text:
\section{Introduction}
Bilayer graphene (BLG) rises as a~top-tier candidate material for the upcoming carbon-based electronics \cite{Avo07,Nov12,Bri12} either due to a~tunable band gap \cite{Mac13} or due to topologically-protected quantum channels along domain walls \cite{Mar08,Zha13,LJu15}, recently proposed to host nonlocal Einstein-Podolsky-Rosen pairs \cite{Sch15}. Low-energy physics of BLG is mainly governed by microscopic parameters describing the coupling between the two layers \cite{Mac13}, some of which are still far from being precisely determined.

In the most common Bernal stacking, the leading tight-binding parameters: the intralayer hopping integral between nearest neighbors $t_{0}=3.16\,$eV, and the nearest neighbor interlayer hopping integral $t_{\perp}=0.38\,$eV, are both link to the basic BLG band-structure characteristics, which are the Fermi velocity
\begin{equation}
  \label{vfedef}
  v_F=  \frac{\sqrt{3}\,t_0a}{2\hbar}\approx{}10^6\,\mbox{m/s}
\end{equation}
and the electron effective mass
\begin{equation}
  \label{mefdef}
  m_{\rm eff}=  \frac{t_\perp}{2v_F^2} \approx{} 0.033\,m_e,
\end{equation}
where $a=0.246\,$nm is the lattice constant in a~single layer and $m_e$ is the free electron mass. For the next-nearest neighbor (or {\em skew}) interlayer hopping integral $t'$, the corresponding characteristic is the Lifshitz energy
\begin{equation}
  \label{elidef}
  E_L = \frac{1}{4}\,t_{\perp}\left(t'/t_0\right)^2,
\end{equation}
which can be defined as a value of the electrochemical potential below which the Fermi surface splits into a~four-element manifold. $E_L$ is difficult to be directly determined in the experiment; values of $t'$ obtained from the infrared spectroscopy covers the range from $0.10\,$eV \cite{Mal07} up to $0.38\,$eV \cite{Kuz09}.

On the other hand, in BLG even a~tiny band structure modification near the Dirac point due to $t'\neq{}0$ may have a significant impact on the minimal conductivity \cite{Cse07a}. For $t'=0$, both the mode-matching analysis \cite{Sny07} and the Kubo formalism \cite{Cse07b} lead to
\begin{equation}
  \label{eq:pseudo}
  \sigma_{0}=2\sigma_{\rm MLG}=\frac{8e^2}{\pi{}h},
\end{equation}
where $\sigma_{\rm MLG}$ denotes the universal conductivity of a~monolayer \cite{Kat06a,Two06,Mia07}. For $t'\neq{}0$, similar theoretical considerations show the conductivity $\sigma(L)$ is no longer universal but size-dependent, and monotonically grows with the system size $L$. Depending on the crystallographic orientation of a~sample, the conductivity approaches $(7/3)\,\sigma_{0}\leqslant\sigma(\infty)\leqslant{}3\,\sigma_{0}$ \cite{Mog09}. The transport anisotropy appears as the secondary Dirac cones, present for $t'\neq{}0$, break the rotational invariance of the dispersion relation \cite{Mac06}. In principle, the experimental study of $\sigma(L)$ for clean bilayer samples should be sufficient to determine $t'$ \cite{Rut14a}, also in the presence of small interaction-induced energy gap \cite{Rtt11,Bao12,Rut14b}. (For instance, the value of $\sigma\approx{}2.5\,\sigma_0$ reported by Mayorov {\em et al.}\ \cite{May11} coincides with the above-mentioned large-$L$ predictions.) Unfortunately, device-to-device conductance fluctuations in real disordered systems may put the effectiveness of such a procedure in question. For this reason, new phenomena, unveiling the influence of $t'$ on transport characteristics which can be measured in a single-device setup, are desired.

In this paper, we consider the Corbino geometry, in which a disk-shaped sample is surrounded from both interior and exterior sides with metallic leads [see Fig.\ \ref{setcoraux}(a)]. Such a choice is motivated by the absence of boundary effects, and irrelevance of the sample crystallographic orientation. It is worth to stress that anisotropic aspects of quantum transport are still present in such a~system as the rotational symmetry is intrinsically broken due to trigonal warping. In effect, the total angular momentum is not conserved, and the standard mode-matching method cannot be applied as in simpler systems studied earlier (namely, the monolayer disk \cite{Ryc10,Kat10} and the BLG disk with $t'=0$ \cite{Rut14d}). Here, we  overcome this difficulty by developing a~numerical transfer-matrix approach in the angular momentum space. Contrary to real-space discretization applied in some related works \cite{Two08,Ihn12,Bah13}, our approach takes benefit from the sparsity of transfer matrix in such a~representation allowing highly-efficient (albeit conceptually simple) transport calculations for physical disk diameters exceeding $10\,\mu$m. Recently developed linear-scaling approach employing the time evolution scheme \cite{Ort13} seems to be a possible counterpart but its efficient adaptation for calculating higher charge-transfer cumulants might be difficult.

The paper is organized as follows. In Sec.\ II we present details of the system considered, the effective Dirac equation for low-energy excitations, and provide a~description of our numerical procedure. Sec.\ III focuses on the size dependence of the conductivity $\sigma$, the Fano factor ${\cal F}$, and the third charge-transfer cumulant for BLG Corbino disk at zero magnetic field. In Sec.\ IV, the effects of perpendicular magnetic field on the above-mentioned transport characteristics are discussed. Also in Sec.\ IV, the numerical results for the disk are compared with analytical ones, obtained for an artificial system [a~rectangular device with periodic boundary conditions, see Fig.\ \ref{setcoraux}(b)], for which the mode-matching analysis is possible, and the universal magnetoconductance characteristics, relevant for large system sizes and high magnetic fields, are identified. The conclusions are given in Sec.\ V.

\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{setcoraux}}
\caption{\label{setcoraux}
  Systems discussed in the paper (schematic). (a) The Corbino disk in Bernal-stacked bilayer graphene. The current flows through the disk-shaped area with the inner radius $R_\mathrm{i}$ and the outer radius $R_\mathrm{o}$ in a~perpendicular magnetic field $\mathbf{B}=(0,0,B)$. The coordinate system and the electrostatic potential $U(r)$ (with $r=\sqrt{x^2+y^2}$) are also shown. The leads (white areas) are modeled as infinitely-doped graphene regions ($|U_\infty|\rightarrow\infty$). (b) An artificial (nanotube-like) system formed of a BLG strip of the width $W$, contacted by two electrodes at a distance $L$ in uniform field $B$, upon applying periodic boundary conditions.
}
\end{figure}

\section{The model and the numerical approach}

\subsection{Effective Dirac equation}
The Corbino disk in Bernal-stacked BLG is depicted schematically in Fig.\ \ref{setcoraux}(a). We start our analysis from the four-band low-energy Hamiltonian for $K$ valley \cite{Mac13}, which is given by
\begin{equation}
\label{hameffu}
H=\left(\begin{array}{cccc}
0 & \pi & t_{\perp} & 0 \\
\pi^{\dagger} & 0 & 0 & \nu\pi \\
t_{\perp} & 0 & 0 & \pi^{\dagger} \\
0 & \nu\pi^{\dagger} & \pi & 0
\end{array}\right)+U(r),
\end{equation}
where $\pi=v_{F}\left(p_{x}+ip_{y}\right)=-i\hbar v_{F}e^{i\varphi}\left(\partial_{r}+i\frac{\partial_{\varphi}}{r}-\frac{eB}{2\hbar}r\right)$, with the gauge-invariant momenta $p_j=(-i\hbar\partial_j+eA_j)$ ($j=1,2$) and the symmetric gauge $(A_x,A_y)=(B/2)(-y,x)$ corresponding to the uniform magnetic field parallel to $z$-axis. We have further defined the dimensionless parameter $\nu=t'/t_{0}$, and introduced the polar coordinates $(r,\varphi)$. The potential energy $U(r)$ depends only on $r=\sqrt{x^2+y^2}$, and the remaining symbols are the same as in Eqs.\ (\ref{vfedef}--\ref{elidef}). As mentioned earlier, the available values of $t'$ following from different experimental \cite{Mal07,Kuz09,QLi09} and computational \cite{Min07} approaches are far from being consistent. Magnetotransport through BLG disk with $t'=0$ were discussed in analytical terms in Ref.\ \cite{Rut14d}. In this paper we take the values of $t'$ varying from $0.1\,$eV up to $0.3\,$eV in order to investigate numerically how it affects the system behavior.

For the disk area, $R_{\rm i}<r<R_{\rm o}$, we set $U(r)=0$ and the effective Dirac equation $H\psi=E\psi$ (with $E$ the Fermi energy) can be written as
\begin{equation}
\left(\begin{array}{cccc}
\epsilon & -f & -it & 0\\
-f^{*} & \epsilon & 0 & -\nu f\\
-it & 0 & \epsilon & -f^{*}\\
0 & -\nu f^{*} & -f & \epsilon
\end{array}\right)\psi\left(r,\phi\right)=0,\label{eq:Dirac}
\end{equation}
where $t=t_{\perp}/(\hbar v_{F})\equiv l_{\perp}^{-1},$ $f=e^{i\varphi}\left(\partial_{r}+i\frac{\partial_{\varphi}}{r}-\frac{1}{2l_{B}^{2}}r\right)$,
$\epsilon=E/(\hbar v_{F})$, and the magnetic length $l_{B}=\sqrt{{\hbar}/{(e|B|)}}$.

In the absence of trigonal warping ($\nu=0$) the system possesses a~cylindrical symmetry and the effective Hamiltonian (\ref{hameffu}) commutes with the total angular momentum operator \cite{Per07b}
\begin{equation}
  \label{jztot}
  J_z=-i\hbar\partial_\varphi + \frac{\hbar}{2}
  \left(\begin{array}{cc}
      \sigma_0 & 0 \\
      0 & -\sigma_0
    \end{array}\right) + \frac{\hbar}{2}
  \left(\begin{array}{cc}
      -\sigma_z & 0 \\
      0 & \sigma_z
    \end{array}\right),
\end{equation}
where $\sigma_0$ is the $2\times{}2$ identity matrix, and $\sigma_z$ is one of the Pauli matrices. In such a case, the wavefunctions are products of angular and radial parts $\phi^{m}\left(r,\varphi\right)=e^{im\varphi}\left[\phi_{1}^{m},e^{-i\varphi}\phi_{2}^{m},\phi_{3}^{m},e^{i\varphi}\phi_{4}^{m}\right]^{T}\left(r\right)$ with $m$ being an integer angular-momentum quantum number.

