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

Currently, the properties of strongly correlated systems
such as quantum spin systems are much better understood
in their respective ground states, i.e., at zero temperature,
than at finite temperatures. Generically, the ground state
is a pure state and the excitations may or may not have unconventional
statistics. But since they occur only in small numbers at zero
temperature their statistics is not exceedingly difficult to describe.

At finite temperatures, however, the physics becomes richer and more
complicated because there is a finite concentration of excitations.
Then their statistics and their mutual interactions play an important role.
Sharp $\delta$-peaks signalling infinitely long-lived excitations
acquire a broadening induced by thermal fluctuations. Spectral
weight is shifted due to variations of  the matrix elements and
due to variations of the available phase space.

For one-dimensional quantum spin systems Essler and collaborators
analytically investigated the changes of line shapes in the dynamic
structure factor \cite{james08,essle08,james09}.
They considered systems where the elementary
excitations are hardcore bosons which are local, i.e.,
they have minimal spatial extension. The main finding is that the line shapes
broaden upon heating
as expected. The positions of the peaks shift depending upon the available
phase space. Rather unexpectedly, the line shapes at finite temperatures
are generically asymmetric, in agreement with earlier numerical evidence
\cite{mikes06}. Recently, the experimental verification of these
findings has been achieved \cite{tenna10a}.
So the usual expectation of Lorentzian broadening does not hold.
This is due to the interplay of the harcore property and the available
phase space.

In the present article, we aim at the investigation of the complementary
effect induced by vertex corrections. We focus on the changes in the
spectral weight of the elementary modes due to heating. We do not
study the broadening of the peaks corresponding to the elementary modes.

The system we are investigating is a two-leg ladder
with localized $S=1/2$ at each vertex, see
Ref.\ \onlinecite{schmi05b} and references therein. The dominant couplings are
the antiferromagnetic Heisenberg exchange couplings between nearest neighbours,
see Fig.\ \ref{leiterJ}. The nearest neighbour exchange couplings
are of similar magnitude. The ground state has zero spin and the elementary
excitations are of triplet character, i.e., they have $S=1$.
Thus we dub them triplons \cite{schmi03c} in order to distinguish them from the
elementary spin wave excitations, magnons,  of systems with magnetic long-range
order.  The centers of the triplons can be seen to sit on the rungs of the
ladder. Further details about the system and its experimental realization
will be given in the following sections.
\begin{figure}[ht]
  \includegraphics[scale=0.25]{fig1.png}
  \caption{(Color online)
    Sketch of the two-leg spin ladder. Circles indicate spins with $S=1/2$.
    Solid lines represent the two-spin exchange couplings on the rungs
    $J_\perp$ and on the legs $J_\parallel$.
    The four-spin cyclic exchange is denoted by $J_\textrm{cyc}$. }
  \label{leiterJ}
\end{figure}

The temperature development of the spectral weight is induced by
two effects. First, the triplons are hardcore bosons so that
they can only be created on a rung if there is no other triplon
on this rung before. Clearly, this effect depends strongly on temperature
because the relative frequency of a successful excitation depends on
the concentration of thermally activated triplons. The higher the
temperature is the less probable is an excitation process. This effect,
due to the hardcore nature, has been dealt with in the
preceding investigations as well \cite{james08,essle08,james09}.

Second, triplons can be excited \emph{if} there are other triplons
on rungs nearby. We call this process a conditional excitation
henceforth. Thereby, the matrix element for the excitation
of a triplon depends on the concentration of thermally activated
triplons. It turns out that the sign of the conditional excitations
is negative so that \emph{higher} temperatures and concomitantly higher
concentrations of triplons imply a \emph{lower} spectral weight.

The conditional excitations become more and more
important the more the triplons are extended. Hence the
vertex corrections due to conditional excitations are significant
for isotropic spin ladders where the  couplings along rungs and legs
of the ladder are of similar magnitude. In the limits investigated
previously the vertex corrections did not play a significant role
\cite{mikes06,james08,essle08}.

We find that the conditional excitations dominate the temperature
dependence of the spectral weight. Our theoretical results agree
very  well with our experimental ones so that evidence for the
relevance of these vertex corrections is provided.

Technically, our theoretical calculations are based first on
continuous unitary transformations \cite{wegne94,kehre06}
which map the original
model onto an effective model which conserves the number of
excitations \cite{uhrig98c,knett00a,knett03a}. This is done
in a perturbative fashion where the unperturbed Hamiltonian
is given by isolated rungs \cite{knett01b,knett03b}.
Alternative realizations of degenerate perturbation theory
are also based on systematic changes of the underlying basis
\cite{trebs00,zheng01a,oitma06}.

Second, we simplify the operators which describe
excitation processes by a mean-field (MF) treatment.
In this step we focus on a description in terms of
single modes.

The paper is set up as follows. In Sect.\ \ref{M}
we present our model in detail. Subsequently, the theoretical approach is
presented in  Sect.\ \ref{A}. The
theoretical results are shown in Sect.\ \ref{TR} and
compared to experimental results in Sect.\ \ref{CE}.
Finally, the paper is concluded in Sect.\ \ref{Conclusions}.

\section{Model}
\label{M}

We study a two-leg $S= 1/2$ ladder as it is depicted in Fig.\ \ref{leiterJ}.
The model is characterized by  Heisenberg exchange couplings on the rungs
$J_\perp$ and on the legs $J_\parallel$ of the ladder as well as by
 a four-spin cyclic exchange $J_\textrm{cyc}$. The
corresponding Hamiltonian in units of $J_\perp$ reads
\begin{subequations}
\begin{eqnarray}
\frac{H}{J_\perp} &=&H_\perp+\frac{J_\parallel}{J_\perp}H_\parallel+
\frac{J_{\rm cyc}}{J_\perp} H_{\rm cyc}
\\
  &=& \sum_i \vec{S}_{i,l} \cdot \vec{S}_{i,r} + x \sum_{i; \tau}
\vec{S}_{i,\tau} \cdot  \vec{S}_{i+1, \tau} + x_{\rm cyc} H_{\rm cyc},
\qquad
\label{hamil}
\end{eqnarray}
\end{subequations}
where $i$ denotes the rungs and $\tau \in \left\lbrace  1,2 \right\rbrace $
the upper and lower leg of the ladder.
The expansion parameters employed are
\begin{subequations}
\begin{eqnarray}
x &:=& J_\parallel/J_\perp
\\
x_\text{cyc} &:=& J_\text{cyc}/J_\perp.
\end{eqnarray}
\end{subequations}
The four-spin ring exchange is given by
 \begin{eqnarray}
H_\textrm{cyc}&=& \sum_{i}  \left[\left( \vec{S}_{i, l}\cdot \vec{S}_{i+1,l}
\right) \left( \vec{S}_{i, r}\cdot \vec{S}_{i+1,r}\right)\right.
\nonumber\\
&&+ \left( \vec{S}_{i, l}\cdot \vec{S}_{i,r}\right)\left( \vec{S}_{i+1, l}
\cdot \vec{S}_{i+1,r}\right)
\nonumber\\
&&\left.-\left( \vec{S}_{i, l}\cdot \vec{S}_{i+1,r}\right)
\left( \vec{S}_{i+1,l}\cdot \vec{S}_{i,r}\right)
\right].
\end{eqnarray}
These four-spin terms  constitute the most important correction
to the commonly considered nearest neighbor
Heisenberg exchange. By now, this is established for planes
\cite{schmi90b,honda93,mulle02a,reisc04} and ladders
\cite{brehm99,mizun99,matsu00a,matsu00b,schmi05b}. The size of $J_{\rm cyc}$
in cuprate ladders was found to be of the order 0.2-0.25 $J_\perp$
\cite{nunne02,schmi03b,schmi05a,notbo07}.
Further corrections like the two-spin
interaction over the diagonal are almost one order of magnitude smaller
\cite{mizun99} so that they are neglected in the following.

The physics of spin ladders is well understood from the limit of
isolated rungs $\{ x=0, x_{\rm cyc}=0\}$.
In this limit the ground state is the product state
of singlets and the excitations are completely local triplets.
The elementary excitations
at finite values $x$ and $x_{\rm cyc}$ are dressed triplets which are
called triplons \cite{schmi03c}. Triplon excitations are gapped and have
total spin one. In the following we will consider triplets on the
rungs as local triplons.

It is natural to represent the rung states in terms of bond
operators \cite{chubu89a,chubu89b,sachd90}.
Within this representation, the four-dimensional
 Hilbert space of each rung $i$ of the ladder is given by the singlet
\begin{equation}
 s^\dagger_{i}\vert 0 \rangle_i\!=\vert s \rangle_{i} \!=\frac{1}{\sqrt{2}}
\left( \vert\! \uparrow\downarrow\rangle -\vert\! \downarrow\uparrow\rangle
\right)
\end{equation}
 and the three triplet states
\begin{subequations}
\begin{eqnarray}
t^\dagger_{-1,i}\vert 0 \rangle_i&\!=\vert t_{-1}  \rangle_{i}
&\!=\frac{-1}{\sqrt{2}}\left( \vert\! \uparrow\uparrow\rangle -
\vert\!\downarrow\downarrow\rangle\right)
\\
t^\dagger_{0,i}\vert 0 \rangle_i&\!=\vert t_0 \rangle_{i} &\!=
\frac{\mathrm{i}}{\sqrt{2}}\left( \vert\! \uparrow \uparrow \rangle +
\vert\! \downarrow\downarrow\rangle\right)
\\
t^\dagger_{+1,i}\vert 0 \rangle_i&\!=\vert t_{+1} \rangle_i
&\!=\frac{1}{\sqrt{2}}\left( \vert\! \uparrow\downarrow\rangle +
\vert\! \downarrow\uparrow\rangle\right).
\end{eqnarray}
\end{subequations}
The state $\vert 0 \rangle = \prod_i \vert 0 \rangle_i$ represents the
completely empty system. This is actually an unphysical state which
is needed here only for formal reasons.

The physically most important state is the ground state at
$\{x=0,x_{\rm cyc}=0\}$
 which will be used as a reference state $|{\rm ref}\rangle$ for our
calculation. It is given by the product of singlets on all rungs
\begin{equation}
 \vert {\rm ref} \rangle:=\underset{i}{\prod}\vert s\rangle_i.
\end{equation}
One can also interpret $\vert {\rm ref}\rangle$ as a singlet
condensate, i.e. singlet operators can be replaced by unity $s=1=s^\dagger$
Physically, $\vert {\rm ref} \rangle$
is the triplon vacuum, i.e., the state without any excitation.

The operator $t^\dagger_{\alpha,i}$ with
$\alpha \in \left\lbrace0, \pm 1 \right\rbrace $ creates an elementary
excitation with $S^z=\alpha$  (flavor) when applied to a singlet state on rung
$i$. Each rung is either occupied by a singlet or by a triplon, resulting in
the so called \emph{hardcore} constraint on all rungs $i$
\begin{equation}
 \mathbbm{1}=s_i^\dagger s^{\phantom{\dagger}}_i +\sum_\alpha
t^\dagger_{\alpha,i} t^{\phantom{\dagger}}_{\alpha,i} \quad \alpha \in
\left\lbrace -1,0,1\right\rbrace,
\label{bedingung}
\end{equation}
which implies particle-number conservation
in terms of singlets \emph{and} triplons.

The Hamiltonian (\ref{hamil}) can be reformulated in terms of
the above bond operators.  One finds
\begin{equation}
 H = H_\perp + T_0 + T_{-2} + T_2
\end{equation}
where the index $n$ in $T_n$ indicates the change of the total triplon number,
e.g., $T_{+2}\propto t^\dagger t^\dagger\! ss$ are pair creation processes and
$T_{-2}\propto tts^\dagger\! s^\dagger$ are pair annihilation processes. In
contrast, $T_0$ does not change the number of triplons in the system. This
term includes triplon hopping processes $t^\dagger ts^\dagger\! s$ and
two-triplon interactions $t^\dagger t^\dagger t t$. Note that we have omitted
the spatial and flavor index here in order to keep the notation light.
Finally, the operator $H_\perp$ counts the total number of triplons in
the system.

The low-energy spectrum of spin ladders is well studied
\cite{barne93,gopal94,greve96,shelt96,oitma96b,sushk98,eder98,jurec00,trebs00,knett01b,schmi05b}.
All excitations can be classified by the parity
with respect to reflections about the centerline of the spin ladder: an odd
number of triplons has odd parity and  an even number of triplons  has even
parity.

