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Cops train disabled kids for Jerusalem Marathon
Officer Oren training with his ADI partner.
Tears welled in his eyes the first time Jerusalem Police Officer Nir met his marathon training partner. That's a typical reaction upon encountering the children at ADI , Israel's largest network of facilities for children with severe physical and cognitive disabilities.
But it wasn't long before Officer Nir regained his composure and bonded wordlessly with his seven-year-old charge. Each Sunday morning, they practice two hours to prepare for the 800-meter "Community Race" at the fifth annual Jerusalem Winner Marathon on March 13.
Physical therapists at ADI's Jerusalem facility taught Officer Nir how to harness himself to the child using an innovative Firefly Upsee, invented by Israeli mother Debby Elnatan to give adult-aided mobility to kids unable to walk on their own.
When ISRAEL21c visited on a Sunday morning in February, about 18 police officers were at ADI, training as they've done each week since early October with their marathon partners, using Upsee harnesses, walkers and other mobility devices.
Their volunteering resulted from a presentation given at police headquarters by ADI at the suggestion of an officer whose sister is an ADI resident. She wanted to get her coworkers involved with the children, and ADI administrators realized the cops could help kids enjoy the thrill of a marathon – even if it takes a couple of hours to complete the non-competitive track.
"We are not doing this because we have to," Officer Nir tells ISRAEL21c as he walks tethered to his young partner. "It's on our own time; we make up the hours later. Everyone here just thought it was something important to do, and a joy."
Officer Nir using an Upsee harness to train with his ADI partner.
Officer Zaloo is partnered with a tiny 12-year-old girl crippled by a rare chromosomal condition. "I was the first to say I would do it, but the first time I came here, it was very hard for me to see children like this," relates the policewoman, who emigrated from Ethiopia in 1992 and knows something about hardship. "But now I'm happy every time I come here. If I have to miss a Sunday, I feel a kind of emptiness."
Awareness and inclusion
From the Jerusalem Marathon's first year, able-bodied runners have participated to raise money for ADI as well as 24 other charitable causes. ADI volunteers pushed some of the kids in wheelchairs around the community track, and each child was rewarded with a medal.
"Our main goal was to raise awareness and model inclusion," says Dov Hirth, head of international public relations for ADI's four facilities. "We have a few hundred people running on behalf of ADI, a ton of volunteers, and a lot of high-profile people who want to be involved." For example, last year supermodel Esti Ginzburg came by to encourage ADI's contingent.
Before the 2014 marathon, one of ADI Jerusalem's physical therapists suggested that some of the children, if properly prepared, could do more than be pushed in a wheelchair. Responding to this challenge, members of the staff worked hard to train a small group of carefully selected children for the Community Race.
Having 18 Jerusalem police officers take over the training has allowed the number of participants to be expanded to 15 kids chosen for their ability to be outdoors for an extended period and to walk with assistance. A few of the kids need more than one adult by their side.
The volunteers had to be trained too, in order to understand how to use the mobility devices and how to communicate with non-verbal children. A blink, a smile or hand gesture can have great meaning. "In our facility we only speak the language of the heart," says Hirth.
Officer Elchanan training with his ADI partner.
Occupational therapist Rivka Keesing, director of academia and research for the organization's four branches, is amazed to see police officers coming even on their days off so as not to miss a practice.
"They use an app to measure speed, time and distance to track their progress from week to week," Keesing tells ISRAEL21c."We see the improvement right before our eyes. The kids are walking more and better than before the training."
One officer is paired with a little girl who doesn't live at the facility but with her family an hour and a half from Jerusalem.
"Her parents can't get her to the marathon, so he said he'll go pick her up at 6 in the morning and bring her," says Keesing. "There are lots of connections like that between the officers and the kids, and it's very special."
Staff and volunteers will bring most of the facility's 106 children – along with medical equipment and support staff — to the track on marathon day, just to experience the excitement.
By Abigail Klein Leichman / Israel21c / March 5, 2015 and The Jewish Voice / March 19, 2015 / The Jewish Press / March 30, 2015
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EMPTY CANVAS is a FUN AND ENERGETIC performance art show that will captivate your audience! Artist / musician Mike McMath, along with his twin brother, musician Scott McMath, completes a painting while singing lead and back-up vocals. The brothers McMath perform great covers and fun originals as a background for an original painting that is given to the client after the show. The performance can be completed in one or two hours. Empty Canvas is totally unique and will bring music, art and excitement to your event.


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The Nostalgia and Romanticism of the Holiday Catalogue
In the age of mail orders.
Writer Kristy Archibald
For many of us, it's hard to imagine a world without online shopping and speedy deliveries right to our doorsteps, especially post-COVID. But the concept of long-distance shopping goes back to the time of the early railroads. Mail-order catalogues were the way to shop from home for more than a century. Now, the time when retailers, and marketers for big-box Canadian retailers, targeted their consumers directly through the mail-order catalogue seems far away.
Even if you grew up in the '90s and early 2000s, you're likely no stranger to window shopping through a thick curated seasonal catalogue filled with vibrant images of every product you could imagine. My favourite was the Sears Wish Book. Kids would live for the arrival of this magazine every year to see what new toys and clothing styles were available and spend hours perusing and highlighting items for their holiday wish list—a tangible version of Pinterest and IG shopping.
These catalogues and their covers are iconic and illustrate milestones in Canadian history that earlier generations can warmly remember and relate to. Even today, the covers hold such a strong sense of national brand power that they instantly take us back to simple moments growing up through Canadian winters and remind us of the sensational holiday magic they sparked in our homes.
Let's take a trip back and explore the catalogues that started a movement, shaped our culture, and branded the Canadian dream throughout the years.
The mail-order catalogue was introduced to the Canadian masses in the late 1800s to solve a specific problem: the distance from the store to its target consumer base due to the sheer size of our country. Many major department stores and retailers began in Halifax, Toronto, and Montreal and later spread to western cities such as Vancouver, Calgary, and Winnipeg, but they had difficulty reaching the growing populations in other urban and even rural areas of Canada. When the mail-order catalogue was finally established, these shops were able to disseminate their brand, products, and ideas into homes across the country.
With this new shopping method, magazine designers aimed to curate customer loyalty. They achieved this by allowing consumers to visualize their hopes and wishes through the vivid images that filled the pages. Holiday covers and product drawings (later photographs) would typically represent the quintessential Canadian dream showcasing the perfect happy family. You'd see images of kids smiling as they played quietly with their toys or laughing with Santa, while Mom and Dad enjoyed a peaceful moment together by the tree. Or they would depict a classic Canadian outdoor adventure, the opposite of your average chaos-filled household but an image that every family aspired to.
According to the Canadian Museum of History, "designers would illustrate visions that consumers had of themselves and a life that they idealized so they could identify with the products in a way that reflected their day-to-day lives, converting them into loyal customers that would continue to come back year-after-year."
The Catalogues
Canadian Museum of History.
The Hudson's Bay Catalogue
The first print publication was in 1881 and was one of the country's first mail-order catalogues. Even though it was short-lived, ending its consumer mail-order publication in 1913 due to its competition with neighbouring stores, it still had a profound effect on Canadian consumerism as a whole. As the oldest Canadian retailer, the Hudson's Bay Company helped to economize the fur trade and make essential goods such as ready-to-wear clothing, footwear, groceries, tobacco, and alcohol more accessible to the public.
Army & Navy
Deemed "Canada's Original Discount Department store," it was established in 1919 on West Hastings Street in Vancouver. The concept of the store was simple: everything was a bargain. Typically the store sold liquidation stock from bankrupt merchants or army surplus goods. Its mail-order catalogue, as you can imagine, was filled with bright bargains, discounts, and sales, and was published from 1925 to 1986. According to the Canadian Museum of History, it was the longest-running Canadian department store catalogue to date.
The Woodward's Department Store Catalogue
The Woodward's mail-order department published its first catalogue in 1897 and continued producing volumes for over 50 years until 1953. Early magazines offered consumers a look into women's and men's fashions, luggage, hardware, and jewellery styles. As Vancouver grew in the 1900s, new departments began to be advertised in its pages, such as groceries, furniture, stationery supplies, and high-end clothing options from across the ocean. Woodward's was also one of the first to introduce a holiday section, printed on pink paper and called Santa Claus's Toyland.
The Simpsons-Sears Catalogue
Located in the heart of the country—Toronto—Simpson's had its sights on the luxury market. The slogan on its fall/winter 1934–1935 catalogue said it all: "A Great City Store in your Home." Simpson's published its first mail-order catalogue in 1894, and by the early 19th century it was the heartbeat and success of the business. In the early '50s Simpson's partnered with U.S.-based Sears Roebuck. By the mid-1970s, the catalogue was often over 700 pages long.
The Wish Book
The Wish Book grew out of the growing popularity of the Simpson-Sears Christmas edition catalogue and by the '90s was typically close to 900 pages long, with over 400 pages devoted to toys alone. True to its name, the Wish Book was a magazine that bred young dreamers, entrepreneurs, scientists, cooks, and more. It had everything from fashion to furniture to toys and gave its readers the sense that they could do or be anything they wanted. Cutting-edge toys and accessories were first seen here, from digital watches, electrical hand held-games (before Game Boy), record players, Barbies, coloured TVs, and more. Sadly the Wish Book published its last edition in 2016.
The Eaton's Catalogue
For 92 years, the Eaton's catalogue was a staple in almost every Canadian household. Originally published in 1884, the catalogue offered consumers the best-quality fashions, jewellery, furniture, horse collars, suitcases, and leather trunks. It was also one of the first catalogues besides Woodward's to publish a special Christmas edition, dating back to the early 1900s. By the 1970s Simpsons-Sears and the Wish Book were flourishing in the market and sadly forced Eaton's to disband its mail-order department in 1977.
Catalogues Today
In some ways, COVID has brought us full circle, posing a similar challenge of distance and connection for businesses and their consumers. To once again extend their reach and shorten the distance, they've strategically influenced their consumers by speaking to them directly through their inbox, social media, and digital advertising. Brands such as Canadian Tire, Indigo, Best Buy, and various grocery stores have even redirected their budgets to intensify their flyers and physical mail-out catalogues. Is the mail-order catalogue on its way back? We aren't upset if it is!
Part of a series on Canadian Fashion History. Read more HERE.
Canadian Fashion History
Castell d'Empordà
Scalawags: Bata Kindai Amgoza ibn LoBagola
CultureSummer 2013
5 Women You May Not Have Heard of Who Shaped Canadian Art and Culture
The Surf Club

redpajama_set_name: RedPajamaCommonCrawl


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

Arguably the most rigorous and fundamental approach to study phase transitions was pioneered by Yang and Lee \cite{YangLee:1952,*LeeYang:1952}.
They related transitions to zeros of model's partition function, so the theory is applicable whatever the nature of order parameter or
symmetry breaking is.  The original analysis of Yang and Lee of the statistical mechanics of the Ising model was further extended for various
models, including those out of equilibrium. See, e.g. papers \cite{Fisher:1965,Fisher:1980,Matveev:2008,Bena:2005,Wei:2014}, and more references in there.
The requirement of the Lee-Yang zero of the partition function at the critical point of continuous thermal transition becomes the condition for zero of the spectrum (gap closure) at the quantum phase transition point. In both cases the zeros must occur in the real range of physical parameters.

In 1970 Stephenson \cite{Stephenson-I:1970,Stephenson-II:1970,Stephenson:1970PRB} found a new type of transitions in classical Ising models which he called ``disorder lines" (DLs). The transition consists in modulation of the monotonic exponential decay of the correlation functions  by incommensurate (IC) oscillations.  At zero temperature similar modulation was later found by Barouch and McCoy \cite{McCoyII:1971} in the $XY$ quantum chain with transverse field. Disorder lines, or  modulation transitions are quite general phenomena occurring in a large variety of models  \cite{Nussinov:2011,Nussinov:2012,Salinas:2012,Ogilvie:2020,Chitov:2021}. Whether one can define the modulation as a transition is a subtle issue \cite{Ruelle:1974}, since DL is a weak feature lacking identifiable divergencies in the derivatives of thermodynamic potential. A non-analyticity is manifested in the behavior of the correlation length, which remains finite but demonstrates a cusp at the DL point.

