text:
\subsection{\large \bf I. Introduction}

Kondo effect\cite{Hewson}, the screening of magnetic impurities by
the conduction electron reservoir, has been intensively
studied over the last half-century.
  Recently, there has been revival interest in Kondo effect in
semiconductor  quantum dot devices due to the progress in
fabracating nanostructures\cite{goldharber}. When there are odd number of electrons
on the dot, Kondo effect overcomes the Coulomb Blockade,
leading to a narrow pronounced resonance peak in local impurity
density of states and enhanced conductance through the quantum
dot\cite{glazman}\cite{goldharber}. The Kondo effect
is characterized by the single energy scale
, the Kondo temperature $T_k\approx D e^{-1/J}$\cite{Hewson},
below which the Kondo
 screening develops. Here, $D$, $J$ in $T_k$
are the conduction electron bandwidth
and the dimensionless Kondo coupling, respectively.
In a finite-sized mesoscopic device, $T_k$ sets the length scale
$\xi_k^0 \approx \hbar v_F/T_k$ associated with the size of the Kondo
screening cloud where the cloud of electrons with the size of order
of $\xi_k^0$ surrounding the magnetic impurity form spin-singlet state
with it\cite{ring1}. Here, $v_F$ is the Fermi velocity.
For typical values of $T_k$, $\xi_k^0$ is around $0.1~ 1\mu m$,
which is comparable to the typical size of the quantum dot device,
leading to the finite size effect of the Kondo screening cloud.
This effect has been investigated in a quantum dot embedded in a
mesoscopic ring threaded by a magnetic field\cite{ring1}\cite{ring2}\cite{ring3}.
The experimentally measurable
persistent current induced by the magnetic flux has been shown to be
sensitive to the ratio of the size of the ring $L$ and $\xi_k^0$
\cite{ring1}\cite{ring2}\cite{ring3}.
As $L \ll \xi_{K}^0$ the Kondo screening cloud does not develop completely,
giving rise to the suppression of Kondo effect and a reduction of the persistent current
even if the temperature is lower than $T_k$; while as for $L\gg \xi_k^0$
the Kondo screening cloud is formed, leading to enhanced persistent current.
By measuring the transport
properties (such as: persistent current)
as a function of the system size, we gain insights on how
the Kondo screening cloud is formed as the system size is increased.
This idea offers another route to realize Wilson's
Numerical Renormalization Group (NRG)\cite{NRG} idea on the Kondo problem.

Very recently, study of the Kondo effect has been extended to
the coupled double-quantum-dot setups via antiferromagnetic
RKKY interactions\cite{Marcus2QD}\cite{chung2QD}\cite{rkkyPascal}
\cite{cornaglia}.
The Kondo effect in such double-dot systems
competes with the RKKY interactions,
giving rise to quantum phase transition in the context of
the well-known two-impurity
Kondo problem between the Kondo and
local spin-singlet ground states\cite{2impkondo}.
Close to the quantum critical point physical
observables at finite temperatures exhibit
non-Fermi liquid behaviors.
The crossover behaviors between these two phases
can be accessed by changing temperatures.
However, at zero temperature but at finite-sizes,
the size of the Kondo screening cloud provides us with
an alternative route to the crossover behaviors in the two-impurity
Kondo problem\cite{pascal}.

In this paper, we study the Kondo effect
 in a side-coupled double quantum dot system
embedded in a finite-sized mesoscopic ring.
A similar
side-coupled double-quantum-dot system has been studied
where one of the dot
is coupled to Fermi-liquid leads of conduction electrons
with continuous spectrum\cite{side-coupled}.
Unlike the two-impurity Kondo system, in the side-coupled
double-dot system the Kondo phase is fragile and
unstable towards the local spin-singlet state for
any infinitesmall antiferromagnetic spin-exchange coupling,
and the transition is of the
Kosterliz-Thouless type. Note that a related setup consisting
of double quantum dots in parallel connected to a mesoscopic
ring shows a quantum phase transition of the pseudogap
Anderson model when the magnetic flux is tuned\cite{dias}.
Here, we are interested in not only
the nature of the
Kondo-to-spin-singlet quantum phase transition in our proposed setup
but also the crossover phenomena via a systematic
study of the system at finite sizes. The inverse of the system size
$1/L$ plays a similar role as temperature $T$. The finite
temperature crossover behaviors between the above two quantum ground states
can therefore be accessed effectively via changing the size of the ring.

We focus on the following three quantities to investigate this issue:
the effective Kondo temperature $T_k$ (inversely proportional to the
size of the Kondo screening cloud $\xi_k$
in the presence of the antiferromagnetic spin exchange coupling),
the local density of states
(DOS) $\rho_{QD}(\omega)$ on the quantum dot, and the persistent current (PC) $I$. The plan of the paper is as follows. In Section II., we introduce the model and present a large-N mean-field treatment of the model. In Section III., we present our results on the effective Kondo temperature, the local DOS on the
dot and the persistent current.
We also give detail explanations of our results. The conclusions are given in Section IV..

\vspace{0.15cm}
\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5cm]{model}
\end{center}
\vspace{0.05cm}
\label{setup}
\caption{Sketch of the model. $t_L / t_R $ is the hopping coefficient
between dot $1$ and its left/right conducting island on the ring,
$\phi$ is the magnetic flux (in unit of $2\pi/\Phi_0$ with $\Phi_0=hc/e$) through the ring, $K$ is antiferromagnetic coupling between dot $1$ and dot $2$.}
\end{figure}

\subsection{\large \bf II. Model}

Our model describes a double-quantum-dot system embedded in electronic
reservoir in a finite-sized ring (see Fig.\ref{setup}).
In this setup, only one of the dot (dot $1$) is coupled to
the ring via hoping; while the other dot (dot $2$) is coupled
only to dot $1$ via antiferromagnetic spin-exchange
interaction with a coupling strength $K$ and is
decoupled from the reservoir of the ring. The ring serves as electronic
reservoir consists of $L-1$ conducting islands. We
consider here the half-filled case ($N_e =L+1$ where $N_e$ is the total
number of electrons on the ring and on the two dots).
The key point in this setup with a
closed geometry
is that the Kondo screening cloud is trapped in the ring and can not
escape into the external leads\cite{ring1}. Nevertheless, one can measure
the transmission probability through the quantum dot $1$ embedded in the ring by measuring the persistent current\cite{ring1}\cite{Buttiker} (see details in Sec. III.).
Here we assume a direct antiferromagnetic spin exchange coupling between
two quantum dots which competes with the Kondo effect.
Note that though it has been known experimentally that
the antiferromagnetic spin-spin coupling (RKKY interaction)
between a double-quantum-dot system can be
induced naturally by a conduction electron island in the middle of the two dots\cite{Marcus2QD}, the direct antiferromagnetic spin exchange
coupling may in principle be generated via the second-order
hoping process directly between the two dots
if the two quantum dots are close
enough to each other.
Since in both cases we expect
suppression of the Kondo effect due to local spin-spin exchange
coupling, to simplify our calculations we assume a direct antiferromagnetic
spin-exchange coupling between the two dots.

The Hamiltonian of our model is given by:

\begin{eqnarray}
&{H}& = H_{A}+ KS_{1} \cdot S_{2}
\end{eqnarray}

where $H_{A}$ represents a quantum dot embedded in a ring by the
Anderson impurity model.

\begin{eqnarray}
{H}_{A}  &=&   -  t\sum_{j=1}^{L-1} \sum_{\sigma}[c_{j}^{\dagger}
c_{j+1}+ H.C] + U\sum_{i=1,2}n_{d_i}^{\uparrow}n_{d_i}^{\downarrow}
+ \sum_{\sigma,i=1,2} \epsilon_{d}d_{i\sigma}^{\dagger}d_{i\sigma}\\ \nonumber
& - & \sum_{\sigma} [t_{L}d_{1\sigma}^{\dagger}c_{1\sigma} + t_{R}\exp
(i\phi) c_{L-1\sigma}^{\dagger}d_{1\sigma}+ H.c.]
\end{eqnarray}
where $c_{j}$, $d_1$, $d_2$ represent the annihilation operators
of electron on site $j$ of the ring, dot $1$ and dot $2$, respectively,
 the phase factor $\phi$ is defined by
$\phi=2\pi \Phi/\Phi_0$ with $\Phi$ being
the magnetic flux going through the ring and $\Phi_0=hc/e$ being
the flux quantum,
$t$ is the electron hoping within the tight-binding ring, and
$t_{L/R}$ are the hopings between dot $1$ and the two neighboring
ring electrons ($c_{L-1}$/$c_{1}$).
Here,  $S_{1}$ and $S_{2}$ represent spin operators of dot $1$
and dot $2$, respectively with $S_{i}=\sum_{\sigma
\sigma'}d_{i\sigma}^{\dagger}\sigma_{\sigma\sigma'}d_{i\sigma'}$.
$L$ is the total number of electrons in the system
excluding dot $2$.
The well-known Kondo limit is reached for $U\gg t$.
Here, we consider a simple limit in the Kondo regime,
$U \rightarrow \infty$ where the slave-boson mean-field (SBMF)
approach\cite{ring2}\cite{ring3} is applicable to simplify the
quartic $U$ term.

Further progress can be made by decoupling the quartic
spin-spin interaction $S_1\cdot S_2$
into qudratic one via the Hubbard-Stratonovish transformation
in the framework of the large-N $SU(N)$ mean-field theory where
the symmetry of the Hamiltonian is generalized from the $SU(2)$ with
spin degeneracy being two to
$SU(N)$ with spin degeneracy being $N\rightarrow\infty$\cite{Hewson}.
In the SU(N) generalization of the
slave-boson representation, we have
$d_{i \alpha}$ = $ f_{i\alpha}b_{i}^{\dagger}$,
$\alpha$ is the flavor of the spin: $\alpha = 1,2,...N$, $i=1,2$ is the
indix for the two quantum dots.
The local constraints to enforce the single occupancy on dot $1$
and $2$ are given by
\begin{eqnarray}
\sum_{\alpha} f_{i}^{\dagger \alpha}f_{i \alpha} + b_{i}^{\dagger}b_{i} = \frac{N}{2}
\end{eqnarray}