The most relevant low-energy excitation has odd parity corresponding to a
single triplon. It is completely characterized by the
one-triplon dispersion
\begin {equation}
 \omega(k)=\nu_0+2\sum_{m=1}  \nu_m \cos(m k),
\label{dispersion1}
\end {equation}
where $k$ is the momentum along the ladder and the $\nu_m$ are the
one-triplon hopping amplitudes. The dispersion takes a minimum at $k=\pi$
defining the one-triplon gap $\Delta$. In this work we are solely interested
in one-triplon energies. Therefore, we do not discuss channels with more
triplons. But we note that there are interesting interaction effects present
in two-leg ladders leading to two-triplon bound and anti-bound states as well
as pronounced continua in their response functions
\cite{uhrig96b,uhrig96be,shelt96,sushk98,damle98,trebs00,jurec00,knett01b,schmi05b}.

The dynamic structure factor $S^{\alpha,\beta}_T (k,\omega)$ is the quantity
of interest for inelastic neutron scattering experiments. It is given by
\begin{equation}
 S^{\alpha,\beta}_T (k,\omega) = \frac{1}{1-e^{-\frac{\omega(k)}{T}}}
\,{\rm Im} \, \chi^{\alpha,\beta}_T (k,\omega) \quad,
\end{equation}
where
\begin{equation}
 \chi^{\alpha \beta}_T (k,\omega)= i\int_{-\infty}^\infty e^{i \omega t}
\sum_r e^{-ik r} \langle [ S_r^\alpha (t), S_0^\beta(0) ]\rangle \theta (t)
\end{equation}
denotes the retarded dynamic susceptibility and the superscripts
$\alpha,\beta\in\{\pm 1, 0 \}$ correspond to
the components of the spin. Since our model is SU(2) invariant, the only
finite components of the dynamic structure factor are the diagonal ones
($\alpha=\beta$). We define
\begin{equation}
 S_T (k,\omega)=S^{0,0}_T (k,\omega).
\end{equation}

The dynamic structure factor at zero temperature is dominated by the
one-triplon contribution \cite{knett01b,schmi05b}.
Theoretically, the general form of the one-triplon
contribution (or more generically of any single mode (SM) approximation) reads
\begin{equation}
  S_{T=0}^{\rm SM}(k, \omega)=a^2(k) \delta(\omega-\omega(k)),
  \label{SMA}
\end{equation}
where the prefactor $a^2(k)$ defines the one-triplon spectral weight which is
known to be strongly peaked at the gap momentum $k=\pi$
\cite{uhrig04a,schmi05b}.

\section{Approach}
\label{A}

In this work we are interested in the effects of thermal fluctuations on the
one-triplon contribution to the dynamic structure factor of the spin ladder.
Physically, one expects that the triplon acquires a finite life-time upon
heating. Then the $\delta$-functions at zero temperature are replaced by
resonances which are found to show asymmetric line-shapes \cite{essle08}.

Our approach is complementary. We do not describe the decay of
triplons due to scattering with other thermally
excited triplons, but we develop an effective single-mode approximation.
That means that the one-triplon contribution to the dynamic structure factor
is still given by a $\delta$-function, but with temperature-dependent spectral
weight. On the level of the triplon dynamics, this is less sophisticated than
the approach by Essler and collaborators \cite{james08,essle08,james09}. In
contrast, we are able to efficiently incorporate vertex corrections which are
also relevant at finite temperatures. This leads to a
temperature-dependent one-triplon spectral weight in our theory.

In the following we give a very short introduction to the method of
continuous unitary transformations focusing on
 the perturbative realization which is used in this work.
The acronym CUT stands for the continuous unitary transformations in general
and PCUT for their perturbative realization in particular.

We concentrate on the vertex corrections which are relevant for the single-mode
 description of the finite-temperature dynamic structure factor. For a
detailed introduction to CUTs we refer to the literature, see below. In the
sequel, we describe the mean-field decoupling of the vertex corrections in
detail. They give access to the effective temperature-dependent description
of the dynamic structure factor at low energies.

 \subsection{(P)CUTs}
The CUT method was invented independently by Wegner \cite{wegne94} and
G\l{}azek  and Wilson \cite{glaze93,glaze94} in 1994.
The CUT itself is defined through the flow equation
\begin{equation}
 \partial_l H(l)=\left[ \eta (l), H(l)\right]
\label{defcut}
\end{equation}
with the continuous parameter $l \in [0, \infty ]$ and the initial condition
$H(l=0)=H$ where $H$ is the given Hamiltonian. The final form of the effective
Hamiltonian $H_{\text{eff}}=H(l=\infty )$ depends  on the choice of the
infinitesimal generator $\eta$.

Here we use (quasi)particle conserving CUTs
\cite{mielk98,uhrig98c,knett00a,fisch10a}
which result in the effective Hamiltonian $H_{\text{eff}}$ being
block-diagonal in the number of elementary excitations $q$. In the
case of the spin ladder studied, $q$ is the number of excited triplons.
In an eigenbasis  $|q_i\rangle$ ($i$ is an additional index necessary due
to degeneracies) of
the particle-number operator $Q$ with eigenvalues $q\in\mathbb{N}_0$, the
generator $\eta^{\rm pc}$ is given by
\begin{equation}
\label{generator-pc}
 \eta^{\rm pc}_{ij}(l)=\mathrm{sgn}(q_i-q_l)H_{ij}(l).
\end{equation}
For the two-leg ladder we choose $J_\perp Q=H_\perp$ giving an effective
Hamiltonian which conserves the number of triplons, i.e.,
$\left[ H_\perp, H_{\text{eff}}\right]=0 $. Consequently, the effective
Hamiltonian $H_\text{eff}$ decomposes into a sum of \emph{irreducible}
$n$-triplon operators $H_n$
\begin{equation}
 H_\text{eff}=\sum_{n=0}^\infty H_n,
\label{Hn}
\end{equation}
and each $n$-triplon block can be treated separately, see Refs.\
\onlinecite{knett03a,fisch10a} for details.

We apply the CUTs in their perturbative version, i.e., the effective
Hamiltonian for the two-leg ladder is obtained as a high-order series
expansion in the small  parameters
\begin{subequations}
\begin{eqnarray}
x &:=& J_\parallel/J_\perp\\
x_{\rm cyc} &:=& J_{\rm cyc}/J_\perp.
\end{eqnarray}
\end{subequations}
The essential one-triplon dispersion $\omega(k)$
has already been determined earlier  \cite{schmi05b,knett03a}. The
one-triplon hopping amplitudes $\nu_m$ have been calculated up to order 11 in
both perturbative parameters. Various extrapolation schemes
can be applied successfully \cite{schmi03d}. Especially the low-energy part
of the dispersion including the one-triplon gap can be determined very
reliably up to realistic values $x\approx 1.25-1.5$ and
$x_{\rm cyc}\approx 0.2-0.25$.

In order to calculate spectral properties of the system such as the dynamic
structure factor $S_T(k, \omega)$ one has to transform observables $O$ by the
same CUT as the Hamiltonian. Hence the same generator $\eta^{\rm pc}$
given in Eq.\ \eqref{generator-pc} is used
\begin{equation}
  \partial_l O(l)=\left[ \eta^{\rm pc}(l), O(l)\right].
\label{defcut2}
\end{equation}
The relevant observable $O(r)$ with odd parity for the dynamic structure
factor \cite{schmi05b} is given locally on rung $r$ by
\begin{equation}
\label{observ0}
 O(r) =S_{r,1}^{z}-S_{r,2}^{z}=t_{0,r}^\dagger
s_{0,r}^{\phantom{\dagger}} + t_{0,r}^{\phantom{\dagger}}s_{0,r}^\dagger.
\end{equation}
The bare observable  creates or destroys a single triplon with flavor
$0$ on rung $r$. After the CUT, the effective observable $O^{\rm eff}(r)$
comprises many channels reflecting the complicated nature of the interacting
triplon problem. One finds
\begin{equation}
 O^{\rm eff}(r) =U^\dagger O (r)  U=\sum_{n,m}  O_{n,m}^{\text{eff}}(r)
\end{equation}
where $O_{n,m}^{\text{eff}}(r)$ stands for a process
where $m$ triplons are annihilated and $n$ are created.
Thus $n,m \in \mathbb{N}_0$ with the constraint that the difference $n-m$ is
an odd number reflecting the odd parity of the observable. At zero
temperature, no triplons are present so that only
the terms  $O_{n,0}$ with $n$ odd are relevant.

Below we need the most relevant one-triplon contribution $O_{1,0}$ which
is defined by the one-triplon spectral weights
$a^2 (k)$. The corresponding real space amplitudes have been calculated
earlier up to order 10 \cite{schmi05b}. Extrapolations of the low-energy part
close to $k=\pi$ are reliable up to realistic values of $x$ and  $x_{\rm cyc}$.
The contributions $O_{n,0}$ with $n\ge 3$ are significantly smaller
in weight than   $O_{1,0}$ \cite{knett01b,schmi05b}.

At finite temperatures, also contributions $O_{n,m}$ with $m>0$ matter. These
contributions give rise to vertex corrections. We will argue below that the
most relevant vertex correction for the finite temperature physics at low
energies of the spin ladder is the contribution $O_{2,1}$. It comprises
processes where a second triplon is created assuming another triplon is already
 present in the system due to thermal fluctuations. Thus we refer
to these processes as conditional excitation processes.

We determined this contribution up to order 9 in the perturbative
parameters $x$ and $x_\text{cyc}$.
For extrapolation, we used the method of internal variables
\cite{schmi03a}. Unfortunately, no reliable Pad\'e or DlogPad\'e
resummation on top of this extrapolation scheme succeeded, in contrast to
the extrapolation of the
one-triplon dispersion and the one-triplon spectral weights. Certainly, this
fact reflects the more complicated nature of conditional excitations.
Nevertheless, we expect that the extrapolation still yields
reasonable values for the physical processes
up to realistic values of $x$ and $x_{\rm cyc}$.

\subsection{Effective single-mode approximation}
\label{StarPoi}

Next we set up our temperature-dependent theory which  results in an
effective single-mode approximation. A single-mode contribution to the
dynamic structure factor (see Eq.~(\ref{SMA})) depends solely on the
dispersion and on the spectral weight. The general expression reads
\begin{equation}
 S_{T}^{\rm SM}(k, \omega)=\frac{1}{1-e^{-\frac{\omega_T (k)}{T}}} \, a_T^2(k)
\, \delta(\omega-\omega_T(k)),
\label{SMAT}
\end{equation}
where we set the Boltzmann constant $k_\text{B}$ to unity, i.e.,
we measure the temperature in units of energy henceforth.

First, we discuss the temperature effects in the effective
dispersion $\omega_T(k)$. Second, we  incorporate vertex corrections
in the spectral weights $a_T^2(k)$.

\subsubsection{Effective dispersion}

The one-triplon contribution is characterized at zero
temperature by $\omega(k)$. In the PCUT calculation the fully-condensed
singlet state $|{\rm ref}\rangle$ with $\langle s^\dagger s \rangle =1 $ is
used as the reference state. Hence no parameter $s < 1$ for the
singlet occupation needs to be introduced as it was used previously
to take the hardcore constraint into account \cite{sachd90,gopal94}.
The CUT takes the hardcore constraint into account.

At finite temperatures, however, the situation is different. Even after
the CUT there will be a finite occupation of triplons which lowers the
singlet occupation. In analogy to the previous
studies at zero temperature, we take this effect into account
by introducing  the mean-field parameter $s(T)$
defined as the condensate value $\langle s^\dagger \rangle =s$
 with $s\in [0,1 ]$. We stress the difference to previous studies
\cite{sachd90,gopal94} where $s<1$ was used to consider quantum fluctuations
while we use this concept only to consider thermal fluctations.
More complex expressions such as $s_j^\dag s_i^{\phantom\dag}$ are
reduced to $s^2$ which focusses  on the average value
neglecting fluctuations.

Then the $s$-dependent effective one-triplon dispersion reads
\begin {equation}
 \omega_T(k,s)=\nu_0+2s^2 (T) \sum_{m=1}  \nu_m \cos(m k)
\label{dispersion2}
\end {equation}
because each non-local hopping
$t_{\alpha,j}^\dag t_{\alpha,i}^{\phantom\dag}$
with $j\ne i$ implies the transfer of a singlet
$s_{i}^\dag s_{j}^{\phantom\dag}$. The local processes
$t_{\alpha,j}^\dag t_{\alpha,j}^{\phantom\dag}$  do not require
such a renormalization.