Important conclusions can be made from the work on the nature of DLs in the Lee-Yang formalism \cite{Chitov:2017PRE,Chitov:2021}. The conventional phase transitions are well understood as occurrence of zero(s) $z$ of the partition function $F(z)=0$ in the \textit{real range of physical parameters}: temperature, field, coupling, wave number, etc.  Here $z$ stands for some conveniently chosen  function of physical quantities, like e.g., fugacity, $e^{ik}$, spectrum of the transfer matrix.  The DL corresponds to the pair of merging complex conjugate roots $F(z)=F(z^*)=0$ occurring at \textit{complex values of physical parameters}. For instance, the infinite cascades of DLs found in the classical Ising chain \cite{Chitov:2017PRE}, correspond to zeros of the partition function $z(h)$ in the range of complex magnetic field $h \in \mathbb{C}$; modulations revealed in various free fermionic models in the spatial dimensions $d=1,2,3$ correspond to zeros of the partition function $z(\mathbf{q})$ in the range of complex wave vectors $|\mathbf{q}| \in \mathbb{C}$ \cite{Chitov:2021}.

An interesting aspect of the critical properties on the disorder line is factorization of the ground state found in the transverse field $XY$ quantum chain \cite{McCoyII:1971,Franchini:2017} leading to constant correlation functions, the hallmark of disentanglement. Since the possibility to interpret the
disentanglement point as a transition of some kind  is actively studied in the literature \cite{Wolf:2006,Amico:2006,Illuminati:2013,WeiGold:2005,*WeiGold:2011}, it is a very pertinent problem to clarify the relation between the zero-temperature limit of the Lee-Yang zeros and disentanglement in other models.

To this end we analyse in this paper the dimerized quantum $XY$ spin chain in the uniform and staggered transverse fields. This is an exactly-solvable free fermionic model with rich phase diagram \cite{Perk:1975,DuttaTIM:2015,Chitov:2019}. The DLs and disentanglement in this model were not studied before. The
DL as a modulation transition is also manifested in the IC oscillations of the wave function of the Majorana edge states, as reported in \cite{Karevski:2000} for the homogeneous transverse $XY$ chain. After the work by Kitaev on that model \cite{Kitaev:2001}, the Majorana modes in quantum condensed matter systems became a very active topic of research, for reviews, see e.g., \cite{Alicea:2012,Valkov:2022}. The existence of those edge modes in quantum chain can be linked to the localization of zeros of the resolvent of the transfer matrix on the complex plane  \cite{DeGottardi:2011,DeGottardi:2013b}. The latter, as follows from the earlier result for the special case \cite{Chitov:2018}, is not an independent property since the eigenvalues of the transfer matrix are zeros of the spectrum.

So the main goal of the present work is to prove from analysis of the exactly solvable model \eqref{XYHam} that zeros of the spectrum $z_\pm$, which are also the zero-temperature limit of the Lee-Yang zeros, control the critical points, modulation transitions, disentanglement, winding numbers, and the Majorana edge modes.

The rest of the paper is organized as follows:
In Sec.~\ref{Model} we introduce the model to be studied and give an account on its spectrum and the ground-state phases.
In Sec.~\ref{DL3} we present our new findings on the disorder lines of the model and give the results on the correlation functions
and other thermodynamic parameters in all regions of the phase diagram. The factorized disentangled ground states of the model occurring
on a subset of the disorder lines, are analyzed in Sec.~\ref{Factor}. The regions on the phase diagram where the zero-energy edge Majoranas
exist and their wave functions are presented in Sec.~\ref{MES}. The results are summarized in the concluding Sec.~\ref{Concl}.

\section{Model, spectrum and its zeros}\label{Model}

In the spin representation the model is the dimerized quantum $XY$ chain in the presence
of uniform ($h$) and alternating ($h_a$) transverse magnetic fields with the Hamiltonian:
\begin{widetext}
\begin{equation}
\label{XYHam}
   H =- \sum_{n=1}^{N}~   J \Big[ (1+\gamma+\delta (-1)^{n} )
 S_{n}^{x }S_{n+1}^{x}
 + (1-\gamma+ \delta (-1)^{n}) S_{n}^{y} S_{n+1}^{y} \Big]
 +  \big[ h+(-1)^{n} h_a \big] S_{n}^{z} ~.
\end{equation}
\end{widetext}
Here the spin operators $S^\alpha_n= \frac12 \sigma^\alpha_n$ are expressed via the standard Pauli matrices, coupling $J>0$ is ferromagnetic. We assume in our formulas
$0 \leq \delta \leq 1$ and $\gamma \geq 0$. The range of negative $\gamma$ is readily available under exchange
$\gamma \leftrightarrow -\gamma$ and $x \leftrightarrow y$.
The Jordan-Wigner (JW) transformation \cite{LiebSM:1961,Franchini:2017} maps \eqref{XYHam} onto the free
Hamiltonian of spinless fermions
\begin{widetext}
\begin{equation}
\label{XYFermi}
   H =- \sum_{n=1}^{N}~  \frac{J}{2}  \Big[ (1+\delta (-1)^{n} )(c_{n}^\dag c_{n+1} +\mathrm{h.c.})+
                                          \gamma (c_{n}^\dag c_{n+1}^\dag +\mathrm{h.c.}) \Big]
             +  \big( h+(-1)^{n} h_a \big) \big( c_{n}^\dag c_{n}- \frac12~ \big)  ~,
\end{equation}
\end{widetext}
sometimes called in recent literature the (modulated) Kitaev chain \cite{Kitaev:2001}. In the fermionic representation \eqref{XYFermi} the
chain has dimerized hopping and modulated chemical potential. So, whether we deal with spins or with fermions, is a matter of convention.
For the comprehensive analysis of the present model at zero temperature, including its phase diagram, local and non-local order parameters for each phase, we refer readers to the recent work \cite{Chitov:2019}. In this paper we follow the notations of \cite{Chitov:2019}.

The model can be brought to the Bogoliubov-de Gennes spinor form with the $4\times 4$ Hamiltonian  matrix (we set $J=1$):
\begin{equation}
\label{Hk}
  \hat{\mathcal{H}}(k) = \left(%
\begin{array}{cc}
  \hat{A} & \hat{B} \\
  \hat{B}^\dag  & -\hat{A} \\
\end{array}%
\right)~,
\end{equation}
where
\begin{equation}
\label{A}
  \hat{A} \equiv  \left(%
\begin{array}{cc}
  h +\cos k & h_a+i \delta \sin k \\
  h_a-i \delta \sin k & h -\cos k \\
\end{array}%
\right)~,
\end{equation}
and
\begin{equation}
\label{B}
 \hat{B} \equiv  \left(%
\begin{array}{cc}
  -i \gamma \sin k & 0 \\
  0 & i \gamma \sin k \\
\end{array}%
\right)~,
\end{equation}
With a unitary transformation, the Hamiltonian \eqref{Hk} can be brought to the block off-diagonal form
\begin{equation}
\label{HD}
  \hat{\mathcal{H}}^\prime(k)=\left(%
\begin{array}{cc}
  0 & \hat{D}(k) \\
  \hat{D}^\dag (k) & 0 \\
\end{array}%
\right)
\end{equation}
used in the following. Here we defined the operator
\begin{equation}
\label{Dk}
    \hat{D}(k) \equiv \hat{A} (k)+\hat{B}(k)~,
\end{equation}
which has two eigenvalues
\begin{eqnarray}
  \Lambda_\pm (k)&=& h \pm  \frac{1}{\sqrt{2}} \Big(1+2 h_a^2+\delta^2-\gamma^2\nonumber  \\
    &+&  (1-\delta^2+\gamma^2) \cos 2k -2 i \gamma \sin 2k \Big)^{1/2}~.
\label{lambdapm}
\end{eqnarray}
The spectrum of this Hamiltonian consists of four eigenvalues  $\pm E_{\pm}$ \cite{Perk:1975}:
\begin{equation}
\label{Epm}
 E_{\pm}(k)= \frac{1}{\sqrt{2}} \Big| \sqrt{\mathfrak{C}_2(k)+ | \Lambda_+ \Lambda_-|} \pm
                                \sqrt{\mathfrak{C}_2(k)- | \Lambda_+ \Lambda_-|} \Big|~,
\end{equation}
with
\begin{equation}
  \label{C2}
  \mathfrak{C}_2(k) \equiv h^2+h_a^2+\cos^2k+(\delta^2+\gamma^2)\sin^2k~.
\end{equation}

The eigenvalues of $\hat{\mathcal{H}}$ and $\hat{D}$ are related by useful formula
\begin{equation}
\label{C4Lpm}
  E_+ E_- =| \Lambda_+ \Lambda_-|~.
\end{equation}

To extend the earlier analysis \cite{Chitov:2019} of the model to incorporate within the same framework the disorder lines (DLs) and the ground-state
factorization \cite{Chitov:2021}, we perform analytical continuation of the wave numbers onto the complex plane as $(e^{i k})^2 \equiv z$. Then the
first Brillouin zone (BZ) $k \in [-\pi/2, \pi/2]$ maps onto unit circle $|z|=1$.
Eq.~\eqref{C4Lpm} allows us to relate the zeros of the spectra of $\hat{\mathcal{H}}$ and $\hat{D}$ as
\begin{equation}
\label{EpmDD}
E_+^2(z) E_-^2(z) =\mathrm{det}\hat{D}(z) \mathrm{det}\hat{D}^\dag (z)~,
\end{equation}
while the zeros of the spectra of $\hat{A} \pm \hat{B}$ are readily found yielding
\begin{eqnarray}
\label{detD*}
  \mathrm{det}\hat{D}^\dag (z) &=& -\frac{(1+\gamma)^2-\delta^2}{4z}(z-z_+)(z-z_-), \\
   \mathrm{det}\hat{D} (z) &=& -\frac{(1-\gamma)^2-\delta^2}{4z}(z-z_+^{-1})(z-z_-^{-1}).
\label{detD}
\end{eqnarray}
Note that $\hat{D} \leftrightarrow \hat{D}^\dag$  when $\gamma \leftrightarrow -\gamma$, and the zeros of determinants of those matrices are mutually reciprocal:
\begin{equation}
\label{zinv}
  z_\pm(-\gamma)=z_\mp^{-1}~,
\end{equation}
with their explicit expressions given by
\begin{equation}
\label{zpm}
  z_\pm= \frac{\big[ (h^2-h_-^2)^{1/2} \pm (h^2-h_+^2)^{1/2} \big]^2 }{(1+\gamma)^2-\delta^2}
  \equiv e^{2ik_\pm}~,
\end{equation}
and
\begin{equation}
\label{hpm}
  h_\pm^2 \equiv  h_a^2+ \frac12 \Big[ 1+\delta^2-\gamma^2 \pm \big[(1+\gamma)^2-\delta^2 \big]^{1/2}
  \big[(1-\gamma)^2-\delta^2 \big]^{1/2} \Big]
\end{equation}
A word of caution: the above notation $h_\pm^2$ is introduced for further convenience only, and in general  $h_\pm^2 \in \mathbb{C}$.

Quantum phase transitions of the model \eqref{XYHam} correspond to the case when roots \eqref{zpm} lie on the unit circle $|z_\pm|=1$.
For reader's convenience we recapitulate below the properties of the model's phase diagram analysed in detail in \cite{Chitov:2019}:

\textit{(i)} At $|h| > \sqrt{h_a^2+1}$
the model is in the polarized or paramagnetic (PM) phase. At its boundary, shown as bold blue solid lines in Figs.~\ref{HaGam},\ref{DelGam},\ref{HaDelGam}, the root $z_+=1$, and the gap in the spectrum vanishes at the center of the BZ $k=0$.

\textit{(ii)}
At $|h| < \sqrt{h_a^2+1}$ the model is in the ferromagnetic (FM)
phase with the spontaneous longitudinal magnetization $m_x \neq 0$ at $\gamma>0$ and $m_y \neq 0$ at $\gamma<0$.

\textit{(iii)}
The circle on the ($h,\gamma$) plane with the radius $R_\circ =\sqrt{h_a^2+\delta^2}$, shown as bold brown solid line in Figs.~\ref{HaGam},\ref{DelGam},\ref{HaDelGam},
corresponds to the boundary of the topological phase $\mathcal{O}_z(\pi/2)$ with the oscillating string order. At this critical line one of the roots $z_\pm=-1$,
and the gap vanishes at the edge of the BZ $k= \pm \pi/2$.

\textit{(iv)}
Two line segments at $\gamma=0$, shown as bold magenta solid lines in Figs.~\ref{HaGam},\ref{DelGam},\ref{HaDelGam},
correspond to the incommensurate (IC) gapless phase with the complex conjugate roots $|z_\pm|=1$, and the gap vanishing at the IC
wavevector
\begin{equation}
 \label{qIC}
  q= \pm \arcsin \sqrt{\frac{1+h_a^2-h^2}{1-\delta^2}}~.
\end{equation}
This wavevector \eqref{qIC} varies continuously from $q=0$ at the intersection of $\gamma=0$ and $h=\pm\sqrt{h_a^2+1}$
to $q= \pm \pi/2$ where the critical segments end at the intersections with the boundary of the topological phase $\mathcal{O}_z(\pi/2)$ .

