CHARLOTTE, NC - The Charlotte Mecklenburg Library is pleased to present four documentaries by Steve Crump in honor of Black History Month. The films will run February 6-27, will take place at Main Library (310 North Tryon Street), and will be introduced Crump who is a WBTV reporter and award-winning filmmaker. For more information contact Sam Shapiro at (704) 416-0252.
Saturday, Feb. 6, 2 p.m.
Tuesday, Feb. 9, 7 p.m.
Seventeen months before Rosa Parks refused to surrenderher seat on an Alabama bus in 1955, a little-known Civil Rights heroine named Sarah Mae Flemming refused to obey a similar ordinance on a bus in Columbia, S.C. This film tells her story, which has largely faded from the public's memory.
Tuesday, Feb. 16, 7 p.m.
This rousing documentary connects jazz greats Dizzy Gillespie, John Coltrane, and Thelonius Monk to their North Carolina roots.
Saturday, Feb. 27, 2 p.m.
Over 300 pets were adopted in April thank to Paws in the Park in Pineville.


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\section{Introduction}

Antiferromagnetic spin ladders are paradigmatic examples of quantum magnetism in low dimensions \cite{Dagotto-Rice-Science,Mila-2016,mila-2016-bis,mila-berry-phase,laflorencie-2011,Totsuka1,NH4CuCl3-magnetization,NH4CuCl3-model,cabra-2001,Mila-magnons-spinons}.
These systems exhibit a plenty of interesting properties, such as absence of long range order, energy gaps, magnetic disordered phases, among others \cite{Mikeska-ch1}.
One particular issue of these systems that captured both experimental and theoretical efforts is the existence of magnetization plateaus \cite{Mila-capitulo-plateaus}. For a quantum spin-S chain, the necessary condition for the occurrence of a magnetization plateau has been established by Oshikawa, Yamanaka and Affleck (OYA)\cite{OYA} as
\begin{eqnarray}
N(S-m) \in \mathbb{Z},
\end{eqnarray}
where  $N$ is the period of the spin state, $S$ the magnitude of spin and $m$ the magnetization per site in units of $g \mu_B$.
The OYA criterium has been successfully applied to different one dimensional magnets \cite{Lamas-Pujol_2010,cabra-2001} and revisited by different techniques\cite{TTH,totsuka-2015,topological-condition,plat-2015}.
In this context several  quasi-one-dimensional compounds has been studied in the past years. As an example we mention the compound  Cu$_2$Cl$_4\cdot$D$_8$C$_4$SO$_2$, which presents a singlet ground state. In order to describe the properties of this material different models have been proposed \cite{Fujisawa-2003,experimental_tube_prb,tube_inelastic_scatering}, however the magnetic properties of this compound has not been completely understood.
Another system with a remarkable magnetic behavior is the $S = \frac12$ zig-zag ladder compound NH$_4$CuCl$_3$. In this material the magnetization curve presents two plateaus at $m =\frac14 $ and $m =\frac34 $ of the saturation magnetization, irrespective of the external field direction, but not at $m =0 $ and $m =\frac12 $ \cite{NH4CuCl3-magnetization}.

In this paper we study an antiferromagnet model in the zig-zag geometry showing that the interplay between geometrical frustration and plaquette dimerization promotes the existence of plateaus at $m =\frac14 $ and $m =\frac34 $.
Although we do not intend to provide a theoretical description of these materials, our results may be relevant for the discussion of the emergence of rational magnetization plateaus and its magnon and spinon excitations.
The model analyzed in this work belongs to a family of frustrated antiferromagnetic systems with a fully dimerized
exact ground state \cite{Brenig-tetrahedral,Matera-Lamas_2014,Lamas-Matera_2015}. Recently it was shown that by imposing certain \emph{exact} condition to the couplings, the exact ground state of the system is robust against the inclusion of weak disorder \cite{Lamas-Matera_2015}. This property provides to the model interesting magnetic properties when the unit cell is increased in order to consider plaquette dimerization.
The exact condition can be imposed both in one or the two different square plaquettes giving a rich plateaus structures, an aspect that will be exploited in this work.

The outline of the paper is as follows. In Section \ref{sec:exgs} we study the zero magnetic field phase diagram. Starting form the line where the ground state is exactly determined we use a bond operator formalism to study the excitation gap. In Section \ref{sec:PlateauPhases} we analyze the effect of an external magnetic field on the model. We study the magnetization process by means of density matrix renormalization group (DMRG) calculations and exploring the parameters space we determine regions where different plateaus are present.
In the highly dimerized plaquette regime we derive low energy effective models and studying the excitations we estimate the critical magnetic fields and the plateau weights. The interplay between dimerization and frustration results in a rich plateaus structure that is captured by the low energy model and is consistent with the numerical results obtained through DMRG.
Finally in Section \ref{sec:conclusions} We present the conclusions. Some relevant formulas and expressions used throughout the work are shown in the Appendix.

\section{Zero magnetic field phase diagram}
\label{sec:exgs}
\subsection{Exact ground state in the absence of magnetic field}
\label{sec:exact-lines}

We consider the following Heisenberg model on a two legs spin-$S$ ladder
\begin{figure}[t]
\includegraphics[width=0.8\columnwidth]{fig-ladder.eps}
\caption{(Color online). Schematic representation of the dimerized zig-zag ladder. }
\label{fig:ladder}
\end{figure}
\begin{eqnarray}
\label{eq:general_Hamiltonian}
\mathbf{H}&=&\sum_{n} J_{0} \left(\vec{\mathbf{S}}_{1,n}\cdot\vec{\mathbf{S}}_{2,n}+\vec{\mathbf{S}}_{3,n}\cdot\vec{\mathbf{S}}_{4,n}\right)\\\nonumber
&+&J \left(\vec{\mathbf{S}}_{1,n}\cdot\vec{\mathbf{S}}_{3,n}+ \vec{\mathbf{S}}_{2,n}\cdot\vec{\mathbf{S}}_{4,n}\right)+J_{d} \, \vec{\mathbf{S}}_{2,n}\cdot\vec{\mathbf{S}}_{3,n} \\\nonumber
&+& K\left( \vec{\mathbf{S}}_{3,n}\cdot\vec{\mathbf{S}}_{1,n+1}+\vec{\mathbf{S}}_{4,n}\cdot\vec{\mathbf{S}}_{2,n+1}\right)+K_{d}\, \vec{\mathbf{S}}_{4,n}\cdot\vec{\mathbf{S}}_{1,n+1},
\end{eqnarray}
where $\vec{\mathbf{S}}_{j,n}$ represents the $j$-th spin on the unit-cell $n$. This ladder is schematized in Fig. \ref{fig:ladder}.
Following the lines scketched in Ref. \onlinecite{Matera-Lamas_2014} we can show that, provided the conditions $J_{d}=2J$ and $K_{d}=2K$,
the ground state of the system corresponds to the fully dimerized state
$ |{\psi}\rangle=\bigotimes_{i=1}^{N_{r}}|s\rangle_{i}$, with
\begin{eqnarray}
\label{eq:dimerstate}
 |s\rangle_{i}= \frac{1}{\sqrt{2 S+1}}\sum_{m=-S}^{S}(-1)^{m+S}|m,-m\rangle_{i},
\end{eqnarray}
where the index $i$ label the rungs in the ladder and $N_{r}$ is the number of rungs. The state
$|m,-m\rangle_{i}$ is a product state such that ${\bf S}^{z}_{2 i-1,n}|m,-m\rangle_{i}=-{\bf S}^{z}_{2 i,n}|m,-m\rangle_{i}=m |m,-m\rangle_{i}$
where the index $i=1,2$ corresponds to the two rungs in the unit-cell $n$.
This state remains the ground state of the system in the range of couplings $J<J_{0}/2$ and $K<J_{0}/2$ .
For this purpose, it is convenient to rewrite the Hamiltonian in the case $J_d=2J$, $K_d=2K$ as follows
\begin{eqnarray}
\mathbf {H}&=&\sum_{n}(\mathbf{h}_{J}(n)+\mathbf{h}_{K}(n)),
\end{eqnarray}
where $\mathbf{h}_{J}(n)$ and $\mathbf{h}_{K}(n)$  are Hamiltonians of the square plaquettes on unit-cell $n$,
given by
\begin{eqnarray}
\label{eq:hJ}
\mathbf{h}_{J}(n)&=&\frac{J}{2}\left[ \left(  \vec{\mathbf{S}}_{1,n}+\vec{\mathbf{S}}_{2,n}+\vec{\mathbf{S}}_{3,n}  \right)^{2} \right. \\\nonumber
&+& \left. \left(  \vec{\mathbf{S}}_{2,n}+\vec{\mathbf{S}}_{3,n}+\vec{\mathbf{S}}_{4,n}  \right)^{2}    \right]\\ \nonumber
&+&\frac{(J_{0}-2J)}{4}\left[ \left( \vec{\mathbf{S}}_{3,n}+\vec{\mathbf{S}}_{4,n}  \right)^{2}   + \left(  \vec{\mathbf{S}}_{1,n}+\vec{\mathbf{S}}_{2,n} \right)^{2}    \right]\\ \nonumber
&-&\frac34(J_{0}+J),
\end{eqnarray}
\begin{eqnarray}
\label{eq:hK}
\mathbf{h}_{K}(n)&=&\frac{K}{2}\left[ \left(  \vec{\mathbf{S}}_{3,n}+\vec{\mathbf{S}}_{4,n}+\vec{\mathbf{S}}_{1,n+1}  \right)^{2} \right. \\\nonumber
&+& \left.  \left(  \vec{\mathbf{S}}_{4,n}+\vec{\mathbf{S}}_{1,n+1}+\vec{\mathbf{S}}_{2,n+1}  \right)^{2}    \right]\\ \nonumber
&+&\frac{(J_{0}-2K)}{4}\left[ \left( \vec{\mathbf{S}}_{3,n}+\vec{\mathbf{S}}_{4,n}  \right)^{2}   + \left(  \vec{\mathbf{S}}_{1,n}+\vec{\mathbf{S}}_{2,n} \right)^{2}    \right]\\ \nonumber
&-&\frac34(J_{0}+K).
\end{eqnarray}
In this way, is easy to see that $\mathbf {H}$ is a semi-definite positive operator if $J<J_{0}/2$ and $K<J_{0}/2$. In the case $S=1/2$, the energy per square plaquette is bounded from bellow by
\small
\begin{eqnarray}
\frac{E_{gs}}{N_s}&\geq&\min_{l_{1,2},l_{3,4}=0,1}   \left\{ -\frac34(J_{0}+J)-\frac34(J_{0}+K)  \right. \\\nonumber
&+&\frac{J}{2}\sum_{n=1}^2\left[ |\frac12-l_{2n-1,2n}|(|\frac12-l_{2n-1,2n}|+1)\right]\\\nonumber
&+& \left. \frac{(J_{0}-2J)}{4}\sum_{n=1}^2\left[ l_{2n-1,2n}(l_{2n-1,2n}+1) \right] \right\}\\\nonumber
&+&\frac{K}{2}\sum_{n=1}^2\left[ |\frac12-l_{2n-1,2n}|(|\frac12-l_{2n-1,2n}|+1)\right]\\\nonumber
&+& \left. \frac{(J_{0}-2K)}{4}\sum_{n=1}^2\left[ l_{2n-1,2n}(l_{2n-1,2n}+1) \right] \right\},
\end{eqnarray}
\normalsize
where $l_{1,2}$ ($l_{3,4}$) is the total spin in the rung between sites $1$ and $2$ ($3$ and $4$). It is straightforward to see that the minimum value correspond to $l_{1,2}=l_{3,4}=0$, given the bounding condition
\begin{eqnarray}
\frac{E_{gs}}{N_s}\geq -\frac32 J_{0}.
\end{eqnarray}
To conclude, writing the Hamiltonian in terms of local operators in each rung
\begin{eqnarray}
\vec{\mathbf{L}}_{1,2}(n)&=&\vec{\mathbf{S}}_{1,n}+\vec{\mathbf{S}}_{2,n}\\\nonumber
\vec{\mathbf{L}}_{3,4}(n)&=&\vec{\mathbf{S}}_{3,n}+\vec{\mathbf{S}}_{4,n}\\\nonumber
\vec{\mathbf{K}}_{1,2}(n)&=&\vec{\mathbf{S}}_{1,n}-\vec{\mathbf{S}}_{2,n}\\\nonumber
\vec{\mathbf{K}}_{3,4}(n)&=&\vec{\mathbf{S}}_{3,n}-\vec{\mathbf{S}}_{4,n},
\end{eqnarray}
is easy to see that
if we restrict the couplings to satisfy the condition $J_d=2J$ and $K_d=2K$, the state $|{\psi}\rangle$ is an eigenstate with eigenvalue $E=-\frac32 J_{0} N_s$.
Therefore we have proved that the state $|{\psi}\rangle$ is the ground state of the system along the lines  $J_d=2J$ and $K_d=2K$ as long as the couplings satisfy  $J<J_{0}/2$ and $K<J_{0}/2$. Notice that these two last constraints represent a sufficient condition for the existence of this singlet ground state, but the true range of couplings can be larger.