The mean-field Hamiltonian for $H_A$ is therefore given by,
\begin{eqnarray}
N H_{A, MF}  &=&   -
t\sum_{j=1}^{L-1}\sum_{\alpha=1}^{N}[c_{j\alpha}^{\dagger}
c_{j+1\alpha} + H.c.] + \sum_{\alpha=1}^{N} \widetilde{\epsilon_{d}}
f_{1 \alpha}^{\dagger}f_{1 \alpha}  + \sum_{\alpha=1}^{N} \epsilon_{d}^{\prime}
f_{2 \alpha}^{\dagger}f_{2 \alpha}\\ \nonumber
& - & \sum_{\alpha=1}^{N} [\widetilde{t_{L}}
f_{1\alpha}^{\dagger}c_{1\alpha}
+ \widetilde{t_{R}}\exp(i\phi)
 c_{L-1\alpha}^{\dagger}f_{1 \alpha}+ H.c] +
 \lambda(b_0^2-1) + \bar{\lambda} (\bar{b}^2-1)
\end{eqnarray}

where $\widetilde{t}_{L/R} = b_0 t_{L/R}$ with $b_0$ being the expectation value
of the $b_1$ boson on dot $1$, $b_0=<b_1>$, $\bar{b}=<b_2>$,
$\lambda$ ($\bar{\lambda}$) is the Lagrange multiplier which enforces the local contraint on dot $1$ ($2$), $\tilde{\epsilon_d}=\epsilon_d-\lambda$, $\epsilon_d^{\prime}=\epsilon_d-\bar{\lambda}$.
The quartic antiferromagnetic spin-spin interaction
can be decoupled via the mean-field variable $\chi$
\cite{chung-largeN}:
\begin{eqnarray}
\chi = \frac{b_0 K}{N} <d_{1 }^{ \dagger \alpha }d_{2, \alpha}>
\end{eqnarray}. Therefore,
\begin{eqnarray}
KS_1\cdot S_2 &=& -\sum_{\alpha=1}^{N}\frac{\chi}{N}
f_1^{\dagger \alpha} f_{2 \alpha}-H.c.+ \frac{\chi^2}{K}
\end{eqnarray}
The full mean-field Hamiltonian is given by
\begin{eqnarray}
H_{MF} = H_{A, MF} -\sum_{\alpha=1}^{N}\frac{\chi}{N}
f_1^{\dagger \alpha} f_{2 \alpha}-H.c.+ \frac{\chi^2}{K}.
\end{eqnarray}

The mean-field energy $E_{MF}$ can be obtained by
diagonalizing the above Hamiltonian and can be expressed as
\begin{eqnarray}
E_{MF} = \sum_{m, \sigma}\epsilon_{m, \sigma}(\lambda, b_0) + \frac{\chi^2}{K}
+ \lambda(b_0^2-1) + \bar{\lambda} (\bar{b}^2-1),
\end{eqnarray}
where $\epsilon_{m,\sigma}$ are the eigenvalues of the Hamiltonian matrix in
$H_{MF}$, the summation over $\emph{m}$ includes all occupied levels of
$H_{MF}$. The values of the mean-field variables
$\lambda$, $b_0$ and $\chi$ are determined by minimizing $E_{MF}$
with respect to $b_0$ and $\chi$:
\begin{equation}
\frac{\partial E_{MF}}{\partial b_0} = \frac{\partial E_{MF}}{\partial \bar{b}} = \frac{\partial E_{MF}}{\partial \chi} =0,
\end{equation}
subject to the constraint $\frac{\partial E_{MF}}{\partial \lambda} = \frac{\partial E_{MF}}{\partial \bar{\lambda}}=0$.
The ground state energy $E_{gs}$
corresponds to the global minimum of
$E_{MF}(\lambda,\bar{\lambda}, b_0, \bar{b}, \chi)$ which satisfies the
mean-field equations.
Note that the advantages of taking the large-$N$ mean-field approach are:
(i) in the large-$N$ limit the solutions from the mean-field equations
are exact though the physical system corresponds to $N=2$,
and (ii) at finite $N$, a systematic $1/N$ correction to the mean-field
results is possible though it is beyond the scope of this paper.
The mean-field phase diagram can be mapped out via the solutions of the
above mean-field equations.

\begin{figure}[h]
\begin{center}
\includegraphics[width=11cm]{phasedia}
\end{center}
\caption{Schematic mean-field
phase diagram of the model for (a) $L=4n, 4n+1, 4n+3$ (case (I))
and (b) $L=4n+2$ (case (II)). The red line indicates the crossover with $K$ being fixed and the green line indicates the crossover with $L$ being
fixed.}
\label{phase}
\end{figure}

\subsection{\large \bf III. Results}

Three physical observables are
identified to investigate
 the Kondo screening effect of our setup at finite sizes:
(i). \emph{Effective Kondo temperature}($T_k$),
(ii). \emph{Local density of states on dot $1$ (LDOS)} ($\rho_{QD}(\omega)$),
(iii). \emph{Persistent current(PC)}. Details are shown below.

Before we present our new results, it is useful to
summarize the behavior of the model at $K=0$ (without
antiferromagnetic spin-spin coupling), which has been intensively
studied\cite{ring1}\cite{ring2}\cite{ring3}.
The Kondo resonance
of a quantum dot embedded in a mesoscopic ring strongly depends on
the finite size $L$ (mod $4$), and the magnetic flux $\Phi$
threading the ring.
In particular, both the Kondo temperature $T_k$ and the persistent current
$I$ follow universal scaling functions
of $\xi_k^0/L$ at a fixed magnetic flux. In the Kondo regime, all physical
observables, such as: Kondo temperature $T_k$, persistent current $I$
 are enhanced as the size $L$ increases, but with different crossover
bahaviors in all four cases of $L=4n$, $L=4n+1$, $L=4n+2$, and $L=4n+3$.
 The magnetic flux dependence of persistent current $I$ exhibits a
symmetry between size $L$ and $L+2$: $I_L(\phi) = I_{L+2}(\phi + \pi)$,
indicating that adding the magnetic flux of $\Phi_0/2$ (or adding a phase
$\pi$)  is equivalent to switch the behavior of the PC from a system with
size $L$ to $L+2$.

With the previous results in mind, we may discuss the general
properties for $ K > 0$. Due to the antiferromagnetic
spin-spin couping, we expect in this case
the competition between the Kondo and the local spin-singlet ground states,
leading to the quantum phase transition. In fact, quantum phase transitions
in double-quantum-dot systems with antiferromagnetic RKKY interactions
have been intensively studied in recent years
\cite{Marcus2QD}\cite{chung2QD} in the framework of two-impurity Kondo problem
\cite{2impkondo} where the quantum dots couple to
the conduction electron Fermi sea with continuous spectrum. Two types of
quantum phase transitions have been identified in these systems:
the phase transition
with a quantum critical point (QCP)
and the one of the Kosterlitz-Thouless (KT) type.
The characteristic behavior near QCP is the observables power-law
dependence on the coupling strength relative to the critical point; while
for the KT transition the crossover energy scale exponentially depend
on the distance to the critical point.
The former (QCP) type of the quantum phase transition
is realized in the double-dot systems where each of the dot
couples to an independent conduction electron
reservoir\cite{Marcus2QD}\cite{chung2QD},
and the critical
point separating the Kondo from the local spin-singlet phase
is the well-known two-impurity Kondo fixed point\cite{2impkondo}. The latter
 (KT) type
exsits in a side-coupled double-dot system where only one of the dots
coupled to the electron reservoir\cite{side-coupled}.
The Kondo resonance becomes more fragile
in the side-coupled system so that an infinitesmall antiferromagnetic
coupling is sufficient
 to suppress the Kondo effect and leads to the spin-singlet ground state.
We expect the similar Kosterlitz-Thouless transition to occur in our
side-coupled double-dot system embedded in a ring. However,
the mesoscopic ring in our setup
consists of a finite number of tight-binding electrons
(instead of a Fermi sea with continuous spectrum),
the details of the transition
might be different from those in
Ref.\cite{side-coupled} (see below).

\subsubsection{\bf A. Mean-field phase diagram}

After solving the mean-field equations, we summarize our main results in the
schematic mean-field phase diagram as shown in Fig. \ref{phase}. We find
indeed the transition between the Kondo and spin-singlet phases is
of the Kosterlitz-Thouless type (see below). However, we find
that the critical point $K_c$
separating the two phases is not at zero as shown in the similar side-coupled
double-dot system studied previously in Ref. ~\cite{side-coupled}
but at a finite value: $K_c> 0$.

There are three regions in the phase diagram, corresponding to different
mean-field solutions:\\

1. For small $K$ we find
$b_0\ne 0$, $\lambda\ne 0$, $\chi=0$. In the thermodynamic limit
this is the Kondo phase studied in Ref.\cite{ring1}\cite{ring2}\cite{ring3}.\\

2. For large $K$, we find $b_0= 0$, $\lambda\ne 0$.
The ground state for $L\rightarrow \infty$
is the local spin-singlet phase where antiferromagnetic spin-spin
coupling completely
suppresses the Kondo effect.\\

3. In the intermediate values of $K$, we find
$b_0= 0$, $\lambda\ne 0$, $\chi\ne 0$.
This corresponds to the
 crossover region between Kondo and spin-singlet phases indicated
in the shaded region in Fig.\ref{phase}. This region is defined
either by $K_{c1}<K<K_{c2}$ for a fixed size $L$ where
$K_{c1}$ and $K_{c2}$ are the boundaries between
the crossover region and the two stable phases or by $L_{c1}<L<L_{c2}$ for a fixed $K$ where
$L_{c1}$ and $L_{c2}$ are defined in a similar way as $K_{c1}$ and $K_{c2}$.

Note that in all the above three cases, we find $\bar{b}=0$, $\bar{\lambda}=\epsilon_d$. We then systematically investigate the crossover behaviors along
the following two different
crossover paths, depending on the finite size $L$ (mod $4$): \\

Case (I) (Fig.\ref{phase} (a)) holds for $L=4n,4n+1,4n+3$ where
we find the crossover occurs for $K<K_c$.\\

Case (II) (Fig.\ref{phase} (b))
occurs for $L=4n+2$ where we find the crossover ranges exist mainly for
$ K> K_c$.\\

 In both cases, we find the crossover energy scale $T^*$ follows
the behavior of the typical Kosterlitz-Thouless transition:
\begin{equation}
T^*= c \widetilde{T_{k}}\exp{[-\pi\widetilde{T_{k}}/(K-K_c)]}
\end{equation}
 where $\widetilde{T_k}=c'T_k$, and both $c$ and $c'$ are non-universal constants.

We investigate the Kondo effect in our setup at finite sizes by either
changing the size $L$ at a fixed $K$ (or following
the red (vertical) line in Fig.\ref{phase})
or changing $K$ at a fixed size
$L$ (or following the green (horizontal) line in Fig. \ref{phase}).
The finite size scaling of $T_k$ indicates that
in the thermodynamic limit $L\rightarrow \infty$,
$K_{c1}$ and $K_{c2}$ converge to a single critical value $K_c$:
$K_{c1}=K_{c2}\rightarrow K_c$ (see below).