\subsection{Outline of the approach}
In the presence of trigonal warping ($\nu\neq{}0$) the cylindrical symmetry is
broken, and the wavefunctions do not correspond directly to eigenstates of $J_{z}$. A generic workaround has been developed for systems with symmetry-breaking potentials (or impurities), where one can still express wavefunctions as linear combinations of eigenfunctions of an ideal system \cite{dattaArt,souma,tit}. Souma and Suzuki \cite{souma} considered quantum transport through Corbino disks in two-dimensional electron gas and showed that the effects of impurities can be studied numerically, starting from truncated wavefunctions in the basis of angular-momentum eigenstates. Here we adapt this method for BLG Corbino disk, as the term proportional to $\nu$ in the Hamiltonian (\ref{hameffu}) can be regarded as a peculiar type of a symmetry-breaking potential.

A general solution of the Dirac equation (\ref{eq:Dirac}) can be written as an infinite linear combination of angular momentum eigenfunctions, namely
\begin{equation}
  \label{eq:cAnM}
  \psi\left(r,\phi\right)=\sum_{m}a_{m}\phi^{m}\left(r,\varphi\right),
\end{equation}
with arbitrary amplitudes $a_{m}$, $m=0,\pm{}1,\pm{}2,\dots$. Multiplying the Dirac equation (\ref{eq:Dirac}) by the factor
$e^{-il\varphi}$ (with $l$ an arbitrary integer) and integrating over the polar angle $\varphi$, we obtain the system of equations
\begin{eqnarray}
\partial_{r}\phi_{1}^{l} & = & -g\left(l,r\right)\phi_{1}^{l}+i\epsilon\phi_{2}^{l}+i\nu t\phi_{1}^{l-3}\nonumber \\
 &  & -i\nu\epsilon\phi_{3}^{l-3}+2\nu g\left(l-2,r\right)\phi_{4}^{l-3},\nonumber \\
\partial_{r}\phi_{2}^{l} & = & i\epsilon\phi_{1}^{l}+g\left(l-1,r\right)\phi_{2}^{l}-it\phi_{3}^{l},\nonumber \\
\partial_{r}\phi_{3}^{l} & = & g\left(l,r\right)\phi_{3}^{l}+i\epsilon\phi_{4}^{l}+i\nu t\phi_{3}^{l+3}\nonumber \\
 &  & -i\nu\epsilon\phi_{1}^{l+3}-2\nu g\left(l+2,r\right)\phi_{2}^{l+3},\nonumber \\
\partial_{r}\phi_{4}^{l} & = & -it\phi_{1}^{l}+i\epsilon\phi_{3}^{l}-g\left(l+1,r\right)\phi_{4}^{l},\label{eq:uklRow}
\end{eqnarray}
where
\begin{equation}
  \label{eq:glr}
  g\left(l,r\right)=\frac{l}{r}+\frac{r}{2l_{B}^{2}}.
\end{equation}
Notice that the terms proportional to $\nu$ correspond to the mode mixing due to trigonal warping. The $D_{3d}$ dihedral symmetry of these terms (coinciding with the BLG lattice symmetry) results in the fact that equation for $\phi^{l}$ is coupled only to $\phi^{l-3}$ and $\phi^{l+3}$, which tremendously simplifies the numerical integration.

Eqs.\ (\ref{eq:uklRow}) and (\ref{eq:glr}), along with the mode-matching conditions for $r=R_{\rm i}$ and  $r=R_{\rm o}$ (we model the leads as heavily-doped BLG areas), allows us to construct a transfer matrix (see Appendix~A for details) which can be utilized in the Landauer-B\"{u}ttiker formalism in order to calculate the conductivity and other charge-transfer characteristics. Typically, the convergence is reached for the wavefunction truncated by taking $|l|\lesssim{}M$ in Eq.\ (\ref{eq:uklRow}), with $M=25 - 500$ depending on the system size, the strength of the trigonal warping, and the applied field (with the upper value corresponding to $R_{\rm o}\approx{}5\,\mu$m, $t'=0.3\,$eV, and $B\approx{}80\,$T). Other computational aspects are also described in Appendix~A.

\section{Quantum transport dependence on the system size}

\subsection{One-parameter scaling}
A particularly intriguing property, arising from the earlier theoretical study of ballistic transport in BLG with skew interlayer hoppings ($t'\neq{}0$), is the one-parameter scaling \cite{Rut14b}. In the absence of disorder and electron-electron interactions the scaling function
\begin{equation}
  \label{betadef}
  \beta(\sigma)=\frac{d\ln\!\sigma}{d\ln\!L},
\end{equation}
which plays a~central role in conceptual understanding of the metal insulator transition \cite{Efe97}, reproduces the scenario predicted for disordered Dirac systems with Coulomb interaction \cite{Ost10}. Here, the discussion is complemented by calculating the Fano factor $\mathcal{F}$ quantifying the shot-noise power, and the factor $\mathcal{R}$ quantifying the third charge transfer cumulant.

Employing the Landauer-B\"{u}ttiker formula for the linear-response regime one can write \cite{Naz}
\begin{eqnarray}
  \sigma & = &
  g_0\Theta_\alpha\mbox{Tr}\,\mathbf{T},
  \label{landsig} \\
  \mathcal{F} & = &
  \frac{\mbox{Tr}\left[\,\mathbf{T}\left(\mathbf{1}-\mathbf{T}\right)\,\right]}{\mbox{Tr}\,\mathbf{T}},
  \label{landff} \\
  \mathcal{R} & = &
  \frac{\mbox{Tr}\left[\,\mathbf{T}\left(\mathbf{1}-\mathbf{T}\right)\left(\mathbf{1}-2\mathbf{T}\right)\,\right]}{\mbox{Tr}\,\mathbf{T}},
  \label{landrr}
\end{eqnarray}
where $g_0=4e^2/h$ is the conductance quantum (with the factor $4$ following from the spin and valley degeneracies), the dimensionless prefactor ($\Theta_\alpha$) in Eq.\ (\ref{landsig}) is equal to $\Theta_{\rm Cor}=\ln(R_{\rm o}/R_{\rm i})/(2\pi)$ for the Corbino geometry of Fig.\ \ref{setcoraux}(a) or $\Theta_{\rm rec}=L/W$ for the rectangular geometry \cite{Ryc09c}, and
$\mathbf{T}=\mathbf{t}^{\dagger}\mathbf{t}$, with $\mathbf{t}$
the transmission matrix determined via the transfer matrix (see Appendix~A).
It was shown in Refs.\ \cite{Rut14a} and \cite{Rut14b} that the conductivity of ballistic BLG with $t'\neq{}0$ scales with the system size. For a~rectangular geometry, we still get $\sigma\approx{}\sigma_0$ for small systems, while for the larger ones ($L\gtrsim{}500\,$nm) the conductivity can be rationalized as
\begin{equation}
  \label{eq:asymp}
  \sigma(L)\simeq\sigma_{\star}\left[\,
    1-\left(\lambda/L\right)^{\gamma}\,\right],
\end{equation}
with typical parameters
\begin{equation}
\label{asymparam}
  \sigma_\star\approx{}3\,\sigma_0, \ \ \ \
  \gamma\approx{}0.5,
\end{equation}
and $\lambda$ depending both on $t'$ and the sample orientation \cite{Rut14b}. The approximating formula given by Eqs.\ (\ref{eq:asymp}) and (\ref{asymparam}) applies for generic crystallographic orientation of the sample. However, for some particular orientations one obtains different asymptotic behaviors, including a slower power-law convergence to $\sigma_\star$ with the exponent $\gamma\approx{}0.25$ if the current is passed precisely along an armchair direction \cite{Rut14a}, or the oscillating conductivity with the lower bound of $\sigma\lesssim(7/3)\,\sigma_0$ if the current is passed precisely along a zigzag direction \cite{Mog09}. Such issues are absent in the Corbino geometry, allowing one to focus on the universal scaling properties of the material.

\begin{figure}
\includegraphics[width=0.9\linewidth]{sigscallR2pla}
\caption{\label{fig:Minimal-conductivity-of}
Minimal conductivity of unbiased BLG Corbino disk with the radii ratio
$R_{\rm o}/R_{\rm i}=2$  as a~function of the radii
difference $R_{\rm o}-R_{\rm i}$, specified in the units of $\ell$
[see Eqs.\ (\ref{elldef},\ref{ellval}) in the main text],
for different values of $t'$.
Triangles, squares and circles represent the data obtained numerically
for $t'=0.3\,$eV, $t'=0.2\,$eV, and $t'=0.1\,$eV (respectively), with the
lines (dot-dashed, solid and dotted) depicting the approximating
Eq.\ (\ref{eq:asymp}) with best-fitted parameters listed in Table
\ref{tab:Comparison-of-pseudodiffusive}.
The inset in the top panel presents the scaling
function $\beta(\sigma)$ [see Eq.\ (\ref{betadef})],
with $L\equiv{}R_{\rm o}-R_{\rm i}$, extracted numerically from
$\sigma\left(R_{i}\right)$ datasets. (Notice that the best-fitted lines
almost overlap each other.)
The bottom panel is a zoom in, for smaller radii differences, which allows
to depict the region where the actual value of the conductivity may deviate
from Eq.\ (\ref{eq:asymp}).
We further notice that
only selected datapoints from the bottom panel are shown in the top panel
for clarity.
}
\end{figure}

\begin{figure}
\includegraphics[width=0.9\linewidth]{sigscallR3o2pla}
\caption{\label{fig:Minimal-conductivity-R3o2}
Same as Fig.\ \ref{fig:Minimal-conductivity-of} but for the radii ratio
$R_{\rm o}/R_{\rm i}=1.5$. For the parameters in Eq.\ (\ref{eq:asymp})
corresponding to the lines depicted, see
Table~\ref{tab:Comparison-of-pseudodiffusive-R3o2}.
}
\end{figure}

\begin{table}[h]
\caption{
  \label{tab:Comparison-of-pseudodiffusive}
  Least-squares fitted parameters in Eq.\ (\ref{eq:asymp})
  corresponding to the lines in Fig.\ \ref{fig:Minimal-conductivity-of}.
  The last column gives the values of