This exact result is a very convenient starting point for dimer expansions. Notice that the existence of a line where the singlet product state is the true ground state of the system allows us to use dimer expansions around a point in the parameter space where the Hamiltonian is maximally frustrated. In the following Section we present some results obtained by different techniques based on dimer expansions.

\subsection{Bond Operators appproach}

Close to the line in the parameter space where the ground state corresponding to Hamiltonian \ref{eq:general_Hamiltonian} is exactly known it is convenient to exploit  the description in terms of bosonic operators, the so called \emph{bond operators} (BO) \cite{SachdevBO}
which label the dimer's singlet-triplet spectrum.
On the exact line the ground state is a direct product of states on each square plaquette. On each plaquette the state corresponds to a product of a singlet between spins located at sites  $1,n$ and $2,n$ and another singlet
between spins at $3,n$ and $4,n$. Within BO theory the four spins $\vec{S}_{i}$ on each
square plaquette are expressed as

\begin{fleqn}
\begin{align}
\label{eq:BO-A}
S_{\stackrel{{\scriptstyle 1,n}}{{\scriptstyle 2,n}}}^{\alpha}&=\frac{1}{2}(\pm
s_{A,n}^{\dagger}a_{\alpha,n}\pm a_{\alpha,n}^{\dagger}s_{A,n}-\sum_{\beta,\gamma}i
\varepsilon_{\alpha\beta\gamma}a_{\beta,n}^{\dagger}a_{\gamma,n}^{\phantom{\dagger}})\\
\label{eq:BO-B}
S_{\stackrel{{\scriptstyle 3,n}}{{\scriptstyle 4,n}}}^{\alpha}&=\frac{1}{2}(\pm
s_{B,n}^{\dagger}b_{\alpha,n}\pm b_{\alpha,n}^{\dagger}s_{B,n}-\sum_{\beta,\gamma}i
\varepsilon_{\alpha\beta\gamma}b_{\beta,n}^{\dagger}b_{\gamma,n}^{\phantom{\dagger}}),
\end{align}
\end{fleqn}
where $s_{A,n}^{(\dagger)}$and $a_{\alpha,n}^{(\dagger)}$ destroy(create)
the singlet and triplet states of the dimer between sites $1,n$ and $2,n$ and greek labels, $\alpha=1,2,3$,
refer to the threefold triplet multiplet. Equivalently operators $s_{B,n}^{(\dagger)}$and $b_{\alpha,n}^{(\dagger)}$ act on the dimer between sites $3,n$ and $4,n$. A hard-core constraint
\begin{eqnarray}
\label{eq:constraint-A}
\displaystyle{s_{A,n}^{\dagger}s_{A,n}}+\sum_{\alpha}a_{\alpha,n}^{\dagger}
a_{\alpha,n}^{\phantom{\dagger}}&=&1,\\
\label{eq:constraint-B}
s_{B,n}^{\dagger}s_{B,n}+\sum_{\alpha}b_{\alpha,n}^{\dagger}
b_{\alpha,n}^{\phantom{\dagger}}&=&1,
\end{eqnarray}
is implied on each sublattice, which renders the algebra of the r.h.s of Equations. (\ref{eq:BO-A}) and (\ref{eq:BO-B})identical to that of spins.
Inserting the BO representation into a spin model leads to an interacting
Bose gas. In the BO-MFT,  singlets are condensed by
$s^{(\dagger)}\rightarrow s\in\,$Re and the constraint in the number of bosons per site is
satisfied on average with a global Lagrange multiplier $\lambda$. In this approach
terms only up to second order in the BOs are retained and the quadratic Hamiltonian can be diagonalized by standard Bogoliubov transformation leading to an energy $E$ \emph{per unit cell} of
\begin{equation}
E={-}\frac{3}{4}J_{0}-\frac{3}{2}s^{2}J_{0}+(5-2s^{2})\lambda+\frac{3}{N}\sum_{k}(\omega^{+}_{k}+\omega^{-}_{k}),\label{wa8}
\end{equation}
with the \emph{triplon dispersion}
\begin{equation}
\label{eq:BO-dispersion}
\omega^{\pm}_{k}=|a|\sqrt{1\pm|g|\frac{s^{2}}{a}\beta_{\delta}(k)},
\end{equation}
where
\begin{eqnarray}
\beta_{\delta}(k)&=&\frac{1}{2}\sqrt{1+\delta^2 + 2\delta \cos(k)},
\end{eqnarray}
$a=\frac{J_{0}}{4}-\lambda$, $g=J_d-2J$, and $s=\langle s_{A,n} \rangle=\langle s^{\dagger}_{A,n} \rangle=\langle s_{B,n} \rangle=\langle s^{\dagger}_{B,n} \rangle$ is a real parameter and we have restricted the study
to the case $K=\delta J$ and $K_d=\delta_d J_d$ with $\delta = \delta_d$ \emph{ie} \emph{homogeneous} dimerization. The general case will be analyzed with other techniques in the following Sections.
In order to obtain the mean field parameters $s$ and $a$  the energy $E$ is extremized, obtaining two
self-consistency equations $\partial E/\partial a=0$ and $\partial E / \partial s=0$.
These two equations can be combined obtaining
\small
\begin{eqnarray}
\label{eq:BO-selfcon1}
d&=&\frac{5}{2J_{0}}-\frac{3}{8 \pi J_{0}}\hbox{sign}(a)\int dk \left( \frac{1}{\gamma_-(k)} +\frac{1}{\gamma_+(k)} \right),\\
\label{eq:BO-selfcon2}
a&=&J_{0}-\frac{3}{16\pi}|g|\hbox{sign}(a) \int dk  \left( \frac{\beta_{\delta}(k)}{\gamma_+(k)} -\frac{\beta_{\delta}(k)}{\gamma_-(k)} \right),
\end{eqnarray}
\normalsize
where $\gamma_\pm(k)=\sqrt{1\pm d |g| \beta_{\delta}(k)}$ and $d=s^{2}/a$. These equations can be solved sel-consistently.

We mention in passing, that the trivial limit, i.e. $g=0$, leads to $d=1/J_{0}$, $s=1$, and $\lambda=-\frac{3}{4}J_{0}$, and therefore to a \emph{singlet-triplet gap} of $\Delta=1$ and a ground state energy of
$E/N={-}\frac32 J_{0}$, which is consistent with two saturated singlets per unit cell.

\subsection{Triplet dispersion and energy gap}

In this Section we will analyze the triplet dispersion in absence of an external magnetic field in the model given by Eq.(\ref{eq:general_Hamiltonian}). For this, we will compare the results obtained by applying techniques of different nature, which complement each other and bring a perspective that none of the techniques can provide separately. The methods used here are on the one hand the numerical technique DMRG, and on the other hand strong coupling dimer and plaquette expansions, as the BO method from the previous Section, and the series expansions technique (SE) considered in the Appendix. \\
The Fig. \ref {fig:gap-BO} summarizes the main result of this Section, obtained by the different techniques mentioned previously, for the triplet gap, \emph{ie} the minimum of the dispersion, as a function of the parameter $ g = J_d-2J $. The case considered in this figure is that of a homogeneous structure without dimerization, \emph{ie} $\delta = \delta_d=1$. The DMRG results, shown with blue dots in Fig. \ref {fig:gap-BO}, for $ J = 0, 0.1, 0.2, 0.3 $ and 0.4 from right to left, and were calculated on finite structures of $ L = 40 $ plaquettes (number of sites $N_s=160$) with periodic boundary conditions (PBC), maintaining up to $ M = 520 $ states in the computation, which has shown to be enough to achieve the required precision. Unless specified otherwise, in the remainder of the work these will be the DMRG parameters used. All the DMRG computations presented in  this work were performed with the open-source code ALPS \cite{ALPS}. \\
On the other hand, the SE results are indicated by red line in this figure with the same $J-$parametrization as DMRG, and were obtained from an expansion in dimers to $O(10)$ by using the method of continuous unitary transformations (CUT) \cite{SE-CUT}. Finally, the BO results are of two types. The first one is the \emph{Holstein-Primakoff} approximation (BO-HP), which is indicated by green line in the Fig. \ref {fig:gap-BO}. In the BO-HP the self-consistent parameters in Eqs.(\ref{eq:BO-selfcon1},\ref{eq:BO-selfcon2}) are solved for $g=0$ (obtaining $s=a=d=1$) and are replaced in the dispersion Eq.(\ref{eq:BO-dispersion}) which, after the re-scaling $k \rightarrow 2k$, imposed by halving the unit cell (dimer basis), takes the simple form $\omega_{HP}=\sqrt{1-|g|\cos k}$. Note that this expression coincides with the treatment of the model under the random phase approximation (RPA) method \cite{link-BO-SE}. The second type of BO result is the self-consistent solution of mean-field BO, Eqs.(\ref{eq:BO-selfcon1}
,\ref{eq:BO-selfcon2}), and is indicated by the orange line in the Fig. \ref {fig:gap-BO}. \\

\begin{figure}[t]
\includegraphics[width=0.9\columnwidth]{compHPSeDMRG.eps}
\caption{(Color online) Spin gap $\Delta$ versus $g=J_d-2J$ obtained by means of different techniques employed in the work. The SE (red lines) and DMRG (blue dots) results, are parameterized for $ J = 0, 0.1, 0.2, 0.3 $ and 0.4 from right to left. Notice that for bond operators approximations, Holstein-Primakoff (BO-HP, green line) and self-conistent (BO-MFT, orange line), the spin gap (and the dispersion) depend only on $g$, which is an artifact of the approximation. The peak shape of the gap maxima indicates a crossing of levels between  $k = 0$ (right) and  $k = \pi$ (left) dispersion branches, detected by SE. Gap right branches together with the peaks shift to the left by increasing $J-$values and tend to collapse into a single g-dependent line for $g <0$, where the methods (except BO-HP) exhibit a very good agreement.}
\label{fig:gap-BO}
\end{figure}
Several comments are in order. First of all note that, apart from the $g-$dependence, exists a \emph{residual} interaction, represented here through the $J-$parametrization, that is captured by SE and DMRG, but not by BO.
That is, BO encapsulates the entire dependence on $J$ and $J_d$ in $g$, which is artifact of the mean field approximation. \\
In addition, as can be observed, there is a very good quantitative agreement between DMRG and SE results, which is maintained until intermediate values of $ J $, according to the perturbative character of the SE approximation.\\
On the other hand, note that the absolute maximum of the gap $(= 1)$ at $g = 0$, as well as the other maxima in Fig. \ref {fig:gap-BO}, are non-derivable (peaks), representing a crossing of the dispersion branches at $k = 0$ (right) and $k=\pi$ (left) of the peak, which are detected by the SE. This in turn reflects a change from a dimerized chain-type to ladder-type of spectrum, respectively.\\
Regarding the role of the residual interaction, it shifts the gap peak to the left $(g<0)$. However its effect on the two dispersion branches is very different. While the branch $k = 0$ (to the right of the peak) moves along with the peak to the left by increasing $J$, the branch $k = \pi$ on the left tends to collapse on a single curve, \emph{ie} its dependence on $J$ and $J_d$ is captured mainly by $g$. These features are detected by means of SE and numerically corroborated by DMRG as can be seen in Fig. \ref {fig:gap-BO}.\\
Finally, with respect to BO, it is worth noting that although the mean field level does not take into account the residual interaction, but only g, it is able to capture some essential aspects of the gap behaviour, not only qualitatively, but also up to a certain quantitative level. Here the two approaches BO describe the gap with different precision on both sides of the gap peak. As can be seen in Fig. \ref {fig:gap-BO}, the BO-HP (green line) adequately describes the right branch of the gap (in particular the $J=0$ line), while the BO-MFT (orange line) agreement for the left branch is very good, as compared with SE and DMRG results. This tendency of the dispersions to collapse with a single $g-$dependence for $g <0$ could be related to the fact that in that region the gap remains large, ie more adiabatically connected to the exact line $g = 0$, as shown in the next Section through a sweep in the $J-J_d$ plane of the gap. Finally let us mention that in quantitative terms, the role of the residual
interaction and the dependence on $g$ is clarified in the Appendix, where it is explicitly shown that at leading order the dispersions obtained by SE and BO-HP coincide and depend only on $g$.