\subsubsection{\bf B. The effective Kondo temperature}

 In the single Kondo dot system embedded in a 2D electron gas (2DEG),
it is well-known that the physical
properties follow universal functions of $(T/T_K^0)$ in Kondo
regime where $T_k^0$ is the Kondo temperature of the single dot
in the thermodynamic limit\cite{goldharber}.
In a single quantum dot embedded in a mesoscopic ring with
a finite size, the effective Kondo temperature
can be defined as\cite{ring2}
\begin{eqnarray}
T_k^{1QD}= \epsilon_d-(E_{gs}-E_{gs}^{o})
\label{Tk1QD}
\end{eqnarray}
, where $\epsilon_d$ is the energy of the dot $1$,
$E_{gs}$ and $E_{gs}^0$ correspond to the energy of the full
system and of the tight-binding ring with the open boundary condition,
respectively. The effective Kondo temperature
$T_k$ is the energy gain due to the coupling between quantum dot and the ring,
which corresponds to the energy associated with the Kondo coupling.
In the thermodynamic limit ($L\rightarrow \infty$),
$T_k$ approaches to $T_k^0$.
For the relevant parameters in the $K=0$
Anderson model of the
single dot-ring setup: $t=1$, $t_L=t_R=0.4 t$, $\epsilon_d=-0.8 t$,
we find $T_k^0\approx 0.0189 t$, $\xi_k^0\sim 106$. Note that for simplicity,
we set $t=1$ throughout the paper as our unit.

In the presence of the antiferromagnetic spin-spin coupling
the effective Kondo temperature $T_k$
at a finite size in the context of our large-N slave-boson mean-field
approach can be generalized from Eq. \ref{Tk1QD} to:
\begin{eqnarray}
T_k= \epsilon_d-(E_{gs}-E_{gs}^{o})-\frac{\chi^2}{K}
\label{Tk2QD}
\end{eqnarray}
Note that the effective $T_k$ defined in Eq.\ref{Tk1QD} and Eq.\ref{Tk2QD}
are slightly different from that defined in Ref.\cite{ring2} where
$T_k$ is measured relative to the hightest occupied level defined
as $\epsilon_F$, corresponding to the Fermi energy in the thermodynamic
limit\cite{ring2}\cite{ring3}.
Though this minor difference in definition
does not affect the results and the physics, $T_k$ defined here
offers an
intuitive understanding of the crossover at finite sizes corresponding
to case (I) and (II) mentioned above--for $K<K_c$, $T_k$
is expected to increase with increasing size $L$ for case (I)
as the system recovers
the Kondo resonance in $L\infty$ limit,
and for $K>K_c$
$T_k$ is expected to decrease
to $0$ as the system approaches to the local spin-singlet
ground state in the thermodynamic limit
 (see details below).
Nevertheless, we have checked that with the definition
in Ref. \cite{ring2} and \cite{ring3} our results for $T_k$ at finite sizes
 indeed reproduce those in Ref.\cite{ring2} and \cite{ring3}.
We have also checked that our results on $T_k$ for $K=0$
are consistent
with the finite-size behaviors in $E_{gs}$ and $E_{gs}^0$
via perturbation theory in Ref. \cite{ring1}.

In the Kondo phase, $T_k$ reduces to the
Kondo temperature for the single quantum dot embedded in a mesoscopic ring
($K=0$). When $L \rightarrow  \infty$, $T_k$ in the Kondo phase
approaches to $T_K^0$.  In the crossover
between Kondo and the local singlet phases, $T_k$ decreases with increasing
$K$, and finally $T_k\rightarrow 0$ when the system is in the local singlet
ground state where the Kondo effect is completely suppressed by the antiferromagnetic spin-spin couping. We analysize below in details
the crossover between the Kondo and the local spin-singlet ground states
from the behaviors of the Kondo temperature.

\subsubsection{ \textbf{ 1. Varies K with fixed L }}
\vspace{5mm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=16cm ]{Tk-k}
\end{center}
\caption{$T_k$ as functions of $K$. (a)Case I: $L=4n$ (b)Case II:
$L=4n+2$.
Other parameters: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$.
Here we set $t=1$ as the unit.}
\label{TK-k}
\end{figure}

We first monitor how the Kondo resonance is destroyed by varying
the antiferromagnetic spin-spin
coupling strength $K$ at a fixed finite size $L$.
Fig.(\ref{TK-k}) shows $T_k$ as a function of antiferromagnetic
coupling strength
$K$ at a fixed size $L=300$ (or $\xi_k^0/L\sim 0.3$).
In both case (I) and (II), $T_k$ vanishes as $K\rightarrow K_{c2}$,
indicating the suppression of Kondo resonance by the antiferromagnetic
spin-spin interaction.
 However, there are minor differences between these two cases
in how fast $T_k$ vanishes
as $K$ close to $K_{c2}$.
In case (II), $T_k$ remains a constant over a wider range of $K<K_{c1}$
compared to that in case (I) before it decays.
This suggests that the Kondo effect
at a finite size
seems more robust in case (II) ($L=4n+2$) than in case (I)
($L=4n,4n+1,4n+3$) so that the crossover
region in case (II) is much narrower than in that in case (I).
This is consistent with our numerics as $T_k$ for $L=4n+2$
is found to have the largest value among $L$ (mod $4$) for a given size.

\vspace{5mm}
\subsubsection{\textbf{ 2. Varies $L$ with fixed $K$}}
\vspace{5mm}

Next, we present the results on the finite-size
dependence of the Kondo resonance
at a fixed antiferromagnetic spin-spin coupling
strength $K$. In analogous to the Numerical Renormalization Group
(NRG) method, the ground state is computed and monitored as
we decrease the energy scale $1/L$ (or equivalently increase
the system size $L$)
until we reach the thermodynamic limit $L\rightarrow \infty$
(or effective zero temperature).

First, we describe the qualitative behaviors of $T_k$ at finite sizes
in the two cases as mentioned above.
For case (I) (represented by $L=4n$, see Fig.\ref{TK4n}(a))
$T_k$ increases with increasing $L$ as the crossover region
is for $K<K_c$ where the Kondo resonance is recovered at large
system size $L\gg\xi_k^0$; while for case (II) ($L=4n+2$, see Fig.\ref{TK4n}(b))
since the crossover
occurs for $K>K_c$, $T_k$ decreases with increasing $L$ and it vanishes
in the thermodynamic limit.
It is clear from the crossover behavior in Fig. \ref{TK4n} that
the finite-size effect appears for $L\l\xi_k^0$ where the Kondo screening
cloud has not yet fully developed for $K<K_c$ and has not yet completely
destroyed for $K>K_c$; while this effect diminishes as the system approaches to
the thermodynamic limit or $L\gg \xi_k^0$.
Note that in case (I) with large $K>K_c$ and case (II) with
small $K<K_c$, the ground state remains at local singlet
and Kondo state, respectively; therefore, no crossover behaviors
are found. It is also worthwhile noting that $T_k$ for $L=4n+2$ (case (II))
for $K=0$ approaches to $T_k^0$ from above as $L$ approaches to the thermodynamic limit. This suggests that the Kondo effect in this case
seems more robust
at finite size than in the thermodynamic limit.
Therefore, at any finite size $L$, it is necessary to apply a larger antiferromagnetic
coupling $K$ compared to $K_c$ which is required in $L\rightarrow \infty$ limit to suppress the Kondo effect. This provides
an explanation why we always find the crossover behavior for $L=4n+2$
for $K>K_c$. On the other hand in case (I) ($L=4n,4n+1,4n+3$) for $K=0$,
$T_k$ approaches to $T_k^0$ from below
as $L$ increases, which explains why the crossover occurs for $K<K_c$
in this case.

\vspace{5mm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=6cm ]{Tk4n}
\end{center}
\caption{$T_K$ as functions of $1/L$: (a) $L=4n$,
$K$ varies from $0.0025t$ to $0.015t$, distance: 0.0025t.
(b) $L=4n+2$, $K$  varies from $0.036t$ to
 $0.04t$, distance: $0.001t$. Other parameters: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t,
 $\phi=0$. Here, we set $t=1$ as the unit.}
\label{TK4n}
\end{figure}
\vspace{5mm}

To investigate the nature of the quantum phase transition
in the thermodynamic limit more closely, we then perform the finite-size
scaling for $T_k$ in the crossover region.
We find $T_k$ in case (I) and (II)
follows its own unique universal scaling function of $1/(T^*L)$
as $K\rightarrow K_c$ (see Fig. \ref{TK4n+2}),
where $T^*$ is the crossover energy scale defined as:
 \begin{eqnarray}
T^*= c \widetilde{T_k}\exp{(-\pi\widetilde{T_k} /\mid K-K_c\mid)}
\label{Tstar}
\end{eqnarray}
with $ \widetilde{T_k} =c' T_k$ has the same form in both case (I) and (II).
Here, $c$, $c'$, $K_c$ are non-universal
fitting prefactors depending on the initial parameters of the Hamiltonian.
For $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$, we find
$c\approx 5.5$, $K_c\approx 0.0271$ (in unit of $t$) in both case (I) and (II);
$c'\approx 0.65 $ in case (I), and $c'\approx 0.5$ for case (II).
Note that the expression for $T^*$ in Eq. \ref{Tstar}
is quite general as $K$ can be either smaller (case (I))
or larger (case (II)) than $K_c$. The universal scaling at finite sizes
and the same exponential form for the crossover energy scale
$T^*$ valid for both cases strongly indicate that in the thermodynamic limit
the system exhibits the
Kosterlitz-Thouless transition at a finite critical
antiferromagnetic spin-spin coupling
strength $K_c>0$. We have checked the consistency of our result
from our finite-size scaling that $K_c$ indeed
reaches to the same value in the thermodynamic limit
for both case (I) and (II) even though the corresponding
crossover regions are on the
opposite side of the transition
($K<K_c$ for case (I) and $K>K_c$ for case (II)).

Note that unlike the similar side-coupled double-dot system studied
previously in Ref.\cite{side-coupled}
where the KT transition between  the Kondo and spin-singlet
phase occurs at $K_c=0$, we find in our setup a finite $K_c>0$
for the same KT transition.
We think this difference might be related to the more singular
DOS of the conduction electron bath in the current setup of a tight-binding ring
compared to that in a continuous Fermi sea with a constant DOS, making the Kondo
resonance more robust against the antiferromagnetic spin-spin
coupling in the current set up and therefore leads to
a finite $K_c$ instead of $K_c=0$. However, it is also possible that
the finite $K_c$ we find here
is the artifact of the large-N slave-boson  mean-field theory.
 Further study is necessary to clarify this issue.

Finally, as a consistency check, as the system approaches to the thermodynamic limit, $L\rightarrow \infty$, $T_k\rightarrow T_k^0$ for $L=4n$ (case (I),
$K<K_c$); while $T_k\rightarrow 0$ for $L=4n+2$ (case (II), $K>K_c$), as expected  (see Fig. \ref{TK4n+2}).