The roots  \eqref{zpm} contain also the information about another class of solutions, corresponding to DLs
\cite{Stephenson-I:1970,Stephenson-II:1970,Stephenson:1970PRB}. These lines, shown in dashed green in Figs.~\ref{HaGam},\ref{DelGam},\ref{HaDelGam},
correspond to the points where the roots $z_\pm$ acquire imaginary parts and become complex conjugate \cite{Chitov:2017PRE,Chitov:2021,Jones:2019}.
See Figs.~\ref{ZpmA} and \ref{ZpmB}.
The regions of the phase diagram bounded by the DLs correspond to the so-called oscillating phases (regimes), where the monotonous exponential decay of the correlation functions is modulated by IC oscillations. In the present model the oscillating regions are localized within the FM phase and smoothly connected to the IC gapless (critical) lines. In addition we found the disorder lines of the second kind when the regime changes abruptly from no oscillations ($q=0$) to the commensurate oscillations ($q=\pi/2$). Our new results presenting the detailed analysis of the model's DLs for three special cases are given in Sec.~\ref{DL3}.

\section{Disorder lines}\label{DL3}
\subsection{Special case: $h_a \neq 0$ and $\delta=0$}\label{HaGamSec}
The boundaries of the oscillating regions \eqref{hpm} (DLs) are quite simple in this case:
\begin{equation}
\label{hpmHa}
 h^2=h^2_\pm =
\left\{
\begin{array}{lr}
1+h_a^2-\gamma^2 \\[0.2cm]
h_a^2  \\
\end{array}
\right.
\end{equation}
The roots \eqref{zpm} are given by:
\begin{equation}
\label{zpmHa}
  z_\pm= \Big[ \frac{(h^2-h_a^2)^{1/2}  \pm (h^2+\gamma^2-1-h_a^2)^{1/2}   }{1+\gamma} \Big]^2~.
\end{equation}
The phase diagram including the DLs for this case is depicted in Fig.~\ref{HaGam}. The complex-valued
roots $z_\pm$ are localized between intersections of the ``disorder circle" of radius $R_{\scriptscriptstyle DL}=\sqrt{1+h_a^2}$ and the lines
$h = \pm h_a$. They are shown in dashed green in Fig.~\ref{HaGam}. The radius of the topological circle $R_\circ=h_a$.
Note that in the limit $h_a \to 0$ one recovers the results of Barouch and McCoy \cite{McCoyII:1971} with a single DL
determined by the circle $h^2+\gamma^2=1$.
\begin{figure}[]
\centering{\includegraphics[width=8.0cm]{PhaseDiagA.pdf}}
\caption{The ground state phase diagram on ($h,\gamma$) plane for the case of zero dimerization $\delta=0$ and non-zero staggered field $h_a \neq 0$. The bold solid lines denote the phase boundaries: PM-FM in blue; FM-$\mathcal{O}_z$ in brown; gapless IC in magenta. The disorder lines are localized within the ferromagnetically-ordered phase and shown in dashed green. The regions with IC oscillations, labeled by (C) are shown in light green; (B) regions shown in light yellow contain no oscillations since $q=0$; in regions labeled by (D) and shown in light blue, the oscillations are commensurate with $q=\pi/2$.}
\label{HaGam}
\end{figure}
With parametrisation of the roots \eqref{zpm} as $z(k)$, where $k \in \mathbb{C}$ and
\begin{equation}
\label{kC}
 k \equiv q+i \kappa, ~z=e^{2(iq -\kappa)}~,
\end{equation}
$q$ yields the wave number of oscillations, while $\kappa$ is the inverse correlation length \cite{Chitov:2017PRE,Chitov:2021}.
Evolution of the roots \eqref{zpmHa} on the complex plane and dependencies of $q$ and $\kappa$ along several paths on the ($h,\gamma$) plane (see Fig.~\ref{HaGam}) are presented in Figs.~\ref{ZpmA} and \ref{KapQA}. As expected from general arguments, the lines
of quantum criticality correspond to the roots $z_\pm$ \eqref{zpmHa} lying on the unit circle on the complex plane and vanishing $\kappa$. DLs are weaker singularities and they are accompanied by cusps in the correlation length.

\begin{figure}[]
\centering{\includegraphics[width=7.5cm]{ZpmA.pdf}}
\caption{The roots $z_\pm$ and $e^{ik_\pm}$  determined by Eqs. \eqref{zpm} and \eqref{zpmHa} for $\delta=0$ and $h_a \neq 0$ in the first quadrant of the phase diagram (see Fig.~\ref{HaGam}). The positions of the roots are shown in the complex plane with respect to the unit circle $|z|=1$ drawn by solid blue line. }
\label{ZpmA}
\end{figure}

\begin{figure}[]
\centering{\includegraphics[width=9.0cm]{KapQA.pdf}}
\caption{The modulation wave number ($q$) and the inverse correlation length ($\kappa$)  along several paths indicated in ($h,\gamma$) plane,  see Fig.~\ref{HaGam} for the case $\delta=0$ and $h_a \neq 0$. The disorder lines bounding the regions of the IC wave numbers $q$ are accompanied by the cusps of $\kappa$, while the quantum phase transitions PM-FM and FM-$\mathcal{O}_z$ correspond to $\kappa=0$.}
\label{KapQA}
\end{figure}

The IC oscillations with complex conjugate roots $z_\pm$ are localized in the regions colored in green in Figs.~\ref{HaGam} and \ref{ZpmA}. For those regions in the first quadrant one can find explicitly:
\begin{equation}
\label{qICHa}
 q=
\left\{
\begin{array}{lr}
 \arcsin \sqrt{\frac{1+h_a^2-h^2-\gamma^2}{1-\gamma^2}},~[h>h_a] \cup [h^2+\gamma^2< R_{\scriptscriptstyle DL}^2 ] \\[0.2cm]
 \arcsin \sqrt{\frac{h_a^2-h^2}{\gamma^2-1}},~[h<h_a] \cup [h^2+\gamma^2> R_{\scriptscriptstyle DL}^2 ] \\
\end{array}
\right.
\end{equation}
and
\begin{equation}
\label{kapHa}
  \kappa = -\frac12 \ln \frac{|\gamma-1|}{\gamma+1}~.
\end{equation}
The DLs found for this case (see Fig.~\ref{HaGam}) are of the first kind, since the wave number of the IC oscillations changes
continuously between the boundaries of commensurate regions, see Fig.~\ref{KapQA}.
Outside of the IC regions (FM phase) the wave number is:
\begin{equation}
\label{qHa}
 q=
\left\{
\begin{array}{lr}
 0~,~~[h>h_a] \cup [h^2+\gamma^2> R_{\scriptscriptstyle DL}^2 ] \\[0.2cm]
 \pi/2~,~~[h<h_a] \cup [h^2+\gamma^2< R_{\scriptscriptstyle DL}^2 ]  \\
\end{array}
\right.
\end{equation}
These two regions are labelled as B ($q=0$) and D ($q=\pi/2$) in Figs.~\ref{HaGam} and \ref{ZpmA}.

The results for the correlation functions in different phases and regimes of oscillations are collected in Fig.~\ref{CFA}.
It has been checked by explicit calculations that the types of behaviors are qualitatively equivalent for all spin-spin and
string-string correlation functions. In Fig.~\ref{CFA} we plot the representative numerical results for the string correlation function
\begin{equation}
\label{Dzz}
  \mathfrak{D}_{zz}(L,R) \equiv
 \Big \langle \prod_{l=L}^{R} \sigma_l^z \Big \rangle =
 \Big \langle  \prod_{l=L}^{R} \big[ i b_l a_l \big]  \Big \rangle~.
\end{equation}
$\mathfrak{D}_{zz}$ is defined in terms of the original spin operators $\sigma_l^z$
or Majorana fermions related to the JW fermions as
\begin{equation}
\label{Maj}
   a_n +i b_n  \equiv 2 c^{\dag}_n~.
\end{equation}
The correlation function \eqref{Dzz} is calculated numerically from the determinant of the block Toeplitz matrix whose elements, i.e., the two-point
Majorana correlators  $\langle ib_na_m \rangle$ are explicitly given in \cite{Chitov:2019}.

As one can see from Fig.~\ref{CFA}, the behavior of the correlation function is easily understood from the structure of the complex roots depicted in
Figs.~\ref{ZpmA} and \ref{KapQA}. The plot (a) corresponds to the polarized phase (PM) where $\mathfrak{D}_{zz}(1,n)$ monotonously decays to the
limiting value $\mathcal{O}_{z}^2$, where $\mathcal{O}_{z}$ can be identified as a single string order parameter of the PM phase \cite{Chitov:2021BR}.

In the panel (b) of Fig.~\ref{CFA} the string correlation function is shown for the region (B) on the phase diagram in Fig.~\ref{HaGam} where oscillations are absent.
This is the ferromagnetic phase where the order parameter is the spontaneous longitudinal magnetization $m_x$, while the string order is absent, and
accordingly, $\mathfrak{D}_{zz}$ decays to zero \cite{Chitov:2019}.  The exponential decay of $\mathfrak{D}_{zz}$ modulated by the IC oscillations, shown in
panel (c) corresponds to the IC region (C) of the FM phase, see Fig.~\ref{HaGam}.

The IC gapless phase shown in magenta in Fig.~\ref{HaGam} is algebraically ordered, $\mathfrak{D}_{zz}$ demonstrates the power-law decaying oscillating behavior:\cite{Chitov:2020}
\begin{equation}
\label{DzzIC0}
  \mathfrak{D}_{zz}(1,n) = \frac{\mathcal{A}}{\sqrt{n}} \cos(k_{\scriptscriptstyle F} n)~.
\end{equation}
The wave number of oscillations in this phase is also the Fermi wave number of the JW fermions $q= k_{\scriptscriptstyle F}$, given by
Eq.~\eqref{qIC}. The explicit formula for the amplitude $\mathcal{A}$ is unavailable. For a particular choice of parameters yielding $k_{\scriptscriptstyle F}=\pi/4$, the results are shown in Fig.~\ref{CFA}(d) with the fit $\mathcal{A} \approx 0.72$. We found an excellent agreement between Eq.~(\ref{DzzIC0}) and direct
numerics.

The panel (e) of Fig.~\ref{CFA} shows the characteristic behavior of $\mathfrak{D}_{zz}$ in the phase labelled as $\mathcal{O}_z(\pi/2)$. The correlation function oscillates with the period of four lattice spacings, and its limiting value yields the string order parameter, see Ref.~\cite{Chitov:2019} for details. In panel (f) we show the correlation function in region (D) of the phase diagram: the four-site oscillations ($q=\pi/2$) are present, but the
string order vanished ($\mathfrak{D}_{zz} \to 0$), since it is the FM phase ($m_x \neq 0$).

\begin{figure*}[t]
\centering{\includegraphics[width=\textwidth]{CorrFunA.pdf}}
\caption{The characteristic behavior of the string-string correlation function $\mathfrak{D}_{zz}$ in different phases and regimes of oscillations
shown in the phase diagram, see Fig.~\ref{HaGam}. The panels (a), (d), and (e) show the correlation functions in the PM, gapless IC, and oscillating string-ordered $\mathcal{O}_z(\pi/2)$, respectively. The panels (b), (c), and (f) correspond to three regions of the FM phase: (b) $q=0$, no oscillations;
(c) gapped IC oscillations; (f) $q=\pi/2$ oscillations. More explanations are given in the text.}
\label{CFA}
\end{figure*}