\section{Magnetic field phases}
\label{sec:PlateauPhases}
\subsection{Plateau phases without plaquette dimerization }
In this Section we will analyze the effect of an external magnetic field on the model given by Eq.(\ref{eq:general_Hamiltonian}). To simplify the analysis we will start by studying the case of homogeneous frustrated plaquettes, \emph{ie} $ \delta = \delta_d = 1 $. Although this case has been previously studied in different regions of the space of parameters and employing different techniques \cite{Shastry-Sutherland81,Tonegawa87,Chitra95,White96,Mikeska96,Yokoyama99}, our analysis presents it in a unified way and contextualizes our subsequent study of the general case with frustration and dimerization between-plaquettes of the following Sections.\\
As it is known the presence of a magnetic field opens the possibility of the emergence of plateau phases in the magnetization curves, where the magnetization is fixed at a certain value, for a finite interval of the magnetic field.\\
In Fig. \ref{fig:phases-hc-delta-1} we show the critical fields $h_c$ delimiting the different possible values of magnetization, determined by DMRG, in function of $ J / J_d $, along the line $ J + J_d = 1 $. The choice of the parameters in this figure is in order to show in a single representation the emergence of intermediate plateaus. As can be observed, apart from the zero field $(M=0)$ and saturation $(M=1)$ plateaus, two additional plateaus at $M=1/3$ and 1/2 are present. Note that the critical line crossing each of this two plateaus represents a small jump that is an artifact due to the open boundary conditions used for the computation.\\
Additionally, in Fig. \ref{fig:phases-hc-delta-1}, the critical line delineating the plateau at $M = 0$, obtained by the SE triplet gap closure condition, is presented with red line. The peak of this curve represents the crossing of the gap closing branches at $ k = 0 $ and $ k = \pi $. As can be observed, the quantitative agreement between DMRG and SE for the line $ M = 0 $ is excellent throughout the range of parameters shown in the figure.\\
\begin{figure}[t]
\includegraphics[width=0.8\columnwidth]{fig-phases-hc-delta-1.eps}
\caption{Critical fields (thin-lines) delimiting the different magnetization sectors on a finite system, determined by DMRG, as a function of $ J / J_d $, along the line $ J + J_d = 1 $. Note the presence of two intermediate plateaus at $M=1/3$ and 1/2. The bold red line indicates $M=0$ critical line (in thermodynamic limit), where SE triplet gap vanishes, showing an excellent agreement with the corresponding DMRG determination.}
\label{fig:phases-hc-delta-1}
\end{figure}

On the other hand, Fig \ref{fig:gap-phases} shows the extension of the plateaus at $M = 0$ (a), $1/3$ (b) and $1/2$ (c),  in the plane $ J-J_d $ calculated on finite systems with the DMRG technique. The determination is not intended to be quantitatively accurate, for which an analysis of a finite-size scaling that goes beyond the objectives of the work should be carried out, but rather to determine the regions in the parameters space where the plateaus are most prominent.\\

Let us first consider Fig \ref{fig:gap-phases}(a) where the width of $M=0$ plateau is considered. The largest plateau width ($\Delta =1$) is at the origin, corresponding to the limiting case of isolated dimers (see Fig. \ref{fig:ladder}).
The maximum width of the plateau follows the line of maximum frustration, $ g = 0 $, indicated in this figure with dotted blue line, up to $ J \simeq 0.2 $, beyond which deviations towards $g<0$ become increasingly marked. In the lower right inset of Fig. \ref{fig:gap-phases} (a) the SE triplet gap is depicted showing the same trend as the DMRG determination. This deviation is consistent with the observed shift of the gap peak in Fig. \ref{fig:gap-BO}, predicted by SE and DMRG, as well as the tendency to collapse into a single $g-$dependent triplet gap curve for $g<0$.\\ Another particular case of our model is represented by the point $ (J = 0, J_d = 1) $, corresponding to the homogeneous Heisenberg chain, which is gapless. Note that both DMRG and SE predict a tendency toward the gap closure (dark zone) approaching that point. The model remains gapless along the line $J_d=1$ up to the point $(J \simeq 0.241, J_d =1)$, indicated by a green circle in the Fig. \ref{fig:gap-phases} (a)). From this point the
system is gapped and for $ (J = 0.5, J_d = 1) $ (Majumdar-Gosh point) the model displays the exact dimer-product zero field ground state, which extends all over the line $ g = 0 $ analyzed in Section \ref{sec:exact-lines}.\\

Regarding the $ M = 1/3 $ plateau, the Fig. \ref{fig:gap-phases} (b), shows that this plateau emerges in a certain reduced and highly frustrated chain limit of the model, where $ J, J_d$, are of the order of the unity, reaching values of plateau width of $ \approx 0.6$. The nature of this plateau is completely different to the others analyzed in this work, having a classical origin \cite{NN-Ising72}, characterized by an "up-up-down" ordering which stabilizes this magnetization value. This state adiabatically evolves from the Ising chain with first and second neighbors interactions, and survives up to the Heisenberg isotropic limit. This plateau has been extensively analyzed in the works of Refs. \onlinecite{Okunishi-Tonegawa03PRB,Okunishi-Tonegawa03JPSJ,AndreasM1/3,Tonegawa04}. In addition, the techniques of strong plaquette expansions that we will use (see next Section) are not specially  suitable to treat this plateau, since it emerges in a region where all the couplings are of the same order. For these
reasons we will not elaborate more on the same in this work.\\

Finally the panel (c) of Fig. \ref{fig:gap-phases} depicts the extension of the $M=1/2$ plateau determined by DMRG. The emergence of this plateau has been analyzed in terms of a dimerized and frustrated ladder from a numerical point of view \cite{Tonegawa-ladder-98, Tonegawa04}. The dashed yellow lines indicate estimations of high field $(h>1)$ limit of $M=1/2$ phase plateau, from an effective low energy dimer model description of the system \cite{Mila-ladder-98,Sen-ladder-99}, indicating a similar trend to our findings.\\

In the following Sections we will develop a low energy plaquette effective model, which will account for present plateaus (except $M = 1/3$, wich would require higher order terms) and predict the emergence of others at $M = 1/4$ and $3/4$ due to the interplay between frustration and inter-plaquette dimerization.

\begin{figure}[h!]
\includegraphics[width=0.8\columnwidth]{densityplot-inset.eps}
\caption{(Color online). Phase diagrams of the plateaus at $M = 0$, $1/3$ and $1/2$ (top to bottom) for $ \delta = \delta_d = 1 $ in the plane $ J-J_d $, up values of the order of one, analyzed by means of DMRG.
The color indicates a decreasing plateau width (from white to black). The blue dotted line in the top panel indicates the line of maximum frustration $ g = 0 $.
In the inset of Panel (a) the SE triplet gap is depicted showing good agreement with the DMRG determination.
}
\label{fig:gap-phases}
\end{figure}

\subsection{Effective model description of dimerized plaquette Hamiltonian  }
\label{sec:effectiveH}

In the previous Sections we have considered the frustrated homogeneous plaquette case, where the frustration
degree is controlled by  $g = J_d - 2 J $, being maximal along the line $g=0$, where the model exhibits an
exact rung-dimer product state. In this Section we will explore the effect of plaquette dimerization
via the parameters $\delta = K/J$ and $\delta_d=K_d/K$. The interplay between on-plaquette frustration and
inter-plaquette dimerization gives rise to a richer plateaus structure that we will analyze in a weak interacting  plaquettes regime, starting from the limit of decoupled plaquettes.
It is worth mentioning that an equivalent
model, although with a different parametrization has been studied analytically using a field theory bosonization approach \cite{Cabra-Grynberg-00}. This FLuttstudy, however, is complementary to ours, since it starts from the strong chain limit.\\

We start by analyzing the Hilbert space corresponding to isolated plaquettes, which is spanned by a basis of sixteen states containing two singlets, nine triplets and five quintuplets, which we identify generically by $|s_i\rangle, |t_j\rangle $ and $|q_k\rangle$, respectively. This states are listed in the Appendix \ref{sec:Effective Hamiltonians} (Table \ref{tab:energies}).
On each plaquette in the absence of magnetic field $h$ the ground state corresponds to the singlet $|s_0\rangle$. When the magnetic field is turned on the energy corresponding to the singlet and a triplet becomes closer until
a critical value where a level crossing occurs and a triplet becomes the ground state. By further increasing the magnetic field we obtain a second level crossing between triplet and the  quintet state $|q_0\rangle$. Here the local structure of the plaquettes couplings becomes important. According to the values of the internal couplings two different types of triplets are involved in the crossings, depending on whether $J_d< \frac{2J}{1+J}$ or $J_d>  \frac{2J}{1+J}$. In the following we refer these two different scenarios as cases A and B, respectively.
In the case A (B) the singlet state $|s_0\rangle$ is degenerate with a triplet state $|t_{A(B)}\rangle$ in the first transition, at the critical field $hc_{1A(B)}$, whereas $|t_{A(B)}\rangle$ is degenerate with the quintet state $|q_0\rangle$ in the second transition at the critical field $hc_{2A(B)}$. These possibilities are represented schematically in eq.(\ref{eq:casesAB})
 \begin{eqnarray}
 \label{eq:casesAB}

    |s_0\rangle \xrightarrow{hc_{1A}} |t_A\rangle \xrightarrow{hc_{2A}} |q_0\rangle, \quad J_d < \frac{2J}{1+J}, \nonumber \\
    |s_0\rangle \xrightarrow{hc_{1B}} |t_B\rangle \xrightarrow{hc_{2B}} |q_0\rangle, \quad J_d > \frac{2J}{1+J}.
 \end{eqnarray}
Around these level crossing points we can derive an effective model. Let us write the Hamiltonian as follows
 %
 \begin{eqnarray}
 \mathbf{H_\mu}=\mathbf{H}_{0,\mu}+\mathbf{H}_{int,\mu},
 \end{eqnarray}
 %
where the extra index $\mu$ indicates the possible cases mentioned, \emph{ie} $\mu=\{1A,2A,1B,2B\}$, being
 %
 \begin{eqnarray}
\mathbf{H}_{0,\mu}&=&\sum_{n} \Big[ \left(\vec{\mathbf{S}}_{1,n}\cdot\vec{\mathbf{S}}_{2,n}+\vec{\mathbf{S}}_{3,n}\cdot\vec{\mathbf{S}}_{4,n}\right) \\\nonumber
&+&J \left(\vec{\mathbf{S}}_{1,n}\cdot\vec{\mathbf{S}}_{3,n}+ \vec{\mathbf{S}}_{2,n}\cdot\vec{\mathbf{S}}_{4,n}\right)+J_{d} \, \vec{\mathbf{S}}_{2,n}\cdot\vec{\mathbf{S}}_{3,n} \\\nonumber
&-&  hc_\mu\sum_{j=1}^{4}\vec{\mathbf{S}}_{j,n} \Big], \\\nonumber
\mathbf{H}_{int,\mu}&=&\sum_{n}\Big[ K\left( \vec{\mathbf{S}}_{3,n}\cdot\vec{\mathbf{S}}_{1,n+1}+\vec{\mathbf{S}}_{4,n}\cdot\vec{\mathbf{S}}_{2,n+1}\right)\\\nonumber
&+&K_{d}\, \vec{\mathbf{S}}_{4,n}\cdot\vec{\mathbf{S}}_{1,n+1}-(h-hc_\mu)\sum_{j=1}^{4}\vec{\mathbf{S}}_{j,n}\Big].
\end{eqnarray}
 %
We construct the effective Hamiltonian via degenerate perturbation theory \cite{Totsuka1,Mila-ladder-98,Sen-ladder-99,mila-10,4-tube-3},
\begin{eqnarray}
\mathbf{H}_{\text{eff},\mu}=\mathbf{H}_{\text{eff},\mu}^{(1)}+\mathbf{H}_{\text{eff},\mu}^{(2)}+\cdots,
\end{eqnarray}
where the superscript indicates the order of the perturbation, although for our purpose in this paper we will only consider the first order of the expansion.