\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=6cm ]{Tk4n+2}
\end{center}
\caption{$T_K$ as functions of $1/(T^*L)$: (a) $L=4n$, $K$
varies from $0.0025t$ to $0.015t$, distance: $0.0025t$.
(b)  $L=4n+2$, $K$
varies from $0.036t$ to $0.04t$, distance: 0.001t.
$T_K$ follows universal scaling functions of
$1/(T^*L)$. Other parameters: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$. Here, we set $t=1$.}
\label{TK4n+2}
\end{figure}

\subsubsection{{\bf C. Local Density of State}}
We now turn our attention to the local density of states (LDOS)
on the dot $1$, given by
\begin{eqnarray}
&\rho_{QD}(\omega)& =-\frac{1}{\pi}Im{G^{R}}_{d_1d_1}(\omega)
\label{rhoQD}
\end{eqnarray}
 where $G^{R}_{d_1d_1}(t)\equiv <d_1(t) d_1^{\dagger}(0)>$
is the retarded Green's function of dot $1$ which directly
couples to the ring.
 Near the Fermi surface, $\rho_{QD}(\omega)$ determines
the transport properties of the system, and it
 can be obtained from the Green's function ${G}_{d_1d_1}(\omega)$ via
equation of motion approach.
First, the mean-field Hamiltonian in momentum space is given by:
\begin{eqnarray}
H_{MF} &=& \sum_{m,\sigma} \epsilon_m c_m^{\dagger}c_m +H.C. + \widetilde{\epsilon_d}f_{1\sigma}^{\dagger}f_{1\sigma}  - \sum_{m} (t_m c_m^{\dagger}
f_1 + H.C.)\\\nonumber
&-& \chi f_1^{\dagger}f_2 - H.C. + \frac{\chi^2}{K} + \lambda (b_0^2-1)
\end{eqnarray}
Where
\begin{equation}
t_m = {\it i} \sqrt{\frac{2}{L}} \sin({\frac{m}{L}\pi}) [\widetilde{t_L} + \widetilde{t_R}\exp(i\phi)(-1)^{m+1}]
\end{equation}
 with $m=1,2,\cdots L-1$\cite{ring2}.

The equation for a general retarded Green's function $G_{ij}^R(\omega)$ is
then given by:
\begin{eqnarray}
(\omega + {\it i} \eta-H_{MF})G_{ij}^R(\omega) = {\it I}
\end{eqnarray}
where ${\it I}$ is the identity matrix,
$i,j$ can be $f_1$, $f_2$, or $m$. We therefore get the following three equations
for $G_{f_1f_1}$, $G_{f_1m}$ and $G_{f_1f_2}$:
\begin{eqnarray}
(\omega + {\it i} \eta-\widetilde{\epsilon_d})G_{f_1f_1}^R(\omega) - \chi G_{f_2f_1}^R - \sum_m t_m G^R _{md1}(\omega) =1
\end{eqnarray}
\begin{eqnarray}
- \chi G^R _{f_1f_1} + (\omega + {\it i} \eta)G^R _{f_2f_1} = 0
\end{eqnarray}
\begin{eqnarray}
- t_m G^R _{f_1f_1} + (\omega + {\it i} \eta - \epsilon_m) G^R _{mf_1} = 0
\end{eqnarray}
The above equations are easily solved and we get:
\begin{eqnarray}
&{G}^R_{d_1d_1}(\omega)& = b_0^2 G^{R}_{f_1f_1} =
\frac{b_0^2}{ \omega  -\widetilde{\epsilon_d} + {\it i} \eta
 \hspace{0.1cm}  - \sum_{m} [ \frac{t_{m}^2{\mid b_0 \mid}^2}{ \omega -\epsilon_m+{\it i} \eta}
] -  \frac{\chi^2}{ \omega +{\it i} \eta}}
\end{eqnarray}
where $\eta\rightarrow 0$. We then obtain $\rho_{QD}(\omega)$ by Eq. \ref{rhoQD}.
In the following we analysize the crossover behaviors in
LDOS $\rho_{QD}(\omega)$.

\subsubsection{1. $K=0$}

\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=12cm ]{dos0}
\end{center}
\caption{ Local density of state for $K=0$ system (in arbitrary unit) (a) $L=4n$
(b) $L=4n+1$ (c) $L=4n+2$ (d) $L=4n+3$. Here, $r=\xi_k^0/L$, the ratio between
$\xi_k^0$ and $L$.
Other parameters:
$\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$. Here, we set $t=1$.}
\label{0dos}
\end{figure}

In the absence of the antiferromagnetic spin-spin
coupling ($K=0$)
the LDOS has been extensively studied where
$\rho_{QD}(\omega)$ depends sensitively
on $L$ (mod$4$) at
finite sizes \cite{ring2}.
As shown in Fig.(\ref{0dos}), our results on LDOS in this case
 at finite sizes are qualitatively in good agreement
with that in Ref.\cite{ring2}.
For the convenience of later discussions we summarize below
the behaviors of LDOS in the four different cases of $L$.
The key features are--
(i). There exsits a main Kondo resonance peak (located either
symmetrically or asymmetrically
with respect to $\omega=0$) followed by pairs of side peaks.
(ii). As shown in Fig. \ref{0dos}, as $L$ increases
the main Kondo peak gets more pronounced and closer to
$\omega=0$; while the side peaks are gradually merged into the
broadened main peak.
In particular,
the LDOS for $L=4n$ is very symmetric
$\rho_{QD}(\omega)= \rho_{QD}(-\omega)$, suggesting the
symmetry between particle and hole excitation energy
in the finite size spectrum. The asymmetric Kondo peaks
for $L=4n+1$ and $L=4n+3$ are located on the opposite
sides (left for $L=4n+1$ and right for $L=4n+3$)
of $\omega$.
For $L=4n+2$, however, the LDOS shows
asymmetric double Kondo peaks with comparable sizes and
a dip at $\omega=0$. These
differences in DOS among the four sizes of $L$ (mod $4$)
can be explained in terms of the energy
levels corresponding to the highest
occupied (HO) and the lowest unoccupied (LU) states relative to the
Fermi level\cite{ring2}: For $L=4n$, both HO and LU levels are
around the Fermi energy $\epsilon_F=0$, leading to a symmetric single
peak in LDOS at $\omega=0$. For $L=4n\pm 1$, HO and LU levels are
both below and above $\epsilon_F$, respectively, giving rise
to an asymmetric single
peak in LDOS below and above $\omega=0$ separately. However, for $L=4n+2$,
LO and HU levels are on the opposite side of the Fermi level,
resulting in splitted double Kondo peaks below and above $\omega=0$.
It should be noted that despite the differences at finite sizes
($L\sim \xi_k^0$), the LDOS
in all four cases indeed merges to a single Kondo peak as
the system reaches the thermodynamic limit $L\rightarrow \infty$
(or $L\gg \xi_k^0$).
(iii). The LDOS on dot $1$ obeys the following symmetries\cite{ring2}:
$\rho_{QD}^{L}(\omega,\phi)= \rho_{QD}^{L+2}(\omega,\phi+\pi)$
and $\rho_{QD}(-\omega,\phi)= \rho_{QD}(\omega,\phi+\pi)$.

We would like to point out here that from the width $D$ of the central
Kondo peak(s), which can be approximately regarded as a quantity
proportional to the effective Kondo
temperature $T_k$, in LDOS
for $K=0$ case one can qualitatively understand the opposite trends
in $T_k$ at finite sizes in case (I) and (II) mentioned above. For
$L=4n,4n+1,4n+3$ (case (I)), the width $D$ becomes larger as $L$
increases, indicating an increase in $T_k$ as the system
approaches to the thermodynamic limit; while as for $L=4n+2$ (case (II)),
$D$ gets smaller as $L$ increases, suggesting a decrease in $T_k$
as $L\rightarrow \infty$.
We analysize in details below the behaviors of LDOS for $K>0$.

\subsubsection{2. $K>0$}

At a finite $K>0$,
the LDOS on the dot shows a crossover between
the Kondo phase and the spin-singlet phase.
For $K<K_{c1}$ at a fixed size $L$, the LDOS remains the same
as that for $K=0$.
For $L=4n,4n+2$ we find a continuous evolution in LDOS from
Kondo to the crossover region near $K= K_{c1}$; while for
$L=4n+1,4n+3$ the LDOS
exhibits a first-order jump at $K_{c1}$.
For $K>K_{c2}$, we find
$\rho_{QD}(\omega) = 0$ as the indicator of the local spin-singlet
phase since $b_0=0$. In the crossover region $K_{c1}<K<K_{c2}$,
the Kondo peak in LDOS splits into two
with respective to $\omega=0$.
The splitting gets wider as $K$ increases
further. Details are shown below.

\vspace{5mm}
\subsubsection{ {\large (i).} \textbf{ Varies $K$ with fixed $L$:}}
\vspace{5mm}

\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=12cm ]{K-dos}
\end{center}
\caption{Local density of state (in arbitrary unit)
with various $K$ for $r=\xi_k^0/L\sim 0.53$.
(a) $L=4n$ (b) $L=4n+1$
 (c) $L=4n+2$ (d) $L=4n+3$. Other parameters: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$. Here, $K$ is in unit of $t$, and we set $t=1$.}
\label{Kdos}
\end{figure}

In Fig.\ref{Kdos}, we show LDOS for various antiferromagnetic spin-spin
coupling strength $K$
at a fixed size $L\approx 200$ ($\xi_k^0/L\sim 0.5$).
The behaviors of LDOS are classified by $L$(mod $4$).
The common features in Fig. \ref{Kdos} are as follows.
In each of the four LDOS plots in Fig. \ref{Kdos}, the Kondo peak
in $\rho_{QD}(\omega)$ splits into two
at a small value of $K>K_{c1}$, indicating the crossover
between Kondo and local spin singlet phases. The two peaks separate
further apart and become more symmetric as $K$ increases. At the
end, $\rho_{QD}(\omega)$ vanishes when $K>K_{c2}$ in the local spin
singlet phase.
Note that the values for $K_{c2}$ depend sensitively
on $L$ (mod $4$). This can be understood as when $L\approx 200$
the effective Kondo temperature $T_k$ in case (II) is much larger
than that in case (I) (see Fig.\ref{TK4n}, Fig. \ref{TK4n+2}) ,
which explains why we need a larger value of $K_{c2}$
in case (II) to suppress the Kondo effect than that in case (I).
 We present details below.

\vspace{5mm}
\subsubsection{ {\large (ii).} \textbf{ Varies $L$ with fixed $K$:}}
\vspace{5mm}

\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=12cm ]{dos-dim}
\end{center}
\caption{Local density of state (in arbitrary unit) with various $L$ and fixed $K$ (in unit of $t$). Here, $r=\xi_k^0/L$. (a) $L=4n$ (b) $L=4n+1$ (c) $L=4n+2$ (d) $L=4n+3$. Other parameters: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$. Here, we set $t=1$.}
\label{dimdos}
\end{figure}

Fig. (\ref{dimdos}) shows how
$\rho_{QD}(\omega)$ changes with the system size $L$.
For $0<K<K_c$, starting from a small size $L<L_{c1}$,
the LDOS changes from the behavior of a local spin-singlet
state with $\rho_{QD}(\omega)=0$ to that in the crossover region
with splitted peaks and finite LDOS at $\omega=0$,
and finally it recovers the Kondo resonance at larger size $L>L_{c2}$.