To determine  the temperature-dependent mean field parameter
$s(T)$ it is inserted in the hardcore
constraint Eq.~(\ref{bedingung}) yielding
\begin{equation}
 1=s^2(T)+3 \left\langle t^\dagger t \right\rangle(T).
\label{norm}
\end{equation}
So the average triplon occupation
$\left\langle t^\dagger t \right\rangle(T)$ determines $s(T)$.
Due to the spin symmetry the expectation values $\left\langle t_\alpha^\dagger
 t_\alpha^{\phantom{\dagger}} \right\rangle (T)$ with $\alpha \in \left\lbrace
-1, 0, 1  \right\rbrace $ are independent of $\alpha$ resulting in the factor
3 in Eq.\ \eqref{norm}.

The average triplon occupation  can be calculated by the sum over all modes
$\langle t_k^\dagger t_k^{\phantom{\dagger}} \rangle$ in momentum space.
At finite temperatures one has
\begin{equation}
 \left\langle t^\dagger_k t^{\phantom{\dagger}}_k \right\rangle=\frac{1}{Z}
\mathrm{Tr}\left(t^\dagger_k t^{\phantom{\dagger}}_k e^{-\frac{H}{T}} \right)=
\frac{\partial F}{\partial \omega_k}
\label{mittelwert}
\end{equation}
where $k$ is the triplon momentum. The free energy $F$ is defined via the
partition function $Z$ by
\begin{equation}
 F= -T \ln(Z)
\label{freieenergie}
\end{equation}
where $k_\textrm{B}$ is set to unity.
Integration over the whole Brillouin zone yields the wanted local
triplon density
\begin{equation}
 \left\langle t^\dagger t \right\rangle\left( T\right)  =\int_{-\pi}^\pi
\frac{\left\langle t^\dagger_k t^{\phantom{\dagger}}_k \right\rangle}
{2 \pi}dk.
\label{mittel}
\end{equation}

If the triplons were bosons without contraint
$\left\langle t^\dagger_k t^{\phantom{\dagger}}_k \right\rangle$
would be given by the standard bosonic occupation $(\exp(\omega(k)/T)-1)^{-1}$.
Unfortunately, there is no equivalent rigorous expression for hardcore bosons.
As an approximate treatment we follow the arguments of Troyer et al.\
\cite{troye94} to reweight the partition function $Z$ yielding
\begin{subequations}
\begin{eqnarray*}
 Z &=& \left(Z_\omega\right)^N
\\
Z_\omega &:=& 1+\frac{3}{2\pi}\int_{-\pi}^\pi e^{- \frac{ \omega_T(k,s)}{ T}}dk
\end{eqnarray*}
\label{partition}
\end{subequations}
where $N$ is the number of rungs. Consequently, the free energy becomes
\begin{equation}
F
=-N T\ln\left(1+\frac{3}{2\pi}\int_{-\pi}^\pi e^{-\frac{\omega_T(k,s)}{T}}dk
\right).
\label{freegie}
\end{equation}
Combining Eqs.\ (\ref{freegie},\ref{mittel},\ref{norm}) yields
\begin{equation}
 s^2(T)=1-\frac{3}{2 \pi}\frac{1}{Z_{\omega}}\int_{-\pi}^\pi
e^{-\frac{\omega_T(k,s)}{T}}dk.
\label{result1}
\end{equation}
Due to the dependence of the dispersion $\omega_T(k,s)$ on $s$
the above equation defines $s$ only implicitly. It has to be evaluated
 self-consistently.

The value of $s^2$ lies in the interval $[0,1]$.
Clearly $\lim_{T\to 0} e^{-\frac{\omega_T(k,s)}{T}} =0$
as long as $\omega_T(k,s)\ge C >0$ which is the physically
reasonable case. Hence $s^2(T=0)=0$ follows. Similarly, the
limit $T\to\infty$ is given by
$\lim_{T\to \infty}e^{-\frac{\omega_T(k,s)}{T}}=1$
so that $s^2(T=\infty)=1/4$ is implied.

For completeness, we mention another effect
which implies a renormalization of the effective dispersion
due to a finite concentration of thermally excited triplons.
Triplon-triplon interactions of the type
$t^\dagger_1 t^\dagger_2 t^{\phantom\dag}_3t^{\phantom\dag}_4$
are known to exist. For instance, they lead to
the occurrence of bound states \cite{windt01,knett01b}.
They also yield some renormalization of the dispersion upon
mean-field decoupling
\begin{eqnarray}
t^\dagger_1 t^\dagger_2 t^{\phantom\dag}_3t^{\phantom\dag}_4 &\approx&
\langle t^\dagger_1 t^{\phantom\dag}_3\rangle t^\dagger_2 t^{\phantom\dag}_4
+
\langle t^\dagger_2 t^{\phantom\dag}_3\rangle t^\dagger_1 t^{\phantom\dag}_4
\nonumber\\
\label{inter-MF}
&& +
\langle t^\dagger_1 t^{\phantom\dag}_4\rangle t^\dagger_2 t^{\phantom\dag}_3
+
\langle t^\dagger_2 t^{\phantom\dag}_4\rangle t^\dagger_1 t^{\phantom\dag}_3
+\text{const.},
\qquad
\end{eqnarray}
which is proportional to the triplon densities
$\left\langle t^\dagger t \right\rangle(T)$.
The inclusion of the above terms will renormalize
the effective dispersion $\omega_T(k)$ slightly.
We do not consider the effect of the
terms in Eq.\ \eqref{inter-MF} here quantitatively for two reasons.

First, an estimate of the quantitative effect
of these corrections indicates that they are not very significant.
For instance, they are of the same order as the effect of the
inclusion of the factor $s^2(T)$ in Eq.\ \eqref{dispersion2}.
If this term is omitted no qualitative changes of our results
will occur because they are dominated by the renormalization
of matrix elements, not by the renormalization of the energies
as we will show in the following.
Second, their systematic treatment
 is very tedious because a quantitative determination
of all interaction elements would be required.

\subsubsection{Vertex corrections and effective spectral weights}
\label{SpinOp}

\begin{figure}
  \includegraphics[scale=0.4]{fig2a.png}\\
  \includegraphics[scale=0.4]{fig2b.png}
  \caption{Propagation of the excitation  included in the calculation.
    Diagram a) shows the \emph{one triplon excitation}; diagram b)
    shows the \emph{conditional triplon excitation} which requires
    the existence of a triplon.}
  \label{condex}
\end{figure}

The second building block for the one-triplon contribution to the dynamic
structure factor is the one-triplon spectral weight.  Its unconditional part,
active also at zero temperature, is captured by $O_{1,0}^{\text{eff}}$  after
the CUT. Its propagation is illustrated diagrammatically in panel a) of
Fig.\ \ref{condex}.
Contributions $O_{n,m}^{\text{eff}}$ with $m>0$ are
only active at finite temperatures; they represent conditional excitations.
Our idea is to include such contributions
as vertex corrections in an effective manner. We identify
$O_{2,1}^{\text{eff}}$ as most important because it requires
only a single triplon to be present. Hence its quantitative contribution
will be proportional to the triplon density.
In second quantization  $O_{2,1}^{\text{eff}}$  contains operators of the form
$t^\dagger_\alpha t^\dagger_\beta t^{\phantom{\dagger}}_\gamma$.
It is illustrated diagrammatically in panel b) of Fig.\ \ref{condex}.

Any term of $O_{m+1,m}^{\text{eff}}$ with $m>1$ requires the
existence of $m$ triplons so that its quantitative contribution will
be proportional  to the $m$-th power of the triplon density. In the regime
of low temperatures not  larger than about the spin gap each
additional power in the triplon density stands for another additional
exponentially small factor.

Restricting to the two parts  $O_{m+1,m}^{\text{eff}}$
with $m\in\{ 0,1\}$, the effective observable is given by
\begin{eqnarray}
  O^{\rm eff} (r)  &=& U^\dagger O(r) U
  \nonumber\\
  &\approx& O_{1,0}^{\text{eff}}(r) + O_{2,1}^{\text{eff}}(r)
  \nonumber\\
  &= &\sum_p \left( a^{\phantom{\dagger}}_p t^\dagger_{0,r+p}+  \textrm{h.c.}
  \right)
  \nonumber\\
  && \hspace*{-1cm}
  +\sum_{\alpha, \beta, \gamma} \sum_{i<j,p} \left( a_{i,j,p}^{\alpha, \beta,
    \gamma} t^\dagger_{\alpha, r+i} t^\dagger_{\beta,r+j}
  t^{\phantom{\dagger}}_{\gamma,r+p} +  \textrm{h.c.}\right).\quad
  \label{Sl}
\end{eqnarray}
The ordering of operators in the above equation is not unique.
To avoid double counting,  the convention $i < j$ is used.
A few excitation processes appearing in
$O_{2,1}^{\text{eff}}$ in low order in the small parameters are illustrated in
Fig.~\ref{obser}.
\begin{figure}[ht]
  \centering
  \includegraphics[scale=0.45]{fig3.png}
  \caption{(color online) Some conditional excitation processes are sketched
    which are of low order in $x$. The arrows indicate the rung on which
    the observable $O_{2,1}^{\text{eff}}(r)$ acts, whereas the cross
    \textbf{x} denotes the rung which is occupied by the
    one-triplon before the conditional excitation. Black filled circles denote
    the rungs occupied by triplons after the conditional excitation process.
    Open circles are in their singlet state after the excitation; the cyan
    (gray) part is shown for completeness.}
  \label{obser}
\end{figure}

Symmetries of the system imply further constraints.
The conservation of the total $S^z$ component for processes of the type
$t^\dagger_\alpha t^\dagger_\beta t_\gamma$ lead to
\begin{equation}
 0=\alpha+\beta-\gamma \quad \quad \alpha, \beta,\gamma \in \left\lbrace 0,
\pm 1\right\rbrace ,
\label{constraint}
\end{equation}
where it is understood that the bare observable is given by
Eq.\ \eqref{observ0}.

For scattering experiments we are interested in the Fourier transform
of $ O^{\rm eff} (r)$ in Eq.~(\ref{Sl}). For the unconditional
contribution $O_{1,0}^{\text{eff}}$ we find
\begin{equation}
\begin{split}
O_{1,0}^{\text{eff}}(k) &= \frac{1}{\sqrt{N}} \sum_{r}  e^{\mathrm i k r}
O_{1,0}^{z, \text{eff}}(r)
\\
&= \frac{1}{\sqrt{N}} \sum_{r,p} \left( e^{ \mathrm i k r }
a^{\phantom{\dagger}}_p t^\dagger_{0,r+p}+ e^{\mathrm i k r }
a^{\phantom{\dagger}}_p t^{\phantom{\dagger}}_{0,r+p}\right)
\\
&= \frac{1}{\sqrt{N}}\sum_{r,p} a^{\phantom{\dagger}}_p\left(
e^{ \mathrm i k (r-p)} t_{0,r}^\dagger+e^{ \mathrm i k (r-p)}
t^{\phantom{\dagger}}_{0,r} \right)
\\
&=  a(k) \left(  t_{0, -k}^\dagger+  t^{\phantom{\dagger}}_{0, k} \right)
\end{split}
\label{rq}
\end{equation}
where we use
\begin{equation}
a(k) := \sum_pe^{ -\mathrm i k p} a_p.
\end{equation}
Note that the coefficients $a_p$ are real. For the symmetric spin ladder
studied here they are also symmetric $a_p=a_{-p}$.
The operators $t_{0, k}^\dagger$ and $t_{0, k}$ are defined by
\begin{subequations}
\label{operatordef}
\begin{eqnarray}
 t_{0, k}^{\phantom\dagger} &:=& \frac{1}{\sqrt{N}}\sum_r
e^{ \mathrm i k r}t^{\phantom\dagger} _{0, r}
 \\
 t_{0, k}^{\dagger} &:=& \frac{1}{\sqrt{N}}\sum_r
e^{ -\mathrm i k r}t^{\dagger} _{0, r}.
\end{eqnarray}
\end{subequations}

The complete treatment of the conditional excitation
is complicated since it involves several sums
and the resulting terms are not diagonal in momentum space.
We do not, however, aim at the exhaustive description of the multi-particle
response, but at the renormalization of the single-mode
response. Therefore, we use a mean-field decoupling
for $O_{2,1}^{\text{eff}}(k)$ to identify the processes belonging
to the creation of a single triplon.