\subsection{Special case: $h_a = 0$ and $\delta \neq 0$}\label{DelGamSec}
In this case the IC oscillating region shown in green in Fig.~\ref{DelGam} is bounded by two DLs
\begin{equation}
\label{hpmDel}
  h^2=h_\pm^2 =  \frac12 \Big[ 1+\delta^2-\gamma^2 \pm \big[(1+\delta^2-\gamma^2)^2-4 \delta^2 \big]^{1/2} \Big]~,
\end{equation}
depicted by dashed green lines.
In ($h,\gamma$) upper half-plane shown in Fig.~\ref{DelGam}, the DLs (dashed green) are located between the paramagnetic phase $h >1$ and the topological circle of radius $R_\circ=\delta$.
\begin{equation}
\label{hpm0}
 \mathrm{At~} \gamma=0:~h_\pm =
\left\{
\begin{array}{lr}
1 \\[0.2cm]
\delta  \\
\end{array}
\right.
\end{equation}
and two curves $h_\pm(\gamma)$ smoothly intersect ($h_+=h_-$) at $\gamma=\pm(1-\delta)$.
\begin{figure}[]
\centering{\includegraphics[width=8.5cm]{PhaseDiagB.pdf}}
\caption{The ground state phase diagram in ($h,\gamma$) plane for the case $\delta \neq 0$ and $h_a = 0$. The bold solid lines denote the phase boundaries: PM-FM in blue; FM-$\mathcal{O}_z$ in brown; gapless IC in magenta. The disorder lines are localized within the FM phase. The disorder lines of the first kind determined by the real solutions of \eqref{hpmDel} are shown in dashed green. They bound the region with IC oscillations (C) shown in green.  (B) regions with no oscillations ($q=0$) shown in light yellow; (D) regions with the commensurate oscillations ($q=\pi/2$) shown in light blue. Regions (B) and (D) are separated by the disorder lines of the second kind given by \eqref{DL2expl} and plotted in dotted green.}
\label{DelGam}
\end{figure}

The inverse correlation length inside the IC oscillating region (C),  shown by green, is
\begin{equation}
\label{kapDel}
  \kappa = -\frac14 \ln \frac{(1-\gamma)^2-\delta^2}{(1+\gamma)^2-\delta^2}~,
\end{equation}
while the wave number of oscillations grows continuously from $q=0$ on the upper boundary of the IC region, to $q=\pi/2$ on its lower boundary,
following the formula:
\begin{equation}
 \label{qDel}
  q= \arcsin \sqrt{\frac{h_+^2-h^2}{h_+^2-h_-^2}}~.
\end{equation}
The oscillating IC phase becomes gapless on the critical line $\gamma=0$ shown by bold magenta on the phase diagram in Fig.~\ref{DelGam}.

\begin{figure}[]
\centering{\includegraphics[width=7.0cm]{ZpmB.pdf}}
\caption{The roots $z_\pm$ and $e^{ik_\pm}$  determined by Eqs. \eqref{zpm} and \eqref{hpmDel} for $\delta \neq 0$ and $h_a =0$ in the first quadrant of the phase diagram (see Fig.~\ref{DelGam}), are shown in the complex plane with respect to the unit circle $|z|=1$ drawn by solid blue line. Two roots on the DL2
\eqref{DL2expl} when the condition \eqref{DL2} is satisfied are drawn for a particular point $\gamma=1.005$, $\delta =0.3$: $|e^{i k_\pm}|=0.389$.}
\label{ZpmB}
\end{figure}

\begin{figure}[]
\centering{\includegraphics[width=9.0cm]{KapQB.pdf}}
\caption{The modulation wave number ($q$) and the inverse correlation length ($\kappa$)  along several paths indicated in ($h,\gamma$) plane,  see Fig.~\ref{DelGam} for the case $\delta \neq 0$ and $h_a = 0$. The disorder lines are accompanied by the cusps of $\kappa$, while the quantum phase transitions PM-FM and FM-$\mathcal{O}_z$ correspond to $\kappa=0$. The wave number is continuous, albeit not smooth on the disorder lines of the first kind, while it undergoes a discontinuity on the lines of the second kind.}
\label{KapQB}
\end{figure}

We have also found the so-called \textit{disorder lines of the second kind} (DL2) \cite{Stephenson:1970PRB}, such that the wave vector of modulations changes discontinuously \cite{Nussinov:2011,Nussinov:2012,Chitov:2021} upon crossing those lines. In the case under consideration the modulation transition with a jump from $q=0$ to $q=\pi/2$
occurs when
\begin{equation}
\label{DL21}
  h_\pm^2 \in \mathbb{C},~~h_+ =h_-^\ast~.
\end{equation}
The DLs of the first kind analysed above, are characterized by continuous appearance of the IC modulation, when the roots \eqref{zpm} expand onto complex values, such that $z_+=z_-^\ast$ and the continuous  evolution of the modulation wave number $q$ follows
the smooth growth of the phase of the complex numbers $z_\pm$, see Figs.~\ref{ZpmB} and \ref{KapQB}.

The mechanism of the discontinuous modulation (DL2) is quite different: one can check that in the range where $h_\pm^2$ are complex conjugate,  one root of \eqref{zpm} $e^{ik_+}=a$ is real, while the second root $e^{ik_-}=ib$ is purely imaginary ($a,b>0$), and the transition of the regime occurs
when
\begin{equation}
\label{DL2}
|e^{ik_+}|=|e^{ik_-}|~,
\end{equation}
see Fig.~\ref{ZpmB}.
At this point the correlation length demonstrates a cusp  as at other DLs (see Fig.~\ref{KapQB}), while $q$ undergoes an abrupt jump following the phase of the root: from $\Im (e^{ik_+})=0$ to $\Im (e^{ik_-})=\pi/2$, see  Fig.~\ref{KapQB}. One can verify that the condition  \eqref{DL2} amounts to $h^2=\Re (h_\pm^2)$, yielding the equation of the DL2:
\begin{equation}
\label{DL2expl}
h^2=\frac12(1+\delta^2-\gamma^2)~.
\end{equation}
These DL2s on $(h,\gamma)$ plane are shown in Fig.~\ref{DelGam} in dotted green: they start at the point of intersection of $h_+$ and $h_-$ at $h=\sqrt{\delta}$ and $\gamma=\pm(1-\delta)$, and reach $h=0$ at  $\gamma=\pm \sqrt{1+\delta^2}$.

One should keep in mind that the roots solving the equation for zeros of the spectrum enter the analytic expressions for correlation functions on the same footing \cite{Chitov:2021}. The asymptotes of correlators are controlled however by the leading contribution from the root corresponding to minimal $\kappa$ which yields the inverse correlation length. In case of DL2, the  roots $e^{ik_\pm}$ resulting in, respectively, the monotonous/oscillating contributions to asymptotes of correlation functions, are close in absolute values near DL2, cf. Eq.~\eqref{DL2}, so their contributions are comparable, and thus appearance and vanishing of oscillations are smeared across DL2. This analytical conclusion is confirmed by direct numerical calculations of various correlation functions.

The qualitative behavior of the correlation functions is controlled by the properties of the roots in different regions of the phase diagram, see Fig.~\ref{ZpmB}. From comparison of Figs.~\ref{ZpmA} and \ref{ZpmB}, one can see that the only difference in the root structure between the case A
of previous subsection and the current case B, is that for the latter the appearance of the real and imaginary parts of $e^{i k_\pm}$ is not synchronized in the regime without oscillations ($q=0$) and with commensurate oscillations ($q=\pi/2$). However, since the asymptotes of correlation functions are controlled by the root with minimal $\kappa$ (i.e. the closest to the unit circle), then the correlation functions in different regions of the phase diagram have the behavior similar to that shown in Fig.~\ref{CFA}.

One can check that the results of the present subsection recover those of Barouch and McCoy \cite{McCoyII:1971} in the limit $\delta \to 0$.

\subsection{General case: $h_a \neq  0$ and $\delta \neq 0$}\label{HaDelGamSec}
The ground state phase diagram for the general case is presented in Fig.~\ref{HaDelGam}. Similarly to the previous case with $h_a=0$,
it contains the central IC oscillating region shown in green, bounded by two DLs $h=h_\pm$ ($h_\pm \in \mathbb{R}$,
cf. definitions \eqref{hpm} ) and located between the paramagnetic phase $h >\sqrt{1+h_a^2}$ and the topological circle of radius $R_\circ=\sqrt{\delta^2+h_a^2}$.
\begin{figure}[]
\centering{\includegraphics[width=8.5cm]{PhaseDiagC.pdf}}
\caption{The ground state phase diagram in ($h,\gamma$) plane for the general case $\delta \neq 0$ and $h_a \neq 0$. The bold solid lines denote the phase boundaries: PM-FM in blue; FM-$\mathcal{O}_z$ in brown; gapless IC in magenta. The disorder lines are localized within the FM phase. The disorder lines of the first kind determined by the real solutions of \eqref{hpm} are shown in dashed green. They bound the region with IC oscillations (C) shown in green. (B) regions with no oscillations ($q=0$) shown in light yellow; (D) regions with the commensurate oscillations ($q=\pi/2$) shown in light blue. Regions (B) and (D) are separated by the disorder lines of the second kind given by \eqref{DL2expl2} and plotted in dotted green.}
\label{HaDelGam}
\end{figure}

In addition, non-zero $h_a$ brings about a possibility of another real solutions for $h_\pm$, engendering two extra IC wings at $|\gamma| \gtrsim 1+\delta$. The detailed shape of those wings and of the DL2 depend slightly on the relations between parameters:

\underline{\textit{(i)} $h_a<\delta$:} In this case the IC oscillating wing is bounded by $h_+ \geq 0$  at  $|\gamma| \geq \gamma_\circ$, where
\begin{equation}
\label{gam0}
  \gamma_\circ \equiv \frac{\sqrt{1+h_a^2}\sqrt{\delta^2+h_a^2}}{h_a}~,
\end{equation}
while  the other branch $h_-$ is not pertinent in this range, since $h_-^2<0$ at  $|\gamma|> 1+\delta$. Note that
\begin{equation}
\label{gams}
 \forall~|\gamma|>1+\delta:~h^2_\pm \in \mathbb{R}~.
\end{equation}
As explained above, the DL2 can occur only in the range where $h^2_\pm$ are complex, which is, according to \eqref{hpm}:
\begin{equation}
\label{HpmCom}
 \forall~ 1-\delta<|\gamma|<1+\delta:~h^2_\pm \in \mathbb{C}~.
\end{equation}
The DL2 equation $h^2=\Re (h_\pm^2)$ yields the parabola
\begin{equation}
\label{DL2expl2}
h^2=h_a^2+ \frac12(1+\delta^2-\gamma^2)~.
\end{equation}
The DL2s shown in dotted green  in Fig.~\ref{HaDelGam}, start at $h=h_+=h_-=\sqrt{h_a^2+\delta}$ and $\gamma=\pm(1-\delta)$ and reach $h=h_+=0$ at  $\gamma=\pm \sqrt{1+\delta^2+2h_a^2}$. This is the case plotted in Fig.~\ref{HaDelGam}.

\underline{\textit{(i)} $h_a>\delta$:} In this case (not shown) not only $h_+^2  \in \mathbb{R}$, but also $h_+^2>0$ at $|\gamma|>1+\delta$. The other
branch $h_-^2>0$ at $1+\delta<|\gamma|<\gamma_\circ$.  The DL2s span through the whole range of complex-valued $h^2_\pm$, cf. condition \eqref{HpmCom}. These lines are located between the points $h=h_+=h_-=\sqrt{h_a^2+\delta}$ at $\gamma=\pm(1-\delta)$ and $h=h_+=h_-=\sqrt{h_a^2-\delta}$ at $\gamma=\pm(1+\delta)$.

\begin{figure}[]
\centering{\includegraphics[width=7.5cm]{ZpmC.pdf}}
\caption{The roots $z_\pm$ and $e^{ik_\pm}$  determined by Eqs. \eqref{zpm} and \eqref{hpm} for $\delta \neq 0$ and $h_a \neq 0$ in the first quadrant of the phase diagram (see Fig.~\ref{HaDelGam}), are shown in the complex plane with respect to the unit circle $|z|=1$ drawn by solid blue line. Two roots on the DL2
\eqref{DL2expl2} when the condition \eqref{DL2} is satisfied are drawn for a particular point $\gamma=1.0$, $\delta =0.3$, $h_a=0.4$: $|e^{i k_\pm}|=0.4518$.}
\label{ZpmC}
\end{figure}

The qualitative mechanism for appearance of the DL2s for $h_a<\delta$ and $h_a>\delta$ is the same as explained in the end of the previous subsection after Eq.~\eqref{qDel}. The DL2 correspond to a jump of the wave vector of oscillations $0 \leftrightarrow \pi/2$ and a cusp of the correlation length, see Figs.~\ref{ZpmC} and \ref{KapQC}. The regions of the IC oscillations are shown in green on the phase diagram in Fig.~\ref{HaDelGam}.
The wave vector $q$ and the inverse correlation length $\kappa$ calculated along several paths shown in Fig.~\ref{HaDelGam}
are plotted  in Fig.~\ref{KapQC}. The qualitative behavior of the correlation functions is similar to that plotted in Fig.~\ref{CFA} for different regions of the phase diagram.