Since at the crossing point the ground state of each plaquette is doubly degenerate (except for $J_d = \frac{2J}{1+J}$ which is triply degenerate and will not be considered here), the original dimerized plaquette model with sixteen states per site is reduced to an \emph{effective} spin-1/2 chain model involving the two mentioned degenerate low energy states at the crossover.

The first order effective Hamiltonian around $hc_\mu$ is obtained by applying the standard degenerate perturbation theory

\begin{equation}
\label{eq:O1-formula}
  \mathbf{H}_{\text{eff},\mu}^{(1)}=\sum_{i,j=1,2}|v_i\rangle\langle v_i| \mathbf{H}_{\text{int},\mu}|v_j\rangle\langle v_j|,
\end{equation}
where states $|v_i\rangle$ and $|v_j\rangle$ span the set $\{|s_0\rangle, |t_{A(B)\rangle}\}$ and
$\{|q_0\rangle, |t_{A(B)\rangle}\}$ for $\mu=1A(B)$ and $\mu=2A(B)$, respectively. In this way is obtained,
up to a constant term,
\begin{eqnarray}
\nonumber
\mathbf{H}_{\text{eff},\mu}^{(1)}&=&\sum_{n}J_{xy,\mu}  \left(\mathbf{S}^{x}_{n}\cdot\mathbf{S}^{x}_{n+1}+ \mathbf{S}^{y}_{n}\cdot\mathbf{S}^{y}_{n+1}\right)  \\
\label{eq:Heff1}
&+&J_{zz,\mu}\mathbf{S}^{z}_{n}\cdot\mathbf{S}^{z}_{n+1}-\tilde{h}_\mu\mathbf{S}^{z}_{n},
\end{eqnarray}
where effective couplings $J_{xy,\mu}, J_{zz,\mu} $ and $\tilde{h}_\mu$ are functions of the original plaquette model couplings $J, J_d, K, K_d$ and $h$. The explicit expressions of these functions are available in the Appendix \ref{sec:Effective Hamiltonians} (Eqs. \ref{eff-J-b}-\ref{eff-J-e}).
In addition, the pseudo spin$-1/2$ operators in Eq. \ref{eq:Heff1} are projectors of the degenerate plaquette basis
\begin{eqnarray}
\label{eq:map1}
  \mathbf{S}^z_n &=& \frac{1}{2} \left(|t_{A(B)}\rangle\langle t_{A(B)}|-|s_0\rangle\langle s_0|\right)_n \nonumber \\
  \mathbf{S}^\dag_n &=& \left(|s_0\rangle\langle t_{A(B)}|\right)_n, \quad \mu = 1A(B),
  \end{eqnarray}
\begin{eqnarray}
\label{eq:map2}
  \mathbf{S}^z_n &=& \frac{1}{2} \left(|q_0\rangle\langle q_0|-|t_{A(B)}\rangle\langle t_{A(B)}|\right)_n \nonumber \\
  \mathbf{S}^\dag_n &=& \left(|t_{A(B)}\rangle\langle q_0|\right)_n, \quad \mu = 2A(B),
\end{eqnarray}
where $\mathbf{S}^\dag_n=\mathbf{S}^x_n + i \mathbf{S}^y_n $.

\subsection{Phase diagram of the spin$-1/2$ chain effective Hamiltonian}
\label{sec:effectiveH}

The effective model given by Eq.(\ref{eq:Heff1}) corresponds to an XXZ spin$-\frac12$ chain and can be solved exactly via the Bethe ansatz \cite{bethe}. Here we briefly review the main characteristics of the Bethe ansatz solution, considering the different phases present, and their implications in the magnetization process.
For simplicity in this discussion we will simplify the notation of Eqs.(\ref{eq:Heff1})by removing supra- and sub- indices and indicating the three principal parameters of the model as $J_{xy}$, $ J_{zz} $ and the magnetic field $h$. The phase diagram in the plane $( \Delta =J_{zz}/J_{xy}, h/J_{xy})$ is shown
in the Fig.(\ref{fig:BA-phases},a). As can be observed there are three phases: Ferromagnetic (F, yellow), N\'eel (N, blue) and Luttinger Liquid (LL, gray). The F and N phases are gapped, with spin$-1$ \emph{magnon} and spin$-1/2$ \emph{spinon} domain wall type of excitations, respectively. On the other hand, the intermediate phase LL is gapless and exhibits quasi-long range order.
Regarding critical lines in Fig. \ref{fig:BA-phases}(a), the straight lines separating LL and F phases are explicitly given in terms of $\Delta$  by
\begin{equation}
\label{eq:BA-L_F}
 h_{L-F}=\pm(1+\Delta),
\end{equation}
whereas the border between LL and N phase is obtained by solving \cite{chain-in-field}
\begin{equation}
\label{eq:BA-L_N}
  h_{L-N}=\pm\sinh (g)\sum_{n=-\infty}^{\infty}\frac{(-1)^n}{\cosh(n g)},
\end{equation}
where $g=\operatorname{arcosh}\Delta$. Note that both, $h_{L-F}$ and $h_{L-N}$, are given in units of $J_{xy}$.

The different phases present in the model translate into distinctive characteristics in the magnetization curves. The gapped phases F and N exhibit a plateau in the magnetization curve, while in the LL phase the magnetization continuously increases with the magnetic field. This dependence is illustrated in Fig.\ref{fig:BA-phases}. In panel (a) of this figure three paths with red, blue and green dotted lines are indicated in the phase diagram, while in Fig. \ref{fig:BA-phases}(b) the magnetization curves corresponding to these three paths are showed schematically. Note that the vertical axis in panel (a) is represented as horizontal axis in panel (b). The red curve represents the simplest case, where the system is always in the F phase and has a stepwise structure with two plateaus, jumping directly from one to another with the sign change of the magnetic field. The blue magnetization curve represents an intermediate case, in which the two plateaus corresponding to the F phase are connected through a region
of continuous growth of the magnetization with the field, whose shape is characteristic of the LL phase. Finally, the green magnetization curve in Fig. \ref{fig:BA-phases}(b) indicates the more general case, which comprises the three phases. In this case, in addition to the plateaus at the extremes due to the F phase, an additional intermediate plateau indicative of the N phase is present. These three plateaus are connected by LL phases, consistently with the path indicated in green in Fig. \ref{fig:BA-phases}(a).\\
Before concluding with this Section we would like to comment on another case where the plateau structure has a simple stepped form. This case is the \emph{Ising} limit of the model, in which $J_{x,y}\rightarrow 0\, (\Delta \rightarrow \infty)$, \emph{ie} the right infinite end of Fig. \ref{fig:BA-phases}(a). Here the magnetization curve exhibits the three plateaus with direct jumps between them, without intermediate LL phase.

\begin{figure}[t]
\includegraphics[width=1.0\columnwidth]{fig-BA.eps}
\caption{(Color online). Panel (a): Phase diagram $h$ vs $\Delta$ (in units of $J_{x,y}$) of the XXZ spin$-\frac12$ chain. The three phases present in the model are ferromagnetic (F, yellow), N\'eel (N, blue) and Luttinger liquid (LL, gray). The F and N phases are gapped, and exhibit spin$-1$ magnon and spin$-1/2$ spinon excitations, respectively, whereas LL phase is gapless and presents quasi-long range order. Panel (b) Schematic representation of the magnetization curves $M(h)$ corresponding to the three paths indicated with red, blue and green dotted lines in panel (a). Note that each gapped phase translates into plateaus in the magnetization curves(see the text for details).}
\label{fig:BA-phases}
\end{figure}

\subsection{Effective model description of magnetization process}
In this Section we will analyze how the effective low-energy spin-1/2 chain model with anisotropy $\Delta$, obtained previously, captures the main aspects of the magnetization process of the original plaquette model in a certain, strong plaquette coupling, regime of the parameters space. The effective model is not only useful to account for the numerical results obtained through DMRG, but also brings a direct physical interpretation of the structure of plateaus that can emerge in the original model. In addition, it provides a tool that allows us to detect some characteristics, such as small intermediate plateaus, which would be difficult to detect by direct numerical scanning.

To apply the results of the effective model to the original plaquette model, note that according to Eq.(\ref{eq:map1}), the effective pseudo-spin operators eigenvalues $S_z = -1/2 (1/2)$ correspond to plaquette $S_z = 0 (1)$, of the singlet $|s_0\rangle$ and triplet $|t_{A(B)}\rangle$ eigenvalues of the first level crossing $1A(B)$. Therefore the three possible magnetization plateaus in the effective chain model at $\{-1,0,1\}$ are mapped onto $\{0,1/4,1/2\}$ magnetization plateaus in the plaquette model, normalized to maximum spin per site and plaquette, respectively. Similarly for the second level crossing $2A(B)$ between the triplet $|t_{A(B)}\rangle$ and quintet $|q_0\rangle$, the Eq.(\ref{eq:map2}) maps $\{-1,0,1\}$ onto $\{1/2, 3/4,1\}$, corresponding to effective chain and plaquette models, respectively.