We separate the discussion here into two different cases:
$K<K_c$ (case (I)) and $K>K_c$ (case (II)).
 Fig.(\ref{dimdos}(a),(b),(d))show finite size
dependence of $\rho_{QD}(\omega)$ at $K<K_c$ (case (I)) for
$L=4n$, $4n+1$, and $4n+3$, respectively.
For small system sizes, the system is in the local spin-singlet state
with vanishing LDOS. At intermediate sizes in the crossover region, however,
LDOS starts to develop peaks with a dip at $\omega=0$. As size further
increases, these splitted peaks either gradually ($L=4n$) or
suddenly ($L=4n+1$ and $4n+3$) merge into a single Kondo peak
located either symmetrically ($L=4n$) or asymmetrically ($L=4n+1$ and $4n+3$)
with respect to $\omega=0$. These Kondo peaks then follow
the evolution at finite sizes for $K=0$ and finally recover the
single symmetric Kondo peak in the thermodynamic limit.

For $K>K_c$ (case (II)), the size dependence of
$\rho_{QD}(\omega)$ is shown in
Fig.(\ref{dimdos}(c)) with $K=0.03t > K_c$.
The LDOS exhibits a crossover from the Kondo to local spin-singlet
state with increasing size $L$.
For $K \sim K_c$ the LDOS changes from
singlet state at small sizes to the crossover regime within
the range $100 <L< 800$ that we investigate. To study the
the ultimate fate of the system one needs to go much larger
system size than $L\sim 800$, which is beyond the scope of
our computational limit.
 Note that our analysis on the LDOS by changing
$K$ with fixed $L$ (changing $L$ with fixed $K$) is consistent
 with the green line (red line) of the
schematic phase diagram shown in Fig. (\ref{phase}).
Finally, the first order jumps seen in LDOS at $K=K_{c1}$
for $L= 4n+1$ and $L=4n+3$ may be due to the artifact of the
mean-field theory. Further investigation is needed to clarify
this issue.

\subsubsection{{\bf D. Persistent Current}}

We now analysize the crossover from the behaviors in persistent current.
The Kondo screening cloud in our closed setup is restricted itself
in the ring. To get an experimental access, one possible way is to measure
persistent current (PC) induced by changing the magnetic flux threading
the ring without attaching leads to it. PC is defined as:
\begin{equation}
I=-\frac{e}{\hbar} \frac{\partial
E_{gs}}{\partial \phi}
\end{equation}
PC can be served as a detector for the Kondo screening cloud since
as a function of $\phi$ PC behaves very differently when $\xi_k \gg L$
than when $\xi_k\ll L$.
Such persistent current experiments have been reported recently on micron sized ring without containing the dot\cite{PCexp}. In our setup, PC can be used
to measure the strength of the Kondo correlation as the antiferromagnetic
coupling $K$ is expected to suppress the Kondo effect, leading to a
weaker PC.

\subsection{K=0}

\vspace{5mm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=13cm]{PC_0}
\end{center}
\caption{Persistent current (in unit of $I_0$)
versus magnetic flux (in unit of
$2\pi/\Phi_0$) with $K = 0$.
Red dashed line is for $r=\xi_k^0\sim 0.53$. Black solid line is for
$\xi_k^0/L\sim 1.06$. (a) $L=4n$ (b) $L=4n+1$ (c) $L=4n+2$ (d) $L=4n+3$. Other parameters: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t. Here, we set $t=1$. }
\label{pc0}
\end{figure}

Before we present our new results on the PC, it proves
useful to summarize the previous
results\cite{ring1}\cite{ring2} for $K=0$.
As shown in Fig.(\ref{pc0}), since the system is in the Kondo regime,
PC increases with increasing system size $L$.
PC as a function of the magnetic flux is a weak sinusoidal wave
when $L \ll \xi_k^0$; while it
behaves like a saw-tooth
when $L \gg \xi_k^0$. Meanwhile, there exists a relation between
PC for $L$ and $L+2$: $I_N(\phi) = I_{N+2}(\phi + \pi)$\cite{ring1}
\cite{ring2} due to the $\pi$ shift in LDOS of the quantum dot from
the system with $N$ sites to $N+2$ sites. Furthermore,
the magnitude of PC for $L=odd$ is much smaller than that for $L=even$.
This can be understood as for $L=odd$ the electrons are fully occupied
at the Fermi level, which suppresses the Kondo resonance; while as for $L=even$
there is an unpaired electron on the Fermi level, leading to the
Kondo resonance. The results for $K>0$ are shown below.

\vspace{5mm}
\subsubsection{\textbf{1. Varies $K$ with fixed $L$:}}
\vspace{5mm}

\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=12cm]{PC-K}
\end{center}
\caption{PC versus $\phi$ with various $K$ for $L\sim 240$
($\xi_k^0/L\sim 0.44$). $K = 0, 0.015, 0.03$
(in unit of $t$) (a) $L=4n$ (b) $L=4n+1$ (c) $L=4n+2$ (d) $L=4n+3$. Other parameters of data: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t. Here, we set $t=1$.}
\label{kpc}
\end{figure}

 PC (in unit of $I_0$
where $I_0=e v_F/L$ is the persistent current of an ideal metallic ring
with $v_F$ being Fermi velocity) versus magnetic flux with different antiferromagnetic
coupling strength $K$ at a finite size
$L\sim 240$ ($\xi_k^0/L\sim 0.44$) is shown in Fig.(\ref{kpc}).
The general trend in all four cases is that the
amplitude of PC gets smaller as $K$ is increased and finally
vanishes for $K>K_{c2}$, in agreement
with the expectation on the suppression of the Kondo effect
by the antiferromagnetic spin-spin
coupling. Also, as shown in Fig. \ref{kpc},
the shape of the PC as a function of $\phi$ changes
from saw-tooth shape at smaller $K$ values
to weak sinusoidal waves at
larger $K$ values, similar to that for $K=0$.
There are detail differences among
the four cases.
In Fig.(\ref{kpc}(a) and (c)), with increasing $K$ the persistent
current $I_ {L=240}$ near $0.5\pi< \phi<1.5 \pi$
decreases more slowly than that for $0\pi <\phi <0.5\pi$
and $1.5\pi <\phi <2\pi$; similarily for $I_{L=242}$ except for the range
of $\phi$ is being interchanged.
This suggests that the Kondo effect is more robust in
$0.5\pi< \phi<1.5 \pi$ for $L= 4n$
and in $0\pi <\phi <0.5\pi$
and $1.5\pi <\phi <2\pi$ for $L=4n+2$.
Note that we find the similar
jumps at $K=K_{c1}$ mentioned previously in LDOS to also
appear in PC for $L=4n+1$ and $L=4n+3$; while PC is continuous
at at $K=K_{c1}$ for the other two cases.
Despite the above detail differences, we
find two common features in Fig. (\ref{kpc})
that remain the same as in the case of $K=0$:
 (i). $I_N(\phi) = I_{N+2}(\phi + \pi)$ and (ii).
PC for $L=even$ is larger than that for $L=odd$.
\vspace{5mm}

\subsubsection{\textbf{2. Varies $L$ with fixed $K$:}}
\vspace{5mm}

Fig.(\ref{pcodd}) and Fig.(\ref{pceven}) show PC versus
magnetic flux at different system sizes.
 In the following the discussion is separated into two cases:
(i). $K < K_c$ (case (I)) where PC increases with increasing size $L$, and
(ii). $K > K_c$ (case (II)) where PC decreases to $0$ as $L$ reaches the thermodynamic limit.

\begin{center}
\bf{(i). $K \le K_c$}
\end{center}

\vspace{5mm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=12cm ]{PC-odd}
\end{center}
\caption{Persistent current with various $L$ and fixed $K$ (in unit of $t$).
(a) $L=4n+1$, $K=0.008$ (b) $L=4n+3$, $K=0.008$ (c) $L=4n+1$, $K=0.015$ (d) $L=4n+3$, $K=0.015$. Here, $r=\xi_k^0/L$. The parameters are: $\epsilon_d$ = -0.8t, $t_R$ = $t_L$ = 0.4t, $\phi=0$. Here, we set $t=1$.}
\label{pcodd}
\end{figure}
We first look at cases with $K<K_c$.
Fig.(\ref{pcodd}) shows PC versus magnetic flux with different size $L$
for $K=0.008$ and $K=0.015$ (in unit of $t$).
For $K=0.008$ (see Fig.\ref{pcodd}(a)(b)),
as $L$ is increased we find the PC exhibits two different
behaviors: near $\phi=\pi$ for $L=4n+1$
and  $\phi=0,2\pi$ for $L=4n+3$ PC changes from
the behavior in crossover region to the Kondo state at $K=0$
(see Fig. (\ref{pc0}(b)(d)); while out of these ranges of $\phi$
its behavior remains the same as in the crossover region but
with an increasing amplitude. A first-order jump
is seen to separate these two regions. We expect at much larger system
size the Kondo phase
will be restored eventually over the entire range of $\phi$.
We have also investigated PC for $L=4n,4n+2$ (not shown here)
and find the same qualitative behaviors except that instead of
jumps we find a continuous change in PC.
Also, the behavior in
$I_{4n+1}^{ \phi \sim \pi}$ (or equivalently $I_{4n+3}^{  \phi \sim 0}$)
is the same as that for $K=0$ for all the sizes we
investigate, suggesting the Kondo phase is very easily
restored in these ranges of $\phi$.

On the other hand, for a larger value of
$K=0.015$ (see Fig.\ref{pcodd}(c)(d)), as $L$ increases from $L\sim 60$ to $L\sim 420$ (or $r=\xi_k^0/L$ changes from $1.77$ to $0.25$)
PC still stays in the crossover region with
 an increasing amplitude. This is expected as for larger value of $K$
one must go to much larger system size to
observe the restoring of the Kondo effect in PC.

\begin{center}
\bf{(ii). $K \ge K_c$}
\end{center}

\vspace{5mm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm, height=6cm ]{PC-even}
\end{center}
\caption{Persistent current with various $L$ and fixed $K$.
(a) $L=4n$ (b) $L=4n+2$. Here, $r=\xi_k^0/L$. Other parameters of data: $\epsilon_d$ = -0.8t, $t_R$ = $t_L = 0.4t$, $K=0.036$ (in unit of $t$). Here, we set $t=1$.}
\label{pceven}
\end{figure}

We now investigate the case where $K>K_c$.
As shown in Fig.(\ref{pceven}), since $K>K_{c}$ in this case,
PC decreases in amplitude with increasing
system size $L$ and finally the system reaches the
local spin-singlet state with vanishing PC.
Note that $I_{4n}^{\phi\sim 0}$ and correspondingly
$I_{4n+2}^{\phi\sim \pi}$
are always vanishingly small for all the sizes we investigate,
suggesting the systems are already
in the local spin-singlet states from the start
for these ranges of  $\phi$, which is consistent with
our mean-field phase diagram. However,
for the remaining ranges of $\phi$
the systems start from the crossover region for smaller sizes and
reach finally to the local spin-singlet state at large size.

