text:
\section{Introduction}
\label{Introduction}

A self-consistent description of electrons, magnons, phonons and other
fermionic and bosonic degrees of freedom in solids is still a challenging
problem for strongly correlated systems where the simplest approximations are
usually not sufficient. One of the classical examples is the theory of
itinerant electron magnetism, where the d-electrons form both the
correlated energy bands as well as the quasi-localized magnetic moments and
the magnon spectrum \cite{Moriya}. In this problem the conventional random phase
approximation (RPA) gives a wrong description of the finite temperature effects
whereas the proper corrections to RPA can be successfully calculated only in
a very close vicinity of the Stoner instability point \cite
{Moriya,Dzyalos_Kondrat}. The main difficulty is the coexistence of
localized (atomic-like) and itinerant features in the behavior of
d-electrons in solids \cite{vons_kats_tref1993a,vons_kats_tref1993b,ourPRL_Fe_Ni2001}.
Recent progress in the theory of correlated electron systems is related to the development of
Dynamical Mean Field Theory (DMFT) \cite{Georges_RMP}, which maps the
problem of fermions with local interactions on a crystal lattice onto
a self-consistent effective impurity problem.
The latter can be solved very efficiently within Continuous Time Quantum
Monte Carlo (CT-QMC) \cite{Gull_RMP_CTQMC} or exact diagonalization (ED)\cite{Georges_RMP} schemes.
As a result, both atomic-like
features such as multiplet formation and band dispersion can be taken into
account simultaneously \cite{Umat_Pu,CoinCu}. The combination of the DMFT
approach with realistic electronic structure calculations \cite
{Kotliar_RMP,LK1998}, in particular, for transition metal systems \cite
{ourPRL_Fe_Ni2001,ourRMP} opens a new way for the description of electronic
degrees of freedom in solids. A generalization of the DMFT approach to
non-local interactions and consistent description of collective bosonic
excitations \cite{Kotliar_RMP,Chitra_Kotliar,Sun_Kotliar_EDMFT,optics} boost first
realistic applications of a so-called GW+DMFT approach \cite{Biermann_GW+DMFT}.

Here we construct  a theory, that is capable to describe systems with
strongly correlated electrons interacting with collective bosonic modes.
These modes can appear explicitly in an original Hamiltonian (for instance,
phonons) or arise from electron-electron interactions (like plasmons or
magnons). We assume that the Bose excitations cannot be considered as an
ideal Bose gas and boson-boson correlations are important. Among numerous
examples of such systems are the cases of weak itinerant ferromagnets \cite{Dzyalos_Kondrat} and, more generally,
any materials near quantum phase transition points \cite{sachdev}.

The Feynmann diagrammatic technique became the most popular tool
in the quantum many body theory as a compact and elegant representation of
a perturbation expansion in powers of interaction \cite{AGD,Mahan}.
The main advantage of the diagrammatic approach is an opportunity to sum up
infinite series of relevant diagrams going beyond the formal perturbation
expansion. The mainstream paradigm is that summing up more sophisticated series
expands the area of applicability.
In some cases leading series of diagrams can be separated and summed up
with the use of small parameters which are not directly related to the interaction
strength. Important examples are the Gell-Mann Bruekner theory of the high-density electron gas \cite{GB},
the gaseous approximations of low density Fermi \cite{Galitskii}  and Bose \cite{Belyaev} systems,
the $\varepsilon$-expansion in theory of critical phenomena \cite{Wilson_Kogut}, and the $1/N$ expansion
in the theory of magnetic impurities \cite{Bickers}. Historically, the DMFT was introduced as a leading
order approximation in a formal parameter $1/d$, where $d$ is the dimensionality of the space.
DMFT is exact in the limit of $d \rightarrow \infty$, at the same time attempts to construct
a regular expansion in   $1/d$ turn out to be not practical \cite{Shiller}. Nevertheless, DMFT
appears to be quite a good approximation for many three- and even two-dimensional systems
\cite{Georges_RMP,Kotliar_RMP,ourRMP}, since it provides reasonable interpolations between
non-interacting and atomic limits for any dimensions. In terms of the diagrammatic approach the DMFT
corresponds to a summation of all local in space contributions to the self-energy \cite{Georges_RMP}.

Recently we developed the dual fermion scheme \cite{Rubtsov_DF,Rubtsov_big_PRB} which
introduces new variables in a path integral generating an alternative diagram technique such that
the bare propagators (free dual fermions) correspond to DMFT. This means that the main local
physics is taken into account by the change of variables while the new diagrams describe only non-local
corrections to the self-energy. Interaction terms in the dual variables look more complicated than
the original ones: the interactions become retarded and contain not only pair terms but also multi-fermion
contributions. However, an advantage of dual fermion transformation is a faster convergence of the diagram
series. For example, a simple ladder summation gives quite accurate description for the single band Hubbard
model in strongly correlated regime \cite{Hafermann_Ladder}. Thus, the dual fermion approach is
an example of an alternative strategy in the quantum many-body theory, when a proper choice of variables
allows one to consider a simpler series of diagrams. Another example of such strategy is the use of auxiliary
fermions and bosons in the Hubbard model which allows one to describe the subtle effect of the growth of
the effective mass near metal-insulator transitions, already on the level of the mean-field  \cite{Kotliar-Ruckenstein}.

The consideration of collective (bosonic) degrees of freedom in systems of interacting fermions
within the conventional technique necessarily involves  summations of sophisticated diagrams.
The dispersion law of bosonic excitations (e.g., magnons or plasmons) is determined by the poles
of the corresponding susceptibilities (the two particle fermionic Green's functions) and the minimal
approximations which give rise to these poles, even in the case of a weak interaction corresponding to the ladder
summation (random phase approximation, RPA \cite{AGD,Bohm_Pines,Khodel_RPA}).
In order to take into account magnon-magnon interactions
one needs to consider more complicated, parquet-like diagrams \cite{Dzyalos_Kondrat,Irkhin_Katanin_Katsnelson}.
For strongly correlated systems one can consider the same procedure within the
dual fermion approach. Up to now this is only done at the level of ladder summations \cite{Hafermann_Ladder}.
As we pointed out, the above considerations of complicated diagrams can be replaced by a proper choice
of variables in the path integral. Following this strategy we propose an extension of dual fermion scheme
explicitly adding bosonic variables. It is important to note that the present theory is not a direct extension of
the dual-fermion approach \cite{Rubtsov_DF}. We will see that, even for the
Hubbard-like models with on-site interactions only, it gives a different
result because of the account of collective modes (magnons).

A simultaneous description of fermionic and bosonic correlations on the
equal footing was started within the Extended Dynamical Mean Field (EDMFT)
framework \cite{Chitra_Kotliar,Sun_Kotliar_EDMFT}. The strong-coupling
generalization of  EDMFT \cite{Stanescu_Kotliar} gives a useful extension
of the Hubbard-I like perturbation expansion for lattice models \cite
{Pairault_Senechal_Tremblay}. EDMFT properly includes the physics of
\textit{local} fermionic and bosonic correlations. At the same time, a dispersion
of collective excitations within EDMFT is determined by
the non-local part of the interactions {\it only}. In particular, to describe the dispersive magnons
within EDMFT one needs to  explicitly add direct Heisenberg exchange
interactions to the Hamiltonian.
Obviously, this is not sufficient since the {\it indirect} exchange interactions
(superexchange, double-exchange, RKKY, etc.) contribute as well to the the magnon dispersion.
Moreover, typically, they give the main contribution.

In the present paper we construct a formally exact diagram technique where
the bare fermionic and bosonic propagators correspond to EDMFT, whereas
the indirect exchange is given by diagrammatic corrections. Therefore, the
direct and indirect interactions enter the magnon dispersion law on equal footing.

\section{Definitions}

We consider a lattice fermionic problem with the intersite hopping and
generic non-local interactions. For simplicity, we present only formulae for
the single-band periodic lattice, but a generalization of all expressions to a
multiorbital case is straightforward. The action is
\begin{equation}  \label{Lattice}
S=\sum_r S_{at}[c^\dag_r, c_r]+\sum_{r,R\neq 0, \omega, \sigma}
\varepsilon_R c^\dag_{r \omega \sigma} c_{r+R \omega \sigma} +\sum_{r,R\neq
0, \Omega} V_{R \Omega} \rho_{r \Omega}^*
\rho_{r+R \Omega}
\end{equation}
Here $c^\dag,\,c$ are Grassmann variables, the indices $r, R$ run over
the lattice sites and translations, $k$ stands for the wave-vector, $\tau$ is
the imaginary time,
$\omega=2\pi (n+1/2) / \beta $ and $\Omega=2\pi n / \beta $ are, respectively, the fermionic and bosonic
Matsubara frequencies ($n$ is integer and $\beta$ is the inverse temperature), and $\sigma$ is the spin index.
Prefactors $\beta^{-1}$ and $N^{-1}$ are implied for sums over frequencies and wavevectors,
respectively  ($N$ is the number of lattice sites).
The ``Bosonic variable'' $\rho$ in general is a vector quantity,
whose components for single band models can be the charge-density or the spin-density
operators: $\rho_j \equiv \sum_{\sigma \sigma^{\prime}} c^\dag_{\sigma}
s^{j}_{\sigma \sigma^{\prime}} c_{\sigma^{\prime}}-\bar{\rho}$, where $s^j$ with
$j=\{0, x, y, z\}$ are Pauli matrices ($1, \sigma_x, \sigma_y, \sigma_z, $), and the quantity $\bar{\rho}$
is introduced to satisfy the condition $<\rho>=0$.

To simplify the notations, we skip the index $j$ throughout the paper, but
one should bear in mind that relevant components of $\rho$ are to be accounted in a practical calculation.
For example,  if the physics is determined by charge fluctuations only (as may be the case
for the Coulomb interaction or an electron-electron attraction mediated by phonons),
only the density-density term
$V^{00} \rho_0^* \rho_0$ is important in the interaction part $V^{j j^{\prime}}_{R \Omega}$ of Eq.(\ref{Lattice}). The Heisenberg isotropic exchange interaction corresponds
to $V^{xx}=V^{yy}=V^{zz}=J$, etc.
An important case of the Hubbard model corresponds to $V=0$, where the spin degrees of freedom $\rho_{x,y,z}$ do not show up in the bare expression part of Eq.(\ref{Lattice}), but will appear in the following formulation, since the
low-energy behaviour of system is governed by the spin fluctuations.

Within the path integral representation the atomic action $S_{at}$ is given
by:
\begin{equation}  \label{Atom}
S_{at}=-\sum_{\omega\sigma} (i \omega +\mu) c^\dag_{\omega\sigma}
c_{\omega\sigma} + \int_0^\beta U c^\dag_{\uparrow} c_{\uparrow}
c^\dag_{\downarrow} c_{\downarrow} d\tau,
\end{equation}
where $U$ is the local Coulomb interaction.

We are interested in the single-electron Green's function
$G_{r \tau}=-<c_{r \tau} c^\dag_{r=0, \tau=0}>$ and the bosonic correlator $X_{r
\tau}=-<\rho_{r \tau} \rho_{r=0, \tau=0 }^*>$. It is an important point that
an account of the bosonic quantity is fundamental: we suppose that
collective modes are relevant for the physics of the system.

\section{Local approximations}

The general idea of effective-medium approximations is to reduce the lattice
problem (\ref{Lattice}) to the so-called impurity model, describing a single
atom (or small cluster) in a Gaussian environment. For a system with bosonic
and fermionic degrees of freedom, a single-site impurity problem is
described by the action
\begin{equation}  \label{Impurity}
S_{imp}=S_{at}+\sum_\omega \Delta_\omega c^\dag_\omega c_\omega+\sum_\Omega
\Lambda_\Omega \rho^*_\Omega \rho_\Omega .
\end{equation}

Obtaining a numerical or analytical solution for the impurity problem
is supposed to be much simpler. We later assume that its Green's functions: $%
g_{\tau}=-<c_{\tau} c^\dag_{\tau=0}>_{imp}$ and $\chi_{\tau}=-<\rho_{\tau}
\rho_{\tau=0}^*>_{imp}$ are known, as well as higher-order correlators.
In practice, they can be calculated by a CT-QMC scheme \cite{Gull_RMP_CTQMC}
or by exact diagonalization if the dimension of the Hilbert space is not too
large.
Normally $\chi$ is called the full polarizability. We will call it the bosonic Green's function, to
emphasize the formal analogy between the equations for $g$ and $\chi$ in our formalism.