\begin{figure}[]
\centering{\includegraphics[width=9.0cm]{KapQC.pdf}}
\caption{The modulation wave number ($q$) and the inverse correlation length ($\kappa$)  along several paths indicated in ($h,\gamma$) plane,  see Fig.~\ref{HaDelGam} for the general case $\delta \neq 0$ and $h_a \neq 0$. The disorder lines are accompanied by the cusps of $\kappa$, while the quantum phase transitions PM-FM and FM-$\mathcal{O}_z$ correspond to $\kappa=0$. The wave number is continuous, albeit not smooth on the disorder lines of the first kind, while it undergoes a discontinuity on the lines of the second kind.}
\label{KapQC}
\end{figure}

\section{Ground state factorization}\label{Factor}
The systematic method for the search of disentangled (factorized) states in quantum chains was proposed by  M\"{u}ller and
coworkers \cite{Muller:1982,*Muller:1985}.
The properties of separable states  in various spin models have been actively studied in the literature,
see, e.g. \cite{Amico:2006,Giorgi:2009,Rossignoli:2009,*Cerezo:2017,*Rossignoli:2020,Illuminati:2008,*Illuminati:2009,Jones:2021}
and more references in there. The problem has many facets and potential applications, and one of the outstanding
physical questions is the nature of the disentanglement points: can it be viewed as a quantum transition of
some kind? See, e.g, \cite{Wolf:2006,Amico:2006,Illuminati:2013,WeiGold:2005,*WeiGold:2011}

Very recently \cite{Chitov:2021} the factorization of the GS of the $XY$ chain in the uniform transverse field
was related to a special class of the complex conjugate zeros $z_+=z_-^*$ of its partition function (in the limit $T\to0$).
In that model the parametric line of factorization coincides with the disorder line $\gamma^2 +h^2 =1$ \cite{McCoyII:1971,Franchini:2017},
where the roots $z_\pm$ merge, since their imaginary parts vanish.
The disorder line (factorization) transition is a very weak feature, and it is not straightforward to classify it in the standard Ehrenfest scheme. For the quantum   $XY$ chain it was rigorously shown \cite{Maciazek:2016} that its GS energy is smooth and even infinitely differentiable function on the disorder (factorization) line. The gap does not close at those points, and probably the only clean-cut nonanalytic feature is its cusp.
\footnote{\label{PTRuelle}Some time ago Ruelle \cite{Ruelle:1974} proposed an alternative definition of phase transition without invoking non-analyticities of thermodynamic potentials. It is defined as a discontinuity in the qualitative properties of correlation functions. Discontinuity in the derivative of the correlation length in case of DLs qualifies then as a transition in Ruelle's classification.
Moreover, for the DL as a modulation transition with the modulation wave vector continuously growing deep into
the IC phase, one can use the notion of the critical index of modulation $\nu_{\scriptscriptstyle L}$, defined as $q \propto |h_{\scriptscriptstyle DL} - h|^{\nu_{\scriptscriptstyle L}}$
(with $ \nu_{\scriptscriptstyle L}=1/2$) by Nussinov and co-workers \cite{Nussinov:2012}.}

The goal of this section is to reveal relation between the DLs of the model \eqref{XYHam} analyzed in Sec.~\ref{DL3} and the factorized states
yet to be detected. To deal with the latter we follow the method of M\"{u}ller and coworkers \cite{Muller:1982,*Muller:1985} with some modifications.
Their original idea was to rotate each spin of the chain in the $xz$ plane to make the transformed Hamiltonian ferromagnetic with the fully
separable (factorized) GS. In the Hamiltonian \eqref{XYHam} with $\delta=0$ we rotate spins by the position-dependent angle $\vartheta_n$:
\begin{equation}
\label{Rot}
\left(\begin{array}{c}
\tilde{S}_{n}^{x} \\
\tilde{S}_{n}^{z}
\end{array}\right)=U_{n}^{-1}\left(\begin{array}{c}
S_{n}^{x} \\
S_{n}^{z}
\end{array}\right)
U_{n}=
\left(\begin{array}{cc}
\cos 2 \vartheta_{n} & \sin 2 \vartheta_{n} \\
-\sin 2 \vartheta_{n} & \cos 2 \vartheta_{n}
\end{array}\right)\left(\begin{array}{l}
S_{n}^{x} \\
S_{n}^{z}
\end{array}\right)
\end{equation}
with the rotation matrix
\begin{equation}
\label{Urot}
  U_{n}=\left(\begin{array}{cc}
\cos \vartheta_{n} & \sin \vartheta_{n} \\
-\sin \vartheta_{n} & \cos \vartheta_{n}
\end{array}\right)
\end{equation}
The transformed Hamiltonian
\begin{equation}
\label{Htil}
  \tilde{H}=U^{-1} H U =H(\tilde{S}),~~U \equiv \prod_{n=1}^{N} U_n
\end{equation}
can be brought to a desired form with the choice of two angles of rotation on the even and odd sites:
\begin{equation}
\label{Thetan}
 \vartheta_n=
\left\{
\begin{array}{lr}
\vartheta_e~, ~~n=2l \\[0.2cm]
\vartheta_0~, ~~n=2l-1\\
\end{array}
\right.
\end{equation}
Assuming an even number of sites $N$ and periodic boundary conditions $S_{N+1}^{\alpha}=S_{1}^{\alpha}$,
the transformed Hamiltonian written in terms of the original spins reads
\begin{widetext}
\begin{equation}
\label{HamRot}
   \tilde{H} =- \sum_{n=1}^{N}~ \big[ (1-\gamma) \big( S_{n}^{x }S_{n+1}^{x} + S_{n}^{y} S_{n+1}^{y} \big)
   + \sin 2\vartheta_e \sin 2\vartheta_o  S_{n}^{z} S_{n+1}^{z}  \big]
   +\sum_{l=1}^{N/2}~ \big[ (h-h_a) \cos  2\vartheta_o S_{2l-1}^{z}
   +  (h+h_a) \cos  2\vartheta_e S_{2l}^{z} \big]~,
\end{equation}
\end{widetext}
where the other terms of $\tilde{H}$ (not written explicitly) vanish in its GS if the conditions
\begin{equation}
\label{CosCos}
  \cos  2\vartheta_e \cos  2\vartheta_o =\frac{1-\gamma}{1+\gamma}
\end{equation}
and
\begin{eqnarray}
\label{CondMix1}
  (1+\gamma) \sin 2 \vartheta_e \cos 2 \vartheta_o &=&(h-h_a) \sin 2 \vartheta_o \\
\label{CondMix2}
  (1+\gamma) \sin 2 \vartheta_o \cos 2 \vartheta_e &=&(h+h_a) \sin 2 \vartheta_e
\end{eqnarray}
are satisfied. The solution of Eqs.~(\ref{CosCos},\ref{CondMix1},\ref{CondMix2}) exists for
$h^2+ \gamma^2=1+h_a^2$, thus the factorization occurs on the DL circle (see Fig.~\ref{HaGam}).
Introducing the factorization field $h_f$, we find
\begin{equation}
\label{hf}
  h_f=h_{\scriptscriptstyle DL}^{(1)}=\pm \sqrt{1+h_a^2-\gamma^2}~,
\end{equation}
while the other DL field $h_{\scriptscriptstyle DL}^{(2)}=\pm h_a$ does not solve Eqs.~(\ref{CosCos},\ref{CondMix1},\ref{CondMix2}),
thus $h_f \neq h_{\scriptscriptstyle DL}^{(2)}$.
For the factorizing field \eqref{hf} we find the explicit expressions:
\begin{equation}
\label{CosEO}
  \cos 2 \vartheta_{e/o}=\Bigg[\frac{(\sqrt{1+h_a^2-\gamma^2} \pm h_a)^2+(1-\gamma)^2}{
  (\sqrt{1+h_a^2-\gamma^2} \pm h_a)^2+(1+\gamma)^2 } \Bigg]^{1/2}
\end{equation}
These angles of spin rotation are related to the roots \eqref{zpmHa} merging on the DL circle as:
\begin{equation}
\label{ThetaZpm}
  z_{+} =z_{-} \equiv z_{\scriptscriptstyle DL} = \cos  2\vartheta_e \cos  2\vartheta_o
\end{equation}

The Hamiltonian \eqref{HamRot} is the $XXZ$ chain with the interaction parameter ($\Delta \equiv J_z/J$)
\begin{equation}
\label{DeltaR}
  \Delta =  \frac{\sin 2\vartheta_e \sin 2\vartheta_o}{1-\gamma}~.
\end{equation}
generated by the transformation of the original non-interacting model \eqref{XYHam}.
This $XXZ$ chain is subjected to the uniform and staggered transverse fields $\tilde{h}$ and $\tilde{h}_a$, respectively,
defined as
\begin{eqnarray}
\label{RenH1}
   \frac{(h-h_a) \cos  2\vartheta_o}{1-\gamma} &\equiv& \tilde{h} -\tilde{h}_a \\
\label{RehH2}
   \frac{(h+h_a) \cos  2\vartheta_e}{1-\gamma} &\equiv& \tilde{h} +\tilde{h}_a~.
\end{eqnarray}
The above equations together with Eq.~(\ref{CosCos}) lead to:
\begin{equation}
\label{Hcr}
  \tilde{h}^2=1 +\tilde{h}_a^2.
\end{equation}
An important conclusion then follows from the above equations and the rigorous results
\cite{Alcaraz:1995} for the $XXZ$ chain: with the fields satisfying \eqref{Hcr} and $\Delta>0$ \eqref{DeltaR}, the GS of $\tilde{H}$
\eqref{HamRot} is ferromagnetic with the state ket:
\begin{equation}
\label{FM}
|\mathrm{FM} \rangle=\prod_{n=1}^{N}|\uparrow\rangle_{n}=\prod_{n=1}^{N}\left(\begin{array}{l}1 \\ 0\end{array}\right)_{n}~.
\end{equation}
From \eqref{Htil} the GS of $H$ is obtained as
\begin{equation}
\label{Psi}
 |\Psi \rangle = U |\mathrm{FM} \rangle ~.
\end{equation}
Note that the expression under the square root in \eqref{CosEO} is positive and less that one, so $\cos 2 \vartheta_\sharp$ is real and
we choose the positive branch of the root for it, as well as for $ \cos \vartheta_\sharp =\sqrt{(1+\cos 2 \vartheta_\sharp)/2}$, while
we allow two signs of $ \sin \vartheta_\sharp =\pm \sqrt{1-\cos^2 \vartheta_\sharp}$ to account for the two-fold degeneracy of the GS.
(Here $\sharp =e/o$ for even/odd sites, respectively.)
Then
\begin{widetext}
\begin{equation}
\label{Psipm}
  {\left| \Psi _{\pm}  \right\rangle} =\prod _{l=1}^{N/2}
  \big( \cos \vartheta_o \left| \uparrow  \right\rangle_{2l-1}  \pm \sin \vartheta_o \left| \downarrow \right\rangle_{2l-1}   \big)
  \big( \cos \vartheta_e \left| \uparrow  \right\rangle_{2l}  \pm \sin \vartheta_e \left| \downarrow \right\rangle_{2l}   \big),
\end{equation}
\end{widetext}
with
\begin{equation}
\label{GSProd}
  \left\langle \Psi _{\pm } \Psi _{\pm } \right\rangle =1,~~
  \left\langle \Psi _{+} \Psi _{-} \right\rangle =\Big( \frac{1-\gamma}{1+\gamma} \Big)^{N/2}~.
\end{equation}
The factorized state \eqref{Psipm} is maximally disentangled. To put it more quantitatively let us invoke the two-site concurrence
$\mathcal{C}$ introduced by Wootters \cite{Wootters:1998} as a measure of entanglement. It can be calculated as \cite{Chitov:2021}
\begin{equation}
\label{C}
 \mathcal{C}=\sum _{m\ne n}{\left\langle \Psi  \right|} i\sigma _{m}^{y} i\sigma _{n}^{y} {\left| \Psi  \right\rangle} ~.
\end{equation}
For the states \eqref{Psipm} one can easily verify:
\begin{equation}
\label{YY}
 {\left\langle \Psi _{\pm}  \right|} \sigma _{m}^{y} {\left| \Psi _{\pm}  \right\rangle} =
  {\left\langle \Psi _{\pm}  \right|} \sigma _{m}^{y} \sigma _{n}^{y} {\left| \Psi _{\pm}  \right\rangle} =0,~~\forall~ m \neq n.
\end{equation}
thus  $\mathcal{C}=0$ on the factorized DL circle \eqref{hf}, as expected \cite{Amico:2006}.
To obtain the longitudinal magnetization we find
\begin{equation}
\label{Mx}
 {\left\langle \Psi _{\pm}  \right|} \sigma _{m}^{x} {\left| \Psi _{\pm}  \right\rangle} =
 \pm \sin 2 \vartheta_\sharp~.
\end{equation}
Thus two degenerate ground states $\left| \Psi _{\pm}  \right\rangle$
can be linked to two possible orientations of the spontaneous order parameter $m_x$. Similarly,
\begin{equation}
\label{Mz}
 {\left\langle \Psi _{\pm}  \right|} \sigma _{m}^{z} {\left| \Psi _{\pm}  \right\rangle} =
  \cos 2 \vartheta_\sharp~.
\end{equation}
A hallmark of the ground state factorization are the constant correlation functions.
For the disentangled state they are found in a closed from:
\begin{equation}
\label{XXZZ}
\left\langle \sigma _{m}^{\alpha} \sigma _{n}^{\alpha} \right\rangle =
\left\langle \sigma _{m}^{\alpha} \right\rangle \left\langle \sigma _{n}^{\alpha} \right\rangle~,
~~\forall~ m \neq n.
\end{equation}
with the average values $\langle \sigma _{m}^{\alpha} \rangle$ given by Eqs.~(\ref{Mx},\ref{Mz},\ref{CosEO}).
The above analytical results are in agreement with the direct numerical calculations of the correlation functions via the Toeplitz
determinants with the explicit formulas provided in \cite{Chitov:2019}. The representative plots of those functions for three points on the line of disentanglement are shown in Fig.~\ref{DisentCorr}. For comparison we also show in Fig.~\ref{haCorr} the spin-spin correlation function on the disorder line $h=h_a$ where the GS is not disentangled.

