70,373 research outputs found
Towards analytic description of a transition from weak to strong coupling regime in correlated electron systems. I. Systematic diagrammatic theory with two-particle Green functions
We analyze behavior of correlated electrons described by Hubbard-like models
at intermediate and strong coupling. We show that with increasing interaction a
pole in a generic two-particle Green function is approached. The pole signals
metal-insulator transition at half filling and gives rise to a new vanishing
``Kondo'' scale causing breakdown of weak-coupling perturbation theory. To
describe the critical behavior at the metal-insulator transition a novel,
self-consistent diagrammatic technique with two-particle Green functions is
developed. The theory is based on the linked-cluster expansion for the
thermodynamic potential with electron-electron interaction as propagator.
Parquet diagrams with a generating functional are derived. Numerical
instabilities due to the metal-insulator transition are demonstrated on
simplifications of the parquet algebra with ring and ladder series only. A
stable numerical solution in the critical region is reached by factorization of
singular terms via a low-frequency expansion in the vertex function. We stress
the necessity for dynamical vertex renormalizations, missing in the simple
approximations, in order to describe the critical, strong-coupling behavior
correctly. We propose a simplification of the full parquet approximation by
keeping only most divergent terms in the asymptotic strong-coupling region. A
qualitatively new, feasible approximation suitable for the description of a
transition from weak to strong coupling is obtained.Comment: 17 pages, 4 figures, REVTe
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
A convergent nonconforming finite element method for compressible Stokes flow
We propose a nonconforming finite element method for isentropic viscous gas
flow in situations where convective effects may be neglected. We approximate
the continuity equation by a piecewise constant discontinuous Galerkin method.
The velocity (momentum) equation is approximated by a finite element method on
div-curl form using the nonconforming Crouzeix-Raviart space. Our main result
is that the finite element method converges to a weak solution. The main
challenge is to demonstrate the strong convergence of the density
approximations, which is mandatory in view of the nonlinear pressure function.
The analysis makes use of a higher integrability estimate on the density
approximations, an equation for the "effective viscous flux", and renormalized
versions of the discontinuous Galerkin method.Comment: 23 page
Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling
We present a general framework how to investigate stability of solutions
within a single self-consistent renormalization scheme being a parquet-type
extension of the Baym-Kadanoff construction of conserving approximations. To
obtain a consistent description of one- and two-particle quantities, needed for
the stability analysis, we impose equations of motion on the one- as well on
the two-particle Green functions simultaneously and introduce approximations in
their input, the completely irreducible two-particle vertex. Thereby we do not
loose singularities caused by multiple two-particle scatterings. We find a
complete set of stability criteria and show that each instability, singularity
in a two-particle function, is connected with a symmetry-breaking order
parameter, either of density type or anomalous. We explicitly study the Hubbard
model at intermediate coupling and demonstrate that approximations with static
vertices get unstable before a long-range order or a metal-insulator transition
can be reached. We use the parquet approximation and turn it to a workable
scheme with dynamical vertex corrections. We derive a qualitatively new theory
with two-particle self-consistence, the complexity of which is comparable with
FLEX-type approximations. We show that it is the simplest consistent and stable
theory being able to describe qualitatively correctly quantum critical points
and the transition from weak to strong coupling in correlated electron systems.Comment: REVTeX, 26 pages, 12 PS figure
HTL Resummation of the Thermodynamic Potential
Starting from the Phi-derivable approximation scheme at leading-loop order,
the thermodynamical potential in a hot scalar theory, as well as in QED and
QCD, is expressed in terms of hard thermal loop propagators. This
nonperturbative approach is consistent with the leading-order perturbative
results, ultraviolet finite, and, for gauge theories, explicitly
gauge-invariant. For hot QCD it is argued that the resummed approximation is
applicable in the large-coupling regime, down to almost twice the transition
temperature.Comment: minor changes, to appear in PRD, 27 pages, 15 eps figure
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