90 research outputs found
Solving a Set of Truncated Dyson-Schwinger Equations with a Globally Converging Method
A globally converging numerical method to solve coupled sets of non-linear
integral equations is presented. Such systems occur e.g. in the study of
Dyson-Schwinger equations of Yang-Mills theory and QCD. The method is based on
the knowledge of the qualitative properties of the solution functions in the
far infrared and ultraviolet. Using this input, the full solutions are
constructed using a globally convergent modified Newton iteration. Two
different systems will be treated as examples: The Dyson-Schwinger equations of
3-dimensional Yang-Mills-Higgs theory provide a system of finite integrals,
while those of 4-dimensional Yang-Mills theory at high temperatures are only
finite after renormalization.Comment: 23 pages, 3 figures, 4 tables, submitted to Comput. Phys. Commun; one
subsection expanded with additional technical details, a few other minor
modifications and updates, version to appear in Comput. Phys. Commu
Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation
We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for phi(4) theory in 2D. It is based on an exact treatment of the functional derivative partial derivative Gamma/partial derivative G of the four-point vertex function Gamma with respect to the two-point correlation function G within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around G = 0 in a HAM control parameter (c0)G, or even a convergent one up to the phase transition point if shifts in G can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving functional integro-differential equations beyond any truncation scheme
Description of nuclear systems with a self-consistent configuration-mixing approach. I: Theory, algorithm, and application to the C test nucleus
Although self-consistent multi-configuration methods have been used for
decades to address the description of atomic and molecular many-body systems,
only a few trials have been made in the context of nuclear structure. This work
aims at the development of such an approach to describe in a unified way
various types of correlations in nuclei, in a self-consistent manner where the
mean-field is improved as correlations are introduced. The goal is to reconcile
the usually set apart Shell-Model and Self-Consistent Mean-Field methods. This
approach is referred as "variational multiparticle-multihole configuration
mixing method". It is based on a double variational principle which yields a
set of two coupled equations that determine at the same time the expansion
coefficients of the many-body wave function and the single particle states. The
formalism is derived and discussed in a general context, starting from a
three-body Hamiltonian. Links to existing many-body techniques such as the
formalism of Green's functions are established. First applications are done
using the two-body D1S Gogny effective force. The numerical procedure is tested
on the C nucleus in order to study the convergence features of the
algorithm in different contexts. Ground state properties as well as
single-particle quantities are analyzed, and the description of the first
state is examined. This study allows to validate our numerical algorithm and
leads to encouraging results. In order to test the method further, we will
realize in the second article of this series, a systematic description of more
nuclei and observables obtained by applying the newly-developed numerical
procedure with the same Gogny force. As raised in the present work,
applications of the variational multiparticle-multihole configuration mixing
method will however ultimately require the use of an extended and more
constrained Gogny force.Comment: 22 pages, 18 figures, accepted for publication in Phys. Rev. C. v2:
minor corrections and references adde
Resolving the QCD phase structure
This thesis discusses the quantitative description of the phase structure of Quantum Chromo-
dynamics (QCD). We find that, in strongly correlated theories such as QCD, even a qualitative
investigation of the phase structure can require highly quantitative methods. Hence, the de-
velopment of a method with systematic error control is essential. In the present work, we use
functional renormalisation group (fRG) method to this aim. This work focusses on three ideas:
Firstly, we identify quantitatively dominating and sub-leading scattering-processes in our
approximations. This allows a formulation of low energy effective theories of the four-quark
interaction, as well as the description of gluon condensation. For the former, we present results
for meson and quark masses. The latter provides an estimate of the Yang-Mills mass gap.
Secondly, we further develop the use of highly precise numerical methods from fluid-dynamics
in the fRG. In particular we use Discontinuous Galerkin methods, which are able to capture
shock-development. Shock-waves are found to play a big role in a possible creation-mechanism
of first-order phase transitions. Lastly, we focus on general RG-transformations (gRGt). For
example, they allow a real time formulation of fRG flows and hence give access to spectral
functions. Furthermore, we use them to formulate complex RG-flows, which enables us to
locate Lee-Yang singularities in the complex plane and extrapolate the position of (real) phase
transitions. Finally, we also use gRGts to formulate significant qualitative improvements of
current fRG approximation schemes by means of dynamical field transformations
Nonequilibrium quantum field dynamics from the two-particle-irreducible effective action.