Now we will describe qualitatively the structure of the magnetization curves in terms of the effective models. To this end, the Fig. \ref{fig:mag-disp} illustrates different representative plateaus structures in the original model obtained by means of DMRG, showed with blue solid line. In particular, the Fig. \ref{fig:mag-disp}(a) shows magnetization curves with a single transition between type-F phases at zero magnetic field and saturation, intermediated by a LL phase. Note that the structure of small steps in the LL portion of the curve is due to the finite size of the numerical calculation. In Fig.\ref{fig:mag-disp}(b) an additional plateau emerges at mid-magnetization. However, according to the previous analysis, this plateau is type-F, like the other two present here. The cases of Figs. \ref{fig:mag-disp}(a,b) are not dimerized, \emph{ie}  $\delta=\delta_d=1$, and illustrate a general feature: in absence of dimerization between plaquettes there can only be plateaus at $ M = 0,1 / 2 $ and $ 1 $ in the
magnetization curve, sharing spin$-1$ magnon excitations.  The origin of these plateaus is intrinsic to each isolated plaquette, as a consequence of the two (1A(B), 2A(B)) possible level crossings, which are only renormalized by the interaction between plaquettes.
\\

On the other hand, the Fig. \ref{fig:mag-disp}(c) show a plaquette-dimerized case, where additional plateaus at $M = 1/4, 3/4$, are present. These plateaus are indicative of the N-type phase which, as we have mentioned,  present spinon-like excitations and are generated by the interaction between plaquettes. A necessary (but not sufficient) condition for their emergence is the presence of plaquette dimerization.\\
Regarding the size of type-N plateaus, we have observed that in general are smaller than those of type-F.  For this reason the effective model is very useful for their detection, since numerically they could easily be overlooked, given the quite large number of parameters involved in the original model.\\
In addition, the effective model not only facilitates the detection of the intermediate N-type plateaus, but also highlights the role of frustration in the emergence of them. In fact it is observed that apart from dimerization, type-N plateaus grow with frustration,
being largest along the line of maximum frustration, \emph{ie} $ g=J_d-2J=0 $ and $ g_k=K_d-2K=0 $, whenever this condition can be satisfied, which depends on the crossing levels (A or B) involved. It is interesting to note that in a certain range of validity frustration-induced type-N plateaus are maximal along the maximum frustration line, where in addition the original model (Eq.(1)) exhibits an exact dimer singlet-product zero field ground state. In some sense it suggests that this condition is not only relevant at zero magnetic field, but also play a role in the magnetization process.

Finally, regarding the dotted red and yellow vertical lines showed in the three panels of Fig. \ref{fig:mag-disp}, they are quantitative estimations of critical fields delimiting the different plateaus based on the dispersion of magnon and spinon excitations of the effective model, and will be discussed in the next Section.

\subsection{Magnetic excitations and critical fields}

Here we will analyze the excitations on the plateau structures in the effective chain model, which in turn will allow us to obtain the critical fields, \emph{ie} the borders of the plateaus and compare with the numerical DMRG results
from the original model. \\First we consider the saturation plateaus at $-(+1)$ in the effective chain model. These correspond to ferromagnetic states, with all the effective spins on each site polarized along the negative and positive direction of the effective field, respectively. The elemental magnon excitation consists of a spin-wave composed by a linear combination of states with a local one-spin inversion, carrying $ \Delta S_z =+(-1) $. The application of the effective Hamiltonian (Eq.(\ref{eq:Heff1})) on this state (subtracting the ground state) gives rise, by a standard calculation, to the magnon (M) dispersion
\begin{equation}
\label{eq:magnon-disp}
  \omega^\pm_{M,\mu}(k)=J_{xy,\mu}\cos(k)-J_{zz,\mu}\mp \tilde{h}_\mu.
\end{equation}
Eq.(\ref{eq:magnon-disp}) describes all magnon-type of excitations present in the strong-plaquette expansion of the original model, via the mapping given by Eqs.(\ref{eq:map1},\ref{eq:map2}), and the effective couplings ((Eqs. \ref{eff-J-b}-\ref{eff-J-e})), namely (\emph{i}): right (high field) borders of $M = 0 \quad (+, \mu = 1A (B))$ and $M=1/2 \quad (+, \mu = 2A (B))$; (\emph{ii}): left (low field) boundaries of $M = 1/2 \quad (-, \mu = 1A (B))$ and $M = 1 \quad (-, \mu = 2A (B))$.

Let us now consider the $M=0$ plateau in the effective model, which is associated to a double degenerate N\'eel state. Here the elemental spinon excitation consists of a spin-wave composed by a linear combination of domain wall states with two consecutive spin inversions, carrying $ \Delta S_z =+(-1/2) $. Following a similar procedure as the magnon case we obtain the spinon (S) dispersion
\begin{equation}
\label{eq:spinon-disp}
  \omega^\pm_{S,\mu}(k)=J_{xy,\mu}\cos(2k)+\frac{1}{2}J_{zz,\mu}\mp \frac{1}{2} \tilde{h}_\mu.
\end{equation}
Equivalently to the previous case, Eq.(\ref{eq:spinon-disp}) describes all spinon-type of excitations, namely (\emph{i}): right (high field) borders of $M = 1/4 \quad (+, \mu = 1A (B))$ and $M=3/4 \quad (+, \mu = 2A (B))$; (\emph{ii}): left (low field) boundaries of $M = 1/4 \quad (-, \mu = 1A (B))$ and $M = 3/4 \quad (-, \mu = 2A (B))$.

From the above magnon $\omega^\pm_{M,\mu}(k)$ and spinon $\omega^\pm_{S,\mu}(k)$ dispersions we can calculate the critical fields, \emph{ie} the edges and the width of the plateaus present in the model. For this, we impose the condition of gap closure (minimum of the dispersion), which allows to determine the effective fields in terms of the effective couplings and thus the critical fields according to the original couplings of the model. Due to the number of parameters of the model, in general we obtain the critical fields numerically which is much simpler compared to the  DMRG or the numerical solution of the Bethe-ansatz equations approaches. \\
The techniques used in this work to determine the critical fields: DMRG, and effective models involve different types of approximation. DMRG suffers finite size effects and the effective models have a perturbative origin.
However, these methods are somewhat complementary and provide consistent results. Although, as we have mentioned, it is not the aim of this work to quantitatively determine the edges of each plateau in the whole parameter
space of the model, it is interesting to compare the results of the different techniques in some representative cases. For this purpose, in Fig. \ref{fig:mag-disp} we show the critical fields determined
by DMRG (full magnetization curve in blue line) and the edges of the different plateaus present by applying the gap closure condition of the corresponding magnon dispersion (red-dashed lines) or spinon dispersions (orange-dashed lines).
As can be observed there is a very good agreement between the different techniques, which is maintained up to intermediate couplings (compared with $J_0$).\\

\begin{figure}[t]
\includegraphics[width=0.8\columnwidth]{magnetizaciones-dispersiones.eps}
\caption{(Color online). Magnetization curves corresponding to $J=K=0.4$ and $J_{d}=K_{d}=0.3$ (panel a)), $J=K=0.2$ and $J_{d}=K_{d}=0.3$ (panel b)), $J=0.8$, $J_{d}=0.6$, $K=0.2$  and $K_{d}=0.45$ (panel c)).
Red-dashed lines and orange-dashed lines correspond to critical fields obtained by means Magnon and Spinon dispersions respectively.
}
\label{fig:mag-disp}
\end{figure}

As previously mentioned and as can be seen in the Fig. \ref{fig:mag-disp}(c) the plateaus at $M = 1/4$ and $3/4$ are small compared to the others, so that they could be difficult to detect numerically by sweeping a fairly large parameter space as in this model using DMRG. It is therefore of particular interest to be able to estimate not only the presence but the maximum possible size of these plateaus as well as their interdependence with frustration and dimerization.\\
From this point of view the effective model offers a direct way of evaluating positions and maximum widths of the plateaus. This will occur when there is no intermediate Luttinger liquid phase between plateaus, \emph{ie} the limit of the effective Ising model where $J_{xy,\mu} = 0$. In this condition the magnon and spinon disperson bands do not propagate (they are flat) and the edges of fields predicted by both merge. From the Eqs (\ref{eq:magnon-disp},\ref{eq:spinon-disp}) we see that the effective critical fields at the Ising limit satisfy
\begin{equation}\label{hc-Ising}
  J_{zz,\mu}=\pm \tilde{h}_\mu, \quad \textmd{for} \quad  J_{xy,\mu}=0.
\end{equation}
Note in passing that this condition is obtained from the effective model exact line $h_{L-F}$  (Eq.(\ref{eq:BA-L_F}))
in the limit $\Delta \rightarrow \infty$.\\

By solving the Eqs.(\ref{hc-Ising}) in terms of the original variables of the model (Eqs. \ref{eff-J-b}-\ref{eff-J-e})) we get the following cases for the widest plateaus $\Delta h$. For case A $(J_d<\frac{2J}{1+J})$, the Ising limit is obtained for $g_k=K_d-2K=0$ with the same $\Delta h=K/2$ for both $M=1/4$ and $M=3/4$ plateaus. \\
On the other hand, for case B $(J_d>\frac{2J}{1+J})$, the plateau widths are not the same, although we can also obtain analytical expressions for both. In the case $M = 1/4$ the Ising condition is obtained for both $g=J_d-2J=0$ and $g_k=K_d-2K=0$ although these conditions are not satisfied for $M = 3/4$. \\

\section{Conclusions}
\label{sec:conclusions}

In the present paper we have studied the magnetization process in a zig-zag quantum antiferromagnet in the presence of both frustration and plaquette dimerization. Due to the complexity of the unit cell, several cases are present giving a rich structure of the magnetization curve. At zero magnetic field the ground state is exactly determined under the conditions $g=J_d-2J=0$ and $g_k=K_d-2K=0$ and corresponds to a product state of spin singlets.
Around this highly frustrated line we investigate the $M=0$ plateau by using dimer series expansions, bond operators mean field theory and numerically by means of density matrix renormalization group.

In order to analyze the magnetization process we complement the numerical results with first order low energy effective models starting from the limit of decoupled plaquettes. From a qualitative point of view, the effective models capture the essential features obtained numerically and bring a simple physical interpretation of the emergent structure of plateaus in the original zig-zag model. In addition, the analysis of magnon and spinon excitations in the efective models allows us to obtain estimations of the critical fields bordering the plateaus, which are in a very good quantitative agrement with the numerical computations
in a strong plaquette dimerization regime.

Our study also suggest that the combined effect of on-plaquette frustration, controlled by $g$ and $g_k$ and inter-plaquette dimerization, via $\delta=K/J$ and $\delta_d=K_d/J_d$, plays a central role in the structure of the magnetization curve. In fact whenever is possible, according to the values of the couplings involved, $M = 1/4$ and $3/4$ plateaus are widest along the line of maximum frustration. In this sense we say that these plateaus are induced by frustration, which together with the dimerization between plaquettes, provide a rich interplay between both aspects that are intrinsic to the model. For this reason, the maximally frustrated $g$ and $g_k = 0$ line is not only fundamental to zero field, but plays a central role beyond zero magnetic field, which is reflected in its influence on the resulting magnetization process.

\section*{Acknowledgments}
We acknowledge useful discussions with M. Matera.
C. A. Lamas is supported by CONICET (PIP 1691) and ANPCyT (PICT 2013-0009)

redpajama_set_name: RedPajamaArXiv


Please note on 5 December 2018, the image, medium and details of this work were updated.
Dominic Beattie lives and works in London. His exhibitions often combine modernist furniture and abstract, pattern-based paintings. Described as having an 'anti-digital aesthetic', Beattie uses non-traditional materials such as marker inks, tape, household paint, spray paint, stickers, collaged paper to create geometric work.
He says of his work, "I tend to make things that approximate the modernist aesthetic in some way. I deliberately avoid trying to express any concepts, I work in a visual realm and I prefer the viewer to decide what they are looking at."
Beattie's exhibitions include, Studio solo at FOLD Gallery, London in 2015, New Order II: British Art Today at the Saatchi Gallery, London in 2014, and Creekside Open 2015 selected by Lisa Milroy and Richard Deacon at APT Gallery in London in 2015. He has also exhibited work at Warrington Museum and The Hospital Club.
Beattie graduated from Camberwell College of Art, London in 2004.