For the so-called local approximations, only the Green's functions of the
impurity model $g_{\omega}, \chi_\Omega$ are taken into account. In a
nutshell, the initial problem (\ref{Lattice}) is replaced by an auxiliary
lattice with Gaussian nodes; those nodes are characterized by the Green's
functions $g_{\omega}, \chi_\Omega$. Such a consideration gives us the well-known
expressions for lattice Green's functions \cite{Sun_Kotliar_EDMFT}, which
are denoted with calligraphic letters in this paper:
\begin{eqnarray}\label{MFG}
{\cal G}_{\omega k}=\frac{1}{g^{-1}_\omega+\Delta_\omega-\varepsilon_k}   \\
{\cal X}_{\Omega K}=\frac{1}{\chi^{-1}_\Omega+\Lambda_\Omega-V_K} \nonumber
\end{eqnarray}
where $k$ is the wave vector and Fourier transforms of the hopping and
interaction parameters are introduced in the obvious way. One can see that
the fermionic and bosonic self-energies of these expressions are $k$%
-independent, that is, local in space.

The hybridization functions $\Delta$ and $\Lambda$ should be defined in such
a way, that the impurity problem provides us the on-site (local) properties
of the solution for the lattice problem. Such a requirement is  physically quite
reasonable. Also, it is closely related to
Feynman's variational approach in dual fermion variables \cite{Rubtsov_big_PRB}.

Most common are so-called static and dynamic mean-field theories. In the
static approach, the hybridization functions $\Delta $ and $\Lambda $ are
chosen to be frequency independent. Their values are defined from the
equalities for the \textit{static parts} of Green's functions:
\begin{eqnarray}\label{MFcondition}
\sum_\omega g_\omega^{MF}=\sum_{\omega k} {\cal G}_{\omega k}^{MF}, \\ \nonumber
\sum_\Omega \chi_\Omega^{MF}=\sum_{\Omega K} {\cal X}_{\Omega K}^{MF}
\end{eqnarray}

In the dynamical EDMFT approach $\Delta_\omega$ and $\Lambda_\Omega$ are
adjusted so that local parts of the lattice Green's functions coincide with
the corresponding quantities of the impurity problem \textit{for all frequency
(or time) arguments}:
\begin{eqnarray}\label{DMFTcondition}
g_\omega=\sum_k {\cal G}_{\omega k}, \\ \nonumber
\chi_\Omega=\sum_K {\cal X}_{\Omega K}
\end{eqnarray}

It is worthwhile to note that the standard definition of the EDMFT \cite
{Sun_Kotliar_EDMFT} introduces an auxiliary bosonic field called $\phi$, and
the second line of Eq.(\ref{DMFTcondition}) is re-expressed as a condition
for the Green's function $\mathcal{X}_\phi$ of that bosonic field. At the
EDMFT level, the conditions for $\mathcal{X}$ and $\mathcal{X}_\phi$ are
equivalent. We discuss this issue in detail in Appendix and show that
similar (although not the same) bosonic field can be introduced in non-local
approximations beyond the EDMFT.

Some ``mixed'' local approximations can also make sense. For instance, if
the hopping is absent (or negligible), it is natural to consider a scheme
with $\Delta=0$ and to adjust bosonic hybridization only. Contrary, $%
\Lambda=0$ corresponds to the ``purely fermionic'' DMFT scheme \cite
{Georges_RMP}.

\section{Dual transformation}

It is easy to make a connection between the lattice and impurity actions:

\begin{equation}  \label{Lattice_imp}
S=\sum_r S_{imp}[c^\dag_r, c_r]+\sum_{\omega, k, \sigma} \left( \varepsilon_k -
\Delta_{\omega\sigma} \right) c^\dag_{\omega k \sigma} c_{\omega k \sigma}
+\sum_{\Omega, K} \left( V_{\Omega K}-\Lambda_{\Omega} \right) \rho_{\Omega K }^* \rho_{\Omega K}
\end{equation}

Further on, we use a general matrix form of the Hubbard-Stratonovich
transformation to decouple the term $c^\dag_{\omega k \sigma} E_{\omega k
\sigma} c_{\omega k \sigma}$ with $E_{\omega k
\sigma}=\Delta_{\omega\sigma}-\varepsilon_{k}$ into the pair of dual fermionic
Grassmann-fields $f^\dag_{\omega k \sigma}, f_{\omega k \sigma} $ and the
term $\rho^*_{\Omega K j} W^{j j^{\prime}}_{\Omega K} \rho_{\Omega K
j^{\prime}} $ with $W^{j j^{\prime}}_{\Omega K} = \Lambda^{j
j^{\prime}}_{\Omega}-V^{j j^{\prime}}_{\Omega K}$ into the pair of dual
bosonic complex-fields $\eta^*_{\Omega K j}, \eta_{\Omega K j}$ (a matrix
notation is used to shorten the expressions):
\begin{eqnarray}  \label{HS}
\begin{array}{l}
\int D\left[c^\dag, c \right]e^{c^\dag E c} = \int D\left[c^\dag, c \right]
\det\left(\alpha^{-1}_f E \alpha^{-1}_f \right) \int D\left[f^\dag, f \right]
e^{-f^\dag \alpha_f E^{-1} \alpha_f f + f^\dag \alpha_f c + c^\dag \alpha_f f } \\
\\
\int D\left[\rho^*, \rho \right]e^{\rho^* W \rho} = \int D\left[\rho^*, \rho
\right] \det\left(\alpha_b W^{-1} \alpha_b \right) \int D\left[\eta^*, \eta \right]
e^{-\eta^* \alpha_b W^{-1} \alpha_b \eta + \eta^* \alpha_b \rho + \rho^* \alpha_b \eta } ,
\end{array}
\end{eqnarray}
where $\alpha_f, \alpha_b$ are arbitrary, but local in space scaling
matrices.

Note that whereas the decoupling of Grassmanian variables is mathematically
straightforward, more comments are needed on the second line of Eq.(\ref{HS}).
The first comment is that $\rho$ is Hermitian, so that $\rho^*_{-\Omega -K
j}=\rho_{\Omega K j}$, and the integral $D[\rho^*, \rho]$ is taken in fact over
the variables from a half-space. The dual variables inherit this property, $%
\eta^*_{-\Omega -K j}=\eta_{\Omega K j}$.
The second comment is that the matrix $\alpha _{b}W^{-1}\alpha _{b}$ is not
positively defined, and integrals over $D[\eta ^{\ast },\eta ]$ should be
re-defined to ensure convergence. Here we describe the corresponding
procedure, following, in particular, Refs. \cite{Rubtsov_ising,Sun_Kotliar_EDMFT}.

For illustration purposes, we suppose $\alpha$ to be just a real constant
factor; the generalization to a complex matrix $\alpha$ is straightforward.
Let us introduce a basis where $W^{-1}$ is diagonal. Let us number the
states in this basis with an index $l$, and consider some matrix element $%
W_{ll} $. For a positive real part of $W_{ll}$, the corresponding part of the
action can be decoupled as follows:
\begin{equation}  \label{HSplus}
e^{\rho_l^{\prime}W_{ll} \rho_l^{\prime}+\rho_l^{\prime\prime}W_{ll}
\rho_l^{\prime\prime}}=\alpha W_{ll}^{-1} \alpha \int d\eta^{\prime}\int
d\eta^{\prime\prime}e^{\eta^{\prime}_l \alpha
\rho^{\prime}_l-\eta^{\prime\prime}_l \alpha
\rho^{\prime\prime}_l-\eta^{\prime}_l \alpha W_{ll}^{-1} \alpha
\eta^{\prime}_l-\eta^{\prime\prime}_l \alpha W_{ll}^{-1} \alpha
\eta^{\prime\prime}_l},
\end{equation}
where we explicitly introduced the Hermitian and anti-Hermitian parts of $%
\rho=\rho^{\prime}+i \rho^{\prime\prime}$. One can see that for $\eta_l^{\prime}$ and
$\eta_l^{\prime\prime}$ being interpreted as real and
imaginary parts of a complex number $\eta$, this decoupling becomes an
explicit form of the second line of Eq. (\ref{HS}), written for a particular
state $l$. The integral limits for positive $W_{ll}$ can be chosen along the
real axis, from $-\infty$ to $+\infty$ for both $\eta_l^{\prime}$ and $%
\eta_l^{\prime\prime}$.

The key observation is that for $W_{ll}$ having a \textit{negative} real
part, the decoupling (\ref{HSplus}) still holds, up to the integration path
along imaginary axis, from $-i \infty$ to $+i \infty$ for $\eta_l^{\prime}$
and in the opposite direction, that is $+i \infty$ to $-i \infty$, for $%
\eta_l^{\prime\prime}$.
Thus, the integration over $%
D[\eta^*, \eta]$ can be considered as a symbolic notation for the
integration over $D[\eta^{\prime}, \eta^{\prime\prime}]$ with $%
\eta=\eta^{\prime}+i \eta^{\prime\prime}, ~ \eta^*=\eta^{\prime}-i
\eta^{\prime\prime}$ substituted in the integrand. The integration path over
each component of vectors $\eta^{\prime}, \eta^{\prime\prime}$ is explicitly
defined for the basis where $\alpha W \alpha$ is diagonal: the integration
goes along the real or imaginary axis, corresponding to the sign of the real part of
the diagonal matrix element  $W_{ll}$.

One can check that such a definition of $\int D[\eta^*, \eta]$ still allows one
to use the standard algebraic manipulations of path integrals. In particular,
to obtain the Green's function of dual bosonic variables $\tilde{G}=-i <\eta \cdot
\eta^*>$ one uses the standsrd procedure of differentiation with respect to
$\alpha W^{-1} \alpha$, as can be explicitly shown from Eq. (\ref{HSplus}).
On the other hand, the statistical properties of dual variables can be
unusual because of the imaginary axis integration path. For instance, similarly to
the fermionic dual Green's function \cite{Rubtsov_big_PRB}, $<\eta \cdot
\eta^*>=<\eta^{\prime}\cdot \eta^{\prime}>+ <\eta^{\prime\prime}\cdot
\eta^{\prime\prime}>$ is not necessary positive-defined and consequently the imaginary part of its Fourier transform
 is not always negative.

