One of the most intriguing matchups in the junior middleweight division occurs on Saturday at the MGM Grand in Las Vegas between Mexican star Saul "Canelo" Alvarez (43-1-1, 31 KOs) and Cuban sharpshooter Erislandy Lara (19-1-2, 12 KOs). The winner of this fight will undoubtedly be acknowledged as the best junior middleweight in the world after Floyd Mayweather. For Alvarez, a victory over the tricky Lara would be the best win of his career and would further reinforce has status as one of the elite young fighters in the sport. Beating Alvarez would give Lara the opportunity to emerge as a real player in boxing, something more substantial than a "B-side."
Alvarez and Lara have two recent opponents in common. Both beat Austin Trout and Alfredo Angulo. However, Alvarez struggled with Trout while Lara dominated him and Lara went down to the canvas twice against Angulo, who wasn't competitive against Alvarez. In analyzing past performances, I'm not sure how much those outings will inform us about Saturday's outcome. Although Trout, like Lara, was also a crafty southpaw, the two lefties have significantly different styles. Trout didn't have much power and stayed in the pocket, winning early rounds against Alvarez with his jab. Lara uses his legs a lot more than Trout does and rarely relies on his jab; his left hand counter is a real weapon. Perhaps Lara, similar to Trout, will be able to reduce Alvarez's punch volume, but he will attempt to do so by different means: limited engagement, movement and hard counters.
Meanwhile, the version of Angulo that Alvarez beat was a far less active and energetic one than Lara faced. Lara was only up one point on two of the scorecards and down on the third before he ended that fight with a straight left hand. Angulo pressured Lara relentlessly and did a very good job of cutting off the ring. However, Canelo isn't a pressure fighter. Although he features a high punch volume, he's actually best at mid-range, where his combinations can flow. It may be instructive for Canelo that Angulo had success in dropping Lara while he was trying to escape with his hands down, but I expect that Saturday's fight will play out much differently than Lara-Angulo did.
With no shortage of bad blood between the fighters, the lead up to Saturday's match has been filled with interesting subplots. But which narrative will take shape in the ring? Will Alvarez's flashy combinations carry the day or will Lara's excellent ring generalship and hard left hands be enough to secure the win? Read below for the keys to the fight. My prediction will be at the end of the article.
Even when Lara is at his best, he rarely lets his hands go. Studying his opponent and waiting for the opportunity to land his counter left hand, Lara can be frustrating to watch. When he throws that shot, it's a surgical strike, a missile that detonates on impact. Between counter lefts, there often can be a lot of circling, feinting and posing.
Canelo is a far more active fighter. Often throwing between 60-80 punches a round, he features practically every punch in the book and unleashes devastating combinations. When he is at his best, punches flow freely and land with precision. Lara will look to reduce Canelo's punch output throughout the fight. He'll tie up during combinations and take steps back as Alvarez approaches. He wants to get Alvarez down to just one punch at a time.
In my estimation, the activity battle will be the biggest factor in determining who wins rounds and puts his stamp on the fight. If there is a slow pace, Lara's hard lefts will have more of an impact on the judges. However, if Canelo can fire off 50+ shots a round, Lara won't be able to match that activity. At that pace, even if Lara connects with a few hard counters each round, he won't win them.
2. Who will be first?
As we saw during Alvarez's fight with Floyd Mayweather, he struggled mightily against the pound-for-pound king's tactical style. By the middle rounds, Alvarez was reticent to throw punches. In a similar vein, Lara will look to frustrate Canelo throughout the fight with a series of tactics aimed at slowing down the action. But Alvarez must stick with his game plan and fire off shots. Even though he might not have his customary accuracy against Lara, he can't get dissuaded. Alvarez's key to winning the fight is letting his hands go. Even if he takes a few hard left counters, he must continue to get off.
For Lara, he can't get trapped into thinking that a few good shots a round will be enough for him to win the fight. I'm sure that his trainer, Ronnie Shields, has impressed upon him the urgency of staying busy for the judges. It will be up to Lara to find time during the dead spaces to initiate offense. Throwing a counter after three of Canelo's shots just won't do it. Although a reduction in action will help Lara, a menu of primarily defense and ring generalship won't be enough for victory.
In a match that could be very competitive, a 10-8 round would be a huge swing. Angulo showed just how vulnerable Lara could be to a well-placed shot. What Angulo did magnificently in that fight was throwing punches in anticipation of Lara's movements. For Alvarez, this means making sure to finish off combinations with left hooks and straight right hands, hoping to catch Lara pulling straight back or moving with his hands down. It may not be the first or second punch that does the most damage, but perhaps the third or fourth. Lara can get careless stepping out of the pocket and moving along the ropes. Elongating punch sequences could do wonders for Canelo in this fight.
Alvarez has shown a very good chin at 154 pounds but it wasn't that long ago (2010 to be exact) when he was stunned badly by light-punching Jose Cotto. In addition, Mayweather, Josesito Lopez and Alfonso Gomez had little trouble landing on him. Although Alvarez's defense has improved, he still leaves a lot of space in between his gloves and I'm not sure if he's ever faced a puncher of Lara's caliber; if Lara lands enough of his left hands, or even just the perfectly placed one, he certainly could send Alvarez to the canvas. In my estimation, both fighters have the potential to go down in this match.
Alvarez was rightly mocked after the Mayweather fight for failing to make adjustments. His trainer, Eddy Reynoso, seemed out of his depth and Alvarez spent most of the fight's second half following Mayweather around without moving his hands or just landing meekly on Mayweather's arms. Alvarez wasn't merely outfought, he was also out-thought. Against a difficult boxer like Lara, will Alvarez and his team have a Plan B that works? Will the fighter be able to execute it? If Alvarez is down to single shots against Lara, what punch should he be throwing? How will he get Lara to open up? These considerations may prove to be very important in the fight.
Like clockwork, there is a moment in practically every Lara bout where Ronnie Shields yells at his fighter to let his hands go. Often these entreaties go ignored and Lara retains his cerebral style, which can often feature a stunning lack of urgency. If Lara falls behind Alvarez early in the fight, what is his recourse? What will he do to change the dynamic? Will he take the types of risks needed to swing the bout in his favor or is he captive to his one style? It will be fascinating to see which corner has the better blueprint for adjustments and which fighter can implement the necessary changes.
Let's face it: judges have not liked Erislandy Lara throughout his career. He was robbed against Paul Williams and should have beaten Vanes Martirosyan on the scorecards instead of getting a draw. The judge who somehow had Martirosyan ahead in that fight, Jerry Roth, will be one of the arbiters for Saturday's match. But these results are not solely to blame on the judges. Lara often has a mentality to do just enough to take a round. It's a calculation that has come back to haunt him. Judges, especially Vegas ones, like activity and aggression, and Lara isn't that guy.
Conversely, Alvarez has fared well with judges. He beat Trout on the scorecards much wider than most had it and he also secured at least three rounds from every judge against Mayweather when two would have been considered overly generous. His punches are very flashy and his combinations can batter around opponents like a piñata.
The other two judges in this contest, Dave Moretti and Levi Martinez, are both fine officials. Roth is the one to watch here. He has slight biases for both "name" fighters and aggressors (irrespective of how effective they are). His horrendous 116-112 card for Brandon Rios over Richar Abril highlighted his dual weaknesses. He could prove to be a very decisive factor in the fight.
I see this fight as being very competitive with a lot of rounds playing out very similarly. Alvarez will be marginally busier while Lara will land a couple of telling blows each round. I think that action will be very contained in the fight, with just a couple of meaningful skirmishes a round. The match will be difficult to score.
However, I'm banking that with the similar characteristics of the rounds that the judges pick a fighter and run with him instead of splitting up the swing rounds. And within those parameters, I believe that the busier fighter will get the benefit of the doubt. I think that Alvarez will prevail by a unanimous decision but the judges will disagree, perhaps significantly, on how many rounds that he actually won.
Saul Alvarez defeats Erislandy Lara by a competitive unanimous decision.