From above results for PC, it implies that by applying magnetic flux,
we may change the ground state
of our system at finite sizes either from the local spin-singlet
state to the crossover region or from the Kondo phase to the crossover
region.

\subsection{IV. Conclusions}
To summarize, we have studied via large-N slave-boson mean-field approach
the Kondo effect in a side-coupled double-quantum-dot system where
one dot is embedded in a mesoscopic ring. The competition between the
Kondo effect and the antiferromagnetic spin-spin
interaction in this geometry gives rise to
the Kosterlitz-Thouless quantum phase transition with a finite critical
value $K_c$ in the thermodynamic limit.
The mean-field phase diagrams of the model depends on the finite size $L$ (mod $4$):
for $L=4n,4n+1,4n+3$ (case (I))
the crossover occurs for $K<K_c$; while for $L=4n+2$ (case (II)) the
crossover exists for $K>K_c$. To further study how the Kondo screening is suppressed
by the RKKY, we have performed a systematic finite-size analysis on the Kondo temperature $T_k$, the
local density of states on the dot $\rho_{QD}(\omega)$ which connects to the ring , and
the persistent current PC induced by the magnetic flux penetrating through the ring.
For a fixed $K<K_c$, we find all the above quantities flow to the Kondo phase (same phase as in $K=0$)
with increasing
system size:
$T_k$ and PC increase in magnitude and $\rho_{QD}(\omega)$ develops a pronounced
single Kondo peak centered at $\omega=0$; while for $K>K_c$, the flow at finite sizes
is towards to the local
spin-singlet phase where all of the three observables decrease in magnitude
to $0$ with increasing size.
From the finite-size scaling of $T_k$, we have shown that $T_k$ is an universal function
of $1/(LT^*)$ for $1/L<T^*$ where $T^*$ has been unambiguously identified the characteristic
 Kosterlitz-Thouless crossover energy scale:
$T^*= c \widetilde{T_{k}}\exp{[-\pi\widetilde{T_{k}}/(K-K_c)]}$.
For a fixed size $L$, we find the ground state remains in the Kondo phase for $K<K_{c1}$ and
it crossovers to the local spin-singlet phase for $K_{c1}<K<K_{c2}$ and finally reaches the
spin-singlet phase for $K>K_{c2}$. For $L=4n+1,4n+3$, we find the first order jumps in
all of the three observables at the phase boundary between Kondo and the crossover region.
Whether these first order jumps are the artifacts of the large-N mean-field theory is yet
to be clarified. Note that unlike the similar system studied previously on the
side-coupled double-dot system embedded in conduction electron Fermi sea with constant density
of states,
the key finding in this paper is the KT transition with
a finite critical value $K_c>0$ in a side-coupled
double-dot system embedded in a mesoscopic ring (in stead of $K_c=0$ for the
system with Fermi sea of continuous spectrum).
Whether this finite $K_c$ is due to the artifact of the mean-field theory or due to the more
singular density of states of the 1D tight-binding ring needs further investigations. Our results on the transport properties of the system
are relevant for future experiments on the side-coupled
double-quantum-dot system embedded in a mesoscopic ring.\\

\acknowledgements

We are greatful for the useful discussions with
G.M. Zhang, C.S. Chu, J.J. Lin
and J.C. Chen.  We also
acknowledge the generous support from the NSC grant
No.95-2112-M-009-049-MY3, the MOE-ATU program, the NCTS of Taiwan, R.O.C.,
and National Center for Theoretical Sciences (NCTS) of Taiwan.

redpajama_set_name: RedPajamaArXiv


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Inquiries Act 2005
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Constitution of inquiry
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The following is the Schedule inserted into the Interpretation Act...
Part 1 Acts of Parliament
Regulation of Railways Act 1871 (c. 78)
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Ministry of Transport Act 1919 (c. 50)
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Road and Rail Traffic Act 1933 (c. 53)
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Agricultural Marketing Act 1958 (c. 47)
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Mental Health Act 1959 (c. 72)
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National Health Service Act 1977 (c. 49)
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Public Passenger Vehicles Act 1981 (c. 14)
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Care Standards Act 2000 (c. 14)
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Education and Libraries (Northern Ireland) Order 1986 (S.I. 1986/594 (N.I. 3))
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Roads (Northern Ireland) Order 1993 (S.I. 1993/3160 (N.I. 15))
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Repeals and revocations

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The Kalergis Family is raising money for our friend Gary Hensley as he prepares to undergo a double bypass heart surgery.
If you have ever enjoyed a day at Sugarday, you can thank a Hensley! Gary and his family have been an instrumental part of the Kalergis household since the mid-80's, and now it's our turn to circle the wagons and be there for them during this difficult time of Gary's surgery and rehabilitation.
Please consider making a donation. Even small contributions can accumulate to make a big difference in this beloved families life!
Gary's Medical Fund has reached 51% of its goal. Thanks to the likes of David, Tammy, Paloma and many more!


WHEN DO YOU NEED TO HIRE A TAX LAWYER IN CT OR NY?
In a world of CPA's, tax preparers, enrolled agents, bookkeepers, brother in laws, and accountants, it can be difficult to understand know when to hire a tax lawyer instead of those "other guys". Normally, you don't use a tax lawyer to prepare your annual income tax return and they probably cost more, so why would you need a tax attorney's help?
The short answer is that it is often best to use a tax lawyer in two situations. First, if the tax authorities are involved and especially if there is any possibility of a criminal issue. Second, if the tax issue is complex or intertwined with legal issues.
Well, you may need a tax lawyer if you have a dispute with the Internal Revenue Service or any State Tax Agency. The State Tax Authorities in New York and Connecticut can be particularly severe and difficult to work with. In addition, they are different. So you have different considerations if you live in Wilton CT or Pound Ridge NY.
Many tax issues disputes come from the audit of one or several past tax returns. If the IRS, New York or Connecticut notifies you of an audit, you should hire a tax attorney immediately.
In an audit, your tax lawyer can talk to the IRS, New York or Connecticut, handle your audit and help negotiate a settlement. Having legal counsel helps ensure that your matter is handled properly and you don't overpay as a result of your audit.
In some instances, taxpayers ignore letters and warnings from the IRS because they're scared or don't know how to respond. In others, there has been some criminal activity. In those cases, the IRS may threaten you with criminal charges. Bottom line, if you learn or suspect that you're the target of an IRS criminal investigation, you'll want to avoid any contact with the tax authorities, especially in New York or Connecticut, and hire a tax lawyer immediately.
Your tax attorney can talk to the IRS and get them to back off. He will work with the IRS in to avoid criminal charges and represent you in court if you are charged with a tax law crime. In all instances, he is best positioned to protect your liberty.
Lawyers are bound by strict confidentiality rules and have an attorney client priviledge, something other tax advisors are not.
Tax problems are usually a serious matter and must be handled appropriately so it's important to that you've hired the best lawyer for your particular situation.


One bite of this delicious sausage made my husband comment "it's just like I used to get at Tante Rose's." Can't get much better than that! Creamy, lightly spiced. These are perfection.
Didn't know what to expect, but we just loved them.
My Dad was from Switzerland and made and sold Swiss Specialty Meats in his Butcher Shop in Queens, NY in the 1950s. I've also had them many times in Switzerland.
Greetings from Gossau, Switzerland. We just ate local Olma Super-Brats with glazed onions, mâche salad mixed with a chopped hard boiled egg, and Thomy mustard.
We are excited to report that there is almost no difference between the sausages we ate and your product.
Bratwurst and other products are of the best quality and taste is just as good as being at the Olma in St.Gallen and purchasing a Bratwurst at the stand.
Excellent. There is just no other like that!
The sausage has the true authentic flavor of the ones we eat every time we go back home to Switzerland. They, along with Rösti and a salad, are our favorite meal. My husband agrees that these sausages are absolutely delicious and he refuses to share them with our kids!!
Thank you for your great service and for offering such wonderful quality products!
Thank - was superfast service and a great Bratwurst! Even though it is made in the USA, it tasted like an authentic Swiss Olma Bratwurst.
SwissFavorties says: this is a natural casing, an animal product, used for suasages all over the world. It is edible. If you grill the sausage, it is supposed to char and explode, which makes the sausage very tasty.
The bratwurst is too soft, not firm enough. It seems to contain too much liquid. Of course, being from St. Gallen I'm somewhat spoiled because I truly had the words best. But I'm sure this product CAN be improved. The texture as others have noticed is important.
SwissFavorites say agreed in this case. As this is a handmade product it varies. Some have firmer and some lesser firm texture. We strive for perfection and hope on average we still beat the competiton.


Ammunition Depot has made a commitment to providing our customers with an option for buying from a trusted provider, the parts they need to keep their rifles in great working order and as failsafe preparation for those who know they are hard on their guns or who shoot in bulk. We offer not only replacement parts and OEM standard offerings for fixing broken or worn components, but also aftermarket and third-party accessory parts and add-on pieces that can bring your rifle into a whole new level of performance, accuracy, and reliability. We carry parts for the most popular makes and models that cover the spectrum from Ruger to Remington to Winchester to savage and beyond. From action parts to barrels, stocks, gas system and feed system components to small parts, pins and springs – Ammunition Depot is doing what we can to help you find the part you need to fix your favorite old hunting rifle and the parts you want to make your go-to shooter the best it can be.


Fun and private beachside community! Completely renovated inside and out! New siding, roof, decking, flooring, full kitchen, paint, electrical, trim, nearly everything! Too much to list! Enjoy entertaining in the large living room, on the wraparound deck, or steps away at the private beach. Outdoor shower. Community also boasts laundry, playground, basketball court. Close proximity to all long islands north fork ammenities. summer time only, open April 15th through October 15th, land is leased.
Floor Plan Level1: Kitchen, Full Bath, 2 Bedrooms, Large Living Room.