Let us continue with the derivation of the dual formalism.
The Hubbard-Stratonovich decoupling in the partition function with the action (%
\ref{Lattice}) gives (in matrix form)
\begin{eqnarray}\label{Smix}
\begin{array}{l}
\int D[c^{\dag }c]e^{-\sum_{r}S_{imp}[c_{r}^{\dag },c_{r}]+c^{\dag }(\Delta
-\varepsilon )c+\rho ^{\ast }(\Lambda -V)\rho }=\det (\alpha _{f}^{-1}(\Delta
-\varepsilon )\alpha _{f}^{-1})\det (\alpha _{b}(\Lambda -V)^{-1}\alpha
_{b})\times \\
\\
\times \int D[c^{\dag }c]\int D[f^{\dag }f]\int D[\eta ^{\ast }\eta
]e^{-\sum_{r}S_{imp}[c_{r}^{\dag },c_{r}]+f^{\dag }\alpha _{f}c+c^{\dag
}\alpha _{f}f+\rho ^{\ast }\alpha _{b}\eta +\eta ^{\ast }\alpha _{b}\rho
-f^{\dag }\alpha _{f}(\Delta -\varepsilon )^{-1}\alpha _{f}f-\eta ^{\ast
}\alpha _{b}(\Lambda -V)^{-1}\alpha _{b}\eta }.
\end{array}
\end{eqnarray}
Integrating out the ``localized'' fermionic Grassmann variables $c^{\dag },c$
with $\alpha =g_{\omega}^{-1} $ for the fermionic and $\alpha =\chi _{\Omega }^{-1}$ for the
bosonic part, we arrive at the following action in dual variables
\begin{equation}
\begin{array}{l}
\tilde S=-\sum_{\omega k}\tilde{\mathcal{G}}_{\omega k}^{-1}f_{\omega K}^{\dag
}f_{\omega K}-\sum_{\Omega k}\tilde{\mathcal{X}}_{\Omega K}^{-1}\eta
_{\Omega K}^{\ast }\eta _{\Omega K}+\sum_{i}\tilde{U}[\eta
_{i},f_{i},f_{i}^{\dag }] \\
\\
\tilde{\mathcal{G}}_{\omega k}=(g_{\omega}^{-1}+\Delta _{\omega }
  -\varepsilon _{k})^{-1}-g_{\omega }\\
\\
\tilde{\mathcal{X}}_{\Omega K}=(\chi _{\Omega}^{-1}+\Lambda _{\Omega }
-V_{K} )^{-1}-\chi _{\Omega } \\
\\
\tilde{U}[\eta ,f,f^{\dag }]=\sum_{\omega \Omega }\left( \lambda _{\omega
\Omega }\eta _{\Omega }^{\ast }f_{\omega +\Omega }^{\dag }f_{\omega
}+\lambda _{\omega \Omega }^{\ast }\eta _{\Omega }f_{\omega }^{\dag }\
f_{\omega +\Omega }\right) +\frac{1}{4}\sum_{\omega \omega ^{\prime }\Omega
}\gamma _{\omega \omega ^{\prime }\Omega }f_{\omega +\Omega }^{\dag
}f_{\omega ^{\prime }-\Omega }^{\dag }f_{\omega  ^{\prime }}f_{\omega}+...,
\end{array}
\label{DB_action}
\end{equation}
where $\gamma$ is a full four-point fermionic vertex of the impurity model
\begin{equation}  \label{Gamma}
\gamma_{\omega \omega^{\prime}\Omega}=\frac{<c_{\omega+\Omega}
c_{\omega^{\prime}-\Omega} c^\dag_\omega
c^\dag_{\omega^{\prime}}>_{imp}-g_\omega g_{\omega^{\prime}}
(\delta_{\Omega+\omega-\omega^{\prime}}-\delta_{\Omega})}{g_{\omega+\Omega}
g_{\omega^{\prime}-\Omega} g_\omega g_{\omega^{\prime}}}
\end{equation}
($\delta$ is the Kronecker symbol), and $\lambda$ is a ``mixed'' quantity
\begin{equation}  \label{Lambda}
\lambda_{\omega \Omega}=\frac{-<c_{\omega+\Omega} c_\omega^\dag
\rho_\Omega>_{imp}-<\rho>_{imp} g_\omega \delta_\Omega} {g_\omega
g_{\omega+\Omega} \chi_\Omega}.
\end{equation}
The second term in the nominator typically equals zero as $<\rho>_{imp}$
vanishes.
The values of $\gamma$ and $\lambda$ are related, since $\rho_\Omega=\sum_{%
\omega^{\prime}}  c^\dag_{\omega^{\prime}\sigma} s_{\sigma \sigma^{\prime}}
c_{\omega^{\prime}-\Omega, \sigma^{\prime}}$,
\begin{equation}  \label{lambdagamma}
\lambda_{\Omega \omega}=\chi_\Omega^{-1}\left(1-\sum_{\omega^{\prime} \sigma \sigma^{\prime}}
\gamma^{\sigma \sigma^{\prime}}_{\omega \omega^{\prime} \Omega} g^{\sigma}_{\omega^{\prime}}
s_{\sigma \sigma^{\prime}} g^{\sigma^{\prime}}_{\omega^{\prime}-\Omega} \right)
\end{equation}
It is important to note that for the Gaussian impurity problem $\gamma$
vanishes, but $\lambda$ takes a finite value of $\chi_\Omega^{-1}$.

The following exact relations between fermionic and bosonic Green's functions of
original and dual variables can be proven:
\begin{eqnarray}
 \label{Exact}
 \begin{array}{l}
G_{\omega k}=(\Delta _{\omega }-\varepsilon _{k})^{-1}g_{\omega }^{-1}\tilde{G}%
_{\omega k}g_{\omega }^{-1}(\Delta _{\omega }-\varepsilon _{k})^{-1}+(\Delta
_{\omega }-\varepsilon _{k})^{-1}; \\
\\
X_{\Omega K}=(\Lambda _{\Omega }-V_{K})^{-1}\chi _{\Omega }^{-1}\tilde{X}%
_{\Omega K}\chi _{\Omega }^{-1}(\Lambda _{\Omega }-V_{K})^{-1}+(\Lambda
_{\Omega }-V_{K})^{-1};
\end{array}
\end{eqnarray}
The relation for the fermionic Green's functions is derived in Ref. \cite
{Rubtsov_big_PRB}. Main idea is to substitute the right-hand side of Eq.(\ref
{Smix}) in the formula $G_{\omega k}=-\frac{1}{Z}\frac{\partial Z}{\partial
\varepsilon _{\omega k}}$. The last term $(\Delta _{\omega }-\varepsilon
_{k})^{-1} $ appears from the differentiation of the fermionic determinant. The
bosonic relation is derived quite similarly, but the signs appearing during
the derivation should be discussed. First, the expression for the bosonic
Green's function has a different sign, $X_{\Omega k}=\frac{1}{Z}\frac{%
\partial Z}{\partial V_{\Omega k}}$. This gives factors of  $-1$ at $X$ and $\tilde{X}$.
Second, since the bosonic Hubbard-Stratonovich transformation
contains an inverse of the determinant, the same different sign appears for
the last term $(\Lambda _{\Omega }-V_{k})^{-1}$, so that the signs of all
the terms are changed, compared to the fermionic relation between the real and
dual fermionic Green's functions. Therefore the entire resulting relationship
between the real and dual bosonic Green's functions obeys exactly the same
form, similar to the results of the supersymmetric Hubbard-Stratonovich
transformation \cite{Stanescu_Kotliar} .

It is useful to introduce the self-energy (for fermions) and
polarization-operator (for bosons) corrections to the effective-medium
approximation, defined as
\begin{equation}  \label{SelfEnergy}
\Sigma^{\prime}=G^{-1}-\mathcal{G}^{-1}; ~~~ \Pi^{\prime}=X^{-1}-\mathcal{X}%
^{-1}.
\end{equation}
We remind that $\mathcal{G}, {\mathcal{X}}$ are given by the expressions (%
\ref{MFG}).

The self-energy and polarization operator of the dual ensemble are given by the
following expressions:
\begin{equation}  \label{SelfEnergyDual}
\tilde{\Sigma}=\tilde{G}^{-1}-\tilde{\mathcal{G}}^{-1};~~~ \tilde{\Pi}=%
\tilde{X}^{-1}-\tilde{\mathcal{X}}^{-1}.
\end{equation}
Being re-expressed in these quantities, the exact relations (\ref{Exact})
have a particularly simple form:
\begin{eqnarray}  \label{Exact2}
\begin{array}{l}
(\Sigma^{\prime}_{\omega k})^{-1}=\tilde{\Sigma}_{\omega k}^{-1}+g_\omega
\\
\\
(\Pi^{\prime}_{\Omega K})^{-1}=\tilde{\Pi}_{\Omega K}^{-1}+\chi_\Omega
\end{array}
\end{eqnarray}
which results in the following expression for the fermionic and bosonic Green's functions:
\begin{eqnarray}
 \label{Exact}
 \begin{array}{l}
G_{\omega k}^{-1}=
(g_{\omega}+g_{\omega} \tilde{\Sigma}_{\omega k}  g_{\omega} )^{-1}+\Delta _{\omega }
  -t_{k}
\\
\\
X_{\Omega K}^{-1}=
(\chi _{\Omega}+\chi _{\Omega}  \tilde{\Pi}_{\Omega K}     \chi _{\Omega}     )^{-1}+\Lambda _{\Omega }
-V_{K} \\
\end{array}
\end{eqnarray}
One can immediately see from these formulas that the effective-medium
expressions (\ref{MFG}) correspond to the Gaussian approximation for the
dual variables: $\tilde{\Sigma}, \tilde{\Pi}$ (and consequently $\Sigma^{\prime}, \Pi^{\prime}$) vanish.

Following the effective medium paradigm we introduce a generalized self-consistent
condition for
$\Delta _{\omega }$ and $\Lambda _{\Omega }$ similar to the Eq.(\ref{DMFTcondition}), but
for "exact" fermionic and bosonic Green's functions:
\begin{eqnarray}\label{DBcondition}
g_\omega=\sum_k { G}_{\omega k}, \\ \nonumber
\chi_\Omega=\sum_K { X}_{\Omega K}
\end{eqnarray}

\section{Basic diagrams and simple approximations}
\label{BareDiagSec}

The simplest non-local physics can already  be described in the EDMFT.
In our formalism this corresponds to a zero-th order approximation
(``no diagrams'') $ \tilde{\Sigma}=0$ and $\tilde{\Pi}=0$, and
condition (\ref{DMFTcondition}) is satisfied. We intend to preserve the
advantages of the EDMFT, but also include  the formation of collective modes and their
interaction with the electrons. For a ``purely fermionic''
Hubbard-like theory this means an account of the ladder series in dual
space \cite{Hafermann_Ladder}.

For the present theory with the dual action of Eq.(\ref{DB_action}),
there are two basic Green's function lines (Fig. \ref{Basic_diagram}):
a fermionic (full-line with arrow) and a bosonic (wavy-line) as well
as two interaction vertices: a triangle (corresponding to the dual
boson-fermion interaction - $\lambda $) and a square (describing
the dual-fermion interaction - $\gamma $). In this paper, we restrict ourselves to a polarization operator, given by
the electron ladder ended by two triangles (Fig. \ref{fig_ladder} ). Equally, one can consider this diagram as
 just a two-lines bubble, with a renormalized vertex (Fig. \ref{fig_pi} ).
\begin{equation}  \label{buble}
\begin{array}{l}
\tilde \Pi_{\Omega K}= { \lambda}^{\rm{eff}}_{\Omega} \cdot \tilde{{\tt X}}_{\Omega K}^0 \cdot \lambda_{\Omega},\\   \\
\lambda^{\rm{eff}}_\Omega=\left(1-\gamma_\Omega \tilde{{\tt X}}_{\Omega K}^0 \right)^{-1} \lambda_\Omega .
\end{array}
\end{equation}
Here a symbolic notation is used, so that $\gamma_\Omega$ is a tensor, and $\lambda_\Omega$ is a vector with components corresponding to different $\omega$, and the dots in the first line
denote a scalar product vector-tensor-vector. The bare dual polarization operator (an empty loop of the dual Green's function)
 $\tilde {\tt X}^{0}$ is a tensor having only diagonal components. Later we also use bare polarization operators for the initial variables and for the impurity model, so that
\begin{equation}
\begin{array}{l}
\tilde{{\tt X}}_{\Omega K}^{0,\omega}=-\sum_{k} {\tilde{\cal G}}_{\omega k} {\tilde{\cal G}}_{\omega+\Omega k+K},\\ \\
{{\tt X}}_{\Omega K}^{0,\omega}=-\sum_{k} {{\cal G}}_{\omega k} {{\cal G}}_{\omega+\Omega k+K},\\ \\
{{\tt x}}_{\Omega}^{0,\omega}=-{{g}}_{\omega} {{ g}}_{\omega+\Omega}.
\end{array}
\end{equation}
Scalar polarizabilities, respectively denoted by ${\tilde {\cal X}}^{0}_{\Omega K}$, ${{\cal X}}^{0}_{\Omega K}$, and $\chi^0_\Omega$, can be found by summating over fermionic frequencies, so that, for example,  ${{\cal X}}^{0}_{\Omega K}=-\sum_{k\omega} {{\cal G}}_{\omega k} {{\cal G}}_{\omega+\Omega k+K}$.

The simplest approximation one can make is just to neglect the vertex part $\gamma$.
In this case
$\lambda^{\rm{eff}}_{\omega\Omega}=\lambda_{\omega\Omega}=\chi^{-1}_\Omega$. Simple manipulations
with the second lines of Eqs. (\ref{MFG}), (\ref{SelfEnergy}) and
(\ref{Exact2}) then give
\begin{equation}  \label{DRPA}
X_{\Omega K}=\frac{1}{\left(\chi_\Omega+\tilde{{\cal X}}^{0}_{\Omega K}\right)^{-1} +\Lambda_\Omega-V_K}.
\end{equation}
A similar non-local contribution to the fermionic self-energy can be introduced to
describe the renormalization of the fermionic Green's function $G$.
The corresponding diagram contains one fermionic and one bosonic line (Fig. \ref{fig_sig}).
We will however concentrate on the analysis of bosonic quantities and
for simplicity use the EDMFT result $\tilde{\mathcal{G}}$ instead of
the renormalized $\tilde G$ propagator. This allows for a further simplification.
Indeed, in the local approximation $\mathcal{G}$ and $\tilde{%
\mathcal{G}}^{dual}$ differ in their local part only \cite
{Rubtsov_DF,Rubtsov_big_PRB}:
\begin{equation}  \label{GGprime2}
\tilde{\mathcal{G}}_{\omega k}=\mathcal{G}_{\omega k}-g_{\omega}.
\end{equation}
Further, since we suppose that $\gamma=0$ and since $<\rho>=0$, the bosonic
Green's function for the impurity problem becomes equal $\chi_\Omega^0$. Substituting
the expression (\ref{GGprime2}) into Eq. (\ref{DRPA}), we obtain
\begin{equation}  \label{RPA2}
X_{\Omega K}=\frac{1}{\left({\mathcal X}^{0}_{\Omega K}\right)^{-1}+ \Lambda_\Omega-V_K }.
\end{equation}
For $\Lambda=0$, this expression is clearly related to the Random Phase
Approximation (RPA). To get the ``textbook'' RPA, one should completely
neglect the interaction terms of the impurity problem, so that $%
g_\omega=(i\omega-\Delta_\omega)^{-1}$ and $\mathcal{G}_{\omega,
k}=(i\omega-\varepsilon_k)^{-1}$. The scheme with an exactly calculated
$g_\omega$ can be called RPA+DMFT.
In this case the RPA is a simple case of Fig. \ref{fig_pi}.
To go beyond the RPA, one should be careful. As we will discuss in Sec. \ref{conserv}
in general it is wrong to work with the renormalized Green's functions but to neglect the vertex $\gamma$.
Because this may lead to a violation of the conservation laws.