text:
Bitcoin 2022: Thiel calls Buffett 'sociopathic,' Mexican billionaire has 60% in BTC
HomeAll Posts...Bitcoin 2022: Thiel calls Buffett...
April 8, 2022 by superadmin in News 0Comments 0Reactions
The Bitcoin 2022 conference was packed with new announcements, but Peter Thiel stole the show by throwing $100 bills into the crowd and calling Warren Buffet a "sociopathic grandpa."
Day two of the Bitcoin 2022 in Miami was as colorful as we've come to expect, with PayPal co-founder and venture capitalist Peter Thiel offering up his Bitcoin (BTC) "enemies list," while Mexico's third-richest man, Ricardo Salinas, revealed he has 60% of his portfolio in Bitcoin.
Thiel, one of the keynote speakers, took to the main Nakamoto stage of the conference in the early afternoon and kicked the speech off in style by throwing a wad of $100 bills into the crowd in an attempt to explain the difference between crypto and government-backed fiat currency.
As the crowd quickly reached for the money, Thiel joked, "I thought you guys were supposed to be Bitcoin maximalists."
He then gave an energizing pro-Bitcoin speech in which he predicted that Bitcoin would increase 100-fold from today's price and that traditional financial markets would eventually crumble.
"The central banks are going bankrupt. We are at the end of the fiat money regime."
Thiel also offered up his enemies list in which he gave legendary Berkshire Hathaway investor Warren Buffett the top spot, labeling him "Enemy No.1." Thiel called Buffett a "sociopathic grandpa" — obviously taking issues with the Berkshire Hathaway CEO's Bitcoin-critical commentary.
Thiel's list also included JPMorgan Chase CEO Jamie Dimon and Blackrock CEO Larry Fink, denouncing the group of billionaires for running a "gerontocracy" — a society ruled by old people — against cryptocurrency.
Later in the day, Mexican billionaire Ricardo Salinas used his 15 minutes on the main stage to take aim at bonds and announce that he currently holds 60% of his entire liquid investment portfolio in Bitcoin — a considerable increase from the 10% that he held in December 2020.
"I definitely don't have any bonds […] I have 60% in Bitcoin and Bitcoin equities, and then 40% in hard asset stocks like oil and gas and gold miners, and that's where I am."
Salinas, whose net worth is estimated to be roughly $12.8 billion, was an early investor in Bitcoin. In an interview with Cointelegraph, Salinas said that he purchased his first Bitcoin in 2013 when the average price was roughly $200 each.
Earlier in the day, Samson Mow, the former chief strategy officer of Blockstream, announced two new jurisdictions would be moving to adopt Bitcoin as legal tender. The Caribbean island of Roatán and Madeira, an autonomous region of Portugal, confirmed that they will be adopting BTC as legal tender. Mexico is also reportedly considering the move.
Related: Mayor unveils 'Miami Bull' statue with laser eyes to kick off Bitcoin 2022
El Salvadoran president Nayib Bukele was scheduled to speak second-last on Thursday. However, the Bitcoin Conference officially announced his withdrawal on Twitter on Thursday, quoting "unforeseen circumstances." There is currently a state of emergency in the country due to spiraling gang violence.
Day three of the Bitcoin 2022 conference will feature more talks from high-profile guests including a fireside talk with famous psychologist Jordan Peterson, a keynote speech from legendary investor Mike Novogratz and a presentation by Thiel Capital director Eric Weinstein.
Ethereum risks 'double-bust' drop despite ETH price rebounding 30% in two weeks
EU Makes Deal on MiCA Legislation to Regulate Crypto Markets

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text:
\section{Introduction}

Carbon-based nanoscale structures holding hexagonal repeating units and different type of
boundaries attract considerable interest by unprecedented application possibilities
in the design of organic nanodevices \cite{Be4,Be6}.
From these, nanoribbons are of
particular interest \cite{Be8}, being studied especially because of the emergence
possibilities of itinerant ferromagnetism \cite{Be15,Be17,Be18},
this subject being also driven
by the aim to produce magnetic behavior in organic materials not containing magnetic
elements \cite{Be13}.

In these systems the shape of the edges is either of zig-zag or armchair type
\cite{Be9}. The former case is
usually associated to localized edge states \cite{Be14} which, due to
their high degeneracy, can lead to flat-band ferromagnetism \cite{Be15}
(in most cases a spin-polarized many particle, but localized ground state).
The localized edge states for zig-zag edges on thick samples have been
observed experimentally by scanning tunneling microscopy \cite{Be16}, but it is
also known
that in the thin
ribbon case (the chain limit) the localized ferromagnetic nature is no more present
in exact terms, and the spin polarized state becomes conducting in the low
concentration limit for mesoscopic samples \cite{TKG}. Nevertheless, the emergence
possibility of the edge localized states for zig-zag boundaries directed the attention
to the armchair edges in the search for itinerant ferromagnetism in these structures.

For armchair carbon-based nanoribbons the bulk states and the localized end states are
considered entangled by the short-range Coulomb repulsion \cite{Be15}.
In these conditions, at the level of theoretical predictions,
itinerant carriers in dispersive bands are supposed to mediate via exchange
coupling the interaction among the local magnetic moments present in flat bands
\cite{Be17,Be18}. Different approximate techniques are used for the description as:
effective field theories combined with variational wave function approach \cite{Be17},
or analytical weak-coupling analysis combined with numerical density matrix
renormalization-group method and first-principles calculations \cite{Be18}.
As a general observation one notes that the deduced results
attract the attention to the
importance of the Coulomb interaction and correlation effects in providing
the physical properties of the systems under study \cite{Be15,Be18,Be18a}. This
information is
important also in a broader context, since for example, potential nanoribbon
applications in building up organic transistors or spin qubits \cite{Be19,Be20}
do not take into account correlation effects at all.