The mean-field decoupling is
motivated by Wick's theorem \cite{wick50}, but represents an approximation
in the present context
because the triplons are hardcore bosons. For concentrations of the triplons
tending to zero, i.e.,
for $T\to 0$, the hardcore constraint becomes ineffective and
the use of Wick's theorem is justified
\begin{equation}
  t^\dagger_{\alpha,i} t^\dagger_{\beta,j} t^{\phantom{\dagger}}_{\gamma,p}
  \approx t^\dagger_{\alpha, i} \langle t^\dagger_{\beta,j}
  t^{\phantom{\dagger}}_{\gamma,p} \rangle + t^\dagger_{\beta, j}
  \langle t^\dagger_{\alpha,i} t^{\phantom{\dagger}}_{\gamma,p} \rangle.
  \label{decoupling}
\end{equation}
Due to the symmetry constraint (\ref{constraint}) the expectation values
are non-zero only for the cases
\begin{equation}
  \begin{split}
    \langle t^\dagger_{\beta,j} t^{\phantom{\dagger}}_{\gamma,p}
    \rangle \neq 0 &\quad\text{if}\quad \beta=\gamma \quad\text{and}
    \quad \alpha=0
    \\
    \langle t^\dagger_{\alpha,i} t^{\phantom{\dagger}}_{\gamma,p}
    \rangle\neq 0
    &\quad \text{if}\quad\alpha=\gamma \quad \text{and} \quad \beta=0.
  \end{split}
\end{equation}
In all other cases the expectation values are zero.
We define the \emph{hopping expectation values} by
\begin{equation}
  \label{hop-exp-val}
  \tau^{\phantom{\dagger}}_T (j-p) := \langle t^\dagger_{\beta,j}
  t^{\phantom{\dagger}}_{\beta,p} \rangle.
\end{equation}
Due to spin rotation symmetry and translation symmetry
this definition is independent of $\beta$,
but depends only on the relative distance of the two rungs $j$ and $p$  as
depicted in Fig.~\ref{leiter}. Consequently, we perform an index shift
$(j-p) \rightarrow j$ yielding $\tau_T(j)$ which is used in the following.

\begin{figure}[htbp!]
  \centering
  \includegraphics[scale=0.18]{./fig4.png}
  \caption{Illustration of the hopping expectation value
    $\tau_T(j-p)$ as defined in Eq.~\eqref{hop-exp-val}.
  }
  \label{leiter}
\end{figure}

Employing the mean-field decoupling \eqref{decoupling} renders the
operator structure of $O_{2,1}^{\text{eff}}$ the same as for
$O_{1,0}^{\text{eff}}$. Therefore,  Fourier transformation makes it
concise
\begin{eqnarray}
O_{2,1}^{\text{eff}}(k)&=& \frac{1}{\sqrt{N}}\sum_{r, i,j,p} \sum_{\alpha,
\beta, \gamma} a_{i,j,p}^{\alpha, \beta, \gamma}  \left( e^{ \mathrm i k r}
  t^\dagger_{\alpha, r+i} t^\dagger_{\beta,r+j}
t^{\phantom{\dagger}}_{\gamma,r+p} \right.
\nonumber\\
&&\left. +  e^{\mathrm i k r} t^{\phantom{\dagger}}_{\alpha, r+i}
t^{\phantom{\dagger}}_{\beta,r+j} t^\dagger_{\gamma,r+p} \right)
\\
&\approx & \frac{1}{\sqrt{N}} \sum_{r, i,j,p}  \sum_{ \beta}
a_{i,j,p}^{0, \beta, \beta}  \left( e^{ \mathrm i k r}  t^\dagger_{0, r+i}
\tau^{\phantom{\dagger}}_T(j-p)
\delta^{\phantom{\dagger}}_{\beta, \gamma}\right.
\nonumber\\
&& \left.+ e^{\mathrm i k r} t^{\phantom{\dagger}}_{0, r+i}
\tau^{\phantom{\dagger}}_T(j-p) \delta^{\phantom{\dagger}}_{\beta, \gamma}
\right)
\nonumber\\
&& + \frac{1}{\sqrt{N}} \sum_{r, i,j,p}\sum_\alpha
a_{i,j,p}^{\alpha, 0, \alpha} \left( e^{ \mathrm i k r} t^\dagger_{0, r+j}
\tau^{\phantom{\dagger}}_T(i-p) \delta^{\phantom{\dagger}}_{\alpha, \gamma}
\right.
\nonumber\\
&&\left.+ e^{\mathrm i k r}  t^{\phantom{\dagger}}_{0, r+j}
\tau^{\phantom{\dagger}}_T(i-p)
\delta^{\phantom{\dagger}}_{\alpha, \gamma}\right).
\end{eqnarray}
Shifting the indices $r \to r-i$ and $j\to j+p$ in the first sum as well as
$r \to r-j$ and $i\to i+p$ in the second sum leads to
\begin{eqnarray}
&&O_{2,1}^{\text{eff}} (k)\ =
\nonumber\\
&&\quad\frac{1}{\sqrt{N}} \sum_{r, i,j,p}
\sum_\beta a_{i,j+p,p}^{0, \beta, \beta}  \left( e^{- \mathrm i k i}
e^{ \mathrm i k r}t^\dagger_{0, r}
\right.
\nonumber\\
&&\ \left.
+ e^{- \mathrm i k i}  e^{ \mathrm i k r}
t^{\phantom{\dagger}}_{0, r}   \right)\tau^{\phantom{\dagger}}_T(j)
\nonumber\\
&&\quad + \frac{1}{\sqrt{N}} \sum_{r, i,j,p}\sum_\alpha
a_{i+p,j,p}^{\alpha, 0, \alpha} \left(e^{- \mathrm i k j}
e^{ \mathrm i k r} t^\dagger_{0, r}
\right.
\nonumber\\
&&\ \left.
+e^{ -\mathrm i k j} e^{\mathrm i k r}  t_{0, r} \right)
\tau^{\phantom{\dagger}}_T(i)
\\
&&\overset{(\ref{operatordef})}{=} \sum_{i,j,p} \sum_\beta
a_{i,j+p,}^{0, \beta, \beta}  \left(   t^\dagger_{0, -k}  +
t^{\phantom{\dagger}}_{0,k}   \right)e^{- \mathrm i k i}
\tau^{\phantom{\dagger}}_T(j)
\nonumber\\
& &\ + \sum_{ i,j,p}\sum_\alpha a_{i+p,j,p}^{\alpha, 0, \alpha}
\left(  t^\dagger_{0, -k} + t^{\phantom{\dagger}}_{0, k}\right)
e^{ -\mathrm i k j} \tau^{\phantom{\dagger}}_T(i)
\\
&& = \sum_{ i,j,p}\sum_\alpha
\left(a_{j,i+p,}^{0, \alpha, \alpha} +a_{i+p,j,p}^{\alpha, 0, \alpha} \right)
\left(  t^\dagger_{0, -k} + t^{\phantom{\dagger}}_{0, k}\right)
e^{ -\mathrm i k j} \tau^{\phantom{\dagger}}_T(i).
\nonumber
\end{eqnarray}

Finally the full effective observable $O^{\rm eff} (k)$ in momentum
space is  given by
\begin{eqnarray}
 O^{\rm eff} (k) &=& O_{1,0}^{\text{eff}}(k)+O_{2,1}^{\text{eff}}(k)
\nonumber \\
&=&  \left[a(k) + A_T(k)\right]
\left(  t_{0, -k}^\dagger+ t^{\phantom{\dagger}}_{0, k} \right),
\label{spec-weight-renorm}
\end{eqnarray}
where we used the successive definitions
\begin{subequations}
  \label{Adefs}
  \begin{eqnarray}
    A_T(k) &:=& \sum_i A_i(k) \tau_T(i)
    \\
    A_i(k) &:=& \sum_j A_{i,j} e^{-\mathrm i kj}\\
    A_{i,j}  &:=& \sum_{p,\alpha}
    \left(a_{j,i+p,}^{0, \alpha, \alpha} +a_{i+p,j,p}^{\alpha, 0, \alpha}
    \right).
  \end{eqnarray}
\end{subequations}
Clearly, \eqref{spec-weight-renorm} shows that the conditional excitations
imply a renormalization of the spectral weight of the triplons
which are the single-mode excitations for the spin ladder.
The temperature dependence of $ A_T(k)$ stems from the temperature
dependence of the hopping expectation values $\tau_T(j)$. Similar to the
calculation of the triplon occupation in \eqref{mittel}, the hopping
expectation values result from the suitably weighted
 averages over the Brillouin zone
 \begin{equation}
   \label{nj}
   \tau_T(j)  = \frac{1}{2 \pi Z_\omega}\int_{-\pi}^\pi
   e^{-\frac{\omega_T(k)}{T}} \cos (j k) dk.
 \end{equation}

The expressions \eqref{Adefs} for the effective spectral weights hold for
all momenta $k$.
But the main effect in the spectral weight is measured at the lowest excitation
energy $\Delta:=\omega(\pi)$, i.e., at the momentum $k=\pi$.
Hence we restrict ourselves to $k=\pi$ in the following.
For $A_i(\pi)$ we obtain explicitly
\begin{eqnarray}
  A_i(\pi) &=& \sum_{p, j}(-1)^j \left(  a_{j,i+p,p}^{0, 0, 0}+
  a_{i+p,j,p}^{0, 0, 0} \right.
  \nonumber \\
  &&\left. + 2 \left( a_{j,i+p,p}^{0, 1, 1}+a_{i+p,j,p}^{1, 0, 1}
  \right)\right),
  \label{grossa}
\end{eqnarray}
where the factor $2$ comes from the fact that
$a_{j,i+p,p}^{0, 1, 1}=a_{j,i+p,p}^{0, -1, -1}$ and
$ a_{j,i+p,p}^{1, 0, 1}=a_{j,i+p,p}^{-1, 0, -1}$, respectively.

Adding the unconditional and the conditional excitation processes yields  the
total effective temperature-dependent one-triplon spectral weight at $k=\pi$
\begin{equation}
 a^2_T (\pi) = \big[ a(\pi) + A^{\phantom{\dagger}}_T (\pi) \big]^2 s^2(T).
\label{transition}
\end{equation}
The factor $s^2(T)$ stands for the reduced availability of rungs for the
creation of triplons if there are already some thermally excited.
Putting all pieces together we generalize \eqref{SMA} to finite
temperatures
\begin{equation}
  S_{T}^{\rm SM}(k, \omega)=a^2_T(k) \delta(\omega-\omega_T(k)),
  \label{SMA-vertex}
\end{equation}
which we use in the present study for $k=\pi$ only.
This is the dynamic structure factor in
single-mode approximation of the dynamic structure factor including
the vertex corrections relevant at finite temperatures.
The physical content of  Eq.\ \eqref{SMA-vertex} will be exploited in the
next section.

\section{Theoretical Results}
\label{TR}

Next we  present our theoretical results obtained by extending the PCUT
result to finite temperature using the mean-field (MF) approach.
We demonstrate that the
intensity decrease in  the dynamic structure factor mainly arises from the
contribution of the conditional excitations.
 The basic energy unit in this theory section is chosen to be the rung
 coupling constant $J_\perp$ allowing for direct comparison
for different parameter sets $\left\lbrace x, x_{\rm cyc}\right\rbrace $.

In Ref.\ \onlinecite{notbo07} the parameters $x=1.5$ and $x_{\rm cyc}=0.2-0.25$
were determined to describe a generic cuprate spin ladder best.
Thus we chose these parameters to compute the dependence of
the self-consistent MF dispersions
shown in Fig.\ \ref{surf} with temperature $T$ and wave vector $k$.
This  provides information on how  the shape of the effective
dispersion changes upon increasing temperature and momentum.
The dispersion becomes flatter on increasing temperature
implying a larger energy gap  $\Delta(T)=\omega(\pi,T)$.
\begin{figure}[htpb!]
  \centering
  \includegraphics[scale=0.29]{fig5.png}
  \caption{Effective mean-field dispersion  for
    $x_{\rm cyc}=0.2$ and $x=1.5$  in dependence on temperature and momentum.}
  \label{surf}
\end{figure}

\begin{figure}[htpb]
  \centering
  \includegraphics[scale=0.6]{fig6.png}
  \caption{(color online) Excitation gap vs.\ temperature $T$
    for $x=\left\lbrace  0.75, 1, 1.25, 1.5\right\rbrace $ and a
    different $x_{\rm cyc}$ in each panel. The red (gray) dots indicate the
    position of the characteristic temperature $T_{\text{char}}$.}
  \label{gapT}
\end{figure}
Figure \ref{gapT} shows the gap as a function of temperature
for various $x=\left\lbrace  0.75, 1, 1.25, 1.5\right\rbrace$
and a different $x_{\rm cyc}$ in each panel. The blue (gray)
dots in each panel
indicate a characteristic temperature $T_{\text{char}}$ above which the spin
gap shows a significant dependency on temperature in form of a steep gap
function $\Delta(T)$ for all $x$ and $x_{\rm cyc}$.
The value of $T_{\text{char}}$ scales with the spin gap $\Delta(T=0)$;
hence it decreases with increasing $x_{\rm cyc}$.