A theory using the single-bubble approximation (\ref{buble}) with
 $\lambda^{\rm{eff}}$ approximated by $\lambda$ (that is, without putting $\lambda$ to
$\chi^{-1}$) is essentially the GW+EDMFT approach
\cite{Sun_Kotliar_EDMFT}. There is only a slight difference in the
self-consistency condition, as we explain in Appendix. It is also
important that for the diagram calculation of the GW+EDMFT approach that the
local part of the bosonic Green's function is excluded \textit{ad
hoc}. As one can see from Eq. (\ref{EDMFTGGprime}), for the EDMFT
choice of the Bose variables, it actually does not vanish exactly. So,
our consideration proposes a certain a modification of GW+EDMFT.
Anyhow, such modification should not strongly affect the results of the
GW+EDMFT. The real advantage of our consideration is the
possibility to go beyond the domain of applicability of the
GW+EDMFT, i.e. in  is the situation when the single-bubble approximation
becomes incorrect. It is considered in the next sections.

The diagrams shown in Figs. \ref{bold3}, \ref{fig_pi}, \ref{fig_sig} form the
minimal consistent set beyond the RPA to describe the formation of collective modes
with electron-boson coupling taken into account explicitly.

\section{Charge conservation}
\label{conserv}

So far, we did not consider whether the constructed approximations conserve particle number,
total spin, etc. However, the conservation laws are particularly important for
bosonic excitations. For example, the charge (particle number) conservation resulting in the equation $\Omega^2 <\rho \rho>_{\Omega K}=K^2 <j j>_{\Omega K} $  immediately leads
to the requirement $ <\rho \rho>_{K=0}=0$ at any finite frequency (provided that the current-current correlator
$<j j>_{\Omega, K=0}$ is finite). For 3D systems with Coulomb interaction the long-wavelength asymptotic
behaviour for the density-density correlator should be
 $ <\rho \rho>_{\omega, K\to 0}\propto \frac{K^2}{\Omega^2+\Omega_p^2}$, where $\Omega_p$ is the plasma frequency. The RPA for free electrons is proven to obey this property, since $X^{0}_{\Omega, K\to0}$ vanishes with ${\cal G}=(i\omega-\varepsilon_k)^{-1}$, but this is not true for the renormalized Green's function \cite{Baym}. Indeed, following the standard proof \cite{AGD, Mahan} one writes  ${{\cal X}}^{0}_{\Omega K}=-\sum_{k\omega} {{\cal G}}_{\omega k} {{\cal G}}_{\omega+\Omega k+K}$ as
\begin{equation}
{\mathcal X}^{0}_{\Omega K}=-\sum_{\omega k} \left( \frac{1}{i\omega-\varepsilon_k-\Sigma_\omega}-
\frac{1}{i(\omega+\Omega)-\varepsilon_{k+K}-\Sigma_{\omega+\Omega}}\right)\frac{1}{i\Omega+\Sigma_\omega-\Sigma_{\omega+\Omega}}.
\end{equation}
For free electrons of dispersionless $\Sigma$ (that is, in static mean-field) the summation
over $\omega$ produces a difference
of occupation numbers, $n_k-n_{k+K}$, in the nominator, that vanishes at   $K=0$. However this
reasoning clearly breaks down for the frequency-dependent $\Sigma_\omega$ in DMFT approximation.
It was known for a long time that the neglection  of vertex corrections together with the use of {\it renormalized} Green's
functions, violate charge and spin conservations \cite{Baym,Baym_hist,Hertz}.
We found that
$X^{0}$ with the DMFT Green's function does not vanish at small $K$  and therefore it follows form Eq. (\ref{RPA2}) that  {\it EDMFT+GW is not a valid description of plasmons}. All the more, {\it plasmons do not appear in simple EDMFT}, as one can easily observe from the second line of Eq. (\ref{MFG}): the local quantity $\chi_\Omega^{-1}+\Lambda_\Omega$ cannot behave like $\Omega^2 K^2$. This is the reason why the GW approach is normally used only for calculating the single-particle Green's function, while any two-particle properties are
investigated within the Bethe-Salpeter equations \cite{GW_RMP}. Nevertheless, such a scheme cannot describe self-consistent effects of collective excitations on the single-particle spectral function.

To construct a conserving approximation, it is useful to consider the following reformulation of the theory. Let us return to the initial lattice action (\ref{Lattice}) and do the Hubbard-Stratonovich decoupling for the bosonic variables only. It will give an analog of  Eq. (\ref{Smix}) with the action
\begin{equation}\label{SFmix}
\begin{array}{l}
S=S^F[c,c^\dag]+\rho^* \alpha_b \eta +\eta^* \alpha_b \rho-\eta^* \alpha_b (\Lambda-V)^{-1} \alpha_b \eta,\\
S^F=\sum_r S_{at}[c^\dag_r, c_r]+\sum_{r,R\neq 0, \omega, \sigma}
\varepsilon_R c^\dag_{r \omega \sigma} c_{r+R \omega \sigma} +\sum_{r, \Omega} \Lambda_{\Omega} \rho_{r \Omega}^*\rho_{r \Omega}.
\end{array}
\end{equation}
Physically, $S^F$ describes a lattice with the interaction being local in space but retarded. Now we formally integrate over the fermionic variables. The Gaussian approximation in dual bosons for such a theory corresponds to the expression
\begin{equation}\label{GreenF}
{X}_{\Omega K}=\frac{1}{\left(X^F_{\Omega K}\right)^{-1}+\Lambda_\Omega-V_K},
\end{equation}
where $X^F$ is an exact $K$-dependent two-particle correlator for the lattice problem $S^F$.
Formally, the calculation of $X^F_{K, \Omega}$ corresponds to a summation of all parts of the diagrams
with fermionic lines in series for Eq.(\ref{DB_action}). Comparing Eqs. (\ref{GreenF}) and  (\ref{Exact}) we obtain
\begin{equation}\label{XPi}
\tilde{\Pi}_{\Omega K}=\chi^{-1}_\Omega X^F_{\Omega K} \chi^{-1}_\Omega-\chi^{-1}_\Omega.
\end{equation}

This way, we need a conserving approximation for $X^F$. Such an approximation is actually known within the framework of DMFT. Because it is a conserving theory in the Baym-Kadanoff sense \cite{Baym}, with the self energy being just the sum of all local diagrams. In such a theory, the conserving approximation for two-particle quantities can be obtained as a Green's function variational derivative with respect to the dispersion law,
\begin{equation}\label{susc4}
<(c_1 c_{1'}^\dag+g_{11'})(c_2 c_{2'}^\dag+g_{22'})>=\frac{\delta G_{11'}}{\delta \varepsilon_{22'}}.
\end{equation}
An important property of the Baym's  criterion of consistency is that the variation should be considered self-consistently, so that the $\Delta$ and $g$ of DMFT vary with $\varepsilon$ to preserve the self-consistency condition (\ref{DMFTcondition}).
One can also explicitly see that such a self-consistent variation preserves the charge conservation. Indeed, the charge conservation follows from gauge invariance under the transformation $c_{\tau, r}\to c_{\tau, r} e^{-i \delta \phi_{\tau,r}},~~c_{\tau, r}^{\dag}\to c_{\tau, r}^{\dag} e^{i \delta \phi_{\tau,r}}$ with an arbitrary dependence of the infinitesimal phase shift $\delta\phi$ on time and space arguments. In an exact theory, such a transformation just results  in similar phase rotations for averages, for example $G_{11'}\to G_{11'} e^{i(\phi'-\phi)}$. A conservative approximation must inherit this property.
DMFT is entirely defined by the self-consistency condition (\ref{DMFTcondition}) and impurity action (\ref{Impurity}). If all the quantities entering the Green's function ${\cal G}_{\omega k}$
(such as $\varepsilon$, $\Delta$, $g$) evolve simultaneously under the gauge transformation,
the self-consistency (\ref{DMFTcondition}) clearly stays fulfilled. The entire Green's function ${\cal G}_{\omega k}$ obeys the gauge phase rotation, which proves that the theory is conserving.

We can prove that the ladder approximation for dual bosons results in such a conserving theory. We introduce a variation of
the local part of the dispersion law
$\delta \varepsilon_{t t' r r}= e^{i (K r-\Omega t)} \delta \varepsilon_{\Omega K} (t-t')$, and corresponding variations of local quantities  $\delta \Delta_{t t' r}= e^{i (K r-\Omega t)} \delta \Delta_{\Omega K} (t-t')$, and $\delta g_{t t' r}= e^{i (K r-\Omega t)} \delta g_{\Omega K} (t-t')$. In the tensorial notation used  previously, the variations $\delta \varepsilon_{\Omega K}, \delta \Delta_{\Omega K}, \delta g_{\Omega K}$ are vectors with a single index  $\omega $ ; their components are  defined as Fourier-transforms: $\delta \varepsilon_{\Omega K}^\omega=\int \delta \varepsilon_{\Omega K}(\tau) e^{i\omega\tau} d\tau$ etc.

The straightforward variation of the self-consistency condition with the DMFT Green's function (\ref{DMFTcondition}) gives:
\begin{equation} \label{suscdraft}
\delta {g}_{\Omega K}={\tt X}^{0}_{\Omega K}
\left(\frac{\delta g_{\Omega K}}{{\tt x}^{0}_{\Omega}} +\delta\Delta_{\Omega K}-\delta\varepsilon_{\Omega K} \right).
\end{equation}
It is worth to remind that ${\tt X}^0_{\Omega K}$ and ${\tt x}^0_\Omega$ are tensors with double indices $(\omega, \omega' )$, as well as the quantities  ${\tt X}^F_{\Omega K}$ and ${\tt x}_\Omega$ to appear in the following formulas.

Now, let us consider the full lattice susceptibility ${\tt X}^F_{\Omega K}$ as a Fourier-transform of Eq.(\ref{susc4}) with pairwise-equal space arguments:
\begin{equation}
{\tt X}^{\omega_1 \omega_2}_{\Omega K}=\sum_{r_1,r_2}\int{\frac{\delta G_{t_1,t_1',r_1,r_1}}{\delta \varepsilon_{t_2,t_2',r_2,r_2}}} e^{i\left(\omega_1 (t_1-t_1')-\omega_2(t_2-t_2') +\Omega (t_1-t_2)-K (r_1-r_2)\right)}
dt_1dt_1'dt_2dt_2'.
\end{equation}
The self-consistency requires $\delta G_{t_1,t_1',r_1,r_1}=\delta g_{t_1,t_1',r_1}$, and therefore the above-introduced variations correspond to
\begin{equation}
{\tt X}^{F}_{\Omega K}=-\frac{\delta {g}_{\Omega,K}}{\delta \varepsilon_{\Omega,K}}.
\end{equation}
Similarly, for the susceptibility of the impurity model
\begin{equation}
{\tt x}_{\Omega}=-\frac{\delta g_{\Omega K}}{\delta \Delta_{\Omega K}}.
\end{equation}
Although the right-hand side formally includes the wavevector $K$, $\tt x$ is of course $K$-independent.