You need to ensure your pool complies (obtain pool safety certificate Form 23) with the new pool safety standard by the 30th November 2015.
From 1st December 2010 - a pool safety certificate Form 23 is required if you sell or lease your property before the time frame above.
You must ensure pool is listed on the state-wide pool register by 4th November 2011.
A pool safety certificate is valid for two years for a non-shared pool (eg. house) or one year for a shared pool (eg. hotel, unit complex).
Shared pool – for shared pool leases, the owner must give the tenant a copy of a pool safety certificate if one is in effect. If there is no certificate in effect, the owner must give the Notice of No Pool Safety Certificate Form 36 to the tenant, to the body corporate and the Department of Infrastructure and Planning. The pool owner must then obtain a pool safety certificate within the 90 days of the accommodation agreement being entered into.
Non-shared pool – before entering into a new or renewed lease agreement the owner must ensure a pool safety certificate is in effect for the pool. The Notice of No Pool Safety Certificate Form 36 cannot be given. Unlike shared pools, there is no 90 day compliance period and there is no phase-in-period.
Before entering into a contract of sale – if no pool safety certificate is in effect before entering into a contract of sale for a property with a pool, the seller must give the prospective purchaser a Notice of No Pool Safety Certificate Form 36.
Before settlement – if a pool safety certificate is in effect, the seller must give the buyer a copy of the certificate before settlement. If a pool safety certificate is not in effect before settlement, the seller must give the purchaser a Form 36 with the settlement date of the form.
After settlement – for non-shared pools, if a certificate is not in effect before settlement, the purchaser has 90 days from settlement to obtain a pool safety certificate.
A pool safety Certificate is only required (before the 30th November 2015) when selling, buying leasing or entering into another type of accommodation agreement for a property with a pool.
For shared pools on properties used exclusively for short-term accommodation , pool owners (eg body corporate) have until 31 May 2011 to obtain a pool safety certificate. For other shared pools, owners have until 30 November 2012 to obtain a pool safety certificate.


text:
Reading Too Much Into 'Mayans MC': Details You May Have Missed From 'Gato/Mis'
Dustin Rowles 10.12.18 9 months ago
Welcome to our weekly breakdown of the minutia of Kurt Sutter and Elgin James' Sons of Anarchy spin-off, Mayans MC. While Kimberly Ricci provides her always excellent coverage of the series (here's her write-up of this episode), here we're going to endeavor to look deeper into the episode and excavate some of the details viewers may have missed, callbacks to Sons of Anarchy, and posit questions explore theories about the direction the series.
Kimberly covered the episode's wildest scenes earlier this week but before we get into the nitty gritty of this episode — and there were some details that may not have been immediately apparent — here's a quick refresher on the storyline:
EZ helps Coco's daughter dispose of a body. Thanks to Chucky's big mouth, Bishop has even more reason to believe that Coco is the rat inside the Mayans. Adelita returns Miguel's baby to him in exchange for $7 million pesos, but Miguel is apprehended for drugs trying to cross back over the border. Adelita, meanwhile, has Miguel's mole killed. Plus, Sons veteran Lincoln Potter delightfully makes an appearance at the end of the episode.
Let's get to it.
1. The episode title, "Gato/Mis," is obviously Spanish and Mayan for "cat," and as is custom, the title animal appears in the first and the last shot.
TOPICS#Sons Of Anarchy
TAGSMAYANS MCSONS OF ANARCHY


text:
Famous Screen Killers Lady Gaga Studied to Play The Countess in 'AHS'
TV / By admin / January 3, 2021 January 3, 2021
Previously American Horror Story: Hotel, Lady Gaga appeared That's City 2: Dame to Kill for and Machete kills, but her role as the Countess in Ryan Murphy's anthology marked her first significant attempt at acting. And, by easily capturing the ruthless, bloodthirsty, sexual vampire he won the Golden Globe for the "Shallow" singer. Gaga prepared heavily for the role, examining a few memorable killers on screen to discover her character.
Lady Gaga | Neilson Barnard / Getty Images
Variety noting that Ryan Murphy wanted Gaga 's character to be "more of a bitch, more sensible, more relaxed." And, Gaga began to make such a part appear "without effort. "So she studied some of the greatest on-screen murderers in cinematic history. She studied Anthony Bkins' Norman Bates from Pyscho and Anthony Hopkins' Lecter Hannibal de The silence of the lambs. She also studied Robert Durst from An Jinx – a six-part documentary series that followed one million suspects (but never charged) of three killings. So let's take a look at each one.
Anthony Perkins as Norman Bates in 'Psycho'
Psycho laying the foundation for the many psychological thrushes with complex endings that would follow the same. Hitchcock's main work was portraying Perkins as a man with Dissociative Identity Disorder (then known as the multiple personality disorder). As the final moments of the film showed, he dressed like his mother and her dead body was still embalmed in the fruit cellar.
Despite the upheaval and murder, Bates was very calm and gathered as a hotel manager. It could be both charming and frightening – moving unnoticed between a sweet youngster and an unexpected aggressive attack. So it made sense that Gaga would study Bates 'performance, since The Countess is a modern and feminine version of a similar duality – switching between courtship and violence.
Anthony Hopkins as Hannibal Lecter in 'Silence of the Lambs'
Anthony Hopkins' s Hannibal Lecter is a cannibalistic serial killer in The silence of the lambs. So there's one immediately obvious connection to Gaga's character AHS. Both the Countess and Lecter feed their victims. And, it is easy to imagine Lady Gaga delivering a scene similar to the fava beans that many remember. Silence, in which the killer talks about preparing food with a victim's liver as the main course.
However, it is Hopkin 's menacing view – a man who seems to be looking through the camera and into the viewer' s soul – that makes his performance inspiring. 'chill. This is a scene that seems to be sitting with Gaga as he discovers the Countess' sense of rest for an oral fight.
Gaga also investigates a three-time murderer suspected of Robert Durst of 'The Jinx'
An Jinx, unlike the films above, it was based on the true story of an American estate heir and the suspected serial killer Robert Durst. Gary Napoli showed a version of the man in HBO miniseries, An Jinx.
In the six-part documentary series The Durst was real, as well as actors portraying the man, and is based on about a decade of exploration. Durst was never convicted. So such a man gave Gaga a different perspective – an outside world full of fiction and into a less cut-off and dry world, guilty of real crime.
Winona Ryder Said She Was 'Embarrassingly Dramatic' After Her Breakup With Johnny Depp
Ben Affleck Says Jennifer Lopez Endured Abuse In the Press Because of Their Relationship
'SNL' Star Pete Davidson Wants His Tattoos 'Burned Off'
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In 2008 the Commission initiated a review of the laws governing the use of assistance animals by people with a disability.
The Victorian Equal Opportunity and Human Rights Commission suggested the review, which was conducted in accordance with section 5(1)(b) of the Victorian Law Reform Commission Act 2000 (Vic). The Act empowers the Commission to initiate inquiries into legal issues of general community concern, provided they are limited in size and scope.
A consultation paper identified inconsistencies and gaps in the regulation of assistance animals and in the laws protecting users of assistance animals. The Commission received 28 submissions and consulted widely. The report on assistance animals was tabled in Parliament on 29 January 2009.
The Commission recommended improving the legal protection for people with disabilities who rely on assistance animals. Amendments were suggested to the Equal Opportunity Act 1995 (Vic) and the Disability Discrimination Act 1992 (Cth) to strengthen the rights of users. The report also recommended establishing a simple regulatory scheme for the training, registration and identification of assistance animals.
At the time the Commission's report was tabled, a review of Victoria's equal opportunity law was under way. The Equal Opportunity Act 2010 (Vic), enacted in response to that review, largely adopted the Commission's recommendation to expand the definition of 'assistance dog' (although it did not adopt the Commission's recommendation on the development of a regulatory scheme for training and accreditation).
The definition of discrimination under the Equal Opportunity Act now clearly specifies that discrimination that occurs because a person uses an assistance dog is discrimination against the person because of a disability. This is in line with the Commission's recommendations and federal law.
The Equal Opportunity Act was also amended to impose express obligations to make 'reasonable adjustments' for a person with a disability in certain areas. For example, whilst not specifically recommended by the Commission, sections 7 and 54 provide that a person must not refuse to provide accommodation to a person with a disability because he or she has an assistance dog.


text:
Q: I made a successful AJAX call to the page it's on, but the php won't execute Maybe the title doesn't make sense...
When loading page.php, it should return content A. But if a user clicks a button 'B' on page.php, that triggers JS which makes an AJAX call to page.php?param=B.
page.php has php code that's supposed to load content B if it gets requested with ?param=B . In the network tab, it shows that a request was indeed sent to page.php?param=B but the code doesn't run.
What essentially I want to do is display data on page.php sorted based on url param.
I hope this makes sense lol.
page.php

<script>
     function sort(selection) {
         //idk y this doesn't work
         let xmlhttp = new XMLHttpRequest();
         xmlhttp.open("GET", "/test.php?sortby=" + selection);
         xmlhttp.send();
     }
</script>

<input type="button" onclick=sort(this.value) value="A"></input>

<?php
if (empty($_GET)) {
//query DB for content, don't sort, display

} else if (isset($_GET['sortby'])) {
if ($_GET['sortby'] === 'A') { //query db for content, sort by A and display }
if ($_GET['sortby'] === 'B') { //query db for content, sort by B and display }
?>

Oh one last note, when I manually enter page.php?sortby=A in the url bar and hit enter, php runs correctly, but when AJAX calls for that same url, php doesn't run.

A: The point of an Ajax request is that the browser does not navigate to a new page, but that the response is handled by JavaScript instead.
You aren't handling the response at all.
You need to wait for the response to load and then do something with it.
function loadHandler() {
    const data = this.response; // https://developer.mozilla.org/en-US/docs/Web/API/XMLHttpRequest/response
    console.log(data);
}

xmlhttp.addEventListener("load", loadHandler);

In the above example I'm just logging it to the console. Normally you would want to insert it into the HTML document.
Generally you wouldn't want to return entire HTML documents from an Ajax request. It is typical to return some data in a JSON structure and use that to update the page.