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Canada Post considering 'legal options' amid protests at sorting facilities
Canadian PressMore from Canadian Press
'UNPRECEDENTED' BLIZZARD: State of emergency in St. John's, N.L.
U.S. sanction law not enough to prove Cdn fraud: Meng's lawyers
U.S. court won't force military to hear Khadr appeal
Que. man charged in alleged threats aimed at PM, Muslims
A fireman works to extinguish a burning barrel at Canada Posts' Gateway Postal Facility, after union groups and activists blocked truck traffic to and from the facility in support of Canada Post workers who were recently legislated back to work by the federal government, in Mississauga, Ont., Saturday, Dec. 1, 2018.Cole Burston / THE CANADIAN PRESS
MISSISSAUGA, Ont. — Canada Post says protesters blocking mail from flowing in and out of one of the country's busiest postal hubs are violating a court order, and the Crown corporation is "considering all available legal options."
The protesters rallying outside the facility in Mississauga, Ont., are largely members of the Canadian Union of Public Employees, acting in solidarity with postal workers who were legislated back to work last week.
The back-to-work legislation was passed on Monday, following weeks of rotating strikes by the Canadian Union of Postal Workers.
CUPE says 19 protests are scheduled this weekend at facilities across the country, and demonstrators will allow workers to enter but not let mail out of the plants.
The Ontario branch of that union says the protesters have been served with an injunction, but they still plan to demonstrate in solidarity with the postal workers.
A copy of the court injunction can be seen at the remains of a makeshift blockade at Canada Posts' Gateway Postal Facility, after union groups and activists blocked truck traffic to and from the facility in support of Canada Post workers who were recently legislated back to work by the federal government, in Mississauga, Ont., Saturday, Dec. 1, 2018. Cole Burston / THE CANADIAN PRESS
In a statement Saturday afternoon, Canada Post says it has contacted police and is considering its options to deal with the demonstrations.
"We have contractual commitments to our customers, as well as statutory obligations to ensure important mail is being received by all Canadians," the Crown corporation said.
"We have asked the police for assistance and we are considering all available legal options."
Injunctions were also granted by courts in Alberta and British Columbia, prohibiting anyone from obstructing or interfering with people or vehicles entering or exiting Canada Post facilities.
Protesters hold brief blockade at entrance of Canada Post facility in Halifax
Senate passes back-to-work bill for postal workers
Jobs and Labor
Labor Strikes and Disputes
Premier Doug Ford refuses to 'roll over' for teacher unions
Anonymous internet posters successfully sued for defamatory comments
Ontario to pay parents impacted by teacher strikes

redpajama_set_name: RedPajamaCommonCrawl


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Wafaa Bilal
Makers Muse Awards: Kindle Project's Makers Muse Award, our founding program which started in 2009, honors artists working in all mediums and forms: traditional and experimental, classic and contemporary.
View All Makers Muse Award Recipients
Iraqi born artist Wafaa Bilal is an assistant professor at New York University's Tisch School of the Arts. He has exhibited and lectured world-wide, promoting awareness of the situation of the Iraqi people and stressing the importance of peaceful conflict resolution. Bilal's 2007 dynamic installation, Domestic Tension, placed him on the receiving end of a paintball gun that was accessible online to a worldwide audience, 24 hours a day. The month-long piece spurred on-line debates and intense conversations. The Chicago Tribune called it "one of the sharpest works of political art to be seen in a long time" and named him Artist of the Year in 2008. Newsweek's assessment was "breathtaking.". In the face of a war that stretches on, the 2004 deaths of his brother and father and the violence in his own history, Bilal nonetheless seeks to imbue his audiences with a sense of empowerment and hope. In fall 2008 City Lights published "Shoot an Iraqi: Art, Life and Resistance Under the Gun," about Bilal's life and the Domestic Tension project. In 2009 Booklist named it a top 10 arts books of the year. In the Spring 2010 ArtFourm named his show Agent Intellect among the best in the northeastern region. His spring 2010 project And…Counting used tattoo as a medium to represent Iraqi and American casualties in the war, creating dialogue around the invisibility of Iraqis killed in the war.
Grantee Information
Website(s): http://www.wafaabilal.com
Year(s) Awarded: 2010
Related Nexus Stories:
NPR Features Wafaa Bilal
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The Law Offices of Owsley & Associates in Houston, Texas is one of the most widely known and respected law firms in Texas for the past 25 years. We focus our attention on Criminal Defense, including Traffic Ticket Defense, and Personal Injury.
Client service, redundant communication, and ability to sort out complex driver's license issues set us apart. Our mission is to reduce your contact with court as much as possible, while dismissing your traffic charges for the least amount possible.
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The second issue of the Dust & Shadow Reader explores the topic of "attunement". We understand attunement as a particular sensitivity (with beings or situations) characterised by a careful, receptive, open awareness. It assumes a willingness to be touched by external circumstances; to be lured, affected and changed by them. Although it is an innate (somatic) ability for many people, it can be developed, enhanced and refined through practice. The practice of attunement can foster experiential, situated learning across different scales. A deliberately hesitant engagement with other entities on their own terms, from quivering butterfly wings to earth-shattering volcanoes. Attuning relies on the human capacity to notice and participate in the (asymmetrically) reciprocal effects that occur in direct, immersive experiences in the world. A kind of "ecological intimacy".
In a state of attunement the world can be experienced teeming with life, where humans are not separate from nature. It's a mutualistic or symbiotic state of being, that accepts human interdependence with the planet as a given. Attunement practices are common in animist cultures, and can be found in many indigenous societies. There are resonances with current philosophical tendencies, including speculative realism or the long winding trails of panpsychism. Yet attunement is a concept that is often difficult to grasp in words, especially words that make sense outside of philosophical, ritual, performative or meditative discourses.
This reader is an attempt to explore attunement from a range of different perspectives. We invite contributions, suggestions and pointers to promising directions from humans (and other non-corvids) from all walks of life. Theoreticians, practitioners, skeptics and enthusiasts, who can help us understand if and how attuning to the wider-than-human world can help adapt to the increasingly uncertain present of the Anthropocene.
If you are interested in contributing to the Dust & Shadow reader on attunement, please send your propositions, questions or suggestions to earth@fo.am by the end of January 2019. Contributions can include short texts (max 500 words), drawings, monochrome photographs, quotes or relevant references. The editorial team will make their selection and collate the reader for publication in early spring of 2019.


To choose a perfect medication for a particular health issue is really a tricky task. We have to take proper care while choosing a medication. It should suit your body and proved to be the best medicine for you. The same concept is used in choosing the best heartworm prevention medication.
Heartworm is basically a parasitic worm that gangs up on the hearts of cat, dog, and other animals. Now, it is scientifically proved that heartworm prevention will easily protect and cure your pet's heart with single heartworm and flea pill. Heartgard is the medicine that protects your pet's heart from heartworm.
Here, in this article, we provide you some tip that helps you in choosing the best heartworm prevention medication for your pets.
It Should Be Convenient: The foremost thing that you have to follow in choosing the best medication is its convenience. If the medicine is available in several forms it is easy to take it as per your requirement. Heartworm is available in both form capsules as well as topically. If it is difficult to give tablets to your pet then you can use its topical form. So, choose the right one for your pet.
Choose the Beneficial Medication: Beneficial medication means it should be convenient and also affordable. Choose the right and unique product for your pet so that it is easy for you to protect it from heartworm and other intestinal parasites. So, choose such medication that is effective for your pet and also suits your budget.
Its Effect Should Be Long-Lasting: A medicine should be more beneficial if its effect is long-lasting. It gives relief to your pet for a long time. Opt that medication which is effective as per your pet's health. We know that each medicine reacts differently to animals depending on several factors such as the lifestyle of your pet. If its effect will remain for a long time it will protect your pet from heartworm for long period.
Give Instant Effect to Your Pet: This is also the key point in choosing the best medication. If it give instant relief to an animal then it will be highly beneficial for them. You have to put proper attention on them so that you can see the changes in its behaviour after medication. If it is done its regular activities normally then the medication you choose for them is suit its body.
Prevent Multiple Pest: Heartworm is not only a single parasite that affects animals. There are many more pests that infect them. So, while choosing a medication to be sure that it will able to rectify multiple pest problems. It is easy for you to give timely and proper dosage to it. Choose heartworm as per the prescription.
Conclusion: From the above points, you definitely get an idea to choose the best heartworm prevention medication. It is highly recommended to consult your pet's doctor before starting any type of medication for animals. So, "be safe and alert".
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This succulent photography features a dreamy collection of hen and chicks succulents captured at the perfect moment. Imagine this succulent photo on your living room walls, in your home office, or maybe your bedroom. Soft pastel green and purple shades enhance any decor. This print also makes a wonderful gift for the nature photography and garden art lover in your life.
If you don't see the size you need, please contact me and I'll be happy to help you!


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Return to the news list
Copenhagen, March 8, 2019
DFDS' largest ferry named and ready for service
On 7 March, DFDS' largest ferry ever was named Ephesus Seaways at a naming ceremony in Istanbul and will soon set course for its first sailing to Trieste fully loaded with 450 trailers.
EVP Peder Gellert, Sponsor Eva Lundstedt and SVP Selçuk Boztepe at the naming ceremony in Istanbul
Eva Lundstedt, who is the wife of Volvo's CEO Martin Lundstedt, gave the ship her name and made a short speech about the importance of this ferry for people in the region and DFDS.
It will be the largest freight ferry ever to service DFDS' customers on routes between Turkey and the EU. Its first departure will be on 9 March from Istanbul with arrival in Trieste on 11 March.
"We are extremely proud to present this new ferry to our customers. With space for 6,700 lane metres of cargo, our customers, the Turkish and European logistics companies, can look forward to greater capacity," says EVP Peder Gellert, Head of DFDS Ferry Division.
"This ferry is the first of an order of six from the Chinese Jinling Shipyard and will be deployed in DFDS' route network. It was reflagged under the Turkish flag, and re-named to Ephesus Seaways in accordance with our principle of naming our Mediterranean ships after UNESCO heritage sites in Turkey," says Selçuk Boztepe, Senior Vice President of DFDS in the Mediterranean. The second of the new mega freight ferries will also join our Turkish fleet when it arrives in late spring.
The new freight ferries are each equipped with a unique ramp system with three independent stern ramps and internal ramps, making it possible to load and unload trailers quickly and efficiently. "This means that the freight ferries can be discharged and loaded in record time to enable efficient and reliable transport, to the benefit of the many logistics operators we service in Turkey as well as in the EU, and to the benefit of industries as well," says Selçuk Boztepe.
Gert Jakobsen
Vice President, Communications
gert.jakobsen@dfds.com


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Section B: Chemical and Biological Sciences
Studies on Coordination Chemistry and Antibacterial Activity of Bidentate NS Schiff Base Derived From SBDTC and 4-Benzyloxybenzaldehyde
M. H. Islam M.Phil Fellow Department of Chemistry Rajshahi University of Engineering & Technology
M. C. Sheikh Post Doctoral Fellow Department of Applied Chemistry University of Toyama
M. A. A. A A. Islam Professor Department of Chemistry Rajshahi University of Engineering & Technology
https://doi.org/10.3329/jsr.v11i1.37863
The condensation of 4-benzyloxybenzaldehyde with S-benzyl dithiocarbazate (SBDTC) in absolute ethanol afforded the Schiff base, S-benzyl-β-N-(4-benzyloxyphenyl)methylene-dithiocarbazate (1). The ligand on further reaction with metal ions in absolute ethanol [for Ni(II), Cu(II), Zn(II)] and in methanol [Pd(II)] resulted in the formation of the corresponding four coordinate complexes. The ligand and its complexes were characterized by physico-chemical techniques, viz., molar conductance, magnetic susceptibility measurement, IR, NMR and solution UV/Vis spectroscopic techniques. The complexes demonstrated stronger antibacterial activity than the free ligand indicating that coordination with these metal ions has enhanced biological potential of the ligand. All the compounds exhibited more or less moderate activity against the test organisms.
M. H. Islam, M.Phil Fellow Department of Chemistry Rajshahi University of Engineering & Technology
Deputy Manager (Production)
Panchagarh Sugar Mills Ltd, Panchagarh
(Bangladesh Sugar and Food Industries Corporation
M. C. Sheikh, Post Doctoral Fellow Department of Applied Chemistry University of Toyama
Post Doctoral Fellow
Department of Applied Chemistry
University of Toyama
M. A. A. A A. Islam, Professor Department of Chemistry Rajshahi University of Engineering & Technology
Rajshahi University of Engineering & Technology
Islam, M. H., Sheikh, M. C., & Islam, M. A. A. A. A. (2019). Studies on Coordination Chemistry and Antibacterial Activity of Bidentate NS Schiff Base Derived From SBDTC and 4-Benzyloxybenzaldehyde. Journal of Scientific Research, 11(1), 121–132. https://doi.org/10.3329/jsr.v11i1.37863