Three further striking points are to be mentioned. First, with increasing
$x_{\rm cyc}$ the curves for $x=\left\lbrace  0.75, 1, 1.25, 1.5\right\rbrace$
approach one another until they lie almost on top of one another for
$x_{\rm cyc}=0.15-0.2$. For  $x_{\rm cyc}=0.25$ they start to spread again.
Second, with increasing $x_{\rm cyc}$ the sequence of the curves in each panel
changes from $0.75, 1, 1.25, 1.5$ to $1.5, 1.25, 1, 0.75$ from top to bottom.
Third, the spin gap $\Delta(T)$ decreases on increasing $x_{\rm cyc}$.
\begin{figure}[htpb]
  \centering
  \includegraphics[scale=0.55]{fig7.png}
  \caption{(color online) The dependence of the Average singlet state
    occupation number $s^2(T)$ with temperature for various
    $x=\left\lbrace  0.75, 1, 1.25, 1.5\right\rbrace $ with a different
    $x_{\rm cyc}$ in each panel.}
  \label{singlis}
\end{figure}

Furthermore, it is interesting how much the average singlet state occupation
number $s^2(T)$ changes with temperature. Its behaviour is illustrated in
Fig.\ \ref{singlis}.  Below the characteristic temperature
$T_{\text{char}}$ the occupation number $s^2(T)$ is almost independent
of temperature  for all
$x_{\rm cyc}$ resulting from the constant gap energy $\Delta(T)$.
Clearly, $T_{\text{char}}$ scales with the energy gap $\Delta(0)$. Above
$T_{\text{char}}$ the  occupation number $s^2(T)$ falls off the steeper the
smaller $x$ for a given $x_{\rm cyc}$.

\begin{figure}[htb!]
  \centering
  \includegraphics[scale=0.6]{fig8.png}
  \caption{(color online) The upper panel shows the MF triplon hopping
    expectation value  $\tau(j)$ for $j=\left\lbrace 0,\dots,6\right\rbrace $
    with $x=1.5$ and $x_{\rm cyc}=0.2$. The lower panel depicts the associated
    conditional MF spectral weight $A_T(\pi)$.}
  \label{njApisT}
\end{figure}

The generic  behaviour of the MF triplon hopping excpectation
values $\tau(j)$ is depicted in Fig.\ \ref{njApisT} (upper panel) for
$j=\left\lbrace 0,\cdots,6\right\rbrace$ vs.\  temperature
for $x=1.5$ and $x_{\rm cyc}=0.2$. For all other pairs of $x$ and
 $x_{\rm cyc}$ the hopping expectation values look very similar.
The associated conditional MF spectral weight $A_T(\pi)$ is shown in the bottom
 panel of Fig.\ \ref{njApisT}. Thus, the shape of $A_T(\pi)$ is dominated by
$\tau(j)$ with $j\geq1$ explaining the minimum in $A_T(\pi)$ at about
$T=0.514 J_\perp$ which is about twice
the spin gap $\Delta(0)$ \footnote{This value corresponds to about
630K for $J_\perp=105.5$meV in order to provide a first quantitative link
to experiment. This temperature corresponds to about twice the spin
gap found in Ref.\ \onlinecite{notbo07}.}.

\begin{figure}
  \centering
  \includegraphics[scale=0.45]{./fig9.png}
  \caption{(color online) One-triplon spectral weight at $k=\pi$, where
    the spin gap $\Delta(0)$ occurs,
    for  $T=0$ vs.\ $x$ for various $x_{cyx}$.
    The  one-triplon spectral weight at $k=\pi$  increases on increasing $x$
    for fixed  $x_{\text{\rm cyc}}$. For fixed  $x$ the one-triplon
    spectral weight  increases also on increasing $x_{\text{\rm cyc}}$.}
  \label{apix}
\end{figure}

\begin{figure}[htpb]
  \centering
  \includegraphics[scale=0.5]{fig10.png}
  \caption{(color online) Conditional MF spectral weight vs.\  temperature $T$
    for $x=\left\lbrace 0.75, 1, 1.25,1.5\right\rbrace $ and different
    $x_\text{cyc}$ in each panel.  }
  \label{spectis}
\end{figure}

The zero temperature one-triplon spectral weight $a^2(\pi)$ vs.\ $x$ for
$x_{\rm cyc}=\left\lbrace  0,0.05, 0.1, 0.15, 0.2, 025\right\rbrace $
is shown in Fig.\ \ref{apix}. With increasing $x$ the one-triplon spectral
weight  at $k=\pi$ grows. This increasing weight on increasing $x$
confirms the conclusion, that the most important contribution to the one
triplon weight and to the conditional triplon excitation weight is found at
$k=\pi$. This is the dominant feature in the dynamic structure factor.

\begin{figure}[htb!]
  \centering
  \includegraphics[scale=0.54]{fig11.png}
  \caption{(color online) The momentum integrated dynamic structure factor
    $I(T)=\int S(k, T) \text{d}k$ for $x=\left\lbrace 0.75, 1, 1.2,
    1.5\right\rbrace$ with a different
    $x_{\text{\rm cyc}}$ in each panel is presented. It is plotted vs.\
    temperature, normalized to the  momentum
    integrated dynamic structure factor $I_0=\int S(k, 0) \text{d}k$ at zero
    temperature.}
  \label{compareJperp}
\end{figure}

Fig.\ \ref{spectis} depicts the conditional MF triplon spectral weight
$A_T(\pi)$ obtained for various
$x=\left\lbrace 0.75, 1, 1.25,1.5\right\rbrace $ and different values
$x_\text{cyc}$ vs.\  temperature.
The conditional MF triplon weight for various $x$ shows the expected almost
constant behaviour below the characteristic temperature $T_{\text{char}}$
where all curves lie indistinguishably on top of one another.
Above $T_{\text{char}}$ a steep fall-off is found.
The steepness is affected by the values of $x$ and $x_{\rm cyc}$.
The fall-off is the larger the larger $x$ and $x_{\rm cyc}$ are.
For  fixed $x_{\rm cyc}$  the fall-off becomes stronger on increasing $x$.

The calculated conditional MF amplitude $A_T(\pi)$ displays a minimum
at about $J_\perp/2$ which corresponds to about twice the spin
gap $\Delta(0)$. We do not see any physical reason
why the weight should increase again. Furthermore, the
justification for the approximations involved is less sound
beyond temperatures of about $2\Delta(0)$. Life time effects
of the modes are expected to become more and more important
\cite{james08,essle08,james09} so that we do not expect
the calculated data to be reliable for temperatures beyond $J_\perp/2$.

The negative conditional MF amplitude $A_T(\pi)$ diminishes the one-triplon
spectral weight $a_T(k)^2$ according to Eq.\ (\ref{transition}) yielding the
normalized momentum-integrated structure factor $\frac{I(T)}{I_0}$ presented
in Fig.\ \ref{compareJperp}. This is the experimentally relevant
quantity. The shape of the normalized momentum-integrated structure factor
$\frac{I(T)}{I_0}$ in Fig.\ \ref{compareJperp}
is very similar to the one of the conditional MF spectral weight
in Fig.\ \ref{spectis} confirming that the conditional excitation
process is the dominating effect on increasing temperature.
This fact demonstrates the importance of vertex corrections
in strongly correlated systems.

\section{Comparison to Experiment}
\label{CE}

In an inelastic neutron scattering experiment the dynamic structure factor is
 measured by the intensity of the scattered neutrons. In order to compare with
theory a series of measurements
were made on a cuprate spin ladder at different temperatures $T$. These
measurements reveal the temperature development of the structure factor.

So far we discussed the theoretical results relative to
the energy scale given by the  coupling constant $J_\perp$.
In order to compare to experimental data, we have to determine
this energy scale.
For each pair $\left\lbrace x, x_{\rm cyc}\right\rbrace $ the  coupling
constant $J_\perp$ is
determined from the experimental spin gap $\Delta(T=0)=27.6$meV
 \cite{notiz2,notbo07}.
The obtained values of $J_\perp$ are shown in Fig.\ \ref{couplings}
as functions of $x$ (left panel) and of $x_{\rm cyc}$ (right panel).

A striking point is found in the right panel of Fig.\ \ref{couplings}
where for $x_{\rm cyc}=0.15$ the value of $J_\perp$ is nearly
the same for all $x$. The sequence of the curves for various values
of $x$ changes at $x_{\rm cyc}=0.15$, i.e., for lower $x_{\rm cyc}$
the curves rise on rising $x$ while they fall for larger $x_{\rm cyc}$.
Thus the curve  for $x_{\rm cyc}=0.15$ in the left panel of
Fig.\ \ref{couplings} is almost flat.
\begin{figure}[htpb!]
  \centering
  \includegraphics[scale=0.33]{./fig12.png}
  \caption{(color online) Rung coupling constants $J_\perp$
    vs.\ $x$ (left panel) and vs.\ $x_{\rm cyc}$ (right panel)
    determined from the experimental energy gap $\Delta(0)=27.6$meV
   \cite{notiz2,notbo07}.}
  \label{couplings}
\end{figure}

\subsection{Sample and INS Experiments}

Single crystals of La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$ were grown using the
`traveling solvent floating zone' method \cite{revco99} at $9$ bar oxygen
pressure. INS measurements were performed on the undoped ladder,
 La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$
using the MAPS time-of-flight spectrometer at ISIS, Rutherford Appleton
Laboratory, U.K. The sample consisted of three co-aligned single crystals
with a total mass of 23g (see  Ref.\ \onlinecite{notbo07} for details).
The crystals were mounted in a closed cycle cryostat with the $(0kl)$
reciprocal lattice plane horizontal and the $c$ axis perpendicular to the
incident neutron beam $k_i$. A Fermi chopper was used to select an incident
neutron energy of $100$meV and $50$meV and was rotated at a speed of $300$ Hz
to give an energy resolution at the elastic line of
$4.1$meV and $1.6$meV, respectively. Close to the gap,
at an energy transfer of $30$meV the
energy resolution is $3.06$meV and $0.93$meV at an incident
neutron energy of $100$meV and $50$meV, respectively.
Data were collected
at temperatures of $15$K, $50$K, $100$K, and $150$K, and of
$15$K, $50$K, and $150$K  with an incident energy of $100$meV and $50$meV,
respectively. Incoherent nuclear scattering
from a vanadium standard was used to normalize the magnetic cross-section.
In the following text wave vectors $Q_c$ and $Q_a$ represent the direction
along the ladder and along the rung respectively.

\subsection{Background Subtraction and Data Analysis}

The ladder signal was extracted by taking a constant-$Q$ cut at $Q_c=0.5$ and
$Q_a=1.2$ (wave vector ranges $0.45<Q_c<0.55$ and
$0.9<Q_a<1.5$) where $Q_c$ and $Q_a$ are given in units of
$2\pi/c$ and $2\pi/a$, respectively.
The wave vector $Q_c=0.5$ corresponds to the position of
the gap and the $Q_a$ range is chosen to
maximise the ratio of magnetic intensity to background due to the rung
modulation which goes as $\approx (1-\cos(Q_a d_\text{rung}))$, where
$d_\text{rung}$ is the rung distance. The background was determined from a
constant-$Q$ cut for the wave
vector range $-0.2<Q_c<0.2 $, and $0.9<Q_a<1.5 $.
This range is appropriate for the determination of the
background, because it contains no magnetic scattering due to the ladder
structure factor. Each temperature run was treated in exactly the same way,
yielding the data shown  in Fig.\ \ref{data}.

\begin{figure}
  \centering
  \includegraphics[scale=0.33]{./fig13.png}
  \caption{(color online) Background subtracted ladder signals measured at
    various temperatures extracted from INS measurements on
    La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$. The lower
    panel shows a zoom of the top panel. }
  \label{data}
\end{figure}

The highest intensity was measured for the coldest run at $15$K. The
experiment does not reveal any temperature dependence between $15$K and $50$K.
A slight decrease in intensity is observed for $100$K and a significant
decrease for $150$K.
The spin excitation gap is found to be at $E_\Delta=(30 \pm 0.5)$meV,
see lower panel of Fig.\ \ref{data},
which lies a little lower than that found in Refs.\
\onlinecite{eccle96,azuma94} and a little higher than found in Ref.\
\onlinecite{notbo07}. Here we define the gap as the
energy below the intensity maximum where the
scattering strength is half of its maximum value.