On the theoretical side the
studies directed to the explanation of ferromagnetism are in a stage
present at other organic systems as well, for example pentagon chains \cite{Be13},
where the initial attempts based on flat band ferromagnetism
\cite{MT} encounter the difficulties of this model. One of these is
the fact that a perfectly flat band is difficult to obtain. The second difficulty is
related to the observation that the connectivity condition needed for the flat
band ferromagnetism, is often not satisfied in experimentally manageable
situations \cite{Be21}, and this case is encountered for armchair nanoribbons
as well \cite{Be18}. The corrected theory in the light of this new input is usually
a some kind of developed flat band theory which
does not need the in-flat-band connectivity
condition. On this line, for example in the case of the pentagon chains, a model
applied in
describing rare earth compounds has been suggested to work \cite{Be21} using strong
hybridization effects \cite{Be22}, while in the
case of the armchair nanoribbons constructed from hexagon cells,
electrons from dispersive bands are
conjectured to interconnect the moments created inside the flat bands \cite{Be17,Be18}.
In fact, this last possibility is also known in the literature. In $D=1$ one
dimensional
case it originates from Ref. \cite{Be23}, whose extended version to $D > 1$ can be find
in Ref. \cite{Be24}.

All the previously mentioned results relating ferromagnetism in organic
nanoribbons suggest that spin-polarized states in such systems are intimately connected
to bare flat bands. This suggestion focuses the research directed to the understanding
of
ferromagnetism (and this is the case also for graphene structures)
exclusively on flat band emergence possibilities, in
condition when it is known that systems built up from hexagonal cells containing
several sublattices, usually
do not contain flat bands \cite{RMP,TKG},
and flat bands are possible to appear only in special conditions, for example
as consequences of edge states. Given by the importance of the application
possibilities of
such organic systems in various fields of nanotechnology, this state of facts naturally
leads to a main question: are indeed flat bands necessary for the emergence of
ferromagnetism in structures built up from repeating hexagonal units ?
Guided by this question, with
the aim to provide
relevant information and advancement in this field, one shows below in exact terms,
that the answer to this question is negative.

In order to do that, one chooses the simplest possible armchair nanoribbon, an armchair
hexagon chain (see Fig.1) which besides the fact that represents a real existing
structure (i.e. polyphenanthrene), represents as well an armchair nanoribbon in the
extreme thin limit. One uses for it a realistic Hubbard type of model which takes into
account as well the non-zero next nearest neighbor hopping amplitudes \cite{RMP}, and
different on-site potentials on different type of sites. Even if the system
is not integrable, using a special technique one deduces exact ground states for it.
These, in the small concentration limit turn out to be itinerant and ferromagnetic in
conditions in which flat bands are completely missing from the system. One shows that
the spin polarization emerging for mesoscopic samples is a consequence of the common
effect of correlations and confinement. Since increasing the ribbon width there is not
present a physical reason to cancel this behavior, we expect similar effect to occur
for thicker ribbons as well.

The method one uses is related to positive semidefinite operator properties.
Based on these,
one transforms the Hamiltonian of the system ($\hat H$) in positive semidefinite form,
the ground state being provided by the most general wave vector which gives the minimum
possible eigenvalue (i.e. zero) of the positive semidefinite operators building up
$\hat H$.
The technique itself works well even in unexpected circumstances from the traditional
point
of view of exact solutions as: three dimensions \cite{GV}, disordered and interacting
systems in two dimensions \cite{GD}, stripes, checkerboards and droplets in two dimensions
\cite{GM}, delocalization effect of the Hubbard interaction in two dimensions \cite{GIM},
ferroelectric systems \cite{fe1},
superconductors \cite{fe2,fe3}, or non-integrable
chain structures even under the action of external magnetic fields \cite{GKV1,GKV2}.
Different procedures of the method can be find in Refs.\cite{g1,g2,g3}, and extreme
details
regarding its application in the case of the chain structures have been presented in
Refs. \cite{GKV1,GKV2}.

One notes that the first exact results for the armchair hexagon chain in the interacting
case are contained in this paper. In order to deduce these, a special technique able to
handle the
emerging extended operators in the presented case has been developed, used,
and described.

The remaining part of the paper is organized as follows. Section II. presents in
details the
studied system, Sect. III. describes the transformation of the Hamiltonian in positive
semidefinite form, Sect. IV. deduces the ground state wave function and presents the
physical
properties of the obtained ferromagnetic phase, Sect. V. containing the summary and
conclusions closes the presentation, and the Appendices A-C contain mathematical details.

\section{The system}

The studied system is an armchair hexagon chain whose cell constructed at the site
${\bf j}$ is presented in Fig.1. One has 8 in-cell positions at the sites
${\bf j}+{\bf r}_{\nu}$, $\nu=1,2,...,8$, ${\bf a}$ represents the Bravais
vector of the chain, and for mathematical convenience one has ${\bf r}_1=0$.
One further underlines that the $\nu$ index
denotes as well eight different sublattices $S_{\nu}$ present in the system.