  $L_{0.01}/\ell$, such that the function given by Eq.\ (\ref{eq:asymp})
  matches the actual conductivity with $1\%$ accuracy.
}
\begin{tabular}{c|c|c|c|c|c}
  \hline\hline
  $\ t'\,$(eV)$\ $ & $\ \ell\,$(nm)$\ $ &  $\ \sigma_{\star}\,(8e^{2}/h)\ $
  & $\ \lambda\,$(nm)$\ $ & $\ \gamma\ $ &
  $\ L_{0.01}/\ell\ $ \\
  \hline
  0.1  &  352  &  0.97  &  97  & $\ \ $0.47$\ \ $ &
  6.21 \\
  0.2  &  176  &  0.97  &  40  &  0.49  &
  5.19 \\
  0.3  &  117  &  0.98  &  21  &  0.48  &
  4.44 \\
  \hline \hline
\end{tabular}
\end{table}

\begin{table}[h]
\caption{
  \label{tab:Comparison-of-pseudodiffusive-R3o2}
  Least-squares fitted parameters in Eq.\ (\ref{eq:asymp})
  corresponding to the lines in Fig.\ \ref{fig:Minimal-conductivity-R3o2}.
  The last column is same as in Table~\ref{tab:Comparison-of-pseudodiffusive},
  but for $R_{\rm o}/R_{\rm i}=1.5$.
}
\begin{tabular}{c|c|c|c|c|c}
  \hline\hline
  $\ t'\,$(eV)$\ $ & $\ \ell\,$(nm)$\ $ &  $\ \sigma_{\star}\,(8e^{2}/h)\ $
  & $\ \lambda\,$(nm)$\ $ & $\ \gamma\ $ &
  $\ L_{0.01}/\ell\ $ \\
  \hline
  0.1  &  352  &  1.01  &  110  & $\ \ $0.48$\ \ $ &
  4.79 \\
  0.2  &  176  &  1.00  &  38  &  0.49  &
  4.29 \\
  0.3  &  117  &  1.02  &  23  &  0.49  &
  3.79 \\
  \hline \hline
\end{tabular}
\end{table}

\begin{figure}
\includegraphics[width=0.9\linewidth]{firCombR2}
\caption{\label{fig:Dependence-of-the}
The Fano factor $\mathcal{F}$ [red solid lines] and the third charge
transfer cumulant $\mathcal{R}$ [blue dashed lines] for the Corbino disk with
$R_{\rm o}/R_{\rm i}=2$ as functions of $R_{\rm o}-R_{\rm i}$.
Skew-interlayer hopping $t'$ is varied between the panels.
For large system size, where the Fabry-Perot oscillations become negligible,
the numerical results obtained in the presence of trigonal warping approach
the pseudodiffusive values of $\mathcal{F}=1/3$ and $\mathcal{R}=1/15$
[horizontal lines].
}
\end{figure}

\begin{figure}
\includegraphics[width=0.9\linewidth]{firCombR3o2}
\caption{\label{fig:Dependence-of-R3o2}
Same as Fig.\ \ref{fig:Dependence-of-the} but for $R_{\rm o}/R_{\rm i}=1.5$.
}
\end{figure}

In the numerical calculations presented in this section, we chose radii ratios $R_{\rm o}/R_{\rm i}=2$ and  $R_{\rm o}/R_{\rm i}=1.5$, for which the number of nonzero transmission eigenvalues $T_l$ is relatively large even at the charge-neutrality point, allowing one to expect some remainders of the pseudodiffusive behavior known for a~monolayer \cite{Two06,Ryc09c}. It is also worth to mention that such radii ratios are close to that of real Corbino device in a~monolayer \cite{Pet14}. A~multimode character of the charge transport, combined with a~mode mixing due to the trigonal warping, and with a~necessity to study large systems in attempt to demonstrate a~one-parameter scaling, provides us with an excellent test case to investigate computational aspects of the numerical approach presented in Section IIB. (For the details, see Appendix~A.)

Zero-magnetic field results are presented in
Figs.\ \ref{fig:Minimal-conductivity-of}, \ref{fig:Minimal-conductivity-R3o2}, \ref{fig:Dependence-of-the}, and \ref{fig:Dependence-of-R3o2}.
In order to present the data for different $t'$ on a~single plot, we have
defined the length $\ell$, related to the distance between primary and
secondary Dirac points in quasimomentum space
\begin{equation}
  \label{elldef}
  \frac{2\pi}{\ell}\equiv{}k_\ell = \frac{\nu}{l_\perp}
  = \frac{2}{3}\sqrt{3}\,\frac{t't_\perp{}}{t_0^2a},
\end{equation}
leading to
\begin{equation}
  \label{ellval}
  \ell{}\,t'=35.2\ \text{nm}\!\cdot\!\text{eV}.
\end{equation}

Briefly speaking, the actual conductivity reaches a close-to-asymptotic behavior, described by Eq.\ (\ref{eq:asymp}) with $L\equiv{}R_{\rm o}-R_{\rm i}$, for radii differences lying in a~relatively narrow interval of $4\lesssim{}L/\ell\lesssim6$ (notice that varying the skew-interlayer hopping from $t'=0.1\,$eV to $t'=0.3\,$eV is equivalent to changing the parameter $\ell$ by a~factor of $3$ between the datasets). For larger $L$, lines in Figs.\ \ref{fig:Minimal-conductivity-of} and \ref{fig:Minimal-conductivity-R3o2}, corresponding to Eq.\ (\ref{eq:asymp}) with the least-squares fitted parameters $\sigma_\star$, $\lambda$, and $\gamma$ listed in Tables~\ref{tab:Comparison-of-pseudodiffusive} and \ref{tab:Comparison-of-pseudodiffusive-R3o2}, are closely followed by the datapoints obtained numerically for $t'=0.1\,$eV, $0.2\,$eV, and $0.3\,$eV. For each case, the specific value of $L_{0.01}$; i.e, the radii difference above which the approximating Eq.\ (\ref{eq:asymp}) matches the actual conductivity with an accuracy better than $1\%$; is also given in Table~\ref{tab:Comparison-of-pseudodiffusive} or \ref{tab:Comparison-of-pseudodiffusive-R3o2}. For smaller $L$, in particular for $L/\ell\sim{}1$ [see bottom panels in Figs.\ \ref{fig:Minimal-conductivity-of} and \ref{fig:Minimal-conductivity-R3o2}], the conductivity becomes nonuniversal (both parameter- and geometry-dependent \cite{lsmallfoo}) approaching $\sigma_0$ for $L\ll{}\ell$.

Although the conductivity strongly deviates from the pseudodiffusive
value $\sigma_0$ (even for the lowest considered value of $t'=0.1\,$eV),
the shot-noise power and the third charge-transfer cumulant are close to their pseudodiffusive values, i.e., $\mathcal{F}\approx{}1/3$ and $\mathcal{R}\approx{}1/15$, which are usually approached for $4\lesssim{}L/\ell\lesssim{}6$ (see Figs.\ \ref{fig:Dependence-of-the} and \ref{fig:Dependence-of-R3o2}).

The approximating Eq.\ (\ref{eq:asymp}) leads, via Eq.\ (\ref{betadef}), to the scaling function of the form
\begin{equation}
  \beta(\sigma)\simeq{}-\gamma{}\left(1-\sigma_\star/\sigma\right).
\end{equation}
In turn, the parametres $\gamma$ and $\sigma_{\star}$ defines the position and the slope coefficient at the attractive fixed point ($\beta(\sigma_\star)=0$, $\beta'(\sigma_\star)>0$) of the renormalization group flow \cite{Rut14b}. The scaling functions $\beta(\sigma)$ [see Eq.\ (\ref{betadef})], calculated numerically for the Corbino disks with $R_{\rm o}/R_{\rm i}=2$ and $R_{\rm o}/R_{\rm i}=1.5$ (see insets in Figs.\ \ref{fig:Minimal-conductivity-of} and \ref{fig:Minimal-conductivity-R3o2}, respectively), $t'=0.1\,$eV, $0.2\,$eV, and $0.3\,$eV, suggest that one-parameter scaling is universal with respect to the strength of the trigonal warping. These numerical results coincide with the corresponding analysis of Ref.\ \cite{Rut14b} for a rectangular sample.

\subsection{Crossover to a quantum-tunneling regime}
It was pointed out that generic MLG billiard with (at least) one narrow opening shows a~crossover to the so-called {\em quantum-tunneling} transport regime  \cite{Ryc09c}, in which charge-transfer characteristics are governed by a~limited number of quantum channels, with transmission probabilities showing a~power-law decay with the system size. In particular, for an {\em undoped} Corbino disk the conductance $G=2\pi\sigma/\ln(R_{\rm o}/R_{\rm i})$ at zero field reads
\begin{equation}
  \label{gmlgdisk}
  \frac{G_{\rm MLG}}{g_0} = \sum_{j=\pm\frac{1}{2},\pm\frac{3}{2},\dots}
  \frac{1}{\cosh^2\left[\,j\,{\ln(R_{\rm o}/R_{\rm i})}\,\right]} \simeq{}
  \frac{8R_{\rm i}}{R_{\rm o}},
\end{equation}
where the asymptotic form applies for $R_{\rm o}\gg{}R_{\rm i}$, and represents contributions from the two channels with angular-momentum quantum numbers $j=-1/2$ and $1/2$.