The FWHM of the observed peak was extracted
by fitting a gaussian to the data. As the real shape of the measured curve
deviates from the ideal gaussian, the fit is adjusted such that the
gaussian and the data points with an energy below the energy of the
maximum intensity match each another.
The FWHM values are found to be $5.3\pm0.5$meV for
$15$K, $5.1\pm0.5$meV for $50$K, $5.3\pm0.5$meV for $100$K and $(8
\pm0.5)$meV for
$150$K for an incident energy of $100$meV and $3.3 \pm0.5$meV for $15$K,
$3.5\pm0.5$meV for $50$K and $(5.8\pm0.5)$meV for $150$K for an incident
energy of $50$meV. Different FWHM values obtained for data collected at the
same temperature but different incident energies was due to changes in the
energy resolution.

A broadening effect due to
damping could therefore only be observed at $150$K. This result confirms
that the broadening with increasing temperature is small in comparison to the
intensity decrease with increasing temperature, which in turn validates the
MF approach presented here. Strikingly the experimental intensity curves in
Fig.\ \ref{data} lie exactly on top of each other above $40$meV for all
measured temperatures, indicating that there is a temperature independent
tail which we attribute to the wave vector resolution.

The data is compared to the temperature dependent
theoretical spectral weight described
previously. To determine the relative changes we exclusively used the
intensity signal in the range $28 \leq \omega \leq 40$meV and calculated the
integrated intensities normalized by the area obtained from the $15$K run.
Fig.\ \ref{compare} shows the comparison between
theory and experiment for the temperature dependent changes of the normalized
integrated intensities. The temperature scale is restricted to the temperature
region that has been probed experimentally.

Possible pairs of $\left( x,x_{\rm cyc}\right) $ matching the INS data are
$\left\lbrace \left( 0.75,0.1\right), \left( 1,0.15\right),
\left( 1.25, 0.15\right),\left( 1.5,0.2\right)\right\rbrace $. However,
previous INS measurments \cite{notbo07} and optical conductivity measurements
\cite{nunne02,schmi05b} showed, that values of $x\approx1.25-1.5$ and
$x_{\rm cyc}\approx 0.20-0.27$ are realistic.
To be consistent with previous experimental results we argue that the best
agreement within the experimental error bars is found for $x=1.5$ and
$x_{\rm cyc}=0.2$ (black solid curve).

\begin{figure}[htb!]
  \centering
  \includegraphics[scale=0.65]{./fig14.png}
  \caption{(color online)
    Temperature dependence of normalized dynamic structure factor
    $\frac{S_T(\pi)}{S_0(\pi)}$ at $k=\pi$ for various pairs of
    $\left( x, x_{\text{\rm cyc}} \right)$. The energy scale
    is set by $J_\perp$ given in Tab.\ \ref{table}.
    The black and red (gray) dots with error bars depict the
    experimental result
    obtained from INS for $100$meV and $50$meV, respectively. }
  \label{compare}
\end{figure}

\begin{table}[h]
  \centering
  \begin{tabular}{| c | c | }
    \hline
    $\left(x,x_\text{cyc} \right)$  & $J_\perp \!\left[ \text{meV}\right] $
    \\ \hline
    $\left(0.75, 0 \right)$ & $54.2$ \\ \hline
    $\left(1.5, 0 \right)$ & $66.8$ \\ \hline
    $\left(0.75, 0.1 \right)$ & $76.4$ \\ \hline
    $\left(1, 0.15 \right)$ & $98.4$ \\ \hline
    $\left(1.25, 0.15 \right)$ & $99.1$ \\ \hline
    $\left(1.25, 0.2 \right)$ & $118$ \\ \hline
    $\left(1.5, 0.2 \right)$ & $115$ \\ \hline
    $\left(0.75, 0.25 \right)$ & $166.8$ \\ \hline
    $\left(1.5, 0.25 \right)$ & $135$ \\ \hline
  \end{tabular}
  \caption{Values of the coupling constant $J_\perp$
    based on a spin gap of $\Delta =30$meV for the
    various pairs of  $\left(x,x_\text{cyc} \right)$ for which data
    is presented in Fig.\ \ref{compare} }
  \label{table}
\end{table}

In Ref.\ \onlinecite{notbo07}  $x=1.5$ and $x_{\rm cyc}=0.25$ with
$J_\perp=124$meV were fitted
to the one triplon dispersion extracted from experiment. This little
discrepancy in $x_{\rm cyc}$ does not really matter because of the experimental
errors and the approximation necessary for computing  $A_T(\pi)$.

Furthermore, if slightly different energy gaps are assumed,
 such as $\Delta= 35$meV (found in Refs.\ \onlinecite{eccle96,azuma94}),
$\Delta= 30$meV (present work) and $\Delta= 27.6$meV \cite{notiz2,notbo07},
slightly
different results are implied as presented in Fig.\ \ref{comp}.  Both a gap of
$30$meV and of  $27.6$meV
match the data for realistic parameters  $\left\{x,x_\text{cyc}\right\}$. In
fact, using the spin gap $27.6$meV consistent with the data
from Ref.\ \onlinecite{notbo07}
leads to convincing agreement with the values for
$\left\{x,x_\text{cyc}\right\}$ found in Ref.\ \onlinecite{notbo07}
to fit the INS data. Hence our findings in this work are fully
consistent with previous results.

\begin{figure}
  \centering
  \includegraphics[scale=0.65]{./fig15.png}
  \caption{(color online)
    Normalized spectral weight based on different spin gaps
    ($\Delta= 35$meV (cyan (light gray) lines), $\Delta= 30$meV (dark lines),
    and $\Delta= 27.6$meV (blue (dark gray) lines)) is shown. Dashed lines
    rely on $\left(x,x_\text{cyc}\right)= \left( 1.5, 0.25\right)$ and
    solid lines on $\left(x,x_\text{cyc}\right)= \left( 1.5, 0.2\right)$.
    The black and red (dark gray) dots with error bars depict the
    experimental result
    obtained from INS for $100$meV and $50$meV, respectively.}
\label{comp}
\end{figure}

At about half the spin gap energy $\Delta(0)$
our MF theory of vertex corrections predicts a decrease
of the spectral weight by $25\%$ for
$x=1.5$ and $x_{\rm cyc}=0.2$ with $J_\perp=115$meV
which  agrees very nicely with experiment, see Fig.\ \ref{comp}.
 We stress that a decrease of $25\%$ is well in the regime where
we consider our approach valid and reliable.

Note, however, that we omitted the line broadening
due to thermal fluctuations in the present analysis
which play a more and more important role on increasing
temperature. In particular, the unique experimental determination
of the weight becomes increasingly difficult.

\section{Conclusions}

\label{Conclusions}

In summary, we derived in this paper vertex corrections
in the dynamic structure factor for a quantum antiferromagnet
without long range order. The model system investigated is the two-leg
spin ladder. The vertex corrections are linked to thermal fluctuations;
they are relevant only at finite temperatures. We computed them
in the framework of continuous unitary transformations which lead
to an effective description of the system in terms of hardcore
triplons as elementary excitations.

The vertex corrections are evaluated on the mean-field level,
i.e., on the level of a single-model approximation. We found that
they induce conditional excitation process which \emph{reduce} the
unconditional, zero-temperature excitation amplitude.
Thereby, the spectral weight of the low-lying excitations
is diminished because its temperature dependence
is indeed found to be dominated by the vertex corrections.
The vertex corrections stem eventually from the hardcore
character of the triplons. Yet we emphasize that they go beyond the
obvious mechanism that no triplon can be excited if its
site is already excited by another triplon.

We compared the calculated spectral weight quantitatively with the
one measured in the undoped ladder La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$
using inelastic neutron scattering for a range of temperatures
up to $150$K, equivalent to  half the spin gap energy.
The theory should still apply to considerably higher  temperatures
as long as these stay below twice the spin gap energy.
Experiments at higher temperatures for undoped ladders could
test this framework further.

Further research, both theoretical and experimental,
 on the vertex corrections in other
 low-dimensional quantum antiferromagnets is called for.
In particular, a challenging project consists in the
combination of approaches studying the combined effect of line broadening
of the single modes \cite{james08,essle08,james09}
and of the vertex corrections.

\begin{acknowledgments}
We thank T.G.\ Perring, M.\ Reehuis and C.\ Gibson for experimental support.
K.P.S.\ acknowledges ESF and EuroHorcs for funding through his EURYI. I.E.\
acknowledges the financial support by the NRW Forschungsschule
\emph{Forschung mit Synchrotonstrahlung in den Nano- und Biowissenschaften}.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
from __future__ import absolute_import
from __future__ import print_function
import argparse

import six
import sys

from aetros import api
from aetros.utils import read_home_config, KeyNotConfiguredException

class IdCommand:
    def __init__(self, logger):
        self.logger = logger
        self.client = None
        self.registered = False
        self.active = True

    def main(self, args):
        import aetros.const

        parser = argparse.ArgumentParser(formatter_class=argparse.RawTextHelpFormatter,
                                         prog=aetros.const.__prog__ + ' id')

        parsed_args = parser.parse_args(args)
        config = read_home_config()

        try:
            user = api.user()
        except KeyNotConfiguredException as e:
            self.logger.error(str(e))
            sys.exit(1)

        print("Logged in as %s (%s) on %s" % (user['username'], user['name'], config['host']))

        if len(user['accounts']) > 0:
            for orga in six.itervalues(user['accounts']):
                print("  %s of organisation %s (%s)." % ("Owner" if orga['memberType'] == 1 else "Member", orga['username'], orga['name']))
        else:
            print("  Without membership to an organisation.")

redpajama_set_name: RedPajamaGithub


text:
LIVE THE APOLLO MOON MISSIONS
is an asymmetrical co-op game about the NASA moon missions.
In it, one astronaut player and up to four mission command players work together to plan missions and experience the historic highlights of the Apollo space program.
*images taken from Apollo's 2019-2020 game demo, made in partnership between USC Games and the Berklee College of Music.
ON THE GROUND, AND IN THE SKY
Accuracy and attention to detail bring the equipment that NASA used over 50 years ago back to life.
Use it to work together, solve historic problems, and retrace humanity's steps to the moon.
DR. KENNETH PHILLIPS JOINS AS ADVISOR
Ken Phillips, PhD, is the curator for aerospace science at the California Science Center and professor of Physics and Astronomy at USC. His expertise in the science and history of the U.S. space program will be invaluable to the Apollo team.
APOLLO NOMINATED AT GAMES FOR CHANGE
Apollo's demo was selected as one of four finalists for Best Student Game at the 2020 Games for Change Festival, the largest industry-facing games event in NYC.
APOLLO DEVELOPMENT BEGINS!
We take our original demo and officially decide to go all the way; not because it's easy, but because it's hard.
Sign up below to follow our progress as we develop Apollo!
OFFICIAL SELECTION AT USC GAMES EXPO
Apollo's demo joins undergraduate and graduate student games to be showcased at the annual USC Games Expo.
This year's expo is held entirely online at uscgamesexpo.com on May 12th.
TRACY FULLERTON JOINS AS ADVISOR
Director Emeritus of the USC Games program and director of the Award-winning NEH grant recipient Walden, a Game joins Apollo as a project advisor! Tracy's experience in highly engaging, historically-inspired gaming will be a huge advantage to the team.

redpajama_set_name: RedPajamaCommonCrawl


Great little town with flair right on the Tri Glove National Park 6 km from Austria and 20 km from Italy. Plenty of accommodations and centrally located. Despite the sunshine you can feel the coolness of the mountains on the beautiful river Pišnica. Here we have stopped off and recovered from the 8-day hike a bit. In the distance you can still see the impressive mountains.
This also seems to be a cycling paradise, ranging from rentable wheels and e-bikes to the excellent signposted bike lanes and tours with km specifications and recommendations for bike types.
Nice place with some restaurants and cafes.
Nice place to stay. Restaurants and cafes are sufficiently available.


text:
Q: Set width and height on BlackBerry BitmapField How do I set width and height of a BitmapField in BlackBerry? The following code works, but I can't show the entire image. The display shows only some portion of my image.
 BitmapField bmp = new BitmapField(connectServerForImage(ImageUrl)) {
       protected void layout(int width, int height)
       {
          setExtent(80, 70);
       }
 };

A: I assume you're saying that your BitmapField is showing the image you retrieve cropped?
And you would like it to scale, perhaps scaled to fit the size of your BitmapField?  There's many ways to do that, but one is to resize the image after downloading, before giving it to the BitmapField:
From this answer on resizing a Bitmap
public static Bitmap resizeImage(Bitmap originalImage, int newWidth, int newHeight) {
    Bitmap newImage = new Bitmap(newWidth, newHeight);
    originalImage.scaleInto(newImage, Bitmap.FILTER_BILINEAR, Bitmap.SCALE_TO_FILL);
    return newImage;
}