\begin{figure*}[t]
\centering{\includegraphics[width=13.0cm]{DisentXX_ZZ.pdf}}
\caption{ Constant spin-spin correlation functions at three points on the circle of disentanglement \eqref{hf}. Panels (a) and (d): $\langle \sigma _m^x \sigma_m^x \rangle$ and $\langle \sigma _m^z \sigma_m^z \rangle$ at a point on the boundary between regions (B) and (C) of the FM phase, see Fig.~\ref{HaGam}; (b): point of intersection of two DLs: \eqref{hf} and $h=h_a$; (c): point on the boundary between (D) and IC oscillations.}
\label{DisentCorr}
\end{figure*}

\begin{figure}[]
\centering{\includegraphics[width=8.0cm]{XXha_line.pdf}}
\caption{ Spin-spin correlation function $\langle \sigma _m^x \sigma_m^x \rangle$ on the disorder line $h=h_a$ where the GS is not disentangled.
Contrary to the case shown in Fig.~\ref{DisentCorr}, the correlation functions are not constant.}
\label{haCorr}
\end{figure}

The behavior of correlators on the generic DLs and on the DLs with the factorized disentangled GSs is  stemming from the analytical
properties of the generating function \cite{Chitov:2021,Jones:2021,Jones:2021B}. The elements of the Toeplitz determinants yielding
various spin-spin and string-string correlation functions are the two-point Majorana correlators:
\begin{equation}
\label{Gab}
  \langle i a_n b_m \rangle = \oint _{\left|z\right|=1}\frac{dz}{2\pi i}  z^{n-m-1} \frac{\mathcal{R}(z)}{E_+(z) E_-(z)}~,
\end{equation}
where the explicit formula for $\mathcal{R}(z)$ can be found in \cite{Chitov:2019}. On the DL two complex conjugate zeros of the spectrum $z_\pm$
merge into a single real degenerate root $z_{\scriptscriptstyle DL}$, rendering its contribution to the integrand of \eqref{Gab} meromorphic:
\begin{equation}
\label{GabDL}
  \langle i a_n b_m \rangle \overset{\mathrm{\scriptscriptstyle DL}}{=} \oint _{\left|z\right|=1}\frac{dz}{2\pi i}  z^{n-m} \frac{\tilde{\mathcal{R}}(z)}{(z-z_{\scriptscriptstyle DL})(z-z_{\scriptscriptstyle DL}^{-1})}~,
\end{equation}
For the limiting case $h_a=\delta=0$ the aforementioned restructuring of singularities of the generating function in \eqref{GabDL} in sufficient to both \textit{(i)} yield the DL as a boundary of the modulated region with IC oscillations, and \textit{(ii)} to render the appropriate Toeplitz determinants size-independent on the whole DL, such that Eqs.~\eqref{XXZZ} hold and the state is disentangled (factorized) \cite{Chitov:2021}.

As we have shown above for the factorizing field $h_f$ \eqref{hf} and the DL field $h_{\scriptscriptstyle DL}^{(2)}=\pm h_a$, the properties \textit{(i)} and \textit{(ii)}
do not coincide in general.
One can also check that there are no solutions of the equations for $\tilde{H}$ with the factorized GS when $\delta \neq 0$.
Thus, in the presence of dimerization the states on the DLs analysed in Sec.~\ref{DL3} for the cases B ($h_a=0$) and C ($h_a \neq 0$) are entangled, in agreement with general arguments of \cite{Illuminati:2008,*Illuminati:2009,Giorgi:2009}.

\section{Zero-energy Majorana edge states}\label{MES}

Let us first find the 1D winding number (or the Pontryagin index) defined as \cite{WenZee:1989,SchnyderRyu:2011}
\begin{equation}
\label{Nw}
  N_w=\frac{1}{2\pi i} \int_{BZ} dk \mathrm{Tr}[\partial_k \ln \hat{D}]~,
\end{equation}
where $\hat{D}$ is introduced in Eq.~\eqref{Dk}. It can be readily written as
\begin{equation}
\label{NwCompl}
    N_w=\oint _{\left|z\right|=1} \frac{dz}{2\pi i} \partial_z \ln  \mathrm{det}\hat{D}~.
\end{equation}
Then the topological number \eqref{NwCompl} is the logarithmic residue of $\mathrm{det}\hat{D}$.
It accounts for the excess of the number of zeros over the number of poles (weighted with their degrees of multiplicity)
of $\mathrm{det}\hat{D}$ inside the unit circle on the complex plane \cite{Whittaker:1915}.
The zeros of $\mathrm{det}\hat{D}$ are also zeros of the spectra, cf. Eqs.~({\ref{EpmDD},\ref{detD}).
Any change of winding number occurs only when a root (roots) crosses the unit circle $|z|=1$, which can happen only at
a point of quantum phase transition \cite{Verresen:2018,Chitov:2021}. By the same token $N_w$ does not change upon
crossing disorder lines.

From  Eq.~(\ref{detD}) and known root positions with respect to the complex unit circle
(see Figs.~\ref{ZpmA},\ref{ZpmB},\ref{ZpmC}), we readily obtain the winding numbers for the three gapped phases of the model \eqref{XYHam}:
\footnote{\label{NwOld} The phase of logarithm is not uniquely defined. In earlier related work \cite{Chitov:2019,Chitov:2020} the limit of the phase of logarithm in Eq.~\eqref{Nw} at the ends of the BZ was  chosen  such that $N_w$ differed from the current result \eqref{NwAll} by 1. Note that \eqref{Nw} is
defined mod 2. Definition \eqref{NwCompl} is robust and more intuitively transparent.}
\begin{equation}
\label{NwAll}
N_w=
  \left\{
    \begin{array}{l}
      \phantom{\pm} 0,~\mathrm{PM~phase} \\[0.2cm]
      \mp 1, ~\mathrm{FM~phase},~ \gamma \gtrless 0\\[0.2cm]
      \phantom{\pm} 0, ~\mathcal{O}_z(\pi/2)~ \mathrm{phase.}  \\
    \end{array}
   \right.
\end{equation}

In the search of the zero-energy Majorana edge states of the model \eqref{XYHam} we will follow the earlier related papers
\cite{Karevski:2000,DeGottardi:2011,DeGottardi:2013b,Chitov:2018}. We introduce two zero-energy Majorana operators $\hat \alpha_0$
and  $\hat \beta_0$ as
\begin{equation}
\label{AB0}
2 \hat \eta_{\scriptscriptstyle 0}^\dag \equiv \hat \alpha_{\scriptscriptstyle 0} +i \hat \beta_{\scriptscriptstyle 0}  =\sum_{n=1}^N \big[\phi_n \hat a_n + i \psi_n \hat b_n \big],~\{ \phi_n,\psi_n \} \in \mathbb{R},
\end{equation}
where $\hat \eta_{\scriptscriptstyle 0}^\dag$ denotes the creation operator of the Bogoliubov fermion with zero energy. (The Hamiltonian is diagonal in terms of
the Bogoliubov fermions). The involution of Majorana operators amounts to the requirement of the wave function normalization in \eqref{AB0}:
\begin{eqnarray}
  \label{ANorm}
  \hat \alpha_{\scriptscriptstyle 0}^2 &=& \sum_{n=1}^N \phi_n^2=1~, \\
  \hat \beta_{\scriptscriptstyle 0}^2 &=& \sum_{n=1}^N \psi_n^2=1~.
 \label{BNorm}
\end{eqnarray}
The Heisenberg equations for $\hat \alpha_{\scriptscriptstyle 0}$ and $\hat \beta_{\scriptscriptstyle 0}$ yield
\begin{equation}
\label{Commut}
  [\hat \alpha_{\scriptscriptstyle 0}, \hat H]=[\hat \beta_{\scriptscriptstyle 0}, \hat H]=0~.
\end{equation}
Writing the Hamiltonian \eqref{XYFermi} with open boundary conditions ($N=2L+1$) via the Majorana operators \eqref{Maj}, one can bring the commutativity condition \eqref{Commut} to the form of the iterative equation for the Majorana wave function
\begin{equation}
\label{phiT}
  \left(\begin{array}{l}
   \phi_{n+1} \\
   \phi_n
  \end{array}\right)
=\hat T_n
  \left(\begin{array}{l}
   \phi_n \\
   \phi_{n-1}
  \end{array}\right)
\end{equation}
with the transfer matrix
\begin{equation}
\label{Tn}
\hat T_n =\left(
          \begin{array}{cc}
            \frac{2h_n}{t_n+ \gamma } & -\frac{t_{n-1}-\gamma}{t_n+\gamma} \\[0.2cm]
            1 & 0 \\
          \end{array}
        \right)
\end{equation}
where
\begin{equation}
\label{hntn}
  h_n \equiv h +(-1)^n h_a,~~t_n \equiv 1 +(-1)^n \delta~.
\end{equation}
For the other wave function we get
\begin{equation}
\label{psiT}
  \left(\begin{array}{l}
   \psi_{n+1} \\
   \psi_n
  \end{array}\right)
=\hat T_n(-\gamma)
  \left(\begin{array}{l}
   \psi_n \\
   \psi_{n-1}
  \end{array}\right)~.
\end{equation}
There are only two distinct matrices $\hat T_n$:
\begin{equation}
\label{Tpm}
  \hat T_{\scriptscriptstyle 2l/2l-1}=\hat T_{\pm}.
\end{equation}
Using the period 2 transfer matrix
\begin{equation}
\label{T2}
  \hat T \equiv \hat T_+ \hat T_- =
  \left(
          \begin{array}{cc}
            \frac{4(h^2-h_a^2)-(1-\delta)^2+\gamma^2}{(1+\gamma)^2-\delta^2}
            &  -\frac{2(h+h_a)(1+\delta-\gamma)}{(1+\gamma)^2-\delta^2} \\[0.2cm]
              \frac{2(h-h_a)}{1-\delta+\gamma} & -\frac{1+\delta-\gamma}{1-\delta+\gamma} \\
          \end{array}
        \right)
\end{equation}
the solution of \eqref{phiT} can be written as
\begin{equation}
\label{phiN}
  \left(\begin{array}{l}
   \phi_{\scriptscriptstyle 2m+1} \\
   \phi_{\scriptscriptstyle 2m}
  \end{array}\right)
=\hat T^m
  \left(\begin{array}{c}
   \phi_{\scriptscriptstyle 1} \\
    0
  \end{array}\right)~.
\end{equation}
Similarly,
\begin{equation}
\label{psiN}
  \left(\begin{array}{l}
   \psi_{\scriptscriptstyle 2m+1} \\
   \psi_{\scriptscriptstyle 2m }
  \end{array}\right)
=\hat T^m(-\gamma)
  \left(\begin{array}{c}
   \psi_{\scriptscriptstyle 1} \\
    0
  \end{array}\right)~.
\end{equation}
To look for the tentative Majoranas localized near the right end of the chain, we write equations \eqref{Commut}
in the different iterative form as
\begin{equation}
\label{phiTR}
  \left(\begin{array}{l}
   \phi_{n-1} \\
   \phi_n
  \end{array}\right)
=\hat T_n(-\gamma, -\delta)
  \left(\begin{array}{l}
   \phi_n \\
   \phi_{n+1}
  \end{array}\right)~,
\end{equation}
leading to the solution
\begin{equation}
\label{phiNR}
  \left(\begin{array}{l}
   \phi_{\scriptscriptstyle 2L-2m+1} \\
   \phi_{\scriptscriptstyle 2L-2m+2}
  \end{array}\right)
=\hat T^m(-\gamma)
  \left(\begin{array}{c}
   \phi_{\scriptscriptstyle 2L+1} \\
    0
  \end{array}\right)~.
\end{equation}
In the same way we obtain
\begin{equation}
\label{psiNR}
  \left(\begin{array}{l}
   \psi_{\scriptscriptstyle 2L-2m+1} \\
   \psi_{\scriptscriptstyle 2L-2m+2}
  \end{array}\right)
=\hat T^m
  \left(\begin{array}{c}
   \psi_{\scriptscriptstyle 2L+1} \\
    0
  \end{array}\right)~.
\end{equation}