For BLG billiards, the conformal mapping technique employed in Ref.\ \cite{Ryc09c} cannot be utilized even in the absence of trigonal warping, and the existence of a quantum-tunneling regime is not obvious. For BLG Corbino disk the conductance, for $t'=0$ and at the charge-neutrality point, can be written as \cite{Rut14d}
\begin{align}
  \frac{G_{\rm BLG}(t'=0)}{g_0} &=
  \sum_{m}\left(\,T_m^{+}+T_m^{-}\,\right), \label{gblg0dsk} \\
  T_m^{\pm} &= \frac{1}{\cosh^2\left[\,
      (m\pm{\cal A})\ln\left(R_{\rm o}/R_{\rm i}\right)\,\right]},
  \label{tmpm0dsk}
\end{align}
where the transmission probabilities $T_m^\pm$ (with $m=0,\pm{}1,\pm{}2,\dots$ being the angular-momentum quantum number) correspond to eigenvalues of the matrix ${\bf T}={\bf t}^\dagger{\bf t}$ in Eqs.\ (\ref{landsig}--\ref{landrr}), and we have further defined
\begin{align}
  {\cal A} =&
  -\frac{ \ln\left(\,\Upsilon-\sqrt{\Upsilon^{2}-1}\,\right)}{2\ln\left(R_{\rm o}/R_{\rm i}\right)},
  \label{caladef} \\
  \Upsilon =&
  \cosh\left[\ln\left(\frac{R_{\rm o}}{R_{\rm i}}\right)\right]
  + \frac{ R_{\rm o}^2\!-\!R_{\rm i}^2 }{ 4l_\perp^2 }
  \sinh\left[\ln\left(\frac{R_{\rm o}}{R_{\rm i}}\right)\right].
  \label{upsidef}
\end{align}

We focus now on the system behavior for $R_{\rm o}\gg{}R_{\rm i}\gg{}l_\perp$ \cite{lpefoo}. In such a~parameter range, Eqs.\ (\ref{caladef},$\,$\ref{upsidef}) lead to
\begin{equation}
  \label{calaappr}
  {\cal A} \approx \frac{3}{2} +
  \frac{\ln\left[\,R_{\rm i}/\left(2l_\perp\right)\,\right]}{\ln\left(R_{\rm o}/R_{\rm i}\right)}.
\end{equation}
For any integer ${\cal A}=q\geqslant{}2$ the conductance, analyzed as a~function of $R_{\rm o}/R_{\rm i}$, reaches a~local maximum with $G_{\rm BLG}\approx{}2g_0$, following from the presence of two ballistic channels with $T_{q}^-=T_{-q}^+=1$ [see Eqs.\ (\ref{gblg0dsk}) and (\ref{tmpm0dsk})], occurring at
\begin{equation}
  \label{maxgblg}
  \frac{R_{\rm o}}{R_{\rm i}} \approx
  \left( \frac{R_{\rm i} }{ 2l_\perp } \right) ^{2/(2q-3)}.
\end{equation}
Similarly, Eq.\ (\ref{maxgblg}) with half-odd integer $q\geqslant{}5/2$ approximates a~local conductance minimum (as ${\cal A}\approx{}q$) with $G_{\rm BLG}\approx{}16g_0R_{\rm i}/R_{\rm o}$; i.e., {\em twice} as large as the MLG disk conductance, see Eq.\ (\ref{gmlgdisk}); and four dominant channels characterized by $T_{q+1/2}^-=T_{-q-1/2}^+\approx{}T_{q-1/2}^-=T_{-q+1/2}^+\approx{}4R_{\rm i}/R_{\rm o}$. In turn, a~quantum-tunneling regime reappears periodically when varying $R_{\rm o}/R_{\rm i}$, near any local conductance minimum. The number of well-pronounced minima can roughly be estimated as $0.72\times\ln[\,R_{\rm i}/(2l_\perp)\,]$, as for $R_{\rm o}/R_{\rm i}\lesssim{}4$ the system enters a~multimode pseudodiffusive transport regime. On the other hand, for $R_{\rm o}/R_{\rm i}\gtrsim{}R_{\rm i}^2/(4\l_\perp^2)$, where the threshold value corresponds to the last conductance maximum following from Eq.\ (\ref{maxgblg}) with $q=2$, charge transport is governed by two equivalent channels, with angular momenta $\pm{}2\hbar$ and $T_2^-=T_{-2}^+$ monotonically decaying with increasing $R_{\rm o}/R_{\rm i}$. This suggests the system may re-enter a~quantum-tunneling limit for  $R_{\rm o}/R_{\rm i}\gg{}R_{\rm i}^2/(4\l_\perp^2)$.

In fact, for a~fixed $R_{\rm i}\gg{}l_\perp$ and $R_{\rm o}\rightarrow{}\infty$ we have ${\cal A}\rightarrow{}3/2$, and such a~limit can be regarded as an additional conductance minimum. The asymptotic form of Eq.\ (\ref{gblg0dsk}) then reads
\begin{equation}
  \label{gblgasym}
  \frac{G_{\rm BLG}(t'=0)}{g_0} \simeq \frac{2R_{\rm i}^3}{l_\perp^2 R_{\rm o}}
  \ \ \ \ (R_{\rm i}\gg{}l_\perp,R_{\rm o}\rightarrow{}\infty).
\end{equation}
Although qualitative features of the quantum-tunneling regime for the Corbino geometry are reproduced [in particular, $G_{\rm BLG}(t'=0)\propto{}R_{\rm o}^{-1}$], the asymptotic conductance is elevated by a~large factor of $R_{\rm i}^2/(4\l_\perp^2)$ in comparison to the MLG disk case \cite{gasymfoo}.

\begin{figure}
\includegraphics[width=0.9\linewidth]{gblgnmod}
\caption{ \label{gblgzerofi}
Radii-ratio dependence of the conductance $G=2\pi{}\sigma/\ln(R_{\rm o}/R_{\rm i})$ (top panel) and the effective number of transmission channels (bottom panel) at a~fixed $R_{\rm i}=15\,\l_\perp\approx{}24\,$nm and varying $t'$ (specified for each line). Vertical lines mark the values of ${\cal A}\approx{}3$ (left), ${\cal A}\approx{}5/2$ (middle), and ${\cal A}\approx{}2$ (right) following from Eq.\ (\ref{calaappr}). The corresponding results for two decoupled MLG disks ($t_\perp=t'=0$) are also shown.

}
\end{figure}

In Fig.\ \ref{gblgzerofi}, we compare the conductance (in the top panel) and the effective number of transmission channels  (in the bottom panel)
\begin{equation}
  \label{nchandef}
  N_{\rm ch}=\frac{G}{g_0\left(1-{\cal F}\right)}=
  \frac{\left(\sum_l T_l\right)^2}{\sum_l T_l^2},
\end{equation}
with the index $l\equiv(m,\pm)$ accounting for angular-momenta and layer degrees of freedom, following from Eqs.\ (\ref{gblg0dsk}--\ref{upsidef}) [red lines] for $t'=0$; as well as for the case of decoupled MLG disks ($t_\perp=t'=0$) [blue lines]; with the numerical results obtained from Eqs.\ (\ref{landsig}) and (\ref{landff}) for $t'=0.1\,$eV, $0.2\,$eV, and $0.3\,$eV [remaining lines]. The inner radius is fixed at $R_{\rm i}=15\,l_\perp$, and the outer radius is varied in the range of $1.5<R_{\rm o}/R_{\rm i}<80$.
The effects of trigonal warping are visible for all radii ratios considered, and become particularly significant when approaching $R_{\rm o}/R_{\rm i}=R_{\rm i}^2/(4l_\perp^2)\approx{}56$, corresponding to ${\cal A}\approx{}2$ [following from Eq.\ (\ref{calaappr})]. In such a~parameter range, $G$ systematically decreases, whereas $N_{\rm ch}$ systematically increases when enlarging $t'$. We also notice that Fabry-Perot resonances, corresponding to integer $k_\ell(R_{\rm o}-R_{\rm i})$, are visible for $t'\neq{}0$, indicating the contribution from secondary Dirac points.

For smaller radii ratios, including $R_{\rm o}/R_{\rm i}=7.5$ (corresponding to ${\cal A}\approx{}5/2$) and  $R_{\rm o}/R_{\rm i}=3.83$ (corresponding to ${\cal A}\approx{}3$) [see vertical lines in Fig.\ \ref{gblgzerofi}], the system is close to the pseudodiffusive charge-transport regime. In the $t'=0$ case, the conductance minimum  is shifted from $R_{\rm o}/R_{\rm i}=7.5$ to $R_{\rm o}/R_{\rm i}\approx{}10$ due to the influence of transmission channels with higher angular momenta. (We further notice that the effective number of channels has a~local maximum $N_{\rm ch}\approx{}4.2$ at $R_{\rm o}/R_{\rm i}\approx{}7.5$, where it also precisely matches the value for two decoupled MLG disks, in good agreement with predictions for a~quantum-tunneling regime reported earlier in this subsection.) The trigonal warping noticeably enhances the conductance for $R_{\rm o}/R_{\rm i}\lesssim{}10$; i.e., the effect is opposite then for larger $R_{\rm o}/R_{\rm i}$; with some exception for the smallest considered $t'=0.1\,$eV and $R_{\rm o}/R_{\rm i}\lesssim{}7$, as the disk diameter $2R_{\rm o}\lesssim{}\ell$ in such a~case.

The evolution of $G$ and $N_{\rm ch}$ with the trigonal warping strength, illustrated in Fig.\ \ref{gblgzerofi}, clearly shows that the role of two transmission channels with angular momenta $\pm{}2\hbar$, prominent for $t'=0$ and large $R_{\rm o}/R_{\rm i}$, is strongly suppressed for $t'\neq{}0$, indicating the gradual crossover to a~quantum tunneling regime characterized by $G\propto{}R_{\rm o}^{-1}$. We attribute it to the fact that a~$d$-wave symmetry of normal modes in leads with $m=\pm{}2$ does not match the $D_{3d}$ dihedral symmetry of the low-energy Hamiltonian for $t'\neq{}0$. Moreover, for the two largest considered values of $t'=0.2\,$eV and $0.3\,$eV, both $G$ and $N_{\rm ch}$ are noticeably amplified in comparison to the relevant characteristics for two decoupled MLG disks, signalling the role of quantum states close to the secondary Dirac points is important when the crossover to a~quantum-tunneling regime occurs.

\section{Magnetotransport characteristics}

\subsection{Analytically soluble disk-shaped systems}
Our discussion of the magnetotransport characteristics starts from pointing out that the influence of a~uniform magnetic field $B$  \cite{unifoo} on analytical results given by Eqs.\ (\ref{gmlgdisk}) and (\ref{gblg0dsk}) can be expressed in a~very compact form: Namely, it is sufficient to replace $j$ and $m$ in arguments of hyperbolic cosine by \cite{Kat10,Ryc10,Rut14d}
\begin{equation}
  \overline{\jmath} = j + \frac{\Phi_D}{\Phi_0} \ \ \ \text{and} \ \ \
  \overline{m} = m + \frac{\Phi_D}{\Phi_0},
\end{equation}
with
\begin{align}
  \Phi_D &= \pi{}B(R_{\rm o}^2-R_{\rm i}^2) \label{phddef} \\
  \text{and} \ \ \ \
  \Phi_0 &= \frac{2h}{e}\ln(R_{\rm o}/R_{\rm i}), \label{ph0def}
\end{align}
denoting the flux piercing the disk area ($\Phi_D$) and the basic period of magnetoconductance oscillations ($\Phi_0$).