(Note that the link I provided has an alternate method for resizing images, if you have to support BlackBerry OS < 5.0).
Then, use
Bitmap img = resizeImage(connectServerForImage(ImageUrl), 80, 70);
BitmapField bmp = new BitmapField(img) {
       protected void layout(int width, int height)
       {
          setExtent(80, 70);
       }
};

However, I believe that if you actually set the size of the Bitmap img to 80x70, then there's no need to also set the extent of the BitmapField to 80x70.  It should default to the size of the image you give it.  So, you could simplify the code to:
BitmapField bmp = new BitmapField(img);

A:  int prefWidth=(30*bitmap.getWidth())/bitmap.getHeight();
 Bitmap temp=resizeBitmap(bitmap,prefWidth,30);

where
 private Bitmap resizeBitmap(Bitmap image, int width, int height)
{

    int rgb[] = new int[image.getWidth()*image.getHeight()];
    image.getARGB(rgb, 0, image.getWidth(), 0, 0, image.getWidth(), image.getHeight());
    int rgb2[] = rescaleArray(rgb, image.getWidth(), image.getHeight(), width, height);
    Bitmap temp2 = new Bitmap(width, height);
    temp2.setARGB(rgb2, 0, width, 0, 0, width, height);
    return temp2;
}

This support bb os 4.5

redpajama_set_name: RedPajamaStackExchange


LCD Concepts, formerly known as P Sea Concepts, has been designing and manufacturing custom LCD display solutions since 1998. Dedication to exceptional customer service, skillful use of leading edge technology and a commitment to delivering superior products on time have created a reputation of excellence for LCD Concepts.
LCD Concepts is a family owned and operated business. Nancy Olson manages operations, marketing and the sales team for the company. Brian Olson specializes in product design and development. From 1981 to 2007 the company evolved from custom designed electromechanical packaging of printed circuit boards and mechanical enclosures to sealed, high bright, rugged displays and Ethernet products. In 2007, Nancy and Brian saw the company was growing in a new direction. Products were being designed and sold to service the Off-shore Seismic, Oilfield Exploration and Production, Entertainment, Yachting, and Shipping industries. This continued expansion into various market segments led Nancy and Brian to the decision to re-tool and create a relevant, fresh image for the company.
In 2011, Nancy restructured operations and developed a comprehensive and targeted marketing strategy. This included a complete re-branding of the organization. She implemented a new logo and website making sure all marketing and sales materials was consistent with the new brand. She developed a targeted sales strategy to become a quality supplier for industrial and commercial industries. Brian continued developing custom products and solutions to a more diverse market. Recognizing the growing need of custom solutions in the workplace, LCD Concepts has expanded their product offering. In addition to custom Panel PC's and LCD displays, LCD Concepts now offers large format interactive, all-in-one multi-touch computer with whiteboard – ideal for the classroom, boardroom or conference room.
LCD Concepts' business philosophy is simple: we deliver the highest quality custom work because our people are the very best at what they do. We consistently deliver superior results because our staff is comprised of only the most qualified and experienced technicians, and every product that leaves our facility is assembled by their hands.


Staying healthy is very easy with the help of practicing a meal prep routine.
Relief of Stress, and much more!!
Grab your copy today to learn all the techniques on how to "Meal Prep" , while enjoying all the Delicious Recipes at the same time.


text:
A Charadrius a madarak osztályának lilealakúak (Charadriiformes)  rendjébe a lilefélék (Charadriidae) családjába tartozó nem.

Rendszerezésük
A nemet Carl von Linné svéd természettudós írta le 1758-ban, az alábbi fajok tartoznak ide:

új-zélandi lile (Charadrius obscurus)
parti lile (Charadrius hiaticula)
kanadai lile (Charadrius semipalmatus)
rövidcsőrű lile (Charadrius melodus)
hosszúcsőrű lile  (Charadrius placidus)
kis lile (Charadrius dubius)
Wilson-lile  (Charadrius wilsonia)
ékfarkú lile vagy hosszúfarkú lile (Charadrius vociferus)
madagaszkári lile  (Charadrius thoracicus)
apró lile (Charadrius pecuarius)
Szent Ilona-szigeti lile (Charadrius sanctaehelenae)
szalagos lile (Charadrius tricollaris)
kongói lile (Charadrius forbesi)
széki lile (Charadrius alexandrinus)
fehérarcú lile (Charadrius dealbatus)
Charadrius nivosus
jávai lile (Charadrius javanicus)
fehérhomlokú lile (Charadrius marginatus)
vörösfejű lile (Charadrius ruficapillus)
pikkelyes lile (Charadrius peronii)
fokföldi lile (Charadrius pallidus)
galléros lile (Charadrius collaris)
andesi lile (Charadrius alticola)
maori lile (Charadrius bicinctus)
falklandi lile (Charadrius falklandicus)
tibeti lile (Charadrius mongolus)
sivatagi lile (Charadrius leschenaultii)
sztyeppi lile (Charadrius asiaticus)
barnaszárnyú lile (Charadrius veredus)
tűzföldi lile (Charadrius modestus)
hegyi lile (Charadrius montanus)
havasi lile (Charadrius morinellus vagy Eudromias morinellus)

Egy 2020 decemberében keletkezett tanulmány szerint a széki lile (Charadrius alexandrinus) egy újabb alfaja érdemelne faji szintű besorolást, a Charadrius alexandrinus seebohmi. Amennyiben ezt széles körben elfogadják, úgy az új faj a Charadrius seebohmi tudományos nevet kapja. A széki liléből lett leválasztva az amerikai alfaj Charadrius nivosus 2011-ben és a fehérarcú lile (Charadrius dealbatus) is.

Jegyzetek

Források
ITIS szerinti rendszerbesorolása

Madárnemek

redpajama_set_name: RedPajamaWikipedia


The Hypnotherapy Practitioner Diploma integrates Hypnotherapy, Counselling, Coaching, NLP and Thought Field Therapy in one course and accredited by The General Hypnotherapy Standards Council. A springboard to a new career or as an addition to your existing therapy skill set.
This course is run as one weekend a month over 10 consecutive months. This training is about quality not quantity so the trainers only come together to run this course once a year.


– Seems like she saying got no BF, but really .. she didn't deny not having one.
So guys!! Watch out for these kinda lies!!
At the end of the day you wouldn't wanna be with someone who can't really say that she's dating with someone okae!


text:
Community reacts to devastating loss of popular leisure centre
Community reacts to devastating loss of Tetbury leisure centre
By Sam Krayem-Wood
Residents, councillors and MP's have expressed their disappointment after Tetbury's only leisure centre suddenly closed earlier this week.
Gym-goers at Sir William Romney's Leisure Centre were stunned to hear of its closure when they arrived for their morning work on Monday.
Tetbury resident Danny Ruddick said: "I arrived at 8am to attend my usual hour or so in the gym and it was closed.
"I was informed by a member of staff that it has been closed to members and the public in general.
"There have been no prior warnings issued concerning the impending closure and the member of staff concerned was also only informed this morning.
"So we now have one of the largest towns in the Cotswolds with no funded provision to aid the health and well being of its residents.
Owners Sir William Romney's Leisure Limited have put the closure down to 'the cessation of grant funding.'
In a statement on their website titled 'closing statement' they said: "After nearly six years of providing leisure facilities for our local community, the Directors of Sir William Romney's Leisure Ltd regretfully announce that the leisure centre will cease to trade from Monday, July 8.
"Following the cessation of grant funding, we have had to take the regretful decision to close all of our facilities, classes and holiday provision.
"We will be contacting all our members and creditors without delay.
"Despite the decision to close SWR Leisure, Sir William Romney's School is exploring opportunities to ensure that our sports centre facilities remain available for our local community, particularly in the form of lettings of the sports hall for local sports groups.
"We would like to thank all of our members and regular users for their support and understanding through this difficult time.
"If you have any queries please contact Mr Jim Goudie, Butcher Woods Ltd, 79 Caroline Street, Birmingham B3 1UP, telephone 0121 236 6001."
The full reaction
Sir Geoffrey Clifton Brown MP said: "The announcement of the closure of these leisure centres is absolutely unacceptable.
"Especially, at a time when sport and leisure are becoming ever more important amongst all ages in our society.
"It's a real tragedy our market towns continue to grow in size but continue to lose local facilities and infrastructure."
"I am in touch with local constituents who are relying on these leisure centres to supply child supervision during the six-week summer holiday.
"The decision to close these leisure centres is having a real impact on the day to day lives of residents and we must do all we can to keep them open.
"I have written today to Joe Harris, the new Liberal Democrat Leader of the Cotswold District Council, asking him to provide a funding package for the two leisure centres to remain open for a year.
"This is so that proper and wide-ranging discussion about their future can take place amongst the people of both towns, rather than residents being confronted with a sudden decision to close them which will have come as a great shock to many."
Cotswold District Council leader Cllr Joe Harris added: "I'm disappointed to hear that the Leisure Centre at Sir William Romney School in Tetbury has closed today and also that the facilities at Farmors school in Fairford are set to close soon.
"My administration are committed to ensuring that all communities have access to a decent health and fitness offer and as part of our comprehensive plan for the Cotswolds we will work with both communities to help realise this even with the gyms closing.
"We need to be proactive in finding a solution."
Conservatives leader on Cotswold District Council Cllr Richard Morgan said: "People have booked children's birthday parties and after school clubs and they have no idea what's happening.
"To suddenly shut down and not give people the opportunity to make other plans is dreadful".
Cotswold District Council Cllr for Tetbury Stephen Hirst said: "This is real heartbreak for local residents.
"Sir William Romney school has decided they need to concentrate on educating children instead of running a leisure centre and we have to understand their position.
"We need to recognise that without ongoing grant support these businesses cannot support themselves.
"I am personally working particularly hard to try and move all the summer activities to St Marys school in Tetbury so working parents are unaffected during the summer holidays.
"I am also working with the school to try and get the sports hall open for local sports groups from September onwards.
"These are not ideal solutions but we have to do something quickly to minimise the damage."
A spokesman for Tetbury Town Council said: "Tetbury Town Council has not met formally to discuss the closure of the leisure centre and the impact of SWR Leisure going into administration, but our members are well aware of local families' concern that they may have lost childcare through the long summer holidays if the Summer Holiday Club is not to take place.
"The town council does not have any role in the funding or running of the leisure centre, nor SWR Leisure as a business.
"However, we are aware of efforts to make alternative arrangements and of course the town council would like those efforts to succeed.
"An extraordinary meeting of the council has been convened for the earliest possible date after hearing the news.
"If required, the town council will consider proposals for how it might help.
"The extraordinary meeting will be held at 6.30pm on Monday, July 15, but may be cancelled at the last moment if our friends and partners are successful with their arrangements."