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Rich: WSL gets 30 million dollars!
No more WSL app, surfing in tank at 2020 Olympics and more!
I apologize deeply for yesterday's silence. I've got a story coming that is going to make you laugh and laugh and laugh and laugh but have to sort out some details first. In the meantime, it's a good thing that BeachGrit's still around because otherwise you'd have to pay Stab 20 cents for its new premium service to learn that the World Surf League will make 30 million dollars from its new Facebook deal.
Thirty million dollars!
And let me cut and paste some bits from Variety for you here and free.
Facebook, which is moving aggressively into the business of live sports, locked up a two-year agreement last week with the World Surf League.
The deal brings more than $30 million in licensing revenues to the league, and also comes at a pivotal moment in the history of professional surfing. Last fall, the WSL held an event at the Kelly Slater Wave Pool, a facility in Lemoore, Calif., that creates artificial waves. The league is now working to build several similar facilities around the world in hopes of making the sport less dependent on ocean currents and more amenable to TV viewing.
Some of the more salient bits of the interview with WSL CEO Sophie Goldschmidt.
-Surfing is not available on American TV, though the league does have agreements in other territories, including Australia and Brazil. Since its rebranding a few years ago, the league has fashioned itself as being "digital first" — offering its events on YouTube and through its mobile app. Last year, the WSL signed a non-exclusive deal with Facebook. The new agreement provides for exclusivity, meaning that live events will no longer be available on the WSL app.
-In addition to the scheduling issue, the wave pool will allow the WSL to create its own stadiums, bringing spectators much closer to the action than they would be if they were standing on the beach. The predictability of the waves will also open up the possibilities for camera angles. The league can also host nighttime events, with floodlighting to illuminate the surfers.
-The WSL's moves have riled some traditionalists who see surfing as a spiritual communion with the ocean. Goldschmidt is careful to stress the league's support for ocean conservation efforts, and also emphasizes that ocean surfing is not going away.
-"It's not about wave systems or the ocean," she says. "I think we can have our cake and eat it too. The values and culture of the sport have never been more important, but we can be innovative and push the boundaries."
-The WSL has broken ground on a second wave facility in Florida, and has plans to build more in Japan and Europe. Surfing will be in the Tokyo Olympics in 2020, which the WSL sees as an opportunity for a broader audience.
-"We're planning to build a wave system (in Tokyo) in time," for the games, Goldschmidt says. "Maybe there's a chance they'll do it in the wave system."
So, no more WSL app, the future of professional surfing is the pool and they 2020 Olympics are going to be its grand showcase. Right? But who are these traditionalists who see surfing as a spiritual communion with the ocean? They sound like bastards to me.
And that's big news right there. Ultra premium for free!
Noted shaper charged after man's death
A sad coda to noted surfer Tony Hardy's life…
If you live around Margaret River, you'll know the shaper Tony Hardy and his brood. Amazing surf family. Tony is pops to surfer-bodyboarders Gene, Ryan, Brett and Josh.
All of 'em except Josh have won a State surfing title. Even his granddaughter Willow scooped one up this year to go alongside her two Small Fries crowns.
But Tony doesn't surf much anymore, blames age, injuries. Hangs out at Nathan Rose's surfboard factory, the joint Maurice Cole built in 1995. Uses the shower when he needs to.
"He's ruled a line through surfing," says one Margaret River surfer.
Tony, who is sixty-seven, is what you'd call, in polite company, an eccentric. You'll see him at Main Break, Margaret River, the wave he owned in the seventies, trimming the pig-face that has grown over the footpath. Or stopped at the side of the road in Margs with his shears trimming the bush.
On Tuesday afternoon, Tony allegedly got into a fight with his ex-wife's husband and belted him around so badly with a paint roller (allegedly) he had to be flown to Perth via the Flying Doc service to treat his life-threatening injuries. Tony's ex-wife was also hospitalised for injuries.
The cops caught Tony hiding in bushland later in the afternoon.
Homicide squad detectives say charges of causing grievous bodily harm and aggravated assault occasioning bodily harm may be upgraded.
Surf Ranch is a Satanic Mirror!
It shows you who and what you are but the worst possible version.
Surf Ranch that fucking Surf Ranch. That God forsaken lake in the middle of California and right near my ex-wife's hometown Surf Ranch. With the Indian casino around the corner and a fence surrounding the environmentally friendly mulch and wood chip landscaping Surf Ranch. The parking lot that still had professional surfers' name placards in front of the spaces Surf Ranch with one or maybe two Tesla power stations Surf Ranch.
Do you want to know where I parked? In Mick Fanning's spot. Do you want to know what happened to me later in the day on wave number four? I dislocated my shoulder for the 25th-ish time in my life and am headed to major surgery in two weeks. Where the doctor saws part of my shoulder bone off, flips it upside down and screws it back in so the muscle runs across the ball joint and helps keep everything in place.
Mick Fanning that fucking Mick Fanning.
Longtom asked in the comments of the initial Surf Ranch post, "Couldn't you fake fire and fury for the people?" Shame washed over me. Shame followed by exhaustion. I am exhausted of thinking about Surf Ranch and looking at Surf Ranch and hearing the damned words Surf Ranch but he is right. Exhaustion is not an excuse to perform surf journalism poorly and so here I am, mustering the bile that rotted in my throat as Derek and I drove south that evening.
Surf Ranch that fucking Surf Ranch is not a pornography, as previously described, nor is it a hooker. Those two are built purely to give you exactly what you want. To reflect back what you imagine yourself to be. Surf Ranch is exactly opposite. It shows you who and what you are but the worst possible version. A horrible, satanic mirror and allow me to explain. I never not once in my entire life imagined myself a "good" surfer.
I surf.
I like to surf.
And that is all. I have also never cared. I love to write and some of the grandest writers of all time are failed bullfighters, boxers, baseball players, dancers. I am a failed surfer but Surf Ranch, oh Surf Ranch, it makes you more than a failure. It makes you a fraud.
Shall I describe? You paddle out with three other friends and line up along a fence. The man in front has priority. The man behind him second priority. Etc. The pool is calm and warm enough and you have one thought banging in your head.
DO NOT BLOW THE TAKE OFF.
You've taken off on thousands of waves in your life. Hundreds of thousands. So why are you even worried? But you are worried because this is not a wave this is Surf Ranch and then the countdown begins. A voice shouting across the water "One minute until wave!" One minute until wave? How often do you get a warning in the ocean like that? Never. Never especially if you are me.
That minute is eternal. You think DO NOT BLOW THE TAKEOFF. And you think DO NOT THINK DO NOT BLOW THE TAKE OFF. And you think DO NOT THINK DO NOT THINK DO NOT BLOW THE TAKEOFF. And then the plow whirs to life. It is not as loud as you would imagine. Not quite industrial but definitely mechanical and it whirs and the wave starts breaking on the shoal at the side of the pool very counter-intuitively. It is breaking over there and then the swell starts gurgling toward you and your heart is pounding DO NOT BLOW THE TAKEOFF as you paddle toward one of the flag poles at the distant end of the pool in the exact opposite way that you would normally paddle. Counter-intuitive.
The wave is there. You paddle. You stand. You did not blow the takeoff but now here it is racing before you. The famed Surf Ranch. The marvelous Surf Ranch and your legs move like they almost should and your arms move like they almost should but it is not right because you are not right. Everything is perfect as far as the wave is concerned but you are a halting, stuttering mess of conservatism and trepidation. You are you but you are the absolute worst possible version of you. The satanic version. The fraudulent version. And you surf the entire wave that way knowing you have five maybe six more chances to fix yourself.
But there is no fixing yourself. Fixing yourself takes years and years and years in heavy therapy. It does not happen in one hour at Surf Ranch.
Every single surf journalist was depressed afterward. Morbidly depressed. You could feel it in their energy. See it on their faces. Every single God forsaken surf journalist except Chris Cote (he surfed the wave better than Kelly Slater) and Vaughn Blakey (he created a new sort of tall man art form).
Every single other one though. Oh they all surfed well enough, Pete Taras and Todd Prodanovich stand out in my memory, but every one depressed and if they say no they are lying. I was there. I saw and felt with my arm dangling loose and dead Surf Ranch. Nick Carroll? You were depressed. Sean Doherty? You were depressed. You were all depressed except Chris Cote and Vaughn Blakey and maybe Pete Taras and Todd Prodanovich.
But would you like to know my proudest moment? I paddled for a left and did not blow the takeoff and was cruising down the line with my legs and arms moving almost like they should. Midway down the pool I got bored, thought "fuck this Surf Ranch" and kicked out. One of the very fine lifeguards, I think he is also a lifeguard for Maverick's, pulled up to me on the ski and asked, "Is everything all right?"
I responded, "Yes."
He said, "You had a lot more wave there."
I told him, "Yeah, I got bored."
He said, "You were the first person I have ever seen kick out."
I smiled and then, one wave later, my arm was taken from its shoulder socket. Of course I was only bored because my lack of skill was fully exposed. My heart beating FRAUD FRAUD FRAUD not only to you but to me but to hell with it all. That's what I think now, months removed.
To hell with it. It would all be kind of fun if you were a billionaire who could build your own or a millionaire who could rent one for the weekend with your four best friends but you are neither and so to hell with Surf Ranch.
Fucking Surf Ranch.
Just in: The Inertia big in Korea!
Such a wonderful surprise!
May I confess a little something? I am horrible with numbers in any of their manifestations. Percentages, prices, simple equations each confound and confuse my feeble mind. I was passed through high school mathematics by the good graces of my teachers. I failed my one university math course and maybe shouldn't actually have a degree because that one math course was a requirement. Since co-founding BeachGrit, though, numbers-based analytics have become a part of my every day.
The hard numbers matter. Who clicked what, where those clicks came from, how long those clicks stick around, where they go next matter and how the competition tries to game them is hilarious. Like, Venice-adjacent Stab magazine's bounce rate is over 70%. That means that a vast vast majority of their "readers" don't ever click on an article and disappear instantly. Even better, Venice-adjacent The Inertia gets its second most visitors from South Korea.
South Korea!
And what do you think these South Korean surfers are engaging with? Which part of The Inertia's offerings? Do you think they enjoy tracing the colorful history of stand-up paddle boarding? Or this dog was towed into a monster wave in Hawaii on a SUP? Maybe the South Koreans are simply big fans of trite social justice and chia coconut pudding recipes.
Whatever the case Zach Weisberg and co. should be very proud but also a little bit scared. One stray finger wiggle from neighboring North Korea's Kim Jong Un and its second largest audience vanishes into the ether.
In other news, I am going to write a real honest piece about the Surf Ranch today. Stay tuned!
Nice man from Surfer peeking around the corner, with Chas and Derek, Surf Ranch, November 3, 2017. So happy! "I was massively depressed for a month after surfing it," says Chas.
No Embargo: Warshaw and Co on Surf Ranch!
"The fact that you can now put those waves on a credit card fucks with everything I love about surfing."
Earlier today, the non-disclosure agreements Chas Smith, Matt Warshaw and I signed with the WSL as part of a deal to surf the Slater-Fincham pool in November expired. These NDAs, which were signed at various points around the lavishly outfitted Surf Ranch on Jackson Avenue, Lemoore, mine as I swam in the jacuzzi with a topless Sal Masakela, promised swift and stiff punishment.
I was thrilled by the notion of not writing and shelved, in the literal and not the drugged, late-night sense, my notepad and pencil.
But, here we are, three months later and no embargo.
What have we got to say for ourselves?
As is the norm in these situations, with no notes and only a few snap shots to revive a memory, a little round-table ensued with me and Chas, who surfed the pool in the first media session on November 3 and Matt Warshaw, who enjoyed the considerable facilities a few weeks later.
DR: Matt, Chas, how did you find out about the invite and what was your mindset?