To advance further \cite{Chitov:2018} we write the transfer matrix $\hat T$
via two orthogonal idempotent operators (projectors)
\begin{equation}
\label{Ppm}
  \hat{\mathcal{P}}_\pm \equiv \pm \frac{ \hat T -\lambda_\mp \mathbb{1} }{\lambda_+ - \lambda_-}
\end{equation}
as
\begin{equation}
\label{Tpm}
  \hat T =\lambda_+ \hat{\mathcal{P}}_+  + \lambda_- \hat{\mathcal{P}}_- ~,
\end{equation}
where $\lambda_\pm$ are two eigenvalues of $\hat T$.
Then
\begin{equation}
\label{Tpower}
  \hat T^m =\lambda_+^m  \hat{\mathcal{P}}_+  + \lambda_-^m \hat{\mathcal{P}}_-
\end{equation}
As follows from Eqs.~(\ref{ANorm},\ref{BNorm},\ref{phiN},\ref{psiN},\ref{phiNR},\ref{psiNR},\ref{Tpower}), for the existence of the normalizable
zero-energy Majorana states in the thermodynamic limit $N \to \infty$, the eigenvalues of the transfer matrix must lie inside the complex unit circle
$|\lambda_\pm|<1$.

The key result is that the eigenvalues of the transfer matrix $\hat T$ \eqref{T2} are zeros of the spectrum \eqref{zpm}:
\begin{equation}
\label{lamz}
  \lambda_\pm= z_\pm~,
\end{equation}
in agreement with the earlier findings \cite{Chitov:2018} for the case $h_a=\delta=0$. Also, according to Eq.~\eqref{zinv}, the eigenvalues
of $\hat T(-\gamma)$ are $z_\pm^{-1}$.
\textit{Thus the zeros of the spectrum $z_\pm$, which are also the zero-temperature limit of the partition function zeros, control the critical points, modulation transitions, disentanglement, topological numbers, and the zero-energy Majorana states.} The properties on the \textit{edge} ($\hat T$) are controlled by the \textit{bulk} parameters ($z_\pm$).

Note that counting of the localized Majorana modes from number of zeros of the resolvent of the transfer matrix $\hat T$ within the unit circle on the complex plane  \cite{DeGottardi:2011,DeGottardi:2013b} is still valid, however it does not provide an independent information after the eigenvalues of the transfer matrix are being identified as zeros of the spectrum \eqref{lamz}.

Since the information on the roots $z_\pm$ \eqref{zpm} is available for all regions of the model's phase diagram
(see Figs.~\ref{ZpmA},\ref{ZpmB},\ref{ZpmC}), we can give the qualitative answers about existence of the Majorana states prior of any calculations:

1). PM and $\mathcal{O}_z(\pi/2)$ phases:  Among two roots $z_\pm$  one is always located inside the unit circle, and the other one outside (the same for their reciprocals $z_\pm^{-1}$), thus none of the four equations (\ref{phiN},\ref{psiN},\ref{phiNR},\ref{psiNR}) has a normalizable solution, and the zero-energy Majorana modes do not exist in those phases.

2). FM phase witn $m_x \neq 0$ ($\gamma >0$): The solution of (\ref{phiN}) for the wave function $\phi_n$ of the Majorana zero-energy state $\hat \alpha_{\scriptscriptstyle 0}$ localized near the left end of the chain and the solution of (\ref{psiNR}) for $\psi_n$
of the Majorana state $\hat \beta_{\scriptscriptstyle 0}$ localized near the right end, exist. The zero-energy Bogoliubov (Dirac) fermion $\hat \eta_{\scriptscriptstyle 0}^\dag$ \eqref{AB0} is a
superposition of two Majoranas localized on the opposite ends of the chain.

3). FM phase with $m_y \neq 0$ ($\gamma <0$): $\phi_n \leftrightarrow \psi_n$ since in this case equations (\ref{psiN}) and (\ref{phiNR}) have normalizable solutions.  The zero-energy Majoranas $\hat \alpha_{\scriptscriptstyle 0}$  and $\hat \beta_{\scriptscriptstyle 0}$ are now localized near the right and left ends of the chain, respectively.

From the above equations we readily find the explicit formulas for the wave function of the $\hat \alpha_{\scriptscriptstyle 0}$ Majorana in the $m_x$-FM phase:
\begin{eqnarray}
\label{phiOdd}
  \phi_{ 2m+1} &=& \left( \hat T^m \right)_{ 11} \phi_{ 1}~, \\
   \left( \hat T^m \right)_{ 11} &=& \frac{(z_+^m-z_-^m) T_{ 11}-(z_+^{m-1}-z_-^{m-1})z_+ z_-  }{z_+ - z_-}~,  \nonumber \\
  \phi_{ 2m} &=& \left( \hat T^m \right)_{ 21} \phi_{ 1}~, \\
  \label{phiEven}
    \left( \hat T^m \right)_{21} &=& \frac{(z_+^m-z_-^m) T_{ 21}}{z_+ - z_-} ~,\nonumber
\end{eqnarray}
where the matrix elements $T_{11}$ and $T_{21}$ are given in \eqref{T2}.
The normalization condition \eqref{ANorm} yields
\begin{equation}
\label{phi1}
  \phi_1 = \Big[1+\sum_{m=1}^L \Big\{ \left( \hat T^m \right)_{ 11}^2+ \left( \hat T^m \right)_{ 21}^2 \Big\} \Big]^{-1/2}
\end{equation}
For the wave function of the $\hat \beta_{\scriptscriptstyle 0}$ Majorana we get
\begin{eqnarray}
\label{psiEven}
  \psi_{ 2L-2m+1} &=& \left( \hat T^m \right)_{ 11} \psi_{ 2L+1}~, \\
  \psi_{2L- 2m+2} &=& \left( \hat T^m \right)_{ 21} \psi_{ 2L+1}~,
  \label{psiOdd}
\end{eqnarray}
with $\psi_{ 2L+1}$ determined by the right hand side of Eq.~(\ref{phi1}).

The wave functions are exponentially decaying with the penetration depth (inverse correlation length) determined by the root $z_\pm$ closest to the unit circle. In the light yellow regions without oscillations
(see Figs.~\ref{ZpmA},\ref{ZpmB},\ref{ZpmC}) the roots are positive, while the negative root closest to the unit circle results in the four-periodic oscillations of the wave functions in the light blue regions of the phase diagram.
In the IC oscillating (light green) regions of the phase diagram where $z_+=z_-^*$ (cf. notations \eqref{kC}) the formulas become particularly simple:
\begin{eqnarray}
\label{phiOddIC}
 \left( \hat T^m \right)_{ 11} &=& e^{-2 \kappa (m-1)} \frac{\sin 2qm}{\sin2q} T_{11} \nonumber \\
 &-&e^{-2 \kappa m} \frac{\sin 2q(m-1)}{\sin2q} \\
  \left( \hat T^m \right)_{ 21} &=& e^{-2 \kappa (m-1)} \frac{\sin 2qm}{\sin2q} T_{21}~.
  \label{phiEvenIC}
\end{eqnarray}
The plots of the wave functions for representative cases are shown in Fig.~\ref{MajoranaWF}

\begin{figure}[]
\centering{\includegraphics[width=7.0cm]{MajoranaABC.pdf}}
\caption{Wave functions $\phi, \psi$ of the zero-energy  Majoranas $a$, $b$ localized near the left and right ends of the chain, respectively.
The functions are  calculated in three regions (B), (C), and (D) of the $m_x$-FM phase (see Fig.~\ref{HaDelGam}) with $N=501$.
Panel (a): region (B) without oscillations; (b) region of the IC oscillations (C) with the wave number $q =0.1583 \pi$; (c): region (D) with four-periodic
oscillations $q=\pi/2$.}
\label{MajoranaWF}
\end{figure}

\section{Conclusion}\label{Concl}

In this work we present a detailed study of the exactly solvable dimerized ferromagnetic $XY$ chain with uniform and staggered transverse fields,
which is equivalent in the fermionic representation to the noninteracting dimerized Kitaev-Majorana chain with modulated chemical potential.
The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase
in the $U(1)$ limit. It is shown that the model along with  the continuous quantum phase transitions possesses weaker singularities known
as disorder lines (DLs) or modulation transitions. The latter, reported for the first time in this model, are localized in the ferromagnetically
ordered regions of the phases diagram.  The DLs are shown to occur in this model in two types: DLs of the first kind with continuous appearance
of the IC oscillations,  and DLs of the second kind (DL2) corresponding to a jump of the wave number which controls the oscillations.

It is also shown that this model hosts the zero-energy Majorana edge modes in its ferromagnetic phases.
In each phase there is a couple of the Majorana states localized on the opposite ends of the chain, which can be viewed as a ``deconfined"
Bogoliubov (Dirac) fermion with zero energy. The exactly calculated wave functions of those edge Majoranas are shown to acquire modulations on the DLs.

It is demonstrated that the DLs are the locus of the separable ground states. More specifically, the ground state is proved to be separable (factorized)
on a subset of the DLs (circle) in the case of zero dimerization $\delta=0$. In this state the model is disentangled, which is remarkably manifested by
constant correlation functions. With the explicit formula for the factorized ground state, the correlation functions are calculated exactly analytically, in
agreement with the numerical calculations of the Toeplitz determinants.

Probably the most important conclusion of the present study is that all the above results are obtained from analysis of the
the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become
complex zeros $z_\pm$ of the spectrum of the Hamiltonian. Those zeros signal a quantum critical point when one of the roots $z_\pm$ (or both simultaneously)
cross the complex unit circle. In this case (only!) the topological winding number changes as well. Merging of the complex conjugate roots $z_+=z_-^*$
signals the DL (modulation transition) which can also be accompanied by the ground state factorization (disentanglement). The roots $z_\pm$ are also shown to be the eigenvalues of the transfer matrix, thus their number inside the unit circle on the complex plane controls also the existence of the localized zero-energy edge Majoranas.

We think a very important direction of future analysis is to get a deeper insight on the analytical properties of generic disorder lines to single out their subsets where the ground state is factorized, the model is disentangled, and the appropriate Toeplitz determinants are size-independent.

~\\

\noindent $^\dagger$ We deeply regret the sudden death of our co-author, Pavel Nikolaevich Timonin (23/11/1948 - 20/03/2022),
during the final stage of the work on this paper. We did our best for the manuscript to accurately represent his vision of the
results and to meet his very high standards.

\begin{acknowledgments}
G.Y.C. thanks N.G. Jones for helpful correspondence and especially C. Bourbonnais for his support.
G.Y.C. gratefully acknowledges financial support from Institut quantique (IQ) of
Universit\'{e} de Sherbrooke and Regroupement qu\'{e}b\'{e}cois sur les mat\'{e}riaux de pointe (RQMP).
K.G. thanks Girish Sharma and Salvatore R. Manmana for useful discussions and gratefully  acknowledges funding by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 217133147/CRC 1073, project B03 and computational
resources provided by the IIT Mandi.
P.N.T. was supported by the Ministry of Education and
Science of the Russian Federation through the state assignment in the field of scientific
activity, project No.~0852-2020-0032 (BAS0110/20-3-08IF).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
The Gospel of Matthew
Matthew (who was also known as Levi – pronounced Leh-vee, as in Levine) was one of the first disciples of Jesus, and he was later appointed by him to be an Apostle or Sent One.
According to ancient Christian writers such as Origen (who wrote during the early 3rd century), Matthew's account was the first recorded Gospel. Origen wrote:
'The first book was written by Matthew. This Gospel was composed in Hebrew near Jerusalem for Hebrew Christians and translated into Greek, but the Greek copy was lost.'
While many of Origin's personal religious views reflect pagan influences (that began entering Christianity in the early 2nd century), our research of Matthew indicates that Origen was probably right about it being the first written Gospel (regardless of what modern critics have said). It was likely written in Aramaic, and then later translated into Greek.
We also think that the first Greek copies of the Gospel of Matthew were lost. The many errors and additions (mentioned in these translator notes) indicate that the existing Greek text was in fact translated again and corrupted sometime in the 2nd century by the same person or group that translated other 1st century books into Greek.
We have partly drawn this conclusion from the use of certain unusual Greek words (such as parousias) that are found only in later Greek translations of the writings of Matthew, Paul, James, Peter, and John (which were likely originally penned in Aramaic).
For more information, please see the commentary Coming, Presence, or Nearness?
> Read more translator notes
Our Bible uses older manuscripts than most Bibles do. Check us out!