The Hamiltonian of the system can be written as
\begin{eqnarray}
\hat H=\hat T_0+\hat T_1+\hat T_2+\hat H_U,
\label{Keq1}
\end{eqnarray}
where the kinetic part $\hat H_0= \hat T_0+\hat T_1+\hat T_2$ contains the contributions
of the one-particle local potentials ($\hat T_0$), the nearest neighbor
($\hat T_1$) and the next nearest neighbor ($\hat T_2$) hopping terms, while $\hat H_U$
represents the Hubbard interaction. One has
\begin{figure}
\centerline{\includegraphics[width=12 cm,height=6 cm]{zzF1.eps}}
\caption{The armchair hexagonal chain with cell constructed at the site ${\bf j}$
(thick line).
The in-cell positions are ${\bf j}+{\bf r}_{\nu}$, where $\nu=1,2,..8$ is
denoting as well eight different sublattices. ${\bf a}$ is the Bravais vector, and one
has ${\bf r}_1=0$. The numbers inside the cell represent the
$\nu$ index of the in-cell sites.}
\label{Kfig1}
\end{figure}
\begin{eqnarray}
\hat T_0 &=& \sum_{\sigma}\sum_{n=0}^{N_c-1} \{\epsilon_0 \:[
\hat n_{{\bf j},\sigma} +
\hat n_{{\bf j}+{\bf r}_4,\sigma} +
\hat n_{{\bf j}+{\bf r}_5,\sigma} +
\hat n_{{\bf j}+{\bf r}_6,\sigma}]
\nonumber\\
&+& \epsilon_1 \:[
\hat n_{{\bf j}+{\bf r}_2,\sigma} +
\hat n_{{\bf j}+{\bf r}_3,\sigma} +
\hat n_{{\bf j}+{\bf r}_7,\sigma} +
\hat n_{{\bf j}+{\bf r}_8,\sigma}]\},
\nonumber\\
\hat T_1 &=& \sum_{\sigma}\sum_{n=0}^{N_c-1}\{t_1 \:[
\hat c_{{\bf j},\sigma}^{\dagger}\hat c_{{\bf j}+{\bf r}_6,\sigma} +
\hat c_{{\bf j}+{\bf r}_5,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_4,\sigma} + H.c.]
\nonumber\\
&+& t_2 \:[
\hat c_{{\bf j}+{\bf r}_6,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_5,\sigma} +
\hat c_{{\bf j}+{\bf a},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_4,\sigma} + H.c.]
\nonumber\\
&+& t_3 \:[
\hat c_{{\bf j}+{\bf r}_2,\sigma}^{\dagger} \hat c_{{\bf j},\sigma} +
\hat c_{{\bf j}+{\bf r}_4,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_3,\sigma} +
\hat c_{{\bf j}+{\bf r}_7,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_5,\sigma} +
\hat c_{{\bf j}+{\bf r }_6 +{\bf a},\sigma}^{\dagger}\hat c_{{\bf j}+{\bf r}_8,\sigma}
+ H.c.]
\nonumber\\
&+& t_4 \:[
\hat c_{{\bf j}+{\bf r}_3,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_2,\sigma} +
\hat c_{{\bf j}+{\bf r}_8,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_7,\sigma} + H.c.]\},
\nonumber\\
\hat T_2 &=& \sum_{\sigma}\sum_{n=0}^{N_c-1}\{t'_1 \:[
\hat c_{{\bf j}+{\bf r}_3,\sigma}^{\dagger} \hat c_{{\bf j},\sigma} +
\hat c_{{\bf j}+{\bf r}_4,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_2,\sigma} +
\hat c_{{\bf j}+{\bf r}_5,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_3,\sigma} +
\hat c_{{\bf j}+{\bf r}_2,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_6,\sigma}
\nonumber\\
&+& \hat c_{{\bf j}+{\bf r}_4,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_7,\sigma}
 +  \hat c_{{\bf j}+{\bf r}_5,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_8,\sigma}
 +  \hat c_{{\bf j}+{\bf r}_7,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_6+{\bf a},
\sigma}
\nonumber\\
&+& \hat c_{{\bf j}+{\bf r}_8,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf a},\sigma}
 +  \hat c_{{\bf j}+{\bf a},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_3,\sigma}
 +  \hat c_{{\bf j}+{\bf r}_2+{\bf a},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_4,
\sigma}
\nonumber\\
&+& \hat c_{{\bf j}+{\bf r}_7,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_6,\sigma}
+  \hat c_{{\bf j}+{\bf r}_5+{\bf a},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_8,
\sigma} + H.c.]
\nonumber\\
&+& t'_2 \:[
\hat c_{{\bf j},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_5,\sigma} +
\hat c_{{\bf j}+{\bf r}_6,\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_4,\sigma} +
\hat c_{{\bf j}+{\bf a},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_5,\sigma} +
\hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma}^{\dagger} \hat c_{{\bf j}+{\bf r}_4,
\sigma} + H.c.]\},
\nonumber\\
\hat H_U &=& \sum_{n=0}^{N_c-1}\{U_0 \:[
\hat n_{{\bf j},\sigma}\hat n_{{\bf j},-\sigma} +
\hat n_{{\bf j}+{\bf r}_4,\sigma} \hat n_{{\bf j}+{\bf r}_4,-\sigma} +
\hat n_{{\bf j} +{\bf r}_5,\sigma} \hat n_{{\bf j}+{\bf r}_5,-\sigma} +
\hat n_{{\bf j}+{\bf r}_6,\sigma} \hat n_{{\bf j}+{\bf r}_6,-\sigma}]
\nonumber\\
&+& U_1 \:[
\hat n_{{\bf j}+{\bf r}_2,\sigma} \hat n_{{\bf j}+{\bf r}_2,-\sigma} +
\hat n_{{\bf j}+{\bf r}_3,\sigma} \hat n_{{\bf j}+{\bf r}_3,-\sigma} +
\hat n_{{\bf j}+{\bf r}_7,\sigma} \hat n_{{\bf j}+{\bf r}_7,-\sigma} +
\hat n_{{\bf j}+{\bf r}_8,\sigma} \hat n_{{\bf j}+{\bf r}_8,-\sigma}]\}.
\label{Keq2}
\end{eqnarray}
In (\ref{Keq2}) $\hat c^{\dagger}_{{\bf i},\sigma}$ creates an electron with spin
$\sigma$ at the site ${\bf i}$, $\hat n_{{\bf i},\sigma}$ is the particle
number operator, $N_c$ represents the number of cells, and during the summation over $n$,
${\bf j}={\bf i}+n {\bf a}$ is considered, ${\bf i}$ being an arbitrary site of the
sublattice $S_{\nu=1}$. The parameters of $\hat H$ describing a realistic system
are presented in Fig.2. The nearest neighbor hoppings are $t_1$ (touching bonds between
hexagons), $t_2$ (horizontal hopping on the armchair), $t_3$ (the oblique
hopping on the
armchair), and $t_4$ (the hopping on the external armchair bond). The next nearest
neighbor
hoppings are $t'_1$ (bonds with only one end on touching points between hexagons),
and $t'_2$ (bonds with both ends on touching points between hexagons).
Furthermore, one
notes that the one-particle on-site potentials ($\epsilon_{\alpha}$) and the on site
Coulomb repulsions $U_{\alpha}$ are denoted by the index $\alpha=0$ on the touching
points between hexagons, and by the index $\alpha=1$ on other
(external) sites. For mathematical
simplicity one considers $U_0=U_1=U > 0$, and periodic boundary conditions are taken
into account.

\begin{figure}
\centerline{\includegraphics[width=12 cm,height=6 cm]{zzF2.eps}}
\caption{The Hamiltonian parameters (see text following (\ref{Keq2})).}
\label{Kfig2}
\end{figure}