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Scram's latest insert coin series is available now from all good skateshops.
This series takes you back to the days of the games arcade, retro gaming systems and button mashing. Artwork by Greg Turra.
Available on all sizes and 5 directional shapes in our standard 7ply Generator made decks as well as 4 sizes in Redline Carbon Fibre decks.


text:
IT: 🏛 Impeachment? 25th Amendment? 'Orderly' transition?
Join the 72,174 people who've taken action on Causes this week
Contact Reps
Welcome to Friday, January 8th, electorate...
A Capitol police officer is dead from injuries he suffered during Wednesday's riots.
Brian D. Sicknick died at 9:30 p.m. Thursday evening, according to the department.
"Officer Sicknick was responding to the riots on Wednesday, January 6, 2021, at the U.S. Capitol and was injured while physically engaging with protesters," USCP said in a statement.
"He returned to his division office and collapsed. He was taken to a local hospital where he succumbed to his injuries. The death of Officer Sicknick will be investigated by the Metropolitan Police Department's Homicide Branch, the USCP, and our federal partners."
The House Appropriations Committee, which controls the Capitol police department's budget, has launched a review into what went wrong.
Amidst ongoing criticism over a lack of preparedness, Capitol Police Chief Steven Sund has said he will resign effective January 16th.
How do you feel about this news?
Impeach?
A growing number of lawmakers, including top Democrats in Congress, are calling for the president's impeachment after a violent mob of pro-Trump supporters stormed the Capitol on Wednesday.
On Thursday, Rep. Ilhan Omar (D-MN) posted photos of the Articles of Impeachment and said: "We need to move quickly to remove this President from office."
Politicians on both sides of the aisle have placed the blame for the riot on President Trump.
Would you support impeaching President Trump?
25th Amendment?
Another option: lawmakers are calling for Vice President Mike Pence and the Cabinet to use the 25th Amendment to remove Trump from office.
Senate Minority Leader Chuck Schumer (D-NY) and House Speaker Nancy Pelosi (D-CA) each called for the use of the 25th Amendment in response to what Pelosi called "an act of sedition" by inciting the riot at the Capitol.
The call for invoking the 25th Amendment has already gained some bipartisan support, including from Rep. Adam Kinzinger (R-IL).
Should the 25th Amendment be used to remove Trump and make Pence acting president?
Orderly Transition?
In his return to Twitter, President Trump committed to an "orderly" transition of power a day after his supporters stormed the Capitol.
"I would like to begin by addressing the heinous attack on the United States Capitol," Trump opened the video message. "Like all Americans, I am outraged by the violence, lawlessness, and mayhem. I immediately deployed the National Guard and Federal law enforcement to secure the building and expel the intruders. America is, and must always be a nation of law and order."
"The demonstrators who infiltrated the Capitol have defiled the seat of American democracy. To those who engage in the acts of violence and destruction, you do not represent our country and to those who broke the law, you will pay."
Trump appeared to acknowledge the incoming Biden administration:
"Now Congress has certified the results. A new administration will be inaugurated on Jan. 20. My focus now turns to ensuring a smooth, orderly and seamless transition of power. This moment calls for healing and reconciliation."
What do you think of Trump's remarks?
And, in the End…
Think we all could use some self-care today. So no matter which climate you're in, break out your best lotion and celebrate National Winter Skin Relief Day.
Hey. We're going to be ok.
—Josh Herman
Talk to us via email at editorial [at] causes.com. And don't forget to keep in touch @Causes.
Birthright: A War Story


text:
Vinnie Guidera & The Dead Birds Dig Deep And Let Go For 'Shedding'
Friday, December 28, 2018 | Sacramento, CA
Bert Johnson / Capital Public Radio
Vinnie Guidera & The Dead Birds started as a solo project and evolved into an indie rock trio. Their first record "Lows" was released in November 2015, featuring Vinnie's acoustic guitar fed through an electric guitar setup for a blend of fuzz and twang. In November, Vinnie Guidera & The Dead Birds will release their second album "Shedding," which they produced with help from a crowdfunding campaign.
They spoke with Insight's Randol White about the art and DIY aesthetic.
You can find the latest show information for Vinnie Guidera & The Dead Birds on their events page.
Vinnie Guidera & The Dead Birds Debut 'Shedding' Live In Studio
Sacramento indie rock band Vinnie Guidera & The Dead Birds debut new music from their second album "Shedding." They're hosting an album release party at Blue Lamp on Nov. 17 with Shotgun Sawyer and Eugene Ugly.


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PlanePhD is only as good as its community. Everyone wins if you share.
- Features that would be useful to you.
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- Aircraft specific reviews and articles. Please send us your experiences on your aircraft.
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Zithromax (azithromycin) really is a component of each and every classification of drugs called macrolide anti-biotics. Zithromax fights bacteria from inside the childs body.
Zithromax can seem to be how to deal with diverse unique variations of difficulties a result of bacteria, like for example inhaling problems, epidermis disease, head difficulties, and venereal illness.


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For Further Information & a Quote.
Hussain Architectural Design Ltd (HAD) is a multi-award winning independent architectural practice and have years of experience in preparing immigration accommodation inspection reports to satisfy the requirements of the UK Border Agency for Entry Clearance into the UK. Unlike other independent property inspection providers, Hussain Architectural Design Ltd are qualified building and property experts and prepare accommodation reports for immigration purposes to ensure special care is taken during every step of the report process so that our customers receive only the best service we can offer. When we receive your request, we will immediately set to work, to ensure you receive a fast and professional service.
What is a property inspection report?
A property/housing inspection report is a document required by the UK Border Agency (UKBA) to demonstrate to the Entry Clearance Officer that you have obtained an assessment of the proposed accommodation for the applicant.
The housing inspection report for immigration purposed will be required for all applications for the purpose of settlement.
Children; Parents, grandparents and other dependent relatives.
What do our reports cover?
Our reports run to 4-5 pages in length and we have been producing accommodation reports for solicitors and local authorities since 2006.
Are we qualified to carry out the accommodation inspection report for UK visa purposes?
"Obtain an independent assessment of the number of rooms and occupants of the proposed accommodation. A report from a suitably qualified body certifying that a personal inspection has been carried out and giving details of accommodation inspected is usually acceptable for this purpose. Above all, the Entry Clearance officer must be satisfied that the accommodation will not become overcrowded within the definition of the Housing Act 1985 or that it will not contravene public health regulations."
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Why should I choose to use HAD rather than someone else?
We are the UK's leading providers of UKVI property survey reports. Unfortunately there are companies, especially estate agents offering this service that are not qualified to undertake these inspections. Clients have come to us with their applications declined as the UKVI Entry Clearance Officer was not been satisfied.
We will guarantee that our report will be accepted in writing for you and that we are fully registered, please give us a call to discuss further. We have a 100% pass rate and more than happy for you to confirm this with the Home office.


text:
require 'devise/strategies/authenticatable'

module Devise
  module Strategies
    class SubdomainLogin < Authenticatable

      def subdomain
        request.subdomain
      end

      def valid?
        subdomain.present? && params["user"]
      end

      def authenticate!
        return fail!('Invalid email or password.') unless account = Mtdevise::Account.find_by(subdomain: subdomain)
        return fail!('Invalid email or password.') unless user = account.users.find_by(email: params["user"]["email"])
        return fail!('Invalid email or password.') unless user.valid_password?(params["user"]["password"])
        success! user
      end

    end
  end
end

Warden::Strategies.add(:subdomain_login, Devise::Strategies::SubdomainLogin)

redpajama_set_name: RedPajamaGithub


Dr. Jonathan Randle is a Professor in the Department of English and Dean of the School of Humanities and Social Sciences. Dr. Randle joined MC's faculty in 1999 and currently teaches World Masterpieces, Survey of British Literature, Medieval British Literature, and The Inklings. His areas of expertise include Old & Middle English Language and Literature, Epic Literature, and C.S. Lewis.
Dr. Randle was recognized as the Distinguished Lecturer for the College of Arts and Sciences in 2011.
Dr. Randle attends First Presbyterian Church in Jackson. He lived in England for five years and was a double-major when he attended MC.


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text:
Thousands head back to school in Fresno and across the Valley
It's back to school at Fresno Unified
FRESNO, Calif. (KFSN) -- Jade Lee and her parents are patiently waited outside of Fig Garden Elementary for the first day of school.
She is excited to see her friends again meanwhile her mom Shawnte Beatty is excited to return to routine.
"We make breakfast in the morning, she had some grits and sausage this morning, we watched Mr. Rogers get our minds right," said Beatty.
Summer vacation is coming to an end for more students from the North Valley to the South Valley, as they head back to school.
This year Jade is in the fifth grade. She admits she is a little nervous to meet her new teacher.
"I don't know who she is, I'm still scared, what if she is mean teacher," said Lee.
She is not the only one. 74,000 students returned to 106 schools in Fresno, and Fresno Unified School District superintendent Bob Nelson said many of them feel the same way.
Administrators say teachers are all loaded up with supplies and ready to go for the fall semester.
Thousands of students from Fresno Unified are getting ready for their first day of class.
"We talk a lot about the first day jitters, it is a wonderful opportunity, we know that some of our kids are embarking on their journey toward a lifetime of learning," said Nelson.
Change can be scary but students have plenty to look forward to across the district. Buses started taking off at the crack of dawn, this year all of them are low emissions, but that is not the only upgrade.
"There is a new device called child checkmate and it makes the driver go back and check the passenger compartment of the bus before departing, so no child is left behind," said Reginald Ruben, director of transportation.
Lunch has never looked better, new options are on the menu including orange chicken and a vegetarian burrito. Parents can even rate the food and give the district feedback on their new Nutrislice app.
"Every year we roll out six to seven new menu items which encourage the kids to try something different," said Jose Alvarado director of food services.
In the district, people will notice cosmetic changes at schools, classroom upgrades, and solar panel installations. Staff also underwent active assailant training.
"We just try to make sure that we are as prepared as we can be for any possible safety emergency," said Nelson.
All of this, creating a better school experience and making the first day of school more exciting for Jade and her peers.
educationfresnoback to schoolfresnofresno city collegefresno unified school district
Mayor Dyer plans to add more officers to Fresno Police Department
Armed men rob 2 Fresno grocery stores within an hour, police say
Where to see holiday lights in Fresno and Clovis


text:
Q: Update the fill colour of each shape immediately after it is changed in a loop I'm trying to get Visio to update the fill colour of each shape immediately after I change it.
I've tried using various methods - screenupdate, showchanges, sendkeys "%^g" but nothing works with the colour.  Only changing the screen size by 0.01% forces the app to change the text, which is at least something.
I can step through the code and it works but when I run it none of the colours change until the end.
I'm changing the colour of each object using:
Application.ActiveWindow.Page.Shapes.ItemFromID(servshape(y - 1)).CellsU("Fillforegnd").FormulaU = "RGB(253, 190, 0)"

The code runs through a list of dates and changes the colour of the objects when required, problem is it only shows the changes at the end.
The loop through each item in the list is approximately one second, long enough to see any change. I was hoping there was a simple refresh command but that only seems to work with datarecordsets.
Is there any way of refreshing an object fill colour immediately after changing it?

A: Should work with DoEvents:
Option Explicit

Sub reColorAll()
    Dim pg As Visio.Page
    'Set pg = Application.ActiveWindow.Page
    Set pg = ActivePage ' Probably what you want

    Dim shp As Visio.Shape
    For Each shp In pg.Shapes
        If True Then 'test if shape is one of the ones you want, replace true with test
            If shp.CellExistsU("Fillforegnd", False) Then 'test if cell even exists
                shp.CellsU("Fillforegnd").FormulaU = "RGB(253, 190, 0)"
                DoEvents' force Application to update
            End If

            'Timer to simulate delay, can be removed for your case
            Dim pauseTime As Long
            Dim start As Long
            pauseTime = 1   ' Set duration in seconds
            start = Timer    ' Set start time.
            Do While Timer < start + pauseTime
            Loop
            'End Timer Code

        End If
    Next shp

End Sub

Timer Source: