of the Lorentz invariant spectral functional renormalisation group is briefly discussed for theories ranging from real scalar and Yukawa theories to gauge theories and quantum gravity.

Propagating-wave approximation in two-dimensional potential scattering

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

Feynman-Vernon influence functional approach to quantum transport in interacting nanojunctions: An analytical hierarchical study

We present a nonperturbative and formally exact approach for the charge transport in interacting nanojunctions based on a real-time path-integral formulation of the reduced system dynamics. For reservoirs of noninteracting fermions, the exact trace over the leads' degrees of freedom results in the time-nonlocal Feynman-Vernon influence functional, a functional of the Grassmann-valued paths of the nanojunction, which induces correlations among the tunneling transitions in and out of the nanojunction. An expansion of the influence functional in terms of the number of tunneling transitions, and integration of the Grassmann variables between the tunneling times, allows us to obtain a still exact generalized master equation for the populations of the reduced density matrix in the occupation-number representation, as well as a formally exact expression for the current. By borrowing the nomenclature of the famous spin-boson model, we parametrize the two-state dynamics of each single-particle fermionic degree of freedom, in the occupation-number representation, in terms of blips and sojourns. We apply our formalism to the exactly solvable resonant level model (RLM) and to the single-impurity Anderson model (SIAM), the latter being a prototype system for studying strong correlations. For both systems, we demonstrate a hierarchical diagrammatic structure. While the hierarchy closes at the second tier for the RLM, this is not the case for the interacting SIAM. Upon inspection of the current kernel, known results from various perturbative and nonperturbative approximation schemes to quantum transport in the SIAM are recovered. Finally, a noncrossing approximation for the hierarchical kernel is developed, which enables us to systematically decrease temperature at each next level of the approximation. Analytical results for a simplified fourth-tier scheme are presented both in equilibrium and nonequilibrium and with an applied magnetic field.

Cosmological functional renormalization group, extended Galilean invariance, and approximate solutions to the flow equations

The functional renormalisation group is employed to study the non-linear regime of late-time cosmic structure formation. This framework naturally allows for non-perturbative approximation schemes, usually guided by underlying symmetries or a truncation of the theory space. An extended symmetry that is related to Galilean invariance is studied and corresponding Ward identities are derived. These are used to obtain (formally) closed renormalisation group flow equations for two-point correlation functions in the limit of large wave numbers (small scales). The flow equations are analytically solved in an approximation that is connected to the 'sweeping effect' previously described in the context of fluid turbulence.

Shear Viscosity at Infinite Coupling: A Field Theory Calculation.

I derive an exact integral expression for the ratio of shear viscosity over entropy density η/s for the massless (critical) O(N) model at large N with quartic interactions. The calculation is set up and performed entirely from the field theory side using a nonperturbative resummation scheme that captures all contributions to leading order in large N. In 2+1d, η/s is evaluated numerically at all values of the coupling. For infinite coupling, I find (η/s)≃0.42(1)×N. I show that this strong coupling result for the viscosity is universal for a large class of interacting bosonic O(N) models.

A non-perturbative pairwise-additive analysis of charge transfer contributions to intermolecular interaction energies.

Energy decomposition analysis (EDA) based on absolutely localized molecular orbitals (ALMOs) decomposes the interaction energy between molecules into physically interpretable components like geometry distortion, frozen interactions, polarization, and charge transfer (CT, also sometimes called charge delocalization) interactions. In this work, a numerically exact scheme to decompose the CT interaction energy into pairwise additive terms is introduced for the ALMO-EDA using density functional theory. Unlike perturbative pairwise charge-decomposition analysis, the new approach does not break down for strongly interacting systems, or show significant exchange-correlation functional dependence in the decomposed energy components. Both the energy lowering and the charge flow associated with CT can be decomposed. Complementary occupied-virtual orbital pairs (COVPs) that capture the dominant donor and acceptor CT orbitals are obtained for the new decomposition. It is applied to systems with different types of interactions including DNA base-pairs, borane-ammonia adducts, and transition metal hexacarbonyls. While consistent with most existing understanding of the nature of CT in these systems, the results also reveal some new insights into the origin of trends in donor-acceptor interactions.