With the above two formulas formulas, it immediately follows from (\ref{suscdraft}) that
\begin{equation}\label{susc}
\frac{1}{{\tt X}^F_{\Omega K}}-\frac{1}{{\tt x}_{\Omega}}=
\frac{1}{{\tt X}_{\Omega K}^{0}}-\frac{1}{{\tt x}_{\Omega}^{0}}.
\end{equation}
Such a self-consistent condition (\ref{susc}) is equivalent to locality of the irreducibly vertex and it is used in the DMFT scheme for calculations of non-local susceptibilities \cite{Georges_RMP}. It is worthwhile to remind that the susceptibilities with a single frequency argument, which we defined previously, can be obtained by summation over the fermionic frequencies, so that
$X_{\Omega K}^F=\sum_{\omega_1, \omega_2} {\tt X}^{\omega_1 \omega_2}_{\Omega K}$ and
$\chi_{\Omega }=\sum_{\omega_1, \omega_2} {\tt x}^{\omega_1 \omega_2}_{\Omega }$

To establish a connection with the dual formalism, we substitute this expression into Eq. (\ref{XPi}) and
express ${\tt X}$ using Eq. (\ref{susc}). Simple algebraic manipulations give
\begin{equation}
\tilde {\Pi}_{\Omega K}=\sum_{\omega_1\omega_2} \left(
\frac{{\tt x}_\Omega}{\chi_\Omega} \frac{1}{{\tt x}_\Omega^{0}}
\frac{ {\tt X}^{0}_{\Omega K}-{\tt x}^{0}_{\Omega}}
{1-\frac{1}{{\tt x}_\Omega^{0}} ({\tt x}_\Omega-{\tt x}^{0}_{\Omega}) \frac{1}{{\tt x}_\Omega^{0}} \left({\tt X}^{0}_{\Omega K}-{\tt x}^{0}_{\Omega}\right)}
\frac{1}{{\tt x}_\Omega^{0}} \frac{{\tt x}_\Omega}{\chi_\Omega}
\right)_{\omega_1 \omega_2}.
\end{equation}
One can recognize that $\frac{{\tt x}_\Omega}{\chi_\Omega} \frac{1}{{\tt x}_\Omega^{0}}$  and $\frac{1}{{\tt x}_\Omega^{0}} ({\tt x}_\Omega-{\tt x}^{0}_{\Omega}) \frac{1}{{\tt x}_\Omega^{0}}$ are, respectively, the
previously introduced quantities $\lambda$ and $\gamma$ with proper indices. Furthermore, it follows from the self-consistency condition (\ref{DMFTcondition}) that
\begin{equation}
{\tt X}^{0}_{\Omega K}-{\tt x}^{0}_{\Omega}=\tilde{{\tt X}}_{\Omega K}^{0}.
\end{equation}
So we conclude that the obtained expression for $\Pi$ is exactly the result of the dual ladder summation (\ref{buble}).

Finally, let us consider the plasmon dispersion given by the equation
\begin{equation}
\chi_\Omega \tilde{\Pi}_{\Omega K} \chi_\Omega+\chi_\Omega=\frac{1}{V_K-\Lambda_\Omega}.
\end{equation}
In the long-wavelength limit, one considers an expansion in $K^2$, in particular $\tilde{\tt X}^{0}_{\Omega K}=\tilde{\tt X}^{(0,0)}_{\Omega}+\tilde{\tt X}^{(0,2)}_{\Omega} K^2+...$. The zeroth-order term in the left-hand side vanishes due to charge conservation. The second-order term results in the dispersion equation for plasmons in a 3D correlated lattice:
 \begin{equation}
\chi_\Omega \left(\lambda_\Omega^{\rm {eff}} \cdot
\tilde{\tt X}^{(0,2)}_{\Omega}  \cdot
\lambda_\Omega^{\rm {eff}}\right) \chi_\Omega =\frac{1}{4\pi e^2}.
\end{equation}
For the 2D case, the right-hand side equals $\frac{K}{2 \pi e^2}$.

While deriving the last equation, we supposed that $\Lambda_\Omega$ stays finite and can be therefore neglected in comparison with $V_{K}$ at small $K$. Note however that this quantity still enters $S^F$ and therefore affects the left-hand side of the dispersion relation, so that the physics of local plasmonic fluctuations is taken into account.

\section{Ladder summation in a strong-coupling limit}

The single-bubble approximation becomes invalid when vertex
$\gamma$ is large. There is a general situation of this kind:
the four-point vertex of an impurity problem with \textit{magnetic
moment formed} appears to have a part displaying a slow dynamics that
give a large contribution at low frequencies. This simply means
that for isolated atoms the magnetic moments are free and, thus,
for {\it almost} isolated atoms (when the energy of interatomic
effective interactions is much smaller than the intraatomic one) they can
be considered as {\it almost} integrals of motion.

To illustrate this statement, let us consider the atomic limit of
the single-band Hubbard model. The single-electron Green's function of the
Hubbard atom at half-filling
\begin{equation}  \label{gatomic}
g_{at}=\frac{-i \omega}{\omega^2+(U/2)^2}
\end{equation}
has a time-scale $\tau_U\propto U^{-1}$, which we call fast.
Contrary, the spin-spin correlator $<s_\tau s_{\tau^{\prime}}>$ is
independent, as one can easily check, of time arguments. So, a
time-scale for $<s_\tau s_{\tau^{\prime}}>$ is the inverse
temperature $\beta$. It means that, apart
from a small time domain $|\tau-\tau^{\prime}|\approx U^{-1}$, the value of $%
<s_\tau s_{\tau^{\prime}}>$ is determined by its non-Gaussian part, which
does not fall as $|\tau-\tau^{\prime}|$ increases. The much slower dynamics of $%
\gamma$ in comparison with $g$ just reflects the fact that a
rotation of the spin does not change the energy of the atom,
whereas the change of the particle number does. Direct calculation
of $\gamma$ for the Hubbard atom does support this observation.
The finite-temperature expressions \cite {Hafermann_Superpert} for
$\gamma_{1234}$ contain a ``singular'' term proportional to $\beta
U^2 \delta_{\Omega0}$. So, the first two time arguments
of $\gamma$ in time-domain must coincide, as well as the last two: $%
\gamma=\gamma_{\tau\tau\tau^{\prime}\tau^{\prime}}$, but the difference $%
\tau-\tau^{\prime}$ can be large.

The properties of the impurity problem are different from those of the isolated
atom. However, for the strong-coupling limit the hybridization is
small. In this case the fast dynamics  roughly stay unchanged, so
that the atomic Green's function (\ref{gatomic}) can be used, and the four-point vertex obeys the form $%
\gamma_{\tau\tau\tau^{\prime}\tau^{\prime}}$. The hybridization can introduce
its own slow timescale $\tau_\Lambda$, so that $\gamma$ does depend only on $%
\tau-\tau^{\prime}$. In the frequency representation, this means
\begin{equation}  \label{gammaSlow}
\gamma=\gamma_{\Omega},
\end{equation}
and at small frequencies $\gamma_\Omega\propto \tau_\Lambda U^2$. Since $%
\gamma$ includes a large factor, we can keep only the irreducible term in the
expressions for $\lambda$ and $\chi$. Note that this is completely opposite
to the case considered in Section \ref{BareDiagSec}.

The terms proportional to $g g$ in Eq. (\ref{Gamma}) are small in $1/U$ and therefore can be neglected.
Then we obtain  the following relation from Eq. (\ref{Gamma}):
\begin{equation}  \label{irreducc}
\chi_\Omega=\chi^{0}_\Omega \gamma_\Omega \chi^{0}_\Omega.
\end{equation}
In the above equation we introduced a ``bare susceptibility'' of the
impurity problem
\begin{equation}\label{chi0}
\chi^{0}_\Omega\equiv -\sum_{\omega \sigma} g_{\omega} g_{\Omega+\omega}.
\end{equation}
Next, we use the relation (\ref{lambdagamma}) and keep only the large factor $\gamma$:
\begin{equation}  \label{irreduc}
\lambda_\Omega=(\chi^{0}_\Omega)^{-1}.
\end{equation}
Note that in the present approximation $\lambda$, like $\gamma$,  only has
single frequency argument.

Let us consider the half-filled Hubbard lattice with the nearest-neighbour
hopping $t$, at large $U$. The low-frequency behaviour of this system is
essentially the  dynamics of the Heisenberg model. The effective-medium
description of spin waves in this case should result in the expression
\begin{equation}  \label{Heizenberg}
\chi_\Omega=\sum_K \frac{1}{\chi^{-1}_\Omega+\Lambda_\Omega-J_K},
\end{equation}
with $J_{ij}=\frac{t^2}{U}$ for nearest neighbors.
Although  this expression is rather simple (in our notation, it is just
$\Pi^{\prime}=J_k$),  it was not reproduced by
any  DMFT-like approximations so far.
The superexchange interaction containing $t^2$ can easily be obtained from a
single loop of the fermionic lines, but the RPA and GW+EDMFT
denominators contain an inverse of this quantity (see
Section \ref{BareDiagSec}), so the structure of the formulas is drastically different.

In our formalism, the desired expression appears from the ladder summation,
that is the calculation for the Eq. (\ref{buble}) and corresponding diagram shown in Fig. (\ref{fig_ladder}). In this
summation, we take into account the ``slow'' vertex (\ref{gammaSlow}) only.
Since $\gamma$ depends only on a single time argument, the calculation is
quite simple. With the exact relation (\ref{Exact2}), we obtain
\begin{equation}
\Pi^{\prime}_{\Omega K}=\left[ \left( \lambda \frac{\tilde{X}^{0}_{\Omega
K}}{1- \gamma_\Omega \tilde{X}^{0}_{\Omega K} }\lambda\right)^{-1}+\chi_%
\Omega\right]^{-1} .
\end{equation}
Finally, we substitute the formulas (\ref{irreducc}), (\ref{irreduc}) for $\chi$ and $\lambda$. This
results in the cancellation of $\gamma$ out of the expressions:
\begin{equation}
\Pi^{\prime}_{\Omega K}= \lambda_\Omega {\tilde{X}^{0}_{\Omega K}} \lambda_\Omega ,
\end{equation}
where $\lambda_\Omega$ from Eq. (\ref{irreduc}) has a meaning of an effective local Stoner
parameter. Both $\chi^{0}_\Omega$ and $\tilde{X}^{0}_\Omega$ obey the
``fast'' dynamics only. To determine the low-energy properties, in which we are
interested, it is enough to calculate it at $\Omega=0$, and to
replace the summation over frequency by an integration. A straightforward calculation of the expression (\ref{chi0}) with
$g=g_{at}$ gives $\chi^{0}_{\Omega=0}=\frac{1}{2 U}$. To calculate
$\tilde{X}^0_{\Omega=0}$, we switch to real-space and use the
strong-coupling limit. It gives the Green's function
$\tilde{\mathcal{G}}= \left(g^{0}\right)^{2} t$ for the nearest neighbors
and $\tilde{\mathcal{G}}=0$ elsewhere. Now, the integration gives
$\tilde{X}_{\Omega=0}=\frac{t^2}{4 U^3}$ for the nearest-neighbors and we
end up with the desired expression (\ref{Heizenberg}). This
derivation can be considered as a generalization of the effective
exchange formula for the magnetically ordered case \cite{Licht_exch,Kats_Licht04}
to the paramagnetic state.

The standard theory of magnons in itinerant-electron systems \cite{Moriya} is based
on the RPA which is supposed to be exact at zero temperature; at finite temperatures,
magnon-magnon and electron-magnon interactions should be taken into account \cite{Moriya,Dzyalos_Kondrat,ourRMP}.
Electron-hole excitations are primary objects in such an approach, spin waves are
coherent superposition of such excitations and, in general, are only well-defined  in
a restricted part of the Brillouin zone, due to Stoner damping. In the case of strong-coupling
Hubbard model, we have an ``anti-RPA'' situation when local magnetic moments are
primary objects. Therefore, completely different approaches should be used, e.g. those based on the
Hubbard X-operators \cite{vons_kats_tref1993b,irir}. It is very non trivial that the dual boson approach
describes  these two opposite cases within the same formalism, thus providing a practical way to interpolate
between weak and strong coupling limits.

\section{Conclusions}

In summary, the general scheme for non-local correlation effects
within the dual boson approach is presented. This method gives a
direct means to obtain the higher-order diagrammatic contributions
beyond the EDMFT scheme free from the double-counting contributions
and it can be useful to describe
the magnon-spectrum renormalization in strongly correlated systems.
The method combines the numerically exact solution of an effective
single-cite impurity model with frequency-dependent
electron-electron interactions as well as an analytical diagrammatic
expansion of non-local fermionic and bosonic self-energies. This
approach is designed to describe of correlated systems with
strong local fermion correlations and fluctuating non-local
bosonic modes on equal footing.