All the considered charge-transfer characteristics $\sigma$ (\ref{landsig}), ${\cal F}$ (\ref{landff}), and ${\cal R}$ (\ref{landrr}), are predicted theoretically to show periodic oscillations with the magnetic flux $\Phi_D$ [see Eq.\ (\ref{phddef})] piercing a~graphene-based Corbino disk \cite{Ryc10,Kat10,Rut14d,Rut15a}, provided the Fermi energy corresponds to the Dirac point or to any other Landau level (LL). The oscillations appear due to a~limited number of transmission channels for $R_{\rm o}\gg{}R_{\rm i}$, and show a~formal analogy with similar effects discussed for a nanotube in a magnetic field applied along the axis \cite{Kol12}. One should notice, however, that the oscillation period $\Phi_0$ [see Eq.\ (\ref{ph0def})] for a~disk in a~uniform, perpendicular field, corresponds to a~physical field of $18\,$mT for a~typical $1\,\mu$m disk ($R_{\rm o}=5R_{\rm i}=500\,$nm), while for a~$1\,$nm diameter nanotube in axial field the period, given by the standard Aharonov-Bohm flux quantum $\Phi_{\rm AB}=h/e$, corresponds to $B\approx{}5300\,$T.

Similar effects  were also considered for graphene disks with strain-induced pseudomagnetic fields \cite{Kha13} and with the spin-orbit coupling \cite{Vil14}, extending the list of different theoretical proposals for producing a valley polarization with graphene-based nanostructures \cite{Rec07,Ryc07,Akh08,Fuj10,Gun11,Gru14} and related systems \cite{Pal11,Cul12,Gon13,HXu15}.

For MLG disks, the oscillations magnitude depends only on the radii ratio $R_{\rm o}/R_{\rm i}$. For the conductivity, we have $\Delta{}\sigma_{\rm MLG}\gtrsim{}0.1\sigma_{\rm MLG}$ if $R_{\rm o}/R_{\rm i}\geqslant{}5$. For BLG disks, the mode-matching analysis for the $t'=0$ case \cite{Rut14d} unveils an interference between the two transmission channels for each angular momentum eigenvalue, following from the coupling between the layers quantified by $t_\perp$. In turn, the oscillations magnitude depends also on the physical system size. For instance, conductivity oscillations are predicted to vanish ($\Delta\sigma=0$) for
\begin{equation}
  \label{condvani}
  \frac{R_{\rm o}}{R_{\rm i}}\simeq{}
  \left(\frac{R_{\rm i}t_\perp}{2\hbar{}v_F}\right)^{4/p}
  \ \ \ \ (\text{for }R_{\rm o}\gg{}R_{\rm i}),
\end{equation}
where $p=1,2,3,\dots$. Eq.\ (\ref{condvani}) with {\em even} $p$ [equivalent to half-odd integer $q$ in Eq.\ (\ref{maxgblg})] gives the condition for maximal oscillations, with the magnitude $\Delta\sigma=2\Delta\sigma_{\rm MLG}$, same in the limit of decoupled layers ($t_\perp\rightarrow{}0$).

If the Fermi energy $E$ is close but not precisely adjusted to the Dirac point, the oscillations in both MLG and BLG disks are still predicted to appear in the limited range of magnetic fluxes, namely
\begin{equation}
  \label{phidmax}
  |\Phi_D|\leqslant{}\Phi_D^{\rm max}\simeq{}-\frac{2h}{e}\ln(k_FR_{\rm i}),
  \ \ \ \ (\text{for }k_FR_{\rm i}\ll{}1),
\end{equation}
with $k_F=|E|/(\hbar{}v_F)$, away from which the conductivity is strongly suppressed. Eq.\ (\ref{phidmax}) can be rewritten to obtain the corresponding energy range for a given field
\begin{equation}
  \label{efermax}
  |E|\lesssim{} \frac{\hbar{}v_F}{R_{\rm i}}
  \exp\left( -\frac{R_{\rm o}^2-R_{\rm i}^2}{l_B^2} \right).
\end{equation}
The limits given by Eqs.\ (\ref{phidmax}) and (\ref{efermax}) essentially applies to higher charge-transfer cumulants as well, albeit the dimensionless characteristics ${\cal F}$ and ${\cal R}$ were recently found to show stable, quasiperiodic oscillations in the high source-drain voltage limit \cite{Rut15a}.

Later this Section, we employ the numerical procedure described in Section II in order to find out how the magnetotransport characteristics of BLG disks are affected by the trigonal warping ($t'\neq{}0$).

\subsection{Rectangular BLG device with periodic boundary conditions}
Before discussing magnetotransport of the Corbino disk, it is instructive to consider a simpler artificial system depicted schematically in Fig.\ \ref{setcoraux}(b). A BLG strip of width $W$, contacted by the electrodes at a distance $L$, in a~uniform field $B$, and with periodic boundary conditions in the transverse direction, was earlier discussed in the $W\gg{}L$ limit \cite{Rut14a}, in which the pseudodiffusive charge transport is predicted to appear near LLs. Here we primarily focus on the $W\lesssim{}\pi{}L$ range (a nanotube-like geometry) which do not seem to have a~direct physical analogue, but can be treated in analytical terms and possesses a discrete spectrum of transmission channels closely resembling the situation in the Corbino disk.

The wavefunctions for a rectangular sample are presented in Appendix~B. Each spinor component can be written as a product of the exponential function and the Airy function, with their arguments scaling as $l_B^{-2}\propto{}B$ for high fields; see Eq.\ (\ref{phiksamp}). In turn, taking the asymptotic form of the Airy function $\mbox{Ai}\left(z\right)\simeq\mbox{exp}\left(-2z^{3/2}/3\right)/\left(2z^{1/4}\sqrt{\pi}\right)$, we find the conductivity in the high-field limit can roughly be approximated by
\begin{multline}
  \sigma_{L\gg{}l_B} \approx \frac{g_0L}{W}\sum_{k,\pm}
  \mbox{cosh}^{-2}\left\{ L\left(k-\frac{L}{2l_{B}^{2}}\right)\right. \\
  \left.\pm\frac{1}{24}\Re\left[\gamma_{k}\left(L\right)
      -\gamma_{k}\left(0\right)\right]\right\}, \label{eq:Gapprox}
\end{multline}
where $k=0,\pm{}2\pi/W,\pm{}4\pi/W,\dots$,
\begin{equation}
  \label{eq:gammak}
  \gamma_{k}\left(x\right)=
  \sqrt{-\frac{i\nu{}l_B}{l_\perp}}\, \left(\frac{8x}{l_B}-8kl_B-\frac{i\nu{}l_B}{l_\perp}\right)^{3/2},
\end{equation}
and $\Re\left(z\right)$ denotes the real part of $z$. In the absence of trigonal warping, Eq.\ (\ref{eq:Gapprox}) is replaced by exact expression for the conductivity, namely
\begin{multline}
  \label{eq:Gexact}
  \frac{\sigma_{\rm rec}(t'\!=\!0)}{\sigma_0} =
  1 + \sum_{n=1}^{\infty}{\Bigg\{} \left(\frac{\pi{}nW}{L}\right)\cos(nk_cW) \\
  \times\left.
  \left[{\sinh\left(\displaystyle\frac{\pi{}nW}{2L}\right)}\right]^{-1}
  \cos\left(\frac{\pi{}ne}{h}LWB\right)\right\},
\end{multline}
where $k_c=(1/L)\,\mbox{arsinh}\left[\,Lt_\perp/(2\hbar{}v_F)\,\right]$.
A~Fourier series on the right-hand side represents periodic (approximately sinusoidal for moderate aspect ratios $W/L\gtrsim{}2$) magnetoconductance oscillations with the period ${\cal T}_0=(2h/e)(LW)^{-1}$ \cite{Jackiw}.

For $t'\neq{}0$, the approximating expression given by Eqs.\ (\ref{eq:Gapprox},\ref{eq:gammak}) is particularly convenient when extracting the beating frequency which governs high-field magnetotransport characteristics of the system. It can be shown that the presence of two relevant transmission eigenvalues for each momentum $k$ in Eq.\ (\ref{eq:Gapprox}) follows directly from the fact that the lowest LL in BLG has an additional twofold degeneracy absent for higher LLs \cite{Rut14a}. We define the two field-dependent effective system sizes determining transmission probabilities in Eq. (\ref{eq:Gapprox}), namely
\begin{equation}
  \label{lenpmdef}
  L_\pm=L\mu_\pm,
\end{equation}
and the corresponding momentum-quantization shifts
\begin{equation}
  \label{delkpm}
  \Delta{}k_\pm=\frac{L}{2l_B^2\mu_\pm},
\end{equation}
where
\begin{equation}
  \mu_\pm=1\pm{}

  \sqrt{\frac{8\nu l_B^2}{9Ll_\perp}}.
\end{equation}

The above reasoning holds true for high magnetic fields and arbitrary $W/L$ ratio. For instance, the conductivity in the $W\gg{}L$ limit can be approximated by
\begin{equation}
  \label{eq:Gapprox2}
  \sigma_{W\gg{}L\gg{}l_B}\approx{}
  \frac{2g_0}{\pi{}}\left(\frac{1}{\mu_+}+\frac{1}{\mu_-}\right)=
  \sigma_0 \left(1-\frac{8\nu l_B^2}{9Ll_\perp}\right)^{-1},
\end{equation}
restoring the $t'=0$ value ($\sigma\rightarrow\sigma_0$) for $l_B\rightarrow{}0$.

\begin{figure}
\includegraphics[width=0.8\linewidth]{sigmarec}
\caption{\label{reclowfi}
  Conductivity of the rectangular BLG sample at the charge-neutrality point as a function of the magnetic field $B$. The sample length is fixed at $L=300\,l_\perp\simeq{}480\,$nm, the width is $W=10L$, and the skew-interlayer hopping integral $t'$ is varied between the panels. {Red solid lines:} exact numerical results obtained from the mode-matching analysis via Eq.\ (\ref{landsig}). {Blue solid lines:} results from  Eqs.\ (\ref{eq:Gapprox},\ref{eq:gammak}) for $L\gg{}l_B$. {Dashed lines:} results from Eq.\ (\ref{eq:Gapprox2}) for $W\gg{}L\gg{}l_B$.
}
\end{figure}

In Fig.\ \ref{reclowfi} we compare exact numerical results obtained from Eq.\ (\ref{landsig}) for $W=10L=3000\,l_\perp$ and different values of skew-interlayer hopping $t'$ [red solid lines] with corresponding results following from the approximating Eqs.\ (\ref{eq:Gapprox},\ref{eq:gammak}) [blue solid lines] and Eq.\ (\ref{eq:Gapprox2}) [dashed lines]. For low magnetic fields, the conductivity monotonically grows with increasing $B$ (for any $t'\neq{}0$), up to the maximal value (at $B=B_{\rm peak}$) which may exceed $3\sigma_0$ for larger $t'$-s. For $B>B_{\rm peak}$, the effect of trigonal warping on the conductivity is gradually suppressed, leading to $\sigma-\sigma_0\propto{}t'/B$, in agreement with the approximating Eq.\ (\ref{eq:Gapprox2}).