The two-particle-irreducible effective action offers a powerful approach to the study of quantum field dynamics far from equilibrium. Recent and upcoming heavy ion collision experiments motivate the study of such nonequilibrium dynamics in an expanding space-time background. For the O(N) model I derive exact, causal evolution equations for the statistical and spectral functions in a longitudinally expanding system. It is followed by an investigation into how the expansion affects the prospect of the system reaching equilibrium. Results are obtained in 1+1 dimensions at next-to- leading order in loop- and 1/N-expansions of the 2PI effective action. I focus on the evolution of the statistical function from highly nonequilibrium initial conditions, presenting a detailed analysis of early, intermediate and late-time dynamics. It is found that dynamics at very early times is attracted by a nonthermal fixed point of the mean field equations, after which interactions attempt to drive the system to equilibrium. The competition between the interactions and the expansion is eventually won by the expansion, with so-called freeze-out emerging naturally in this description. In order to investigate the convergence of the 2PI-1/N expansion in the 0(N) model, I compare results obtained numerically in 1+1 dimensions at leading, next- to-leading and next-to-next-to-leading order in 1/N. Convergence with increasing N, and also with decreasing coupling are discussed. A comparison is also made in the classical statistical field theory limit, where exact numerical results are available. I focus on early-time dynamics and quasi-particle properties far from equilibrium and observe rapid effective convergence already for moderate values of 1/N or the coupling strength
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Non-equilibrium dynamics of artificial quantum matter
The rapid progress of the field of ultracold atoms during the past two decades has set new milestones in our control over matter. By cooling dilute atomic gases and molecules to nano-Kelvin temperatures, novel quantum mechanical states of matter can be realized and studied on a table-top experimental setup while bulk matter can be tailored to faithfully simulate abstract theoretical models. Two of such models which have witnessed significant experimental and theoretical attention are (1) the two-component Fermi gas with resonant -wave interactions, and (2) the single-component Fermi gas with dipole-dipole interactions. This thesis is devoted to studying the non-equilibrium collective dynamics of these systems using the general framework of quantum kinetic theory.Physic
Electronic correlations in inhomogeneous model systems: numerical simulation of spectra and transmission
Many fascinating features in condensed matter systems emerge due to the interaction between electrons. Magnetism is such a paramount consequence, which is explained in terms of the exchange interaction of electrons. Another prime example is the metal-to-Mott-insulator transition, where the energy cost of Coulomb repulsion competes against the kinetic energy, the latter favoring delocalization. While systems of correlated electrons are exciting and show remarkable and technologically promising physical properties, they are difficult to treat theoretically. A single-particle description is insufficient; the quantum many-body problem of interacting electrons has to be solved. In the present thesis, we study physical properties of half-metallic ferromagnets which are used in spintronic devices. Half-metals exhibit a metallic spin channel, while the other spin channel is insulating; they are characterized by a high spin polarization. This thesis contributes to the development of numerical methods and applies them to models of half-metallic ferromagnets.
Throughout this work, the single-band Hubbard Hamiltonian is considered, and electronic correlations are treated within dynamical mean-field theory. Instead of directly solving the lattice model, the dynamical mean-field theory amounts to solving a local, effective impurity problem that is determined self-consistently. At finite temperatures, this impurity problem is solved employing continuous-time quantum Monte Carlo algorithms formulated in the action formalism. As these algorithms are formulated in imaginary time, an analytic continuation is required to obtain spectral functions. We formulate a version of the N-point Padé algorithm that calculates the location of the poles in a least-squares sense. To directly obtain spectra for real frequencies, we employ Hamiltonian-based tensor network methods at zero temperature. We also summarize the ideas of the density matrix renormalization group algorithm, and of the time evolution using the time-dependent variational principle, employing a diagrammatic notation.
Real materials never display perfect translational symmetry. Thus, realistic models require the inclusion of disorder effects. In this work, we discuss these within a single-site approximation, the coherent potential approximation, and combine it with the dynamical mean-field theory, allowing to treat interacting electrons in multicomponent alloys on a local level. We extend this combined scheme to off-diagonal disorder, that is, disorder in the hopping amplitudes, by employing the Blackman–Esterling–Berk formalism. For this purpose, we illustrate the ideas of this formalism using tensor diagrams and provide an efficient implementation. The structure of the effective medium is discussed, and a concentration scaling is proposed that resolves some of its peculiarities. The limit of vanishing hopping between different components is discussed and solved analytically for the Bethe lattice with a general coordination number. We exemplify the combined algorithm for a Bethe lattice, showing results that exhibit alloy-band-insulator to correlated-metal to Mott-insulator transitions.
We study models of half-metallic ferromagnets to elucidate the effects of local electronic correlations on the spectral function. To model half-metallicity, a static spin splitting is used to produce the half-metallic density of states. Applying the Padé analytic continuation to the self-energy instead of the Green’s function produces reliable spectral functions agreeing with the zero-temperature results obtained for real frequencies. To address transport properties, we investigate the interface of a half-metallic layer and a metallic, band insulating, or Mott insulating layer. We observe charge reconstruction which induces metallicity at the interface; quasiparticle states are present in the Mott insulating layer even for a large Hubbard interaction.
The transmission through a barrier made of such a single interacting half-metallic layer sandwiched by metallic leads is studied employing the Meir–Wingreen formalism. This allows for a transparent calculation of the transmission in the presence of the Hubbard interaction. For a strong coupling of the central layer to the leads, we identify high intensity bound states which do not contribute to the transmission. For small coupling, on the other hand, we find resonant states which enhance the transmission.
In particular, we demonstrate that even for a single half-metallic layer, highly polarized transmissions are achievable
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