\section{The transformation of the Hamiltonian in a positive semidefinite form}

\subsection{The defined block operators}

In order to transform the Hamiltonian (\ref{Keq1},\ref{Keq2}) in a positive
semidefinite form one defines eight block operators $\hat A_{p,{\bf j},\sigma}$,
$p=1,2,...,8$ for each site ${\bf j}$ of the sublattice $\nu=1$, as follows

\begin{eqnarray}
\hat A_{1,{\bf j},\sigma} &=&
a_1 \hat c_{{\bf j},\sigma} +
a_2 \hat c_{{\bf j}+{\bf r}_2,\sigma} +
a_3 \hat c_{{\bf j}+{\bf r}_3,\sigma} +
a_4 \hat c_{{\bf j}+{\bf r}_4,\sigma}
\nonumber\\
\hat A_{2,{\bf j},\sigma} &=&
b_1 \hat c_{{\bf j},\sigma} +
b_4 \hat c_{{\bf j}+{\bf r}_4,\sigma} +
b_5 \hat c_{{\bf j}+{\bf r}_5,\sigma} +
b_6 \hat c_{{\bf j}+{\bf r}_6,\sigma}
\nonumber\\
\hat A_{3,{\bf j},\sigma} &=&
d_0 \hat c_{{\bf j}+{\bf a},\sigma} +
d_4 \hat c_{{\bf j}+{\bf r}_4,\sigma} +
d_5 \hat c_{{\bf j}+{\bf r}_5,\sigma} +
d_6 \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma}
\nonumber\\
\hat A_{4,{\bf j},\sigma} &=&
e_5 \hat c_{{\bf j}+{\bf r}_5,\sigma} +
e_6 \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma} +
e_7 \hat c_{{\bf j}+{\bf r}_7,\sigma} +
e_8 \hat c_{{\bf j}+{\bf r}_8,\sigma}
\nonumber\\
\hat A_{5,{\bf j},\sigma} &=&
f_0 \hat c_{{\bf j}+{\bf a},\sigma} +
f_3 \hat c_{{\bf j}+{\bf r}_3,\sigma} +
f_4 \hat c_{{\bf j}+{\bf r}_4,\sigma} +
f_5 \hat c_{{\bf j}+{\bf r}_5,\sigma}
\nonumber\\
\hat A_{6,{\bf j},\sigma} &=&
g_0 \hat c_{{\bf j}+{\bf a},\sigma} +
g_2 \hat c_{{\bf j}+{\bf r}_2+{\bf a},\sigma} +
g_4 \hat c_{{\bf j}+{\bf r}_4,\sigma} +
g_6 \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma}
\nonumber\\
\hat A_{7,{\bf j},\sigma} &=&
h_0 \hat c_{{\bf j}+{\bf a},\sigma} +
h_5 \hat c_{{\bf j}+{\bf r}_5+{\bf a},\sigma} +
h_6 \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma} +
h_8 \hat c_{{\bf j}+{\bf r}_8,\sigma}
\nonumber\\
\hat A_{8,{\bf j},\sigma} &=&
k_4 \hat c_{{\bf j}+{\bf r}_4,\sigma} +
k_5 \hat c_{{\bf j}+{\bf r}_5,\sigma} +
k_6 \hat c_{{\bf j}+{\bf r}_6,\sigma} +
k_7 \hat c_{{\bf j}+{\bf r}_7,\sigma} .
\label{Keq3}
\end{eqnarray}

As  seen from (\ref{Keq3}) the used block operators $\hat A_{p,{\bf j},\sigma}$
are linear combinations of the starting fermionic operators $\hat c_{{\bf i},\sigma}$
acting on the sites of the block $p$ defined at the site ${\bf j}$ (see
Figs.\ref{Kfig3}-\ref{Kfig10}). The numerical prefactors
$a_i, b_i,d_i,...,k_i$ are unknown at the moment, and will be deduced further on.

\begin{figure}
\centerline{\includegraphics[width=7 cm,height=6 cm]{zzF3.eps}}
\caption{The block $p=1$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{1, {\bf j},\sigma}$. The thick line shows
the block, while the numbers are indicating the $\nu$ index of the sites
whose position is ${\bf j}+{\bf r}_{\nu}$.}
\label{Kfig3}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=7 cm,height=6 cm]{zzF4.eps}}
\caption{The block $p=2$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{2, {\bf j},\sigma}$. The thick line shows
the block, while the numbers are indicating the $\nu$ index of the sites
whose position is ${\bf j}+{\bf r}_{\nu}$.}
\label{Kfig4}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=8 cm,height=6 cm]{zzF5.eps}}
\caption{The block $p=3$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{3, {\bf j},\sigma}$. The thick line shows
the block, while the numbers are indicating the $\nu$ index of the sites
connected exclusively to a single cell.}
\label{Kfig5}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=8 cm,height=6 cm]{zzF6.eps}}
\caption{The block $p=4$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{4, {\bf j},\sigma}$. The thick line shows
the block, while the numbers are indicating the $\nu$ index of the sites
connected exclusively to a single cell.}
\label{Kfig6}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=8 cm,height=6 cm]{zzF7.eps}}
\caption{The block $p=5$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{5, {\bf j},\sigma}$. The thick line shows
the block, while the numbers are indicating the $\nu$ index of the sites
whose position is ${\bf j}+{\bf r}_{\nu}$.}
\label{Kfig7}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=10 cm,height=6 cm]{zzF8.eps}}
\caption{The block $p=6$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{6, {\bf j},\sigma}$. The thick line shows
the block, the number four indicates the $\nu=4$ site. Other block sites (placed in the
following cell) are indicated by their position
${\bf j}+{\bf r}_{\nu} +{\bf a}$.}
\label{Kfig8}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=10 cm,height=6 cm]{zzF9.eps}}
\caption{The block $p=7$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{7, {\bf j},\sigma}$. The thick line shows
the block, the number eight indicates the $\nu=8$ site. Other block sites (placed in the
following cell) are indicated by their position
${\bf j}+{\bf r}_{\nu} +{\bf a}$.}
\label{Kfig9}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=7 cm,height=6 cm]{zzF10.eps}}
\caption{The block $p=8$ depicted in the cell defined at the site ${\bf j}$
providing the block operator $\hat A_{8, {\bf j},\sigma}$. The thick line shows
the block, while the numbers are indicating the $\nu$ index of the sites
whose position is ${\bf j}+{\bf r}_{\nu}$.}
\label{Kfig10}
\end{figure}