The value of $B_{\rm peak}$ is related to the quasimomentum shifts given by Eq.\ (\ref{delkpm}). For a finite aspect ratio $W/L$, maximal conductivity appears when the average shift is of the same order of magnitude as the distance between primary and secondary Dirac points [given by Eq.\ (\ref{elldef})], namely
\begin{equation}
  \left(\frac{\Delta{}k_++\Delta{}k_-}{2}\right)_{\rm peak}\sim{}\,k_\ell,
\end{equation}
what can be rewritten as
\begin{equation}
  \label{bpeakrect}
  B_{\rm peak}\sim{}\frac{2h}{e}\frac{1}{L{\ell}}\equiv{}B_L,
\end{equation}
where  we have defined the crossover {\em Lifshitz field} $B_L$. It is also visible in the top panel of Fig.\ \ref{reclowfi} that the approximating Eqs.\ (\ref{eq:Gapprox},\ref{eq:gammak})  [blue dashed lines] reproduces the peak position (albeit not the maximal conductivity) with a~good accuracy for $t'=0.3\,$eV.

The conductivity maximum at $B_{\rm peak}\sim{}B_L\neq{}0$ can be regarded as the first effect of the trigonal warping, appearing for samples with finite aspect ratios, but still well-visible for $W/L=10$.

\begin{figure}
\includegraphics[width=0.9\linewidth]{colrecT}
\caption{\label{rectbeats}
  Magnetoconductance oscillations for the rectangular BLG sample of the length
$L=45\,l_\perp\simeq{}72\,$nm and the width $W=2L$ at the charge-neutrality
point. Top panel: The conductivity as a function of magnetic field for three
values of the skew-interlayer hopping integral $t'$ (specified for each line).
The lines for $t'=0.3\,$eV and $t'=0.2\,$eV are shifted by $1$ and $1/2$
for clarity.
Bottom panel:
Consecutive periods of the beating envelope extracted from the data
shown in the top panel, for $t'=0.1\,$eV (circles), $t'=0.2\,$eV (squares),
and $t'=0.3\,$eV (diamonds).
Solid lines correspond to the approximating Eq.\ (\ref{eq:envper}).
}
\end{figure}

The second effect, present in systems with $W\lesssim{}\pi{}L$, is the emergence of beating patterns. In the top panel of Fig.\ \ref{rectbeats} we display the conductivity, as a~function of magnetic field, for $W=2L=90\,l_\perp$. Quasiperiodic beatings are characterized by the field-dependent envelope period, which can be approximated by
\begin{equation}
  \label{eq:envper}
  \mathcal{T}(B) \approx \frac{4h}{eLW}\frac{\mu_{+}\mu_{-}}{\mu_{+}-\mu_{-}}
  \simeq  \frac{3h}{e}\frac{1}{LW}\left(\frac{B}{B_L}\right)^{1/2},
\end{equation}
while the period of internal oscillations remains the same as in Eq.\ (\ref{eq:Gexact}).  The comparison between the actual envelope periods refined from the numerical data [datapoints] and the values following from Eq.\ (\ref{eq:envper}) [lines] is provided in the bottom panel of Fig.\ \ref{rectbeats}.

For high magnetic fields, Eq.\ (\ref{eq:envper}) leads to $\mathcal{T}(B)\propto\sqrt{B}$. In turn, the $t'=0$ behavior characterized by single-frequency, sinusoidal magnetoconductance oscillations with a~size-dependent amplitude [see Eq.\ (\ref{eq:Gexact})] cannot be restored for any $t'\neq{}0$ in the high-field limit. We interpret this effect as a~finite-system analog of the zero-temperature conductance instability (the parameter-driven Lifshitz transition)  in bulk BLG samples \cite{Cse07a}.

\begin{figure}
  \includegraphics[width=0.8\linewidth]{rowCorbR2tpAll}
  \caption{\label{pdiffdisk}
    Conductivity $\sigma$ [$\,$specified in the units of $8e^2/h\,$], the Fano
    factor $\mathcal{F}$, and the $\mathcal{R}$ factor for BLG Corbino disk
    with $R_{\rm i}=300\,l_\perp$, $R_{\rm o}=2R_{\rm i}$, and different
    skew-interlayer hopping integrals $t'$ (specified for each panel).
    Vertical lines correspond to $B=B_L$ [see Eq.\ (\ref{bpeakcorb})],
    the horizontal lines mark the pseudodiffusive values
    ${\cal F}=1/3$ and ${\cal R}=1/15$, the dashed lines
    represent the approximating Eq. (\ref{eq:przewBC}).
  }
\end{figure}

\begin{figure}[h]
  \includegraphics[width=0.9\linewidth]{corcol48}
  \caption{\label{qbeatdisk}
    Charge-transfer characteristics $\sigma$, $\mathcal{F}$, and $\mathcal{R}$
    for BLG disk with $R_{\rm i}=15\,l_\perp$, $R_{\rm o}=4.84\,R_{\rm i}$.
    Each of the main panels compares the numerical results for $t'=0$
    (blue line) and $t'=0.3\,$eV (red line).
    The vertical grey line in the top panel marks the value of $B_L$ obtained
    from Eq.\ (\ref{bpeakcorb}) for $t'=0.3\,$eV.
    The inset: Period of the beating envelope for three different
    values of $t'=0.3\,$eV (circles), $0.2\,$eV
    (orange squares), $0.1\,$eV (triangles).
    Solid lines correspond to Eq. (\ref{eq:okrCor}).
  }
\end{figure}

\subsection{BLG Corbino disks}
Our numerical study of magnetotransport through BLG Corbino disks focuses on two different systems: First one, characterized by $R_{\rm i}=300\,l_\perp$ and $R_{\rm o}=2R_{\rm i}$ is in the pseudodiffusive charge-transport regime, whereas the second one, with $R_{\rm i}=15\,l_\perp$ and $R_{\rm o}=4.84\,R_{\rm i}$, shows the beating patterns. In the latter case, the parameters are chosen such that the magnetoconductance oscillations vanish for $t'=0$ (see Ref.\ \cite{Rut14d}), in order to illustrate the role of trigonal warping more clearly.

The numerical results for $\sigma$, ${\cal F}$, and ${\cal R}$ are presented, as functions of magnetic field $B$, in Figs.\ \ref{pdiffdisk} and \ref{qbeatdisk}.

In the pseudodiffusive transport regime (see Fig.\ \ref{pdiffdisk}) some irregular fluctuations, visible for all the discussed charge-transfer characteristics (and all three values of $t'$), are suppressed for magnetic fields $B\gtrsim{}0.5\,$T. We attribute these fluctuations rather to a~Fabry-Perot interference than to the angular momentum quantization. A striking feature of the data presented in Fig.\ \ref{pdiffdisk} is that a distinct conductance peak appears at each panel near the field
\begin{equation}
  \label{bpeakcorb}

  B_L= \frac{4}{3}\sqrt{3}\,
  \frac{\hbar{}\,t't_\perp}{e\,t_0^2a\left(R_{\rm o}-R_{\rm i}\right)},
\end{equation}
closely resembling the phenomena described above for a~rectangular sample. [Notice that the rightmost equality in Eq.\ (\ref{bpeakrect}) and Eq.\ (\ref{bpeakcorb}) are equivalent provided that $L\equiv{}R_{\rm o}-R_{\rm i}$]. Above the crossover field $B_L$, the conductivity can be approximated by the formula
\begin{equation}
  \label{eq:przewBC}
  \sigma_{B\gg{}B_L}\approx{}

  \sigma_0\left(1-0.5\,\frac{B_L}{B}\right)^{-1},

\end{equation}
which is visualized with  dashed lines in Fig.\ \ref{pdiffdisk},
and can be regarded as a version of Eq.\ (\ref{eq:Gapprox2}) for BLG disk.

For the disk with larger radii ratio (see Fig.\ \ref{qbeatdisk}) all three magnetotransport characteristics exhibit quasiperiodic beating patterns for $t'=0.3\,$eV. This is not the case for $t'=0$, when each characteristic shows approximately sinusoidal, single-frequency, oscillations with a~constant amplitude. As the system no longer possesses a~rotational symmetry in the presence of trigonal warping, the magnetotransport cannot be simply rationalized by defining two effective system sizes in analogy to Eq.\ (\ref{lenpmdef}). Comparing to the rectangular system case (which can be regarded, due to the periodic boundary conditions, as a~nanotube-like, i.e., possessing the rotational symmetry also for $t'\neq{}0$) illustrated in Fig.\ \ref{rectbeats}, the beatings are slightly less regular now, clustering in the groups of three. This certain feature of the data displayed in Fig.\ \ref{qbeatdisk} suggests the presence of four, rather than two, quasiperiodic components determining the transmission probabilities.

It is worth to stress here that the main features of the magnetotransport characteristics still resemble the rectangular (or a~nanotube-like) system case. In particular, we found that the beating-envelope period can now be approximated by
\begin{equation}
  \label{eq:okrCor}
  {\cal T}(B)\approx
  \frac{6.9}{\pi(R_{\rm o}^2-R_{\rm i}^2)}\,\frac{h}{e}
  \left(\frac{B}{B_L}\right)^{1/2},
\end{equation}
see the inset in Fig. \ref{qbeatdisk}. On the other hand, the period of the internal oscillations, ${\cal T}_0=\Phi_0/\left[\pi({R_{\rm o}}^2-{R_{\rm i}}^2)\right]$, remains the same as the basic period for MLG disks or BLG disks with $t'=0$ \cite{Ryc10,Kat10,Rut14d}. Also, a high-field behavior of the conductivity averaged over consecutive intervals, each one of the ${\cal T}_0$ width, can be approximated by Eq.\ (\ref{eq:przewBC}).