In this paper we focused on the applications to the description of bosonic
degrees of freedom. Additional contributions to the electron self energy also arise,
in comparison with the dual fermion approach \cite{Rubtsov_DF}, e.g., the diagram shown in
Fig. \ref{fig_sig}. Similar diagrams are important in the cases of coexistence of disorder
with electron-electron \cite{altar,altar1} or electron-phonon \cite{anokhin} interactions.
It is important to calculate these additional contributions within
the Hubbard model and make a detailed comparison with dual fermion theory. This
will be the subject of our further studies.

\section*{Acknowledgments}

This work was supported by DFG Grant No. 436 113/938/0-R and RFFI 11-02-01443.
MIK acknowledges a support from the EU-India FP-7 collaboration under MONAMI.
AIL acknowledges a support from SFB 668 and FOR-1346.

redpajama_set_name: RedPajamaArXiv


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Viral picture: Woman says her dog's ear looks like President Trump's face
Updated: 7:04 AM CST Nov 15, 2017
Donald Trump has been discovered in the ear of a beagle.Jade Robinson was taking pictures of her dog's ears to give to the vet because the pup had an ear infection. A friend first noticed the resemblance. "She was like, no, no. There's something in the inner ear," Robinson recalled. "She was like, it's Donald Trump. His face is in his ear."Robinson posted the picture to Facebook and it quickly went viral. Watch the video above for a side by side comparison and decide for yourselves
Donald Trump has been discovered in the ear of a beagle.
Jade Robinson was taking pictures of her dog's ears to give to the vet because the pup had an ear infection. A friend first noticed the resemblance.
"She was like, no, no. There's something in the inner ear," Robinson recalled. "She was like, it's Donald Trump. His face is in his ear."
Robinson posted the picture to Facebook and it quickly went viral.
Watch the video above for a side by side comparison and decide for yourselves

redpajama_set_name: RedPajamaCommonCrawl


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Gregory W. Meeks & Glenn Grothman
Compare the voting records of Gregory W. Meeks and Glenn Grothman in 2017-18.
Gregory W. Meeks
Represented New York's 5th Congressional District. This is his 11th term in the House.
Glenn Grothman
Represented Wisconsin's 6th Congressional District. This is his 2nd term in the House.
Gregory W. Meeks and Glenn Grothman are from different parties and disagreed on 70 percent of votes in the 115th Congress (2017-18).
July 25, 2018 — Restoring Access to Medication Act
Feb. 27, 2018 — To place requirements on operational risk capital requirements for banking organizations established by an appropriate Federal banking agency
Feb. 14, 2018 — Protecting Consumers' Access to Credit Act of 2017
Dec. 20, 2017 — Corporate Governance Reform and Transparency Act
Dec. 1, 2017 — Preserving Access to Manufactured Housing Act
June 8, 2018 — Takano of California Part B Amendment No. 16
Sept. 14, 2017 — Ellison of Minnesota Amendment No. 201
Sept. 7, 2017 — Ben Ray Lujan of New Mexico Amendment No. 31
Sept. 7, 2017 — King of Iowa Part B Amendment No. 80, as modified
Sept. 7, 2017 — Hunter of California Part B Amendment No. 77
On Ordering the Previous Question
March 21, 2017 — Providing for consideration of the bill (H.R. 372) to restore the application of the Federal antitrust laws to the business of health insurance to protect competition and consumers.


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Gulder Ultimate Search is Coming Back!
Popular TV Show Gulder Ultimate Search is set to return to TV screens soon. Although Screening dates are yet to be announced, the gist was hinted on their Social Media page.
Gulder NG says it would be having reruns of the greatest moments of the Gulder Ultimate Search show on TV. Taking advantage of the stay at home order caused by the Covid-19 pandemic, the producers have plans to relive the experiences of the past editions.
Read: Multichoice kicks-off their Reality TV Show; Ultimate Love | Everything you need to know
GUS was a survival type reality TV show that highlights the struggle of contestants between the ages of 10-30. They contested with themselves and with the jungle (nature).
The show first premiered in 2004 and thrived all through to 2012 before going into a long hiatus. In 2016, the Gulder Ultimate Search show hinted on returning after disclosing this to journalists at a 'Meet and Greet' in Lagos.
Nigerian Breweries Corporate Affairs Director, Mrs Sade Morgan, said the show was paused due to the cost of running it but stated that plans are on the way to return it to the delight of Nigerians.
Some Nigerians are of the opinion that the show has greater value both in Content and economy than the Big Brother Naija show.
This is because the Gulder Ultimate Search show was the very first 100% local content reality TV show in Nigeria. It was hosten by Nollywood actor Chidi Mokeme and at another time Bob-Manuel Udokwu.
The last season of the show came in 2012 in Aguleri forest in Anambra state which saw the Contestants search for 'The General's Helmet'. The winner was Chinedu Ubachukwu and he was rewarded with 10million Naira and a brand new Ford Explorer car.
Some of its other winners include Kunle Remi, Dominic Mudabai and Michael Nwachukwu.
Read: Everything You Need To Know About BBNaija season 4
Netflix is making their Documentaries Free on YouTube
Nicole Asinugo is Screenwriter for 'Rattlesnake' remake


Good news! We're going to hold a bunch of random events between now and Christmas so you can have your chance. We have plenty of other goodies too!
We'll announce on Facebook and Twitter at least 30 minutes prior to the start of each event, so keep an eye on there, or just stay logged in to Xsyon 24/7!
I'm looking forward to catching up with you all soon for some holiday fun!
Last edited by GuideMichael; 12-19-2013 at 01:03 PM.
I want one of those candy cane thingies.
Can we get some Christmas events during the Australian time zone? I don't see why the Aussies have to stay up between 2am and 7 am for these events and tire ourselves out for Christmas day.
We are doing our best guys, it's difficult during the holiday season to find time to run them when you have so much going on IRL.
Keep an eye on facebook/twitter and in-game chat!


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Home » OUR PARKS » Parks » Scenic Hudson RiverWalk Park at Tarrytown
It's hard to believe that this beautiful shoreline park was once the site of a riverfront asphalt plant.
riverwalk-tarrytown_760.jpg
Upriver view from RiverWalk Park. Robert Rodriguez, Jr.
Park activities
TARRYTOWN, Westchester County (5.2 acres)—Scenic Hudson partnered with the village, county and developer Natural RE/sources to turn this longtime industrial site into a waterfront park offering magnificent views stretching from the Manhattan skyline to the Tappan Zee Bridge. Features include a 0.6-mile riverside esplanade, a series of grass terraces and lawns, and an "eco-corridor" filled with native plants. A segment of the planned 51-mile Westchester County RiverWalk, the park is adjacent to Tarrytown's Pierson Park.
Scenic Hudson appreciates Friends of the RiverWalk-Tarrytown for helping keep this park shipshape. This energetic corps of community volunteers has contributed mightily to the park and has developed a model for the stewardship of the RiverWalk as the founding chapter of its Friends of the RiverWalk.
The park is open from a half-hour before sunrise to a half-hour after sunset. For inquiries about Scenic Hudson RiverWalk Park at Tarrytown, contact the Village of Tarrytown at 914-631-8347.
Park is just north of Pierson Park on West Main Street in Tarrytown.
More to Explore — Nearby Parks


CALGARY, ALBERTA--(CCNMatthews - Jan. 11, 2007) - Eagle Rock Exploration Ltd. (TSX VENTURE:ERX) ("Eagle Rock") is pleased to announce that it has exceeded its 2006 target production rate of 800 boe/d. Eagle Rock, which commenced operations in December 2005 with production of 40 boe/d, exited 2006 with a production rate of 820 boe/d.
Eagle Rock derives 420 bbls/d of oil production from its properties at Antelope Lake in Southwest Saskatchewan, with the remaining 400 boe/d of production coming from its assets at Coutts/Red Coulee in Southern Alberta. Production from the Coutts/Red Coulee properties is comprised of approximately 80% oil and 20% natural gas.
During the first quarter of 2007, Eagle Rock will be focused on maximizing its fluid through-put in its new oil battery at Red Coulee, which has the capacity to handle up to 10,000 bbl of oil/water emulsion per day.
ERX is an emerging junior oil and gas company engaged in the acquisition, exploration and development of oil and natural gas reserves in Western Canada. The common shares of ERX trade on the TSX Venture Exchange under the symbol "ERX". Further information is available on the ERX website at www.eaglerockexploration.com.
This press release includes forward-looking statements and assumptions respecting ERX's strategies, future operations, expected financial results, financial sources, commodity prices, costs of production and quantum of oil and natural gas reserves and discusses certain issues, risks and uncertainties that can be expected to impact on any of such matters. By their nature, forward-looking statements are subject to numerous risks and uncertainties that can significantly affect future results. Actual future results may differ materially from those assumed or described in such forward-looking statements as a result of the impact of issues, risks and uncertainties whether described herein or not, which ERX may not be able to control. The reader is therefore cautioned not to place undue reliance on such forward-looking statements. The forward-looking statements contained in this press release are made as of the date hereof and ERX undertakes no obligation to update publicly or revise any forward-looking statements, whether as a result of new information, future events or otherwise, except as required by applicable securities laws. The forward-looking statements contained in this press release are expressly qualified by this cautionary statement. In addition, the term BOE may be misleading, particularly if used in isolation. A BOE (barrel of oil equivalent) conversion ratio of 6 Mcf per one (1) BOE is based on an energy equivalency conversion method primarily applicable at the burner tip and does not represent a value equivalency at the wellhead.


Okay, so let's do a wind-up of Round Four of the V8 Supercar Championship for 2007.
One word does this round justice. Yuk!
I've never really enjoyed watching a round from Winton's low-speed, tight-twisty track and when it rains at this Ned Kelly land racetrack the racing becomes more of a sedate Sunday drive in the country rather than a cut & thrust round of motorsport.
I'll be succinct. I didn't enjoy the round and I seriously doubt any of the teams and drivers – other than Jamie Whincup and maybe Garth Tander – did either. Congrats to Whincup, and again my belief that V8 Supercar racing is only for the hungry and ruthless, is borne out. It seems that under the current formula unless you're young, barely blooded and willing to risk all in a bid to make champion, then you're unlikely to perform at your best. Winton seemed to bare me out with a plethora of lesser teams and drivers coming to the fore in very adverse conditions over the weekend, while too many of the elite drivers spent too much time on the infield mud. Mark Skaife alone was noted to have gone off the track more than seven times on Saturday alone.
Consistency still rewards its adherents, as displayed by the round points score following today's mud bath.
Murphy (DNF'd race two), Richards, Johnson (who DNF'd race three with a blown motor), Ingall (who lingered at the tail of the field all weekend), and Owen (who??) all managed to keep things together for at least two of the weekend's three races, sufficient to garner points. This is the name of the game. It's no longer a cut-and-thrust fight to the front and hasn't been for at least three years. Toll/HSV might be top dogs at the moment, but even one or two bad rounds, and they too can fall into neverland. Jamie Whincup hit the nail right on the head when he said after race three, that everyone bleeds. No-one is unbeatable.