\subsection{The positive semidefinite form of the Hamiltonian}

Based on the block operators defined in (\ref{Keq3}), the starting Hamiltonian presented
in (\ref{Keq1},\ref{Keq2}) is transformed (i.e. is rewritten in exact terms) in the
following positive
semidefinite form
\begin{eqnarray}
\hat H = \sum_{\sigma} \sum_{p=1}^8 \sum_{{\bf j}=1}^{N_c}
\hat A_{p,{\bf j},\sigma}^{\dagger} \hat A_{p,{\bf j},\sigma} + \hat H_U + K\hat N,
\label{Keq4}
\end{eqnarray}
where $\hat N$ represents the operator of the total number of electrons and $K$ is a
constant. The numerical prefactors in (\ref{Keq3}) and the constant $K$ are expressed
by i) effectuating the products $\hat A_{p,{\bf j},\sigma}^{\dagger}
\hat A_{p,{\bf j},\sigma}$ and the prescribed sums in the first term of (\ref{Keq4}),
and ii) equating the obtained result term by term to the expression of the starting
Hamiltonian (\ref{Keq1},\ref{Keq2}). The obtained equations connecting the parameters
of the starting Hamiltonian to the parameters of the block operators and the constant
$K$ in the transcribed Hamiltonian (\ref{Keq4}) are the following ones: The hopping
terms become expressed as
\begin{eqnarray}
t_{{\bf j}+{\bf r}_5,{\bf j}+{\bf r}_4} &=& t_1
= b_5^*b_4 + d_5^*d_4 + f_5^*f_4 + k_5^*k_4, \quad
t_{{\bf j}+{\bf r}_1,{\bf j}+{\bf r}_6} =
t_1 = b_1^*b_6 + g_0^*g_6 + h_0^*h_6 + d_0^*d_6,
\nonumber\\
t_{{\bf j}+{\bf r}_6,{\bf j}+{\bf r}_5} &=& t_2 = b_6^*b_5 + h_6^*h_5 + k_6^*k_5, \quad
t_{{\bf j}+{\bf r}_1+{\bf a},{\bf j}+{\bf r}_4} = t_2 = d_0^*d_4 + f_0^*f_4 + g_0^*g_4,
\nonumber\\
t_{{\bf j}+{\bf r}_2,{\bf j}+{\bf r}_1} &=& t_3 = a_2^*a_1 + g_2^*g_0, \quad
t_{{\bf j}+{\bf r}_4,{\bf j}+{\bf r}_3} = t_3 = a_4^*a_3 + f_4^*f_3,
\nonumber\\
t_{{\bf j}+{\bf r}_6+{\bf a},{\bf j}+{\bf r}_8} &=& t_3 = e_6^*e_8 + h_6^*h_8, \quad
t_{{\bf j}+{\bf r}_7,{\bf j}+{\bf r}_5} = t_3 = e_7^*e_5 + k_7^*k_5,
\nonumber\\
t_{{\bf j}+{\bf r}_3,{\bf j}+{\bf r}_2} &=& t_4 = a_3^*a_2, \quad
t_{{\bf j}+{\bf r}_8,{\bf j}+{\bf r}_7} = t_4 = e_8^*e_7,
\quad
t_{{\bf j}+{\bf r}_3,{\bf j}+{\bf r}_1} = t'_1 = a_3^*a_1,
\nonumber\\
t_{{\bf j}+{\bf r}_5,{\bf j}+{\bf r}_3} &=& t'_1 = f_5^*f_3, \quad
t_{{\bf j}+{\bf r}_4,{\bf j}+{\bf r}_2} = t'_1 = a_4^*a_2, \quad
t_{{\bf j}+{\bf r}_2,{\bf j}+{\bf r}_6} = t'_1 = g_2^*g_6,
\nonumber\\
t_{{\bf j}+{\bf r}_8,{\bf j}+{\bf r}_1+{\bf a}} &=& t'_1 = h_8^*h_0, \quad
t_{{\bf j}+{\bf r}_5,{\bf j}+{\bf r}_8} = t'_1 = e_5^*e_8, \quad
t_{{\bf j}+{\bf r}_4,{\bf j}+{\bf r}_7} = t'_1 = k_4^*k_7,
\nonumber\\
t_{{\bf j}+{\bf r}_7,{\bf j}+{\bf r}_6} &=& t'_1 = k_7^*k_6, \quad
t_{{\bf j}+{\bf r}_7,{\bf j}+{\bf r}_6+{\bf a}} = t'_1 = e_7^*e_6, \quad
t_{{\bf j}+{\bf r}_1+{\bf a},{\bf j}+{\bf r}_3} = t'_1 = f_0^*f_3,
\nonumber\\
t_{{\bf j}+{\bf r}_2+{\bf a},{\bf j}+{\bf r}_4} &=& t'_1 = g_2^*g_4, \quad
t_{{\bf j}+{\bf r}_5+{\bf a},{\bf j}+{\bf r}_8} = t'_1 = h_5^*h_8, \quad
t_{{\bf j}+{\bf r}_6+{\bf a},{\bf j}+{\bf r}_4} = t'_2 = d_6^*d_4 + g_6^*g_4,
\nonumber\\
t_{{\bf j}+{\bf r}_1,{\bf j}+{\bf r}_5} &=& t'_2 = b_1^*b_5 + h_0^*h_5, \quad
t_{{\bf j}+{\bf r}_6,{\bf j}+{\bf r}_4} = t'_2 = b_6^*b_4 + k_6^*k_4,
\nonumber\\
t_{{\bf j}+{\bf r}_1+{\bf a},{\bf j}+{\bf r}_5} &=& t'_2 = d_0^*d_5 + f_0^*f_5,
\nonumber\\
t_{{\bf j}+{\bf r}_6+{\bf a},{\bf j}+{\bf r}_5} &=& 0 = d_6^*d_5 + e_6^*e_5, \quad
t_{{\bf j}+{\bf r}_4,{\bf j}+{\bf r}_1} = 0 = a_4^*a_1 + b_4^*b_1,
\label{Keq5}
\end{eqnarray}
while the on-site potentials are given by
\begin{eqnarray}
\epsilon_{{\bf j}+{\bf r}_1} &=& \epsilon_0 = |a_1|^2 + |b_1|^2 + |d_0|^2 +
|f_0|^2 + |g_0|^2 + |h_0|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_2} &=& \epsilon_1 = |a_2|^2 + |g_2|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_3} &=& \epsilon_1 = |a_3|^2 + |f_3|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_4} &=& \epsilon_0 = |a_4|^2 + |b_4|^2 + |d_4|^2 +
|f_4|^2 + |g_4|^2 + |k_4|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_5} &=& \epsilon_0 = |b_5|^2 + |d_5|^2 + |e_5|^2 +
|f_5|^2 + |h_5|^2 + |k_5|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_6} &=& \epsilon_0 = |b_6|^2 + |d_6|^2 + |e_6|^2 +
|g_6|^2 + |h_6|^2 + |k_6|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_7} &=& \epsilon_1 = |e_7|^2 + |k_7|^2 + K,
\nonumber\\
\epsilon_{{\bf j}+{\bf r}_8} &=& \epsilon_1 = |e_8|^2 + |h_8|^2 + K .
\label{Keq6}
\end{eqnarray}
For clarity, in (\ref{Keq5}) one specifies also the hopping matrix elements
$t_{{\bf i}_1,{\bf i}_2}$, as the component of $t_{{\bf i}_1,{\bf i}_2}
\hat c^{\dagger}_{{\bf i}_1,\sigma} \hat c_{{\bf i}_2,\sigma}$ kinetic energy term
which leads to the respective equation. Again for clarity, at the level of the
on-site potentials in (\ref{Keq6}), in the index of $\epsilon_{\bf i}$, the site ${\bf i}$
where the analyzed potential is considered is precisely indicated.