Warshaw: Dave Prodan emailed me, and I gratefully accepted. I ran to the medicine cabinet to check our Ativan stash, then emailed my GP for a beta-blocker script. I was excited to a point of nausea. I'm such a fucking head case. I was going to lock in a 15-second tuberide. I was going to sit in the pool and barf into my lap and miss my wave. Both possibilities felt so real. Drugs, I figured, would put a floor beneath me. I kept the vial of beta-blockers on my desk for the next six weeks, and that was comforting. I didn't surf once during that time. I bought a cheap skateboard from Amazon, and rode the schoolyard near my house on the days when it wasn't raining.
Chas: I believe I was texting with Dave Prodan at the time and oh how I love him. If I recall I had just written something horrible about the Surf Ranch and my never ever ever wanting to surf it. He replied, "That will make the following offer a little awkward." And I responded, "I thrive in the awkward!" I adore being proven wrong much more than being proven right. Who wants to be right? I was happy but mostly happy that Derek was invited too and hoping/dreaming/wishing Matt would be in our crew.
DR: I felt no such nerves, ironic considering what would follow. Weak two-foot rights are my bread and butter. My invitation came, first, via Chas, and then, as expected, in a text message from Dave Prodan. I remember walking outside and into a spring morning and feeling weak with joy. The next day I interviewed Matt Biolos on the subject of board design for wave pools with no other objective than to determine what surfboard to take. Now, tell me, first impressions of the joint.
Warshaw: The place was just about exactly how I pictured it. The huge wooden fence, the signage, the food spread, coffee urns, locker area, the whole medium-upscale country club feel. Part of me loved it. Zeroed right in on a locker, poured a coffee on top of my various sedatives. The first wave you see in person is shocking. Miraculous and surreal. The wave seems half-again bigger in person than it does on a computer screen, and it is of course every bit as perfect. But this is also where the experience gets complicated. You have no free agency. You can't do shit at this point. You can't jump in a pick off a couple of insiders. You can't run down the beach and warm up at a different, lesser break. You just watch and wait your turn. The only goal I had was to not blow the takeoff—to not miss my wave, to not catch a rail dropping in. But the first ridden wave I saw, it was obvious that the takeoff is a chip shot, totally easy if you follow instructions—they tell you exactly where to sit, and when to paddle — so I just watched a couple more then went back in the lounge and zoned out. Not watching was better for my nerves.
DR: Talk to me about the size. The joint is bigger than anything I could've imagined. Seven hundred yards long! And that takeoff, that chip shot, I found overwhelmingly troublesome. The tense wait, the way the wave stands up and bulges like it's going to throw and then backs off radically. Very…very…easy to miss. Gerry Lopez fell off on his first wave there. I'm not one for nerves, generally, but I felt like I'd blow a valve and, consequently, found it very hard to enjoy.
Warshaw: No, the size was about what I expected. Although I thought the left was located in a separate pool running alongside the right. It was strange when I realized they just send the locomotive back down the track going the other direction. So if it's your turn, you go right, wait four minutes, then go left back the way you came and end up where you started. I didn't understand that until we arrived and I saw it in action. The whole day we were there, I don't think anybody missed a wave or shanked a takeoff. Almost nobody made one all the way through, maybe one in 10 rides. But somehow we all had the takeoff figured out.
Watch DR's first wave here. Commentary by Grant Ellis and Vaughan Blakey.
As far it being hard to enjoy, yeah, I don't know how many sessions you'd need, how many waves you'd have to ride, before it became enjoyable. For me, anyway. But surfing was rarely enjoyable back when I did it a lot. I loved the intensity, the obsession, the whole huge never-ending project of it all. But no way have I ever plugged into a new spot and felt anything like joy during my first half-dozen waves. Which is how many waves I caught at the Surf Ranch. In the afternoon, on my second and final right, I managed a tube section, the first time I've been tubed in years, and that felt great. Then I came out, drifted high, and got pitched, and was furious with myself. So maybe five second of enjoyment, just before and during that little tube. But the overwhelming feeling with regard to the pool, the takeaway, was just . . . I need more! Tons more. I would have opened a vein for another dozen waves.
Watch Matt Warshaw here!
DR: …open a vein for more. How much would you pay for a dozen waves?
Warshaw: We haven't touched on deep-down existential crisis the Surf Ranch has thrown me into. In the surfing universe where I live, it is so profoundly wrong that you could buy waves of that caliber, at the date and time of your choosing. Perfect surf is something you dream about and aim for. You plot and steal and suffer, over years, decades, to get yourself in the of perfect surf. It should be more or less as difficult and rewarding as Buddhist enlightenment. Either that, or you get so lucky it's like hitting the Lottery. Either way, we're shaped and defined as surfers by the way we have or have not hit upon perfect waves. The fact that you can now put those waves on a credit card fucks with everything I love about surfing. But to answer your question . . . $1,350.00.
"Whereas surfing, real surfing, each wave is a conversation. Action and reaction. I love all the decision-making. How many choices for each ride? Beyond that, how many choices for each session? Where to surf, where to line up, who to surf with, sit inside, outside, down the beach, steal position from that guy, cold-shoulder the other guy, decide to get out after one more than change your mind. God I missed all that." Matt Warshaw
Chas: First impression? Much more surreal than I thought it would be and also so well designed. The "Ranch" theme carried out very well from the lockers to the bathrooms to the picnic tables. I was stricken by the attention to detail around the pool itself. Size, it was massive. Much more impressive in person than in video but also quintessentially American. Like a goldfish filling out its tank. Also quintessentially American in that it was, and felt like, a modern marvel. The pool, train, etc. all pointing to the mid-century aesthetic that man can conquer all through since.
DR: Matt, let's talk a little bit more about your existential crisis. Does it really fuck with everything you love about surfing? And where did you get the $1350 figure?
Warshaw: Yes. If for no other reason than we're throwing the tuberide into the discount bin. Kooks will figure out how to ride the tube over the weekend. Ten-second barrels won't even go into your scoreline at the Surf Ranch Open.
DR: And where did you get the $1350 figure?
Warshaw: Cause I want to ride a few more tubes before I die, and I'm too old and lazy to chase 'em down in the wild.
DR: How about you, Chas?
Chas: I would pay 1200 for 12 waves. A hundred bucks a pop. Not necessarily because I love but because I think it is totally silly to imagine it is worth less. It was much more powerful than I expected. I thought it was going to be weak but that first wave I felt my fins were going to rip out of my board mid-face. Knowing what that damned wave is going to do, each and every time, was… rattling. I found myself pumping awkwardly down the line waiting for the barrel. That damned barrel.
DR: I was so overwhelmed by stress, and a terrible hangover, my mind wandered into some quite sinister places. Not the pools fault, I'll add. You'll like this story, Matt. I was suited up an hour before my session and thought a spell in the jacuzzi would cure my hot and cold flashes. I was seated next to Sal Masakela, a gorgeous man from a television show who was eating protein bar after protein bar. No wonder his physique was such. A few minutes before the sesh was about to start, as everyone lined up in the pool, my suit, which had been pulled around my waist, had somehow twisted itself into the neck gusset. The pool's about to turn on and I can't work out how to get my suit on! I've flown from Australia, I've drunk whiskey all night, I'm there, and I might miss my wave 'cause of a wetsuit malfunction. Dave Prodan coolly pulled it up for me. Now. Question. How would you describe the wave and what wave in the ocean would you liken it to. I tell everyone it's like three-foot Little Marley, at Rainbow.
Warshaw: The wave is smooth, fast, and incredibly easy to ride in that it doesn't throw anything weird at you; no chops, no boils, no sections. For me, going left at the Surf Ranch was like small, speeded-up Macaronis. I rode my lefts all the way across, but way out in front of the lip, cause I can't grab rail. The difference between the pool and Indonesia, in terms of how it feels — and this is pretty disconcerting — is that you only get maybe 15-yards-worth of information ahead of you. So you're not reacting to the wave, you're following a set of instructions in your head. "Don't go high, don't cut back, don't tap the brakes." There's nothing instinctive or responsive about the ride. The whole thing in fact reminded me of skating pools when I was a kid. Pools are static. Each one is a puzzle that you figure out. Whereas surfing, real surfing, each wave is a conversation. Action and reaction. I love all the decision-making. How many choices for each ride? Beyond that, how many choices for each session? Where to surf, where to line up, who to surf with, sit inside, outside, down the beach, steal position from that guy, cold-shoulder the other guy, decide to get out after one more than change your mind. God I missed all that.
DR: How's this park gonna make money? Did you do the mathematics in your head? …and…is it about making money? Or something else?
Warshaw: I never think about the business part as much as I probably should. My only thought is that 98% of the world's surfers want to ride Kelly's wave. That wasn't the case with the other pools. If they can't figure out how to turn a profit this year, they will next year or the year after. The demand is too high. Olympics and money, in that order.
"The pool workers, who were all exceptional, talked about how they created a happiness Factory a la Willy Wonka. They were right about the Willy Wonka part but none of us are Charlie. We're all Mike TV and Augustus Gloop. Horrible horrible gluttons. I suppose I'm ok with that now. It's fine and the same as any other very good wave, more or less." Chas Smith
Chas: It only works as a loss leader. As part of some bigger shopping/living development. As a stand alone pool the cost would have to be over 1k per person per hour with a four person per hour cap and that would be running with no real profit. But what do I know? We're looking for bar gigs!
DR: One thing I noticed at the pool was that everyone felt they were …expected… to scream, Best Day Ever and wave their arms in the air etc, but the undercurrent was, maybe a little sadness. It was like a pack of johns who went to a men's club, fucked everything, the gorgeous gals, active transgender dolls, kinky as anything, everything in the erotic ball park, better than anything you'd be fed at home, but were left with an emptiness. Did you feel?
Warshaw: No, the analogy is off in two ways. At the Chicken Ranch you'd get everything you came for. Twice, maybe. Not true at the Surf Ranch, where I think you'd always be left wanting. Also, the day after whoring it up I'd be suicidal. Again not true for the pool. Afterwards I was just . . . deflated. The sport that made me is remaking itself at the most fundamental level. I accept that this is happening. Just as I would accept a return invitation to the Surf Ranch. But I'm nonetheless mourning the period—and surfing only has two periods, Before and After Kelly's wave — where a 10-second barrel could change your life.
DR: Did you notice the lack of power at the base of the wave? That if you're caught behind a section, even by half a foot, y'can't get out in front?
Warshaw: Nah, I found other ways to mess up. But what you say, yeah, I heard other people saying the same.
DR: And why does paid-sex make you suicidal?
Warshaw: Everything good in life comes from my family. Including the plan ticket to Fresno to ride Surf Ranch.
DR: Happy y'went?
Warshaw: Very happy I went, yes. Apart from the nerves beforehand, the overriding thought was that surfing hadn't throw me anything really totally new for years, maybe decades. The Surf Ranch was a huge rush just for being so completely different than anything else I've done in surfing. I'm really grateful I got a chance to try it!
DR: Chas, you wrote that wavepools were terrible things before going. Did you change your mind, perhaps overwhelmingly change your mind?
Chas: I said terrible: And now I am completely indifferent. I was massively depressed for a month after surfing it. The pool exposes your greed and weakness. The pool workers, who were all exceptional, talked about how they created a happiness Factory a la Willy Wonka. They were right about the Willy Wonka part but none of us are Charlie. We're all Mike TV and Augustus Gloop. Horrible horrible gluttons. I suppose I'm ok with that now. It's fine and the same as any other very good wave, more or less. But I'll die happy never surfing one again. Fuck em.
And here's Chas! (Shortly before dislocating his shoulder.)
Noted shaper charged after man's death