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Competition Details
Competition News
During Phase 3, subject matter experts from academia, industry, and government provided expertise in sensors, platform engineering, mission operations and program management, the World Magnetic Model, and geomagnetism to participating teams.
Jacob Bortnik
Professor of Space Physics, Department of Atmospheric and Oceanic Sciences, UCLA
Dr. Jacob Bortnik is a Professor of Space Physics in the Department of Atmospheric and Oceanic Sciences at UCLA, researching the high-energy particles and waves in the Earth's near-Earth space environment. He uses a variety of tools including numerical modeling, data analysis from satellite and ground-based measurements, laboratory plasma devices, and applications of machine learning to predicting the space environment (and more broadly geophysical systems).
Jacob earned his PhD in Electrical Engineering from Stanford University in basic electromagnetic theory and space physics. He regularly acts as a consultant, and has interests in various topics ranging from Earthquake forecasting to quantitative finance.
Arnaud Chulliat
Senior Research Scientist, University of Colorado, Boulder; NOAA NCEI
Dr. Arnaud Chulliat is a Senior Research Scientist at the Cooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado Boulder. He leads the World Magnetic Model team at NOAA's National Centers for Environmental Information (NCEI). Dr. Chulliat's research interests include geomagnetic field modeling, data analysis and observation techniques. Prior to joining CIRES in 2014, Dr. Chulliat was the Director of Magnetic Observatories at Institut de Physique du Globe de Paris, where he led the installation of new observatories in Asia and the Pacific Ocean. He has authored and co-authored over 75 publications.
Associate Professor, Aerospace Engineering, University of Michigan
Dr. James Cutler's research interests center on space systems — a multidisciplinary approach to enabling future space capability with particular emphasis on novel nanosatellite missions. He also has expertise in sensor systems. He is developing next-generation communication capability and robust space computing infrastructure. He is Co-PI on the first NSF space mission, the Radio Aurora Explorer (RAX). Prof. Cutler's teaching interests are in all things space related.
Jesper Gjerloev
Principal Professional Staff, Johns Hopkins University Applied Physics Laboratory
Dr. Jesper Gjerloev is Principal Professional Staff at the Johns Hopkins University Applied Physics Laboratory, author of some 100 scientific papers of which most are focused on some aspect of magnetosphere-ionosphere physics, and has led a list of projects including NASA mission proposals, science centers and steering committees. His doctoral thesis at the Technical University of Denmark focused on auroral electrodynamics from work performed at NASA-Goddard Space Flight Centre. He is also principal investigator for the SuperMAG collaboration, project scientist for the EZIE mission, PI for the Valkyries mission and project scientist ACES sounding rockets.
Mark Moldwin
Professor, University of Michigan
Dr. Mark Moldwin is an Arthur F. Thurnau Professor of Climate and Space Sciences and Engineering and Applied Physics within the University of Michigan's Department of Climate and Space Sciences and Engineering within the College of Engineering and affiliated with the Space Physics Research Laboratory, the Engineering Education Research program, the African Studies Center, the Michigan Institute for Plasma Science and Engineering, and the Robotics Institute. He is the Faculty Director of the M- Engin program within the university's M-STEM Academies and Executive Director of NASA's Michigan Space Grant Consortium. His areas of research include magnetospheric, ionospheric, and heliospheric plasma physics, and pre-college space science education and outreach.
Phase 1 Submission Form
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As part of its mission, NASA's Center of Excellence for Collaborative Innovation (CoECI) provides assistance in the use of crowdsourcing across the federal government. CoECI provides the NASA Tournament Lab (NTL) as a contract platform for running crowdsourced challenges for NASA and other federal agencies. See all government challenges at challenge.gov.
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BullsEye Telecom is a leading managed telecom services provider.
We deliver customized and integrated phone, data and internet services to multi-location companies nationwide.
BullsEye makes telecom management easier by consolidating invoices and points of contact, which results in containing costs, optimizing performance and continually evolving your enterprise level communication systems.
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Tag Archives: Katie Arrington
Ohio Senate, Maine Gubernatorial Races Tight
By Jim Ellis — June 3, 2022
Ohio: First Post Primary Poll Tight — The Ohio primary was May 3, and now we see the first public general election poll testing US Rep. Tim Ryan (D-Warren/Youngstown) and Republican best-selling author J.D. Vance. Suffolk University surveyed the Ohio electorate (May 22-24; 500 likely Ohio general election voters; live interview) and finds Vance jumping out to a slight three-point lead, 42-39 percent. The poll's tight results is not unusual for an Ohio race, which typically are rated as toss-ups until the final two weeks.
In other questions, 42.6 percent of the respondents answered that either the economy or inflation was their most important issue, with abortion registering third at 11.6 percent. Even though he was leading the race, Vance's favorability index was surprisingly upside down at 35:38 percent positive to negative. Rep. Ryan held a 40:23 percent positive ratio. President Biden fell to 39:56 percent. A total of 49 percent said they want to change the direction in which President Biden is leading the nation, while 24 percent said they want to support the President's leadership.
Maine: Gov. Mills Holds Tepid Edge — Maine Gov. Janet Mills (D), who has seen her approval ratings drop from strong heights of late, still tops former Gov. Paul LePage (R) in a combined new statewide survey from Fabrizio Lee & Associates (R) and Impact Research (D) for AARP (May 10-13; 1,050 likely Maine voters with a representative sample of 500 likely voters; live interview & text), but the respondents have a sour outlook regarding the future. While Gov. Mills holds a 51-46 percent edge on the ballot test against ex-Gov. LePage, her lead drops to just one point, 44-43 percent, among those who say they are definitely committed to one of the candidates.
By a whopping margin of 18:82 percent, however, the respondents believe the country is on the wrong track. The state of Maine is also viewed negatively in a 43:56 percent ratio. President Biden's job approval is upside-down at 45:54 percent. Gov. Mills' job approval ratio barely remains in positive territory at 49:47 percent favorable to unfavorable.
MI-3: Rep. Meijer Trails in New Survey — Michigan freshman Rep. Peter Meijer (R-Grand Rapids) fared poorly in redistricting, taking his Grand Rapids-anchored district from a R+9 rating according to the FiveThirtyEight data organization to a D+3 with 50 percent new territory. A new Public Policy Polling survey (May 25-26; 676 registered MI-3 voters; interactive voice response system) shows Rep. Meijer falling behind Democrat Hillary Scholten, his 2020 general election opponent, by a 39-37 percent clip. The change in district lines and the new partisan complexion certainly makes this result believable. The 2022 MI-3 race will be rated a toss-up with no clear favorite.
SC-1: GOP Primary Tightening — A new Trafalgar Group survey (May 26-29; 556 likely SC-1 Republican primary voters; multiple data collection sources) sees the Republican primary challenge of former state representative and 2018 congressional nominee Katie Arrington coming within potential upset range of freshman Rep. Nancy Mace (R-Charleston). The Trafalgar organization sees the race closing to 46-41 percent, which is much different than a Basswood Research poll taken around the same period (May 21-22; 400 likely SC-1 Republican primary voters). The latter poll found the congresswoman taking a commanding 44-24 percent lead. The South Carolina primary is scheduled for June 14.
Posted in 2022, Election Analysis, Governor, House, Polling, Primary, Senate and tagged FiveThirtyEight, Gov. Janet Mills, Gov. Paul LePage, Hillary Scholten, J.D. Vance, Katie Arrington, Maine, MI-3, Michigan, Ohio, President Biden's job approval, Rep. Nancy Mace, Rep. Peter Meijer, Rep. Tim Ryan, SC-1 on June 3, 2022 by Jim Ellis. Leave a comment
House Incumbent Primaries, Part I: Republicans
Feb. 18, 2022 — As the states complete their individual redistricting processes and candidate filing deadlines appear on the political horizon, some incumbents find themselves facing serious primary challenges. Today, we look at Republican nomination situations in states where redistricting is complete, and Monday next week, we'll look at the Democrats.
CA-5: Rep. Tom McClintock
Primary: June 7 (Jungle)
• McClintock Cash-on-Hand (Dec. 31, 2021): $372,569
FiveThirtyEight Statistical Rating: R+17
Dave's Redistricting App Historical Voting: 56.6% R
The California Citizens Redistricting Commission made significant changes to the Golden State congressional map. As a result, veteran northern California Rep. Tom McClintock (R-Elk Grove) had his choice of two districts, one less Republican that contained more of his home area, and the other more strongly favoring the GOP but stretched from the Sacramento suburbs all the way to the Fresno area. McClintock chose the latter.
The congressman's most serious opponent is Fresno County Supervisor Nathan Magsig (R). California imposes a jungle primary system meaning that the top two finishers in the June qualifying election advance to the general election. Considering the Republican trends in this district, it is wholly possible that both Rep. McClintock and Supervisor Magsig will advance into the general election, especially with three Democratic candidates dividing the liberal base.
IL-15: GOP Pairing
Primary: June 28
Rep. Rodney Davis (R-Taylorville)
Rep. Mary Miller (R-Oakland)
• Davis Cash-on-Hand (Dec. 31, 2021): $1,234,171
• Miller Cash-on-Hand (Dec. 31, 2021): $414,795
The Illinois Democratic gerrymander created a new uber-safe Republican 15th District that attracted both Reps. Rodney Davis and Mary Miller. Therefore, the new member here will be chosen in the June 28 Republican primary.
The race is shaping up as a clear GOP establishment versus movement conservative contest. Virtually all of the state and national Republican leaders, including House Minority Leader Kevin McCarthy, have endorsed Davis. All of the movement right-of-center groups such as the Club for Growth and Freedom Works, along with former President Donald Trump, have endorsed Rep. Miller.
Davis also has a major fundraising advantage. Miller, on the other hand, sees 31 percent of her constituents carrying over to the new 15th, versus 28 percent for Davis. Rep. Darin LaHood (R-Peoria) has the largest contingent of current constituents in the new IL-15 (36 percent) but he is running for re-election in the new 16th CD.
MI-4: GOP Pairing
Primary: Aug. 2
Rep. Bill Huizenga (R-Holland)
Rep. Fred Upton (R-St. Joseph)
• Huizenga Cash-on-Hand (Dec. 31, 2021): $1,141,056
• Upton Cash-on-Hand (Dec. 31, 2021): $1,467,055
FiveThirtyEight Statistical Rating: R+9
With Michigan losing a seat in reapportionment, two of the state's western members were destined to be paired. The new 4th District features a potential contest between Reps. Huizenga and Upton, though the latter man has not yet decided whether to seek re-election. A third candidate, state Rep. Steve Carra (R-Kalamazoo), who carries former President Trump's endorsement, is also in the race.
Carra is not likely to be a major factor because he represents very little of the new 4th Congressional District constituency in the state legislature. This race will come down to Rep. Upton's decision whether to seek a 19th term in the House or retire. If he runs, this will be a major summer primary contest. Should he retire, Rep. Huizenga becomes the prohibitive favorite in the primary, with the inside track for the general election, though the new 4th is more competitive than his current 2nd CD.
MS-4: Rep. Steven Palazzo
Primary: June 7 | Runoff: June 28
• Palazzo Cash-on-Hand (Dec. 31, 2021): $ 385,211
A congressional ethics investigation into Rep. Steven Palazzo's (R-Biloxi) use of campaign funds is an obvious negative as he strives to win re-nomination for a seventh term.
The investigation prompted state Sen. Brice Wiggins (R-Ocean Springs) and Jackson County Sheriff Mike Ezell to launch a primary challenge against the congressman, in addition to four others. Local business owner Carl Boyanton has thrown $550,000 of his own money into his campaign, making him a factor, too. This field could grow or retract as the March 1 candidate filing deadline looms on the political horizon.
It remains to be seen whether the investigation hinders Rep. Palazzo to the point of forcing him into a runoff — he has survived other tough primary challenges with larger than expected percentages — but the possibility of going to a secondary vote is certainly real. Should Palazzo be forced into a runoff, his re-nomination could be in serious jeopardy.
Posted in Election Analysis, House, Redistricting, Voting Rights and tagged California Citizens Redistricting Commission, Illinois, Katie Arrington, Michigan, Mississippi, Nathan Magsig, Nebraska, President Trump, Rep. Alex Mooney, Rep. Bill Huizenga, Rep. Dan Newhouse, Rep. Darin LaHood, Rep. David McKinley, Rep. Fred Upton, Rep. Jamie Herrera Beutler, Rep. Jeff Fortenberry, Rep. Liz Cheney, Rep. Mary Miller, Rep. Nancy Mace, Rep. Rodney Davis, Rep. Steven Palazzo, Rep. Tom McClintock, Rep. Tom Rice, South Carolina, Washington, West Virginia, Wyoming on February 18, 2022 by Jim Ellis. Leave a comment
Do Democrats Have a Chance in SC-1?
South Carolina 1st District Republican nominee, state Rep. Katie Arrington (R-Summerville)
June 18, 2018 — A day after US representative and former governor Mark Sanford (R-Charleston) was denied re-nomination, becoming the second incumbent in this election cycle to fall before the national vote even begins (in May, North Carolina Rep. Bob Pittenger, R-Charlotte, was the first to lose) the chairman of the South Carolina Democratic Party was claiming that his political organization is now staring at a conversion opportunity.
"It is a perfect storm and an opportunity for us to pick that seat up," said Trav Robertson, the SC Democratic chairman in an interview last week with the Huffington Post.
Robertson and the local Democrats are claiming that the 1st District is now competitive because Rep. Sanford is out. The 1st stretches from Daufuskie and Hilton Head Islands along the Atlantic coast in the state's southeastern corner and moves up State Route 17 to the Santee Coastal Reserve while taking in the southern tip of Charleston before jutting up Interstate 26 and State Highway 52 to capture the Goose Creek, Summerville, Moncks Corner, and Bonneau communities.
But it seems the chairman may have it backwards. His "perfect storm" might actually have formed if his candidate, attorney Joe Cunningham, were now facing a weakened Rep. Sanford, who might have won re-nomination with a 50.5 – 46.5 percent spread, instead of losing to state Rep. Katie Arrington (R-Summerville) by such a margin.
Posted in Election Analysis, House, Primary and tagged Barack Obama, Elizabeth Colbert-Busch, Jim DeMint, Joe Cunningham, John McCain, Katie Arrington, Mark Sanford, Mitt Romney, Nikki Haley, President Trump, SC-1, South Carolina, Tim Scott, Trav Robertson on June 18, 2018 by Jim Ellis. Leave a comment
Virginia, South Carolina Filings;
Majority of States Done
April 3, 2018 — As March ended, the two most recent filing states of Virginia and South Carolina reached their declaration deadline. We now see a majority of domains (28) posting a final set of political contenders in preparation for the coming primary season.
In Virginia, we again see a familiar pattern, one that has often emerged in the preceding states. That is, a large number of Democrats filing against incumbent Republican House members, the overwhelming majority of whom have not previously run for office.
Against 1st District Rep. Rob Wittman (R-Montross/Fredericksburg), in what should be a safe Republican district, five Democrats filed — including Prince William County School Board chairman Ryan Sawyers — and will be on the primary ballot.
To the southeast, six Democrats, none of whom have ever previously run for public office, are challenging freshman Rep. Scott Taylor (R-Virginia Beach). Democratic leaders are pushing retired Navy Commander Elaine Luria as their preferred nominee. James County former supervisor, Mary Jones, is challenging Rep. Taylor in the Republican primary, but she is not expected to be a major force.
Rep. Bobby Scott (D-Newport News) is the only incumbent in the 11-member Virginia delegation who will be running unopposed both for his party's nomination and in the general election.
Fourth District freshman Rep. Donald McEachin (D-Henrico) draws two minor Republicans and a Libertarian candidate. He will have little trouble securing a second term in a court-mandated district that was reconfigured before the 2016 election.
The 5th District Democrats are taking advantage of Virginia's unique election laws that allow party leaders in each CD to choose whether they nominate via primary or convention. Six Democrats, all first-time candidates, will battle for delegate support to determine which of them advances to the general election to face freshman Rep. Tom Garrett (R-Scottsville/ Charlottesville). The 5th District Democratic convention will meet on May 5.
Both parties in the open 6th District (Rep. Bob Goodlatte retiring) will meet in convention to produce nominees. Republicans are scheduled for May 19, while Democrats have yet to announce a schedule. Interestingly, for the first time, the Republicans are adopting a plurality format instead of voting multiple times to ensure the winner receives majority delegate support. The western Virginia 6th District is the safest Republican seat in the state, so the eventual nominee becomes the prohibitive favorite in November.
Posted in Election Analysis, Governor, House and tagged Abigail Spanberger, Ben Cline, Catherine Templeton, Chaz Haywood, Cynthia Dunbar, Dmitri Cherny, Elaine Luria, Gov. Henry McMaster, Gov. Mark Sanford, Jennifer Wexton, Katie Arrington, Lt. Gov. Kevin Bryant, Rep. Barbara Comstock, Rep. Bob Goodlatte, Rep. Bobby Scott, Rep. Dave Brat, Rep. Don Beyer, Rep. Donald McEachin, Rep. Gerry Connelly, Rep. Jeff Duncan, Rep. Jim Clyburn, Rep. Joe Wilson, Rep. Morgan Griffith, Rep. Ralph Norman, Rep. Rob Wittman, Rep. Scott Taylor, Rep. Tom Garrett, Rep. Tom Rice, Rep. Trey Gowdy, South Carolina, Virginia, Yancey McGill on April 3, 2018 by Jim Ellis. Leave a comment
Sanford's Next Challenge
Sept. 1, 2017 — We all remember former Gov. Mark Sanford's ignominious exit from the South Carolina political scene in 2010, a year after his international extra-marital affair became worldwide news. His political exile did not last long, as he was able to return to the US House in 2013, winning a special election for the Charleston-anchored congressional district after then-Rep. Tim Scott (R-Charleston) was appointed to the Senate after incumbent Jim DeMint (R) resigned from office.
Considering the way in which Sanford left office, which before engaging in the affair was widely regarded as a successful governorship to the point of him being mentioned as a potential 2012 presidential candidate, his quick return to elective politics was surprising. And, his re-election to the district he previously represented from 1995-2001 proved rather extraordinary.
After winning the 2013 special, Sanford did not even draw a Democratic opponent in the 2014 regular election, capturing 93 percent of the vote against only minor party opposition.
Posted in Election Analysis, House and tagged Charleston, Gary Simrill, Jay Lucas, Jenny Horne, Jim DeMint, Jim Ellis, Katie Arrington, Mark Sanford, President Trump, South Carolina, Tim Scott on September 1, 2017 by Jim Ellis. Leave a comment