\section{Conclusions}
We have investigated, by means of numerical transfer-matrix approach in the angular-momentum space, the effects of the skew-interlayer hopping integrals (the trigonal warping) on selected transport characteristics of bilayer-graphene (BLG) Corbino disks. Additionally, the analytical mode-matching  for an artificial (nanotube-like) system, formed of a~BLG strip upon applying the periodic boundary conditions, was briefly presented and the analogies between these two systems were put forward.

If the Fermi energy is close to the charge-neutrality point, both the scaling behavior at zero magnetic field (which would require a comparison between devices of different sizes in an experimental study) and the single-device magnetotransport discussion unveils several phenomena, in which transport characteristics, such as the conductivity, the Fano factor, and the third charge-transfer cumulant, are noticeably affected by the trigonal warping.

In the pseudodiffusive transport regime, corresponding to the disk radii ratios $R_{\rm o}/R_{\rm i}\lesssim{}2$, the conductivity shows a~one-parameter scaling, in agreement with predictions of Ref.\ \cite{Rut14b} for a~rectangular sample. In the Corbino geometry, however, the role a~crystallographic orientation is eliminated, and the zero-field minimal conductivity can be approximated by
\begin{equation}
  \sigma_{\rm min}\approx{}3\sigma_0\left[\,1-\left(
      \frac{\lambda}{R_{\rm o}\!-\!R_{\rm i}}\right)^{0.5}\,\right],
\end{equation}
where $\lambda=\lambda(t')$ depends only on the skew-interlayer hopping $t'$, and varies from $\lambda\approx{}20\,$nm for $t'=0.3\,$eV to $\lambda\approx{}100\,$nm for $t'=0.1\,$eV. In the uniform magnetic field $B$, the conductivity increases reaching the maximal value $\sigma_{\rm max}\gtrsim{}3\sigma_0$ near the so-called Lifshitz field $B_L$, for which the magnetic length follows the relation
\begin{equation}
  \left(R_{\rm o}\!-\!R_{\rm i}\right)\ell = 4\pi{}l_B^2,
\end{equation}
where $\ell=\sqrt{3}\,\pi{}at_0^2/(t_\perp{}t')$ is defined by $t'$ and other microscopic parameters: the lattice spacing $a$ as well as the nearest-neighbor intra- and interlayer hoppings $t_0$ and $t_\perp$.
Above $B_L$, the conductivity gradually decreases showing the asymptotic behavior $\sigma-\sigma_0\propto{}B_L/B$ [see Eq.\ (\ref{eq:przewBC})]. The second and third charge transfer cumulants stay close to their pseudodiffusive values (${\cal F}=1/3$, ${\cal R}=1/15$) when varying the system size or the magnetic field.

In the opposite, quantum-tunneling regime (corresponding to $R_{\rm o}/R_{\rm i}\gtrsim{}4$), the charge-transfer characteristics are also sensitive to $t'$. At zero field and $t'=0$, the transport is governed by two quantum channels with angular momenta $\pm{}2\hbar$. For $t'\neq{}0$, the backscattering is enhanced for these channels, as the related wavefunctions no longer match the symmetry of the low-energy Hamiltonian. At high magnetic fields, all the charge-transfer characteristics show quasiperiodic beating patterns, with the envelope period ${\cal T}(B)\propto{}\sqrt{B/B_L}$ [see Eq.\ (\ref{eq:okrCor})]. Most remarkably, the beating patterns, triggered by the trigonal-warping ($t'\neq{}0$), remain well-pronounced in the Landau quantization regime. (Unlike the average conductivity enhancement, which is usually eliminated by a~few Tesla field.) It seems this finite-system version of the Lifshitz transition can be related to numerous phenomena appearing in different branches of physics, starting from semiconducting heterostructures \cite{Gui04}, via strongly-correlated electron systems \cite{Kor95}, to neutrino physics \cite{Pon67}, in which scattering the particles between quantum states with different effective masses leads to oscillations in relevant counting statistics, although this time the interference occurs between the evanescent waves.

We stress here that finding the Lifshitz field $B_L$, via the asymptotic behavior of the conductivity, or via the beating period, may allow one -- at least in principle -- to determine the value of $t'$ from a single-device magnetotransport measurement.

Apart from the possible verification of tight-binding parameters in BLG Hamiltonian, we believe the effects we describe, when confirmed experimentally, will provide a~thorough insight into the interplay between massless- and massive-chiral states ruling the quantum transport through BLG devices near the charge-neutrality point. For instance, the conductivity enhancement for $B\sim{}B_L$ may dominate the signatures of interaction-related magnetic catalysis phenomenon \cite{Gus94} (particularly in finite-size systems), and one should precisely distinguish single- and many-body aspects when searching for this intriguing phenomenon in BLG.

As we have focused on clean ballistic systems, several factors which may modify the transport properties of graphene-based devices, including the disorder \cite{Mac13}, lattice defects \cite{Dre10}, or magnetic impurities \cite{Cor09,Sza11,Hon13}, are beyond the scope of this work. Some experimental \cite{Mia07,May11} and numerical \cite{Ort13,Gat14} findings suggest that charge-transfer characteristics in the pseudodiffusive transport regime are quite robust against such factors. For the opposite, quantum-tunneling regime, we put forward the following reasoning: In the presence of trigonal warping, the rotational symmetry supposed in earlier studies of MLG \cite{Ryc10,Kat10} or BLG \cite{Rut14d} disks, no longer applies. In spite of this fact, the basic oscillation period [$\,\Phi_0=2\,(h/e)\ln(R_{\rm o}/R_{\rm i})$, in terms of magnetic flux piercing the disk area] remains unaltered, allowing one to believe that oscillations and beating patterns would appear in a more general situation as well.

\section*{Acknowledgements}
We thank Stephan Roche for the correspondence.
The work was supported by the National Science Centre of Poland (NCN) via Grant Nos. 2014/14/E/ST3/00256 (AR) and 2014/15/N/ST3/03761 (GR).
Computations were partly performed using the PL-Grid infrastructure.

redpajama_set_name: RedPajamaArXiv


HenPower gives people a role and responsibility. It builds relationships and moves away from passive care you so often see in care settings. It harnesses people's imagination and interests to empower themselves. HenPower is about combining hen-keeping and creativity to improve wellbeing. Older people work with professional artists to explore art forms each week. In the North East those taking part in the project had their artwork exhibited at Newcastle University at the back end of last year (URL below).