One notes that
Eqs.(\ref{Keq5},\ref{Keq6}) are the matching conditions of the solution since
the transformation from (\ref{Keq1},\ref{Keq2}) to (\ref{Keq4}) is valid only if
Eqs.(\ref{Keq5},\ref{Keq6}) are satisfied.

\subsection{The explicit expression of the operators $\hat A_{p,{\bf j},\sigma}$}

The explicit expression of the transformed Hamiltonian in (\ref{Keq4}), hence also the
explicit form of the $\hat A_{p,{\bf j},\sigma}$ operators, can be find by solving the
matching conditions (\ref{Keq5},\ref{Keq6}). These are representing 36 coupled nonlinear
complex algebraic equations containing 33 unknown variables, namely the numerical
prefactors $a_i,b_i,d_i,...,k_i$ in (\ref{Keq3}), (in total 32 variables), and the
constant $K$
in (\ref{Keq4}).
Based on the solution deduced in the Appendix A, see (\ref{EquA13}), the introduced
block operators have the expression
\begin{eqnarray}
\hat A_{1,{\bf j},\sigma} &=& \sqrt{t_4} \: [ \:
(\hat c_{{\bf j}+{\bf r}_2,\sigma} + \hat c_{{\bf j}+{\bf r}_3,\sigma}) +
\frac{t'_1}{t_4} (\hat c_{{\bf j},\sigma} + \hat c_{{\bf j}+{\bf r}_4,\sigma}) \: ],
\nonumber\\
\hat A_{2,{\bf j},\sigma} &=& \sqrt{t_4} \: [ \:
(\frac{t'_2}{t'_1} - \frac{t_1+2t'_2}{2t_3}) ( \hat c_{{\bf j}+{\bf r}_5,\sigma} -
\hat c_{{\bf j}+{\bf r}_6,\sigma}) + \frac{t'_1}{t_4} (\hat c_{{\bf j},\sigma} -
\hat c_{{\bf j}+{\bf r}_4,\sigma}) \: ],
\nonumber\\
\hat A_{3,{\bf j},\sigma} &=& \sqrt{t_4} \: [ \:
(\frac{t'_2}{t'_1} - \frac{t_1+2t'_2}{2t_3}) ( \hat c_{{\bf j}+{\bf a},\sigma} -
\hat c_{{\bf j}+{\bf r}_4,\sigma}) + \frac{t'_1}{t_4}
(\hat c_{{\bf j}+{\bf r}_5,\sigma} - \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma}) \: ],
\nonumber\\
\hat A_{4,{\bf j},\sigma} &=& \sqrt{t_4} \: [ \:
(\hat c_{{\bf j}+{\bf r}_7,\sigma} + \hat c_{{\bf j}+{\bf r}_8,\sigma}) +
\frac{t'_1}{t_4} (\hat c_{{\bf j}+{\bf r}_5,\sigma} +
\hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma}) \: ],
\label{Keq7}\\
\hat A_{5,{\bf j},\sigma} &=& t'_1 \sqrt{\frac{t_1+2t'_2}{2t'_1t_3}} \: [ \:
(\hat c_{{\bf j}+{\bf a},\sigma} + \hat c_{{\bf j}+{\bf r}_5,\sigma}) +
\frac{2t_3}{t_1+2t'_2} \hat c_{{\bf j}+{\bf r}_3,\sigma} + \frac{(t_3-t'_1)}{t'_1}
\hat c_{{\bf j}+{\bf r}_4,\sigma} \: ],
\nonumber\\
\hat A_{6,{\bf j},\sigma} &=&
t'_1 \sqrt{\frac{t_1+2t'_2}{2t'_1t_3}} \: [ \:
(\hat c_{{\bf j}+{\bf r}_4,\sigma} + \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma})+
\frac{2t_3}{t_1+2t'_2} \hat c_{{\bf j}+{\bf r}_2+{\bf a},\sigma} +
\frac{(t_3-t'_1)}{t'_1} \hat c_{{\bf j}+{\bf a},\sigma} \: ],
\nonumber\\
\hat A_{7,{\bf j},\sigma} &=&
t'_1 \sqrt{\frac{t_1+2t'_2}{2t'_1t_3}} \: [ \:
(\hat c_{{\bf j}+{\bf a},\sigma} + \hat c_{{\bf j}+{\bf r}_5+{\bf a},\sigma}) +
\frac{2t_3}{t_1+2t'_2} \hat c_{{\bf j}+{\bf r}_8,\sigma} +
\frac{(t_3-t'_1)}{t'_1} \hat c_{{\bf j}+{\bf r}_6+{\bf a},\sigma} \: ],
\nonumber\\
\hat A_{8,{\bf j},\sigma} &=&
t'_1 \sqrt{\frac{t_1+2t'_2}{2t'_1t_3}} \: [ \:
(\hat c_{{\bf j}+{\bf r}_4,\sigma} + \hat c_{{\bf j}+{\bf r}_6,\sigma}) +
\frac{2t_3}{t_1+2t'_2} \hat c_{{\bf j}+{\bf r}_7,\sigma} + \frac{(t_3-t'_1)}{
t'_1} \hat c_{{\bf j}+{\bf r}_5,\sigma} \: ] .
\nonumber
\end{eqnarray}
The parameter space domain where this solution of the matching conditions is valid
is not severely restricted, being given by
$t_4 > 0$,  $t_3 t'_1/(t_1+2t'_2) > 0$, and  (\ref{EquA11}), see Appendix A.
The constant $K$ present in the transformed Hamiltonian from (\ref{Keq4}) is given
in this case by
\begin{eqnarray}
 K=\epsilon_1-t_4 - \frac{2t_3 t'_1}{(t_1+2t'_2)}.
\label{Keq8}
\end{eqnarray}

One notes that the $\hat A_{p,{\bf j},\sigma}$ operators present in (\ref{Keq7})
are Fermi operators,
since $\hat A^{\dagger}_{p,{\bf j},\sigma} \hat A^{\dagger}_{p,{\bf j},\sigma} =
\hat A_{p,{\bf j},\sigma} \hat A_{p,{\bf j},\sigma} =0$ is satisfied. But these operators
are not canonical Fermi operators, since $\{ \hat A_{p,{\bf j},\sigma},
 \hat A^{\dagger}_{p',{\bf j}',\sigma'} \} \ne \delta_{p,p'} \delta_{{\bf j},{\bf j}'}
\delta_{\sigma,\sigma'}$ holds.