Non-perturbative renormalization scheme for the C P -odd three-gluon operator

We define a regularization-independent momentum-subtraction scheme for the C P -odd three-gluon operator at dimension six. This operator appears in effective field theories for heavy physics beyond the Standard Model, describing the indirect effect of new sources of C P-violation at low energies. In a hadronic context, it induces permanent electric dipole moments. The hadronic matrix elements of the three-gluon operator are non-perturbative objects that should ideally be evaluated with lattice QCD. We define a non-perturbative renormalization scheme that can be implemented on the lattice and we compute the scheme transformation to overline{mathrm{MS}} at one loop. Our calculation can be used as an interface to future lattice-QCD calculations of the matrix elements of the three-gluon operator, in order to obtain theoretically robust constraints on physics beyond the Standard Model from measurements of the neutron electric dipole moment.

Cumulant t-Expansion for Strongly Correlated Electrons on a Lattice

A systematic nonperturbative scheme for calculating the ground state energy is adapted for studying systems of strongly correlated electrons on a lattice. It includes a method for calculating the cumulants of the Hamiltonian and a method, using the t-expansion, for constructing successive approximations to the ground state energy by these cumulants. The scheme is applied to spinless fermion and Hubbard models, and a method is proposed to overcome the problems found in previous attempts to use this scheme to study the Hubbard model.

Inelastic meson-meson scattering in hadronic matter

We review studies of inelastic meson-meson scattering. In nonperturbative schemes with chiral-perturbation-theory Lagrangians and in models based on effective meson Lagrangians, inelastic meson-meson scattering leads to the successful identification of resonances in meson-meson reactions, adequate inclusion of final state interactions in particle decays, and so on. For mesons of which each consists of a quark and an antiquark, inelastic meson-meson scattering may be caused by quark-antiquark annihilation, quark-antiquark creation, quark interchange, and quark-antiquark annihilation and creation. In transition amplitudes for meson-meson scattering mesonic quark-antiquark relative-motion wave functions depend on hadronic matter, and transition potentials are given in perturbative quantum chromodynamics. Via transition amplitudes the cross sections for inelastic meson-meson scattering depend on the temperature of hadronic matter. Some prominent temperature dependence of the cross sections has been found. Inelastic meson-meson scattering becomes even more significant in proton-proton collisions and lead-lead collisions at the Large Hadron Collider.

Projection of the gravitational dynamics on a subspace of probability distributions: Curl-free Gaussian Ansatz

We present a new approach to model the gravitational dynamics of large-scale structures. Instead of solving the equations of motion up to a finite perturbative order or building phenomenological models, we follow the evolution of the probability distribution of the displacement and velocity fields within an approximation subspace. Keeping the exact equations of motion with their full nonlinearity, this provides a nonperturbative scheme that goes beyond shell crossing. Focusing on the simplest case of a curl-free Gaussian Ansatz for the displacement and velocity fields, we find that truncations of the power spectra on nonlinear scales directly arise from the equations of motion. This leads to a truncated Zeldovich approximation for the density power spectrum, but with a truncation that is not set a priori and with different power spectra for the displacement and velocity fields. The positivity of their autopower spectra also follows from the equations of motion. Although the density power spectrum is only recovered up to a smooth drift on baryon acoustic oscillation (BAO) scales, the predicted density correlation function agrees with numerical simulations within 2% from BAO scales down to $7{h}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{Mpc}$ at $z\ensuremath{\ge}0.35$, without any free parameter. Thus, this parameter-free extension of the Zeldovich approximation is not competitive with other schemes for the power spectrum, but it provides a good prediction for the correlation function. However, the improvement over the standard Zeldovich approximation remains rather modest. This means that including non-Gaussianities will be essential to significantly improve analytical predictions within this general framework.