text:
The 18th man column: 'Escare can be Wigan Warriors' pocket rocket in their Grand Final bid'
Morgan Escare got a starting role last week
Published: 10:14 Thursday 27 September 2018
Our 18th man columnists discuss Escare's role, rule changes and which stadiums they would love to see Wigan play at...
How would you use Morgan Escare in Wigan's remaining matches?
Sean Lawless: As much as I would like to see Escare play longer minutes, I think the key for Escare in the semi-final and hopefully final is the short spells we have seen him play recently.
It is frustrating to see such a talented player being held back on the bench for so long but against tiring players on big pitches like the DW and hopefully Old Trafford, the little French man could be the pocket rocket we need for the last 15 minutes in a game. For the Hull game, however, if Tomkins needs resting then I have no qualms with seeing Escare given the full 80 minutes again.
Darren Wrudd: I like Escare and would love to see him spelled for longer at full-back. Perhaps come off the bench for the last 10 minutes of the first half and then leave him in place.
Swapping Tomkins from full-back to halves in the process. I would be happy to have him on the field for 80 minutes if need be as his energy is infectious.
Next year should be interesting for Morgan though. A utility back with no permanent place but plenty of scope, who knows what will happen but for now give the lad some game time.
Jon Lyon: I'm sorry Morgan, but I wouldn't change a thing right now. It's too close to the Grand Final to start making major changes to the team and the way we play, so would happily see Escare brought off the bench when teams are tired to have the impact he has been having. Sam is playing too well to mess around and Leuluai and Powell have settled into their job swap, with Manfredi and Davies doing well on the wings, so lets leave well alone.
We have to accept though that if we don't give Morgan more game time next year then he will be off, so if the plan is to play Zak at fullback in 2019 then Adrian Lam has some thinking to do.
Robert Kenyon: I'd use him at full-back and put Sam to 7. Morgan Escare is our most exciting and potent player, every time he gets the ball he's dangerous and a joy to watch in a dull game. He's probably not the best defensively but we have a very good defence anyway, he wouldn't have to tackle many people breaking through, but him at full-back running wide would scare the life out of defenders.
David Bailey: Despite his man of the match performance at Huddersfield from full-back, Escare's impact when coming on late against tired defences seems to have cemented his role for the remainder of Shaun Wane's tenure. It was fantastic to hear Morgan has committed to 2019 with Wigan and he may get a more active role under Adrian Lam as the guy is obviously a quality player and worries defences whenever he takes the field. There's always the worry of an injury to a key player but having the likes of Escare and Tomkins versatility available means one less concern for the Warriors.
Super League clubs are considering changes for next year such as Golden Point, reduced subs and a stop-clock – what changes would you welcome and are there any you wouldn't?
Sean Lawless: The reduced interchanges and stop clock are changes for the good of the game. They are aimed at speeding the game up and ensuring that there will be more tired bodies on a field later in a game, both would be realty exciting changes for the game and fit well with the new era for Super League. One rule I can't see working in Super League is two referees, although I would support it – we just simply don't have the talent pool of referees in this country to make it work like the NRL.
Here is how to plan training for an ultramarathon
Darren Wrudd: I would welcome the stop clock and perhaps a harder line on certain time delaying tactics. It's amazing how when under pressure, a Warrington player gets cramp to stop a drop out for a few minutes.
The game is fast enough for the players as it is, but fitter teams with better training are almost penalised by not being able to keep up the pace they would like.
Diving and play acting to get penalties should see a yellow card. Sin bin them and it will stop pretty quickly.
Not really a fan of golden point in the rounds but it is dramatic in the knock out games. But please, please, lets have a proper play the ball. Regain your feet, foot on the ball or penalty the other way. If a tackler is prevented from getting out of the way to glean a penalty, let the game go on as the situation is manufactured by the dummy half.
Too much play acting has crept into the game and that needs to stop. I would not welcome a reduction in substitutions at all though. The game is getting harder and faster and the substitutions can protect player welfare when need be.
Jon Lyon: I think reduced substitutions are necessary if we are to compete at international level. Our forwards need to get used to competing more intensely for more minutes for us to finally overcome the Aussies. The same applies for the stop clock; drop outs are becoming a joke with the time taken to restart the game and we need to get used to not getting an extended breather every time the ball is out of play.
I would be dead set against golden point being brought into league games. Draws are few and far between but when battled for, a point is just reward for a hard fought game, and extra time is usually just a drop goal competition, which isn't fascinating for anyone to watch, and an awful way to lose.
After the Billy Slater saga this week, I think its time to bring back the shoulder charge. As always, foul play should be punished appropriately, but it seems wrong the player with the ball can charge a defender but not the other way round. As long as its front on and contact isn't above shoulder height then let's bring back a bit of contact.
Robert Kenyon: I'd reduce the substitutions but increase subs on the bench. At the start of Super League the bench would consist of two forward and two backs, then along the line it went to three forwards and a utility player, nowadays there are four props on the bench. I rather see more subs on the bench but fewer substitutions so that they would be used tactically and not just spelling players and subbing by the clock. Also, you'd get tired props and halfbacks taking advantage of that.
I'd also only allow three goes with the video ref in which the captain of either team could ask the ref to review either an incident or a try.
Stop the mascots, I might sound like a killjoy but I think it makes us look cheap as a sport to have a load of kids dabbing and flossing in front of the players. I'd prefer to go like the NRL when the players run out into the field with the camera behind them with the players pumped up rather than having a player walk out with a lad who's the same height as him holding hands.
Reduce the commentators on Sky to Eddie Hemings and one other, there's too many involved and it stops the flow of the commentary. Get rid of Stuart Cummings, nobody wants to hear what the ref thinks. Take the microphone off the ref, nobody wants to hear hold/set/move all game, the only thing I want to hear the ref say all game is 'play on'.
Clear up the obstruction rule, whereby the onus is on the defending player to select the correct runner, rather just run into the runner and dive on the floor as a result and ruling out a try, or maybe not allow the ref to look for obstruction on the video ref. I don't know, it needs sorting because rugby league has become one dimensional because teams are too scared to have dummy runners.
Time limit on play the balls, players now seem to want to go down injured or tired and stop the game rather than playing the ball then being treated as they used to, which slows the game down massively.
They should give the ball for someone else to play a couple of meters away from them as soon as possible if they need treatment.
I'd allow people to tap and go themselves if there are no markers, I'd allow striking at the play the ball too, we have become sterile as a sport and I think this would stop this.
I'd also make teams go back eight meters, so that attacks were running from deeper rather than being flat with everyone doing the same move.
I could go on and people may call me a Luddite but I like change when it's for the better and not for the sake of it.
David Bailey: I'll be honest. I'm not a fan of the golden point. I mean I know some things are better decided on the day, Grand Finals, Million Pound games but games with a golden point by and large end as a drop goal-fest. I'd be happier if it was decided by the first try as teams would have to produce some rugby to triumph and there would be less milking of penalties. I think reduced substations would help the fitter teams as the big men wouldn't get as much rest, and I'm all for the shot clock.
Wigan could be playing at the Nou Camp next year. A hypothetical: if you could see Wigan play at any stadium in the world, where would it be and why?
Sean Lawless: If Wigan could play in one stadium in the world, it would have to be the Aviva Stadium in Dublin. Of all the stadiums in the world why that one? Well can you imagine a few thousand Wiganers in Dublin for a weekend – what could possibly go wrong?! Perhaps my favourite stadium though and one that would be great to see Wigan play at would be the US Bank stadium in Minneapolis, home of the Vikings. By far the best stadium I have ever been to!
Darren Wrudd: I love the thought of pushing boundaries in sport and if we were to push, I would go all the way. It would be easy to pick Lang Park, Brisbane which you may know as the Suncorp Stadium after its sponsors, but I would go that step further if we could hypothetically generate a big crowd. Century Link Field, home to the Seattle Seahawks is possibly one of the most fantastic stadium atmospheres out there.
Having held the Guinness record for the loudest roar from a crowd, the set up is most dramatic and capped at one end by the Seattle skyline. I would love to go and experience that and with a Wigan team on the field it would be fantastic.
Jon Lyon: A trip to the Nou Camp would be spectacular, although I've heard the views from the top tier are much like watching ants from a shed roof, not that we're likely to fill the near 100,000 capacity.
I would love to see Wigan play at Suncorp Stadium, Brisbane. I wasn't there in 1994 to see Wigan's greatest moment beating a Brisbane team full of internationals in their own back yard at ANZ Stadium, and would love the chance to see us play there again one day, hopefully with the same result.
Robert Kenyon: I'd love to see Wigan playing at The Aviva Stadium in Dublin, or Lansdowne Road as it used to be called. That way we could all have a session on the Guinness and go and watch Wigan. I've watched Wigan in London, Edinburgh and Cardiff so it would be nice to watch them in Dublin.
David Bailey: Wigan have already played at some iconic venues – Anfield, Twickenham, Murrayfield, The ANZ Stadium, The Allianz Stadium, The County Stadium Milwaukee, The Douglas Bowl and Vetch Field. If I had to choose a stadium out of any in the world for Wigan to play at though I'd be edging towards the Maracana in Mexico because of the history of the stadium. That being said if I could choose one stadium from the past for one more game it would be Central Park obviously, with the night against Manly living long in the memory especially.

redpajama_set_name: RedPajamaCommonCrawl


They need real answers, just like everyone else—and those answers are found in God's Word and revealed to us by His Spirit.
The Lord is my light and my salvation; whom shall I fear? The Lord is the stronghold of my life; of whom shall I be afraid? When evildoers assail me to eat up my flesh, my adversaries and foes, it is they who stumble and fall.
One of my favorite ways to hear from God is through conventional wisdom and common sense. Wisdom discerns truth in a situation, while common sense provides good judgment regarding what to do with the truth. I consider wisdom supernatural because it isn't taught by men; it is a gift from God.
Many sophisticated, intelligent people lack wisdom and common sense. God's Word says, If any of you is deficient in wisdom, let him ask of the giving God [Who gives] to everyone liberally and ungrudgingly, without reproaching or faultfinding, and it will be given him (James 1:5 AMPC).
I am amazed how many people think they must stop using common sense in order to be "spiritual." Spiritual people don't float around all day on clouds of glory; they live in the real world and deal with real issues in real ways. They need real answers, just like everyone else—and those answers are found in God's Word and revealed to us by His Spirit.
We do the seeking, and God does the speaking, but He is the Spirit of Wisdom and will not tell us to do anything that is unwise. Many times we ask God to speak to us and lead us, but if He does not give us a specific word from Scripture or speak a word in our hearts, we still have to live our daily lives and make decisions.
God is not going to dictate every little choice we make, but He does give us wisdom and common sense—and those two make a winning combination.
Prayer Starter: Father, I ask for Your wisdom to live my life and make good decisions that I will be happy with in the future. Give me a heart to know and love Your Word so I can be even more sensitive to Your daily guidance. In Jesus' Name, Amen.


Penguin Classics has published an English translation of Turkish novelist Sabahattin Ali's best-selling novel "Kurk Mantolu Madonna" under the title "Madonna in a Fur Coat." Ali's novel was translated into English by Maureen Freely and Alexander Dawe. The book is the second Turkish novel issued by the prestigious publishing house following the release Ahmet Hamdi Tanp?nar's "The Time Regulation Institute.". The book is about a shy young man who leaves his home in rural Turkey to learn a trade in 1920s Berlin. English today is an essential tool in preparing various English examinations and tests required to study and work abroad. Containing the adventures of five fellow apartment and two thematic programs dedicated to the cinema, music, sports, and cuisine and currency, as in a real TV network.
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The Charente area is a district renowned for its very special light at sunset, splendid Romanesque churches, ruined fortresses, dolmens and charming red-roofed farmhouses – set in the valley folds or locked between vineyards and deep woodland.
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An FCC ID is the product ID assigned by the FCC to identify wireless products in the market. The FCC chooses 3 or 5 character "Grantee" codes to identify the business that created the product. For example, the grantee code for FCC ID: M82-261ACNBT is M82. The remaining characters of the FCC ID, -261ACNBT, are often associated with the product model, but they can be random. These letters are chosen by the applicant. In addition to the application, the FCC also publishes internal images, external images, user manuals, and test results for wireless devices. They can be under the "exhibits" tab below.
Modular Approval. Output power is conducted. This module may be installed only by the OEM or an OEM integrator. This module may be installed only by the OEM or an OEM integrator. This transmitter has been approved for use in mobile RF exposure category configurations with the antenna(s) installed to provide a separation distance of at least 20 cm from all persons. The antenna(s) used for this transmitter must not be collocated or operating in conjunction with any other antenna or transmitter within a host device, except in accordance with FCC multi-transmitter product procedures. Grantee must coordinate with OEM integrators to determine applicable host configurations to ensure RF exposure compliance, including simultaneous transmission.
Modular Approval. Output power listed is maximum combined conducted. This device supports 20, and 40 MHz bandwidths. This module may be installed only by the OEM or an OEM integrator. This transmitter has been approved for use in mobile RF exposure category configurations with the antenna(s) installed to provide a separation distance of at least 20 cm from all persons. The antenna(s) used for this transmitter must not be collocated or operating in conjunction with any other antenna or transmitter within a host device, except in accordance with FCC multi-transmitter product procedures. Grantee must coordinate with OEM integrators to determine applicable host configurations to ensure RF exposure compliance, including simultaneous transmission.
Modular Approval. Output power listed is maximum combined conducted. This device supports 20, 40, and 80 MHz bandwidths. This module may be installed only by the OEM or an OEM integrator. This transmitter has been approved for use in mobile RF exposure category configurations with the antenna(s) installed to provide a separation distance of at least 20 cm from all persons. This device supports client-only operating modes in the 5150-5350 MHz, 5470-5725, and 5825-5850 MHz bands. The antenna(s) used for this transmitter must not be collocated or operating in conjunction with any other antenna or transmitter within a host device, except in accordance with FCC multi-transmitter product procedures. Grantee must coordinate with OEM integrators to determine applicable host configurations to ensure RF exposure compliance, including simultaneous transmission.