This week Zavvi have yet another Steelbook to add to their repertoire; this time, Gremlins on limited edition Blu-Ray Steelbook, the comedy-horror from 1984 has a release date of the 7th of October 2013 and will cost £19.99 £14.99.
Considering the age of the film and quality of the artwork, is the price too much? Let us know your thoughts in the comments or tweet us @Steelbook_UK. Pre-orders for Gremlins Steelbook will be live from 6PM Saturday night.
Price had now dropped to £14.99.


This second novel in the Montana Adventure Series follows young "Montana" to Fort Worth, Texas in 1886, where he hires on at a major cattle ranch and ultimately saves it from foreclosure. Montana Series Novels are best suited for young adults who seek good adventures and positive role models.
I'd been sufferin' for three days now and I couldn't take it much longer. Turns out there are few things in a cowboy's life more uncomfortable than a toothache. It started out as little more than an itch, but now it'd pretty much taken me over. I threw my blanket down late last night thinkin' I might sleep better, but it didn't turn out that way. Every time my face touched the ground, a lightening bolt of pain lit me up. Fortunately, Spirit and I should make Fort Worth by noon and I'll be quick to find my way to the nearest dentist. I've only been to Fort Worth once, and while I didn't know where any were, I suspected a city that size would be full of dentists who'd be happy to help me out.
I can't complain though. The truth is I haven't been sick much since leavin' Montana some three years ago, and this ache in my jaw's made me grateful for that.
I was also noticin' that the further south we rode, the hotter it was. Seemed to be hotter than usual for early June, but then again I'd only been this far south once before and that was last winter.
Maybe this was normal for Texas and I know it bothered me more than it did Spirit, but I was concerned for him. We hadn't seen a water hole since right after dawn and I knew he must be gettin' thirsty.
As much as I was in a hurry to get to town, when the trail passed by a small grove of Mesquite trees I pulled up to a shady spot, grabbed my canteen from my saddle bag and took a break. I had to be careful not to add to my pain, so I was slow in takin' small, careful sips. When I was done I poured some in my cupped hands and held them up so Spirit could get a taste, too.
Spirit seemed to understand as he took a few licks and it was time to get back on the trail.
As we rode on, to get my mind off the pain, I thought back to how I got here. Three years ago I left home to be a cowboy – somethin' I couldn't be on our farm. My first real job was at a big ranch in Wyoming. That job was important to me for two reasons. First, it was where I learned to handle cattle, and second, it's where I found Spirit. He was just a yearling when my old horse broke his leg and had to be put down. Spirit was sired by my boss's horse and Dan planned to give him to his own son, Drew. But Drew was killed in an accident just before I hired on and when he decided it was time, Dan offered to sell him to me. Spirit was the most beautiful horse I'd ever seen and I was quick to take him up on the offer.
It took almost two years of hard work to pay off the loan but when I finally did, Spirit was all mine. Not long after that I left Wyoming for Kansas and got a job with one of the big cattle drives. Most those drives that end up in Dodge City start out in Fort Worth. After the first drive I stayed up there with friends for about a month 'til it was time to head back to Texas to catch another. The tooth ache started just after I left. If it'd started a day earlier I'd have had it taken care of in Dodge, but here I was out on the hot prairie with all the pain I could handle. Dang…this was supposed to be keeping my mind off the pain!


text:
Category: Press » Entertainment
O2 Media TV Shows Address the Issues that Are Most Important to the American Viewing Audience
Kay Renz
Deerfield Beach, FL - Reality TV and crime dramas consistently generate high ratings, but Americans are also tuning in to television shows that help them cope with modern concerns such as the economy, the environment, health and family issues. O2 Media TV shows, which include Designing Spaces, Kitchen Spaces and the Balancing Act are addressing those concerns. Designing Spaces has launched a number of special edition episodes that range from "Green Spaces", covering environmental issues, to "Kid Spaces", for family and young people. The goal of the O2 Media TV shows is to educate viewers and continue to be a reliable source of information for viewers based on their interests, circumstances and needs. The company's television shows air nationally on the Lifetime, WE and TLC networks as well as on regional stations.
"The O2 Media TV shows are both entertaining and enlightening," says Mark Alfieri, founder of O2 Media. "Our goal is to educate viewers on ways in which they can improve their lives, while keeping in mind that today's economy means that most viewers are on a tight budget."
The O2 Media TV Shows are part of O2 Media which is a Florida-based production company that is nationally renowned for creating award-winning educational programming. Their informative shows are geared towards disseminating innovative ideas and practical solutions to everyday challenges. Guests on Designing Spaces have included representatives from companies such as Microsoft, Disney, MTV, Samsung, Chase Bank, Procter & Gamble, Wal-Mart, Celebrity Cruise Lines and many others.
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Sheraton Bali Kuta Resort Introduces a Special Daily Pass Program At Sheraton Fitness By Core Performance™ (Popularity: ): Sheraton Bali Kuta Resort aims fitness enthusiasts in Bali to trial the hassle-free work out experience at Sheraton Fitness by Core Performance™ with an all-inclusive daily pass rateBALI, INDONESIA – 6 February 2014 – Sheraton Bali Kuta Resort launches a special daily pass program for fitness enthusiasts to work out while being indulged by the refreshing oceanic view at Sheraton Fitness by Core Performance™, a membership lifestyle program of Sheraton ...
Free Entertainment With BT Office Furniture (Popularity: ): As a result of a successful relationship with Merlin Entertainments, BT Office Furniture Group is now giving away free Merlin Annual Passes to the UK's top attractions. A Merlin Annual Pass gives unlimited entry* to top attractions including: Legoland(R) Windsor, THORPE PARK and the Alton Towers Resort as well as London attractions such as: Madame Tussauds, The London Eye and the Dungeons.Depending on the total order value, BT Office will ...
The Cable Company Launches Abyss Headphones at the New York Audio Show (Popularity: ): Pennsylvania, US (30 April, 2013) – The Cable Company, a leading provider of specialty audio cables and high-performance sound equipment, launched their new Abyss headphones at the New York Audio Show.The show was held at the New York Palace Hotel between the 12th and 14th of April, and gave visitors the chance to experience the latest cutting-edge audio equipment, as well as attend live entertainment events and seminars. The Cable ...
Sound Solution Announce Its Much Awaited New Products Arrival (Popularity: ): Victoria, British Columbia- April 13, 2012- Sound Solution, a premier supplier of home entertainment and home theatre installation in Western Canada is very excited to announce that all new products are finally in their stock and now ready to be introduces soon this 2012."All New 2012 Products are here, and they are HOT! Save big on Massive Audio's Subwoofer and Amplifier Packages. Starting as low as $499.00. Give us a ...
Tickets Compared Launches New Toolbar for Quick Ticket Links (Popularity: ): Sarasota, FL USA (23rd February, 2012) – Tickets Compared, a leading discount sports, theater and concert tickets comparison website, have launched a new browser toolbar.Available for use on Internet Explorer, Firefox and Google Chrome, the Tickets Compared toolbar allows users to gain instant access to tickets for upcoming music, theater and sporting events. The toolbar provides quick links to events happening all over the USA, and allows users to read ...
International Remote Viewing Association (Popularity: ): The International Remote Viewing Association (IRVA)promotes the topic of remote viewing in an accurate and scientific way to the public and the media. IRVA holds yearly conferences and links to a self-test website where individuals can test their remote viewing abilities.
Accuracy in Media (Popularity: ): A non-profit grassroots citizens watchdog of news media that critiques botched news stories and sets the record straight on important issues that have received slanted coverage.
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To me, there's nothing better than a prime rib steak. Especially the end cut. Just looking at that makes my mouth water. Using A-1 or Worcestershire or Tabasco sauce on it would be an insult to the cook! Maybe dip a couple of bites into a liiiittle bit of horseradish. MMMMMMMMMMMMMMmmmmmmmmm.
Old joke - "Vegan, native American word for bad hunter"
Is she doing it for health or ethical reasons? Have you become one by proxy since she does the shopping and cooking?
Hey HH -- I like that joke, I'll use it!
Started with health reasons, morphing into ethical... But I'm holding the line, dang it! Baloney and cheese quesadilla tonight!
Same as before, so sad!
Chicken Cordon Bleu with pom fritts. Just gonna have to do. Au revoir.


text:
Can't catch a break
Rain moves out this morning, giving way to sun and highs in the upper 80s. Sunny, with high temperatures just shy of 90 tomorrow.
Baltimore added three more homicides on Monday morning. In separate incidences, three men were shot and killed in east Baltimore. In south Baltimore, Curtis Bay Elementary was closed after a body was discovered on a nearby street (no cause of death identified yet).
Yesterday, police identified Sunday's murder victim found in a burning home as 16-year-old Arnesha Bowers.
There have been 13 homicides in the first 8 days of June (though the official count includes two deaths from May).
Larry Lomax, the curfew protester seen pepper sprayed and thrown to the ground by police in a viral video clip (VIDEO), is expected in court this morning, facing assault charges. Lomax is charged with second-degree-assault, inciting a riot, and disorderly conduct because he allegedly approached officers while walking quickly "with his fists balled up tight in an aggressive manner yelling obscenities." After being arrested on May 2nd, Lomax spent the next 4 weeks in jail.
A Circuit Court judge struck down State's Attorney Marilyn Mosby's request for a gag order in the criminal case against six Baltimore officers charged in the death of Freddie Gray. The strike down is a procedural issue; the SA filed the request while the officers' case was still in District Court. Mosby can file another gag order request – her team has been clear that they want to avoid litigating the case in the press.
+ Mayor Stephanie Rawlings-Blake announced $4.2 million in additional funding for after-school and summer programs, saying that the funds were freed up by previous budget cuts.
+ At yesterday's City Council meeting, member Carl Stokes called for the city to stop turning off water to homes until all delinquent business accounts have been dealt with. In the last few weeks, water service was cut off for about 2000 customers, 1300 have made payments and had service restored. A few hundred business accounts owe the city millions in unpaid bills.
The proposed renovation for Federal Hill's Cross Street Market includes plans for a greenhouse, presumably to protect plants from weekend vomit.


What is my purpose behind this fundraising?
I am ALina, doing Bsc in Psychology. I have a strong passion for kids who are specially capacitated. I call them God gifted. I have real love for such kids and am ready to devote all my time and energy to these little angels.
Well, there is an institute in Delhi Malviya Nagar called Institute for Child Development. I was looking for a platform where i could help the disabled kids and I very recently came across their website and I thought why not? Firstly these folks are very famous and eminent in Delhi and genuinely involved in this. They have won the National Education Leadership Award in 2015 for excellence in Paediatric Neurodisbaility Management.
Nonetheless, what i am looking forward to is a drawing competition amongst the disabled kids. There will be 30 kids in totality. To conduct the competition i would require paper, colouring material and gifts that will be given to each and every child. There will be no winners. I don't want any child to get disheartened. So this is the reason why i am raising funds.
I plan to buy good quality, eco friendy drawing sheets, pastel colors, paint, brushes and little goodies such as dolls, teddy bears and eatables for the kids.
Why would I get involved in such a thing?
Everyone has their personal perferances and hobbies. This is my way to emancipation. In future too i would continue with this cause. Let's make a difference. I am not after money. None of this money would go inside my pocket. All of it would be utilized for the kids only. How about gifting them a muffler as a goodie? Give your suggestions.
God has created them and it is our responsibility to make them smile. All the best.
My sister is in Delhi. Let me know if any help is needed. Will be happy to help.
you are doing grt job..