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Anjem Choudary Associate Michael Coe Faces Jail For Knocking 16-Year-Old Boy Unconscious
He was one of the people Choudary was banned from contacting while on bail.
Jack Sommers
Assistant News Editor, HuffPost UK
A Muslim convert and associate of convicted hate preacher Anjem Choudary is facing a substantial jail term after knocking a schoolboy unconscious for cuddling his girlfriend.
Michael Coe, 35, was driving through east London when he spotted the two 16-year-olds hugging on the pavement.
He pulled over to confront the pair, demanding to know if they were Muslims, before calling the girl a "whore", The Press Association reports.
He then grabbed the boy by the throat and threw him to the ground before kicking his head as he lay on the floor, leaving him unconscious and bleeding from two head injuries.
Michael Coe
Metropolitan Police/PA Wire
When passing schoolteacher Boutho Siwela tried to come to the teenager's aid, he then attacked him.
Coe, who is 6ft 1in and weighs 16 and a half stone, admitted "shoving" the boy, who is half his size, but claimed he was acting in self defence.
He was convicted after a trial at Southwark Crown Court of assault occasioning actual bodily harm and battery in Wilson Road, Newham, on April 15.
The jury of six men and six women returned unanimous guilty verdicts after one and a half hours of deliberation.
Coe was convicted of a similar offence in May 2013 after getting out of his car to confront a group of youngsters about their "inappropriate language" on an estate in east London.
He allegedly called a young girl a "slut" and the others "kafir scum" during the incident.
Judge Michael Gledhill described Coe as a danger to the public and warned him he will be facing a "substantial" term of imprisonment.
Sentencing was adjourned until September 21 for further reports.
Coe has a long record of violent offences starting when he was 16, including assaults, burglary, robbery and violent disorder.
He was radicalised in prison by al Qaida terrorist Dhiren Barot in 2007, while serving an eight-year term for firing a shotgun at police during an arrest.
Coe, since known as Mikaeel Ibrahim, has attended a number of extremist demonstrations, including protests over the banning of niqabs in France and the values of Sharia law.
Coe was a close associate of Anjem Choudary, who was recently convicted of inviting support for Islamic State
Stefan Rousseau/PA Wire
He became a close associate of convicted hate preacher Anjem Choudary - founder of the banned organisation Al-Muhajiroun, of which Coe was a member.
Coe was on a list of individuals Choudary was banned from having contact with while he was on bail charged with inviting support for IS, of which Choudary has since been convicted.
He has also been photographed at a demonstration on the 10th anniversary of the September 11 attacks.
In December 2013 he took part in a demonstration in Brick Lane, east London, against Muslim shopkeepers selling alcohol.
The married father of two was jailed for 15 months after being found in the back of a lorry with false ID documents at the Kent port of Dover in November 2014.
He was accused of plotting to travel to Syria to join IS but was acquitted of engaging in conduct in preparation of terrorist acts.
Judge Gledhill said the two teenagers were behaving "perfectly normally" but Coe took offence "because he thought they were Muslims and to his way of thinking Islam prohibits such behaviour".
He added: "He stopped to reprimand them, acting as a self-appointed enforcer of the way he thinks Islam should be conducted."
Judge Gledhill said "substantial" custody was "inevitable" and continued: "He deliberately intervened in something perfectly normal with the intention to force his religious views on others."
He added that another aggravating feature was that Coe was involved in a "strikingly similar" incident in 2013.
uk newsuk crimeUK PoliceMichael Coe


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The Psychiatry Clerkship is a five-week experience that provides third- (MS3) and fourth-year (MS4) medical students with a broad view of behavioral health care through experiences in both inpatient and ambulatory settings. Students participate in formative standardized patient encounters to develop higher-level skills in psychiatric assessment and patient management.
In addition, the highly integrated teaching sessions allow students to learn the fundamentals of evidence-based psychiatric practice focused on a number of common mental health conditions.
For more information about these experiences, please contact Kathy Molter.


A Saltwater Pool IS a Chlorine Pool!
Today's short post is a rant about people thinking that a Saltwater pool is somehow not a Chlorine pool. And I believe it comes from pool builders, who present it as a choice of saltwater or chlorine, without offering much explanation between the two differences.
Earth Day Friendly pool equipment that can reduce your reliance on chlorine, eco-friendly pool heating, and natural chemicals by Natural Chemistry and SeaKlear.
Salt Cell troubleshooting - how to know if the salt cell is working, and how to test salt cell voltage and amperage. Is your salt cell bad, or is it the circuit board? These questions and many more salt cell troubles discussed.
Pool Pumps can be energy wasters, especially when inefficient single speed pump motors are used. Or when the pool pump runs for too long each day, or when the water is not balanced and sanitized properly.
Today's blog post is about pool pump energy usage, and how you can save pump electricity, energy and money, while making your pool more ecofriendly at the same time.
Swimming Pools don't have to be water hogs! How to conserve pool water, saving pool water is easy when you take these simple steps to curtail your pool water needs.
Natural pools are those which use a natural filter of stratified media and aquatic plants to filter the water, without sanitizers, disinfectants, or anything un-natural. Would you like a natural pool in your backyard?
It's Earth Day! ReDuce, ReUse and ReCycle your swimming pool! How to save energy, reduce consumption and recycle your old pool products.
How to operate your pool without using chlorine. Use minerals and ozone, along with MPS and good filtration, and you can say goodbye to buying shock and tablets. Really!
5 myths about solar pool heating - DEBUNKED! If you've been a reader of this blog for some time, you know that we are partial to solar pool heaters. I have installed many pool solar heaters over the years, in all places both north and south. There are a few misconceptions about solar pool heaters, mostly discussed by people that have never installed or even seen a solar pool heater in action.
Alternative pool sanitizers, there is really only one complete system that replaces chlorine - completely, that is Biguanides. But, you can use two other alternate purifying technologies to reduce reliance on chlorine, reducing the amount of chlorine you use, or that you need to produce with a salt chlorinator. Helper Sanitizers, I'm calling them.


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Church praises Prez for halting Norochcholai coal-fired power plant expansion
The Catholic Church has thanked President Gotabaya Rajapaksa for stopping further expansion of the Norochcholai coal-fired power plant complex.
The following is the text of a statement issued by Rt. Rev. Dr. Valance Mendis, Bishop of Chilaw: The diocese of Chilaw including the Clergy, Religious and Laity are very much pleased to receive the news that His Excellency the President has decided not to allow the expansion of Lakvijaya Coal Power Plant in Norochcholai. It is indeed a very timely and a valuable decision.
It has always been our view that Coal Power is rejected in many parts of the world because of the damage it causes to the environment, including human life.
As it is informed, the CEB has planned the expansion of Norochcholai Power Plant by way of further 600 MW.
The 1st phase is supposed to be a 300 MW expansion.
We are pleased about the decision of His Excellency the President. This decision goes well in line with the vision for the country that the government has expressed to have 80% of the energy generation through renewable sources by 2030. Sri Lanka being a country with a tropical climate, should maximize the use of wind and solar power as it is already reiterated by the government. The country is blessed with the water resources which contribute substantially to the grid already by way of Hydro-Power.
We are a developing country and the need of the hour is sustainable development. It is not enough to talk about it. We need to walk the talk in building up a 'Green Economy'. We need to think about tomorrow and the future generations. There is no point in developing a country where people cannot live because of pollution. Life comes first and then development."
UK reiterates commitment to accountability issues, won't comment on ex-CID Director's case
Police probe attacks on four Catholic churches in Mannar town on Tuesday


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European Data Centre - ..lished in February 2018
European Data Centre - News published in February 2018
News published in February 2018
Funding for citizens' dialogue on EU issues
The Ministry of Foreign Affairs and Cooperation, through the State Secretariat for European Affairs, announces a funding call for the year 2018, intended to finance debates on the European Union, taking place during 2018, from the funding allocation date until 30 September 2018, with the sole and primary motto: "More and better Europe. Citizens' debate on the future of Europe"
The purpose of this funding is to promote the citizens' participation in debates on the aforementioned goal and to promote the concept and practice of European citizenship in and with Spanish civil society. The funding will be granted under the principles of publicity, transparency, concurrency, objectivity and non-discrimination.
Any natural or legal person, not belonging to the public sector, legally responsible who, in any case, meet and be able to accredit the following requirements:
•Development of non-profit activity.
•Compliance with tax and social security obligations, in accordance with the provisions of article 22 of Royal Decree 887/2006, of 21 July.
•Have sufficient means and capacity to carry out the activities and guarantee compliance with the goals set forth in said activities.
•Not be included in any of the cases included in article 13, sections 2 and 3 of Act 38/2003, of 17 November, on the restrictions to obtain public funding.
* Application deadline: From 16 February 2018 to 09 March 2018
Call for graduate assistantships based on specialised knowledge in the European Union
Agreement on 6 February 2018, of the Spain's lower house of parliament, by which three graduate assistantships are offered for university graduates with specialised knowledge in the European Union and the Spanish constitutional and parliamentary system.
Application deadline: within the fifteen calendar days following the publication of this call on the Official Journal of the Government of Spain (BOE).
Call for grants from the Secretary of State for European Affairs 2018
A decision of 9 February 2018, from the Secretariat of State for European Affairs, announces a call for funding for the celebration of communication actions and information activities, on matters relating to the scope of its powers.
Euroscola 2018 Contest invites applications
The European Parliament Information Office in Spain, in collaboration with the Representation in Spain of the European Commission, the Secretariat of State for European Affairs (Ministry of Foreign Affairs and Cooperation) and, on this occasion, with the Secretary of State for Culture (Ministry of Education, Culture and Sports) announces the 24th Euroscola Contest, focused on the celebration of the European Year of Cultural Heritage in 2018, whose motto is: "Our heritage: where past meets future".
Secondary and vocational students from Spanish centres aged between 15 and 18 are invited to apply in teams  10 students each led by a teacher of the centre.
The competition will consist of the production of a video with a maximum duration of one minute and thirty seconds (1'30¨) whose theme is based on any of the goals included in the European Year of Cultural Heritage.
The winning teams will participate in a Euroscola day at the European Parliament venue in Strasbourg, living an unforgettable experience with other young people from all over the European Union.
European Union Film Competition
Do you like European films? Test your knowledge and join the competition! You will enter a drawing to win one of the 10 trips to Cannes Film Festival that will be raffled in May 2018, with travel and accommodation expenses included. It is your chance to get to know professionals in the European film and audio visual industry.
Deadline: 18 March 2018
Newsletter on European programmes and initiaves focused on education, training and youth.
InternationalizationCooperation and VolunteerismCompetitions and PrizesCounselling and Work InsertionInformation about congresses, courses, lectures...Scholarships and Funding
Thursday, 8
European Year of Cultural Heritage 2018
The European Year of Cultural Heritage focuses on Europe's rich cultural heritage, highlighting its role in fostering a shared sense of identity and building the future of Europe.
SEMINARIO DE INVESTIGACIÓN: Procesos de radicalización, conflicto y acomodo en contextos migratorios españoles y europeos
Este Seminario ofrece la oportunidad de conocer y dialogar, desde perspectivas interdisciplinares complementarias, sobre los desafíos que los contextos migratorios plantean al Estado secular en Europa. De igual modo, se debatirá sobre los problemas que genera la gestión de la diversidad religiosa y de creencias en el contexto europeo y, en particular, español. Se suma a esta visión la reflexión de expertos y agentes de socialización, con aportaciones desde enfoques sociales y culturales que permiten orientar un futuro de convivencia desde el conocimiento de las mentalidades y cultura religiosa.
Dirigido a: alumnado del Programa de Doctorado en Derecho, Grupo Jóvenes Investigadores de la Facultad de Derecho UA, alumnado de la Facultad de Derecho UA, alumnado de Sociología, y todas aquellas personas interesadas en la temática.
Viernes 9 de febrero de 2018, de 9 a 14:15 horas en la Sala de Juicios de la Facultad de Derecho.
Monday, 5
The Road Trip Project es una experiencia de viaje en carretera en un minibús por Europa para ocho personas en cuatro rutas por todo el continente entre la primavera y el verano de 2018.
Estos cuatro viajes abarcan el continente, desde los países bálticos hasta el punto más occidental de Europa, la longitud del Danubio, alrededor del Mediterráneo y junto al Atlántico. Los viajeros de Road Trip compartirán lo que hayan aprendido. Filmarán y publicarán sus historias. Al final de cada viaje, su experiencia se presentará en una road movie y en una guía de viaje.
La participación está abierta a ciudadanos de la UE o residentes entre las edades de 18 y 30 años, inclusive. Para solicitar ser un viajero de Road Trip Project, los participantes deben grabar un video de 60 segundos de presentación y decir algo sobre su motivación para unirse al proyecto.
La fecha de cierre de las solicitudes es el 11 de febrero de 2018.
El Road Trip Project es una iniciativa del departamento de Política Regional y Urbana de la Comisión Europea.
1st European Summit on Education
The 1st European Summit on Education, held in January 2018 in Brussels, was organised by Commissioner for Education, Culture, Youth and Sports Tibor Navracsics.
This Summit follows the Gothenburg Summit in November 2017, in which the Commission presented its vision to create a European Education Area by 2025 at the latest, and the European Council in December, in which the Member States expressed their willingness to take more actions in the field of education. The purpose of this summit is to discuss the future of education in Europe.
Under the theme "Laying the foundations of a European Education Area: for an inclusive, value-based and innovative education", the Summit focused on answering questions such as how high-quality, inclusive and values-based education can contribute to the success of Europe, what skills will be in the coming decades, and how we can promote basic, digital and entrepreneurship skills.
The programme dealt with a variety of issues, such as the way to involve disadvantaged students, the best way to train teachers or how to promote learning through sports. The relevance of science, technology, engineering and mathematics in education, as well as the role of children in the transformation of societies was also highlighted.
On the basis of the conclusions of the Summit, the Commission will launch new initiatives in the spring of 2018. Among those new actions, proposals related to the mutual recognition of degree certificates, language learning, a common frame for education quality, a European agenda for culture and a new EU strategy for the youth will be presented. Once in Gothenburg, President Juncker invited EU leaders to double the number of young people in the Union who will participate in Erasmus + until 2025, which will require a 29.4 billion-euro budget for the period 2021-2027. In addition, the Commission will continue to work on its project to create a network of European universities and a new EU student card. The card will allow students to study abroad easier than ever.
Academic Regulations News