Many-Body Perturbation Theories for Finite Nuclei

In recent years many-body perturbation theory encountered a renaissance in the field of ab initio nuclear structure theory. In various applications it was shown that perturbation theory, including novel flavors of it, constitutes a useful tool to describe atomic nuclei, either as a full-fledged many-body approach or as an auxiliary method to support more sophisticated non-perturbative many-body schemes. In this work the current status of many-body perturbation theory in the field of nuclear structure is discussed and novel results are provided that highlight its power as a efficient and yet accurate (pre-processing) approach to systematically investigate medium-mass nuclei. Eventually a new generation of chiral nuclear Hamiltonians is benchmarked using several state-of-the-art flavours of many-body perturbation theory.

Simple non-perturbative resummation schemes beyond mean-field II: Thermodynamics of scalar ϕ4 theory in 1 + 1 dimensions at arbitrary coupling

Recently, non-perturbative approximate solutions were presented that go beyond the well-known mean-field resummation. In this work, these non-perturbative approximations are used to calculate finite-temperature equilibrium properties for scalar [Formula: see text] theory in two dimensions such as the pressure, entropy density and speed of sound. Unlike traditional approaches, it is found that results are well-behaved for arbitrary temperature/coupling strength, are independent of the choice of the renormalization scale [Formula: see text], and are apparently converging as the resummation level is increased. Results also suggest the presence of a possible analytic cross-over from the high-temperature to the low-temperature regime based on the change in the thermal entropy density.

Time-Dependent Magnons from First Principles

We propose an efficient and non-perturbative scheme to compute magnetic excitations for extended systems employing the framework of time-dependent density functional theory. Within our approach, we drive the system out of equilibrium using an ultrashort magnetic kick perpendicular to the ground-state magnetization of the material. The dynamical properties of the system are obtained by propagating the time-dependent Kohn–Sham equations in real time, and the analysis of the time-dependent magnetization reveals the transverse magnetic excitation spectrum of the magnet. We illustrate the performance of the method by computing the magnetization dynamics, obtained from a real-time propagation, for iron, cobalt, and nickel and compare them to known results obtained using the linear-response formulation of time-dependent density functional theory. Moreover, we point out that our time-dependent approach is not limited to the linear-response regime, and we present the first results for nonlinear magnetic excitations from first principles in iron.

Convergence of Nonperturbative Approximations to the Renormalization Group.

We provide analytical arguments showing that the "nonperturbative" approximation scheme to Wilson's renormalization group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of nonperturbative methods.

Geometric decoherence in diffusive open quantum systems

Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing non-unitary, non-cyclic and non-adiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realisation of quantum gates due to its robustness against dynamical errors, however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a non-perturbative control scheme.

Valence parton distribution function of pion from fine lattice

We present a lattice QCD study of valence parton distribution inside the pion within the framework of Large Momentum Effective Theory. We use a mixed action approach with 1-HYP smeared valence Wilson clover quarks on 2+1 flavor HISQ sea with the valence quark mass tuned to 300 MeV pion mass. We use $48^3 \times 64$ lattice at a fine lattice spacing $a=0.06$ fm for this computation. We renormalize the quasi-PDF matrix element in the non-perturbative RI-MOM scheme. As a byproduct, we test the validity of 1-loop matching procedure by comparing the RI-MOM renormalized quasi-PDF matrix element with off-shell quark external states as computed in the continuum 1-loop perturbation theory with the lattice results at $a=0.04$ and 0.06 fm. By applying the RI-MOM to ${\bar{\rm MS}}$ one-loop matching, implemented through a fit to phenomenologically motivated PDFs, we obtain the valence PDF of pion.

Systematic uncertainties in parton distribution functions from lattice QCD simulations at the physical point

We present a detailed study of the helicity-dependent and helicity-independent collinear parton distribution functions (PDFs) of the nucleon, using the quasi-PDF approach. The lattice QCD computation is performed employing twisted mass fermions with a physical value of the light quark mass. We give a systematic and in-depth account of the salient features entering in the evaluation of quasi-PDFs and their relation to the light-cone PDFs. In particular, we give details for the computation of the matrix elements, including the study of the various sources of systematic uncertainties, such as excited-states contamination. In addition, we discuss the nonperturbative renormalization scheme used here and its systematics, effects of truncating the Fourier transform and different matching prescriptions. Finally, we show improved results for the PDFs and discuss future directions, challenges and prospects for evaluating precisely PDFs from lattice QCD with fully quantified uncertainties.