Sets the instant represented by this object. The instant is represented by a float whose value should be an integer number of milliseconds since the epoch (1 Jan 1970 00:00:00.000 UTC), ignoring leap seconds. All the field values will be recalculated accordingly.
An instant represented by the number of number of milliseconds between such instant and the epoch, ignoring leap seconds.


SPIELBERG, AUSTRIA – Max Verstappen claimed victory at the Red Bull Ring as Lewis Hamilton retired from the race and Sebastian Vettel retook the championship lead.
Over the past few races there has been a quiet resurgence of Max Verstappen. The Dutchman has quietly gone about his business without any glaring errors on track and had managed to secure two podium places. Although this weekend was not the easiest for Red Bull, Verstappen made the most of an okay situation by turning around a probable non-podium finish into race victory at the Red Bull Ring.
Verstappen only managed to put his car fifth on the grid but was promoted to fourth after Sebastian Vettel was awarded a three-place grid penalty for impeding Carlos Sainz during qualifying. The start of the race on Sunday was chaotic to say the least with the Mercedes duo battling alongside Kimi Räikkönen. However, by the end of the first lap Verstappen had made a crucial move to get in front on Räikkönen.
From then on Verstappen drove a controlled race, in which he constantly had to concentrate and expect the unexpected. When Valtteri Bottas' car broke down on lap 14, Red Bull decided to double stack their drivers in a pit-stop – a move which proved to be decisive later on when Lewis Hamilton had to pit for fresh rubber. By the end of the Grand Prix, Verstappen was managing his tyres and trying to stay ahead of Räikkönen and Vettel who were closing in.
This was Red Bull's first victory in the Austrian Grand Prix at the Red Bull Ring. Despite, team-mate Daniel Ricciardo not being able to finish the race because of an engine issue; it ended up being a good haul of points as Mercedes failed to score.
At the end of the first lap everything seemed to be in control for Mercedes, albeit the drivers had switched position on the track but things only went from bad to worse. It had started out as a good weekend with the Mercedes cars showing their raw pace in the speed trap and Hamilton topping the timing screens. Qualifying saw Bottas take pole position by 0.019 ahead of his team-mate and with Vettel's penalty, it seemed as though there would be a swing in the championship but strategy and unreliability reared their heads.
The race initially went wrong when Bottas had a loss of hydraulic pressure on lap 14. As the Finn parked his car a virtual safety car was launched. Red Bull and Ferrari took the opportunity to make an early pit-stop and double stack their cars, while Mercedes kept Hamilton out on track. However, that proved to be a mistake as when Hamilton did have to make his pit-stop, he re-entered the race in fourth place and James Vowles admitted later to the Brit that it was his fault.
Things went from bad to worse when Hamilton's left rear tyre badly blistered and he needed to come in for another pit-stop on lap 53. The final nail in the coffin came just ten laps later when the Brit felt a loss of power, attributed to a loss of fuel pressure, and Hamilton's race was over. Mercedes, who were in such a strong position at the start of the race, had now lost the lead in both the drivers' and constructors' championships.
Sebastian Vettel had a lucky escape this weekend. The four-time World Champion could have been even further behind the Brit in this year's drivers' championship but instead he leaves Spielberg with a one point lead over his closest rival. Seemingly it was the turn of Mercedes for some bad luck.
In qualifying, Vettel could not break the Mercedes lockout and finished in third place over three tenths of a second behind the duo. After the session had finished, the German was called into the stewards for impeding Carlos Sainz whilst he was on his flying lap. Although there was no intention in the move, the stewards decided to give Vettel a three-place grid penalty dropping him down to sixth.
Sunday was a mixed bag for Ferrari. Vettel began the race in sixth but had dropped down to eighth by the end of the first lap. The Ferrari driver had to make his way through some traffic but then came Bottas' retirement and the virtual safety car. Vettel benefitted from fresher rubber and overtook Kevin Magnussen to take fifth place. When Hamilton emerged from the pits in front of the German, it took Vettel over ten laps to pass the Brit but he eventually succeeded.
The podium position was secured, especially with Hamilton retiring, and it looked as though the Ferrari pair may have been able to catch leader Verstappen. However, it was just a few laps too many as tyre management became crucial to secure their hard-earned points.
It has been a difficult season for Romain Grosjean. Before this race the French driver had not scored any points during 2018 while team-mate Magnussen had secured 27 for the team. However, Grosjean finally made amends with a fourth place finish just ahead of Magnussen in fifth. On a difficult day it was a controlled race for the Haas duo who stayed out of trouble and benefitted from the likes of Hamilton, Bottas and Ricciardo all retiring.
This haul of 22 points has surged Haas up the leaderboard in both the drivers' and constructors' championships. Kevin Magnussen now sits in seventh place in the drivers' championships, essentially the head of the rest of the pack while Haas F1 have climbed to fifth in the constructors' ahead of McLaren and behind Renault.
The retirements of Lewis Hamilton, Valtteri Bottas and Daniel Ricciardo meant that there were opportunities for other teams to add some points to their total. Three teams made the most of this with Haas claiming fourth and fifth, Force India's duo managed to finish in sixth and seventh with Sauber locking out the final two points-finishing positions. Fernando Alonso also gained some more points after finishing eighth.
Force India managed to make not pitting under the safety car work. Sergio Perez and Esteban Ocon extended their first stints but unlike Hamilton, they were able to look after their tyres and claim some well-earned points on a weekend that did not look too promising. While, Sauber recorded their first double points finish since the 2015 Chinese Grand Prix. Marcus Ericsson scored his first points since the Bahrain Grand Prix; whereas, Charles Leclerc has now scored points at five of the last six races.
Alonso began the Austrian Grand Prix from the pit-lane but came away with an eighth place finish. Despite McLaren not looking their best this weekend, the Spaniard has continued his run and scored points at every race this season and the 2018 Le Mans winner will hope that he can extend that at Silverstone.
Lewis Hamilton will be hoping to bounce back in the final part of F1's triple header when he heads home to the British Grand Prix this weekend.


text:
Q: Union query on same table multiple time Select c.id
from Customer c
     inner join Orders o on (o.cid=c.id)
     where c.companyid=?
union
     Select c.id
     Customer c
     inner join RecentOrder ro on (ro.cid=c.cid)
     where c.companyid=?

How can be performance of the query improved? My customer table has 50k data for one "companyid" and so order and RecentOrder table has also huge data. How can I improve the query performance?

A: I would suggest using or:
select c.id
from Customer c
where exists (select 1
              from Order o
              where o.cid = c.id
             ) or
      exists (select 1
              from RecentOrder o
              where o.cid = c.id and c.companyid = ?
             );

For the subqueries, you want indexes on Orders(cid) and RecentOrders(cid).

A: My suggestion would be to use union all if you are sure of these queries will return identical records.  It will have a improvement on performance as you have higher number of records. Union all has a performance hit as it is selecting unique records in participants.
Your query has a syntax error which i have corrected in this answer. If ou want to filter records of the union go for a derived table after getting the records in union all query.
      Select
        c.id
      from Customer c
       inner join Order o
            on o.cid=c.id
       where c.companyid=''

       union  all

       Select
        c.id
      from Customer c
        inner join RecentOrder ro
          on ro.cid=c.cid
         where c.companyid=''

A: Add a non-clustered Index to [Customer] table, it will be improved by seeking and reducing the field lookup. However, it can not avoid scanning on [Order] and [RecentOrder]. If you have set the relations between those tables, the sql optimizer may work fine.
    CREATE NONCLUSTERED INDEX [IX_Companyid_Customerid] ON [dbo].[Customer]
    (
        [companyid] ASC
    )
    INCLUDE ([id])
    GO

It is a sample code as mssql.

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