\section{The deduction of the ground state wave function}

\subsection{The deduction method}

Based on (\ref{Keq7},\ref{Keq8}) now one knows the explicit expression of the
transformed Hamiltonian (\ref{Keq4}), hence one starts the deduction of the
ground state wave function for it, which is written in the form
\begin{eqnarray}
|\Psi_g\rangle = [\prod_{\alpha \in I} \hat B_{\alpha}^{\dagger} ] |0\rangle.
\label{Keq9}
\end{eqnarray}
In (\ref{Keq9}), $|0\rangle$ represents the bare vacuum, $\hat B^{\dagger}_{\alpha}$
are linear
combinations of the starting canonical Fermi creation operators $\hat c^{\dagger}_{
{\bf i},\sigma}$, $\alpha$ represents an index which labels different
$\hat B^{\dagger}$ operators, and $I$ represents the domain from which the $\alpha$
indices must be choosen in order to build up the ground state wave function.

The strategy in deducing the $\hat B^{\dagger}_{\alpha}$ operators is based on the
positive semidefinite structure of $\hat H$ from (\ref{Keq4}), namely on the fact
that $\hat H'=\hat H - K \hat N = \hat H_A + \hat H_U$ is a positive semidefinite
operator, where $\hat H_A = \sum_{\sigma} \sum_{p=1}^8 \sum_{{\bf j}=1}^{N_c}
\hat H_{A,p,{\bf j},\sigma}$ and $\hat H_{A,p,{\bf j},\sigma}=
\hat A_{p,{\bf j},\sigma}^{\dagger} \hat A_{p,{\bf j},\sigma}$. Consequently,
$|\Psi_g \rangle$ is the most general wave vector satisfying the requirement
\begin{eqnarray}
(\hat H_A + \hat H_U) |\Psi_g \rangle = 0,
\label{Keq10}
\end{eqnarray}
which, because both $\hat H_A$ and $\hat H_U$ are independently positive semidefinite,
is satisfied only if both $\hat H_A |\Psi_g\rangle =0$, and $\hat H_U |\Psi_g\rangle = 0$
hold. But since the kernel $ker(\hat O)$, of an operator $\hat O$, is a Hilbert subspace
spanned by all vectors $|\phi\rangle$ satisfying $\hat O |\phi\rangle = 0$, it results
that
in order to deduce $|\Psi_g\rangle$ we must deduce the most general wave vectors placed
inside
\begin{eqnarray}
ker(\hat H')= ker(\hat H_A) \cap ker(\hat H_U).
\label{Keq10a}
\end{eqnarray}

Concentrating first on $ker(\hat H_A)$, one observes that since
$\hat H_{A,p,{\bf j},\sigma}$
are all positive semidefinite operators for all values of all indices, it results that
\begin{eqnarray}
&&ker(\hat H_A)=ker(\hat H_{A,p=1,{\bf j}_1,\uparrow}) \cap ker(\hat H_{A,p=1,{\bf j}_1,
\downarrow}) \cap ker(\hat H_{A,p=2,{\bf j}_1,\uparrow}) \cap ker(\hat H_{A,p=2,{\bf j}_1,
\downarrow}) \cap ....
\nonumber\\
&&\hspace*{2cm}...\cap ker(\hat H_{A,p=8,{\bf j}_{N_c},\uparrow}) \cap
ker(\hat H_{A,p=8,{\bf j}_{N_c},\downarrow}),
\label{Keq11}
\end{eqnarray}
where, in the right side of the equality, all indices $p,{\bf j},\sigma$ occur.

Furthermore, one knows that for fixed
$p=p',{\bf j}={\bf j}',\sigma=\sigma'$ indices, $ker(\hat H_{A,p',{\bf j}',\sigma'})$ is
spanned by vectors of the form $|\Psi_{A,p',{\bf j}',\sigma'}\rangle = \prod_{\alpha \in
I_{A,p',{\bf j}',\sigma'}} \hat B^{\dagger}_{\alpha} |0\rangle$ where
$I_{A,p',{\bf j}',\sigma'}$ collects all $\alpha$ indices for which the anti-commutation
relation $\{\hat A_{p',{\bf j}',\sigma'},\hat B_{\alpha}^{\dagger}\} = 0$ holds
\cite{GKV2}.
Indeed in this case
$\hat H_{A,p',{\bf j}',\sigma'} |\Psi_{A,p',{\bf j}',\sigma'}\rangle =
\hat A^{\dagger}_{p',{\bf j}',\sigma'} \hat A_{p',{\bf j}',\sigma'}
\prod_{\alpha \in I_{A,p',{\bf j}',\sigma'}} \hat B_{\alpha}^{\dagger} |0\rangle=0$
is satisfied,
since based on the prescribed anti-commutation relation $\hat A_{p',{\bf j}',\sigma'}
\hat B_{\alpha}^{\dagger} |0\rangle= - \hat B_{\alpha}^{\dagger}
\hat A_{p',{\bf j}',\sigma'} |0\rangle=0$, which given by
$\hat A_{p',{\bf j}',\sigma'} |0\rangle=0$, is indeed a true relation.

It results that based on (\ref{Keq11}), the vectors contained in  $ker(\hat H_A)$ can be
deduced by determining $\hat B^{\dagger}_{\alpha}$ from the relation

\begin{eqnarray}
\{\hat A_{p,{\bf j},\sigma},\hat B_{\alpha}^{\dagger}\} = 0,
\label{Keq12}
\end{eqnarray}
where (\ref{Keq12}) must be satisfied for all values of all indices. One notes that
different $\hat B^{\dagger}_{\alpha}$ solutions of (\ref{Keq12}) must be linearly
independent.

After obtaining the $\hat B^{\dagger}_{\alpha}$ operators from (\ref{Keq12}),
the domain $I$
present in (\ref{Keq9}) is such fixed to push $|\Psi_g\rangle$ also in
$ker(\hat H_U)$. This
last kernel is simply the Hilbert subspace containing all wave vectors with zero double
occupancy.

\subsection{The calculation of $\hat B^{\dagger}_{\alpha}$ operators}

\subsubsection{The $\hat B^{\dagger}_{\alpha}$ expression}

In deducing the $\hat B^{\dagger}_{\alpha}$ operators one starts from (\ref{Keq12}) and
the already deduced expressions of the $\hat A_{p,{\bf j},\sigma}$ block operators
presented in (\ref{Keq7}). One considers
\begin{eqnarray}
\hat B^{\dagger}_{\alpha} = \sum_{{\bf i} \in {\cal{D}}_{B_{\alpha}}} x_{\alpha,{\bf i}}
\hat c^{\dagger}_{{\bf i},\sigma_{\bf i}},
\label{Keq13}
\end{eqnarray}
where two kind of unknown variables must be determined, namely i) the domain ${\cal{D}}_{
B_{\alpha}}$ on which the operator $\hat B^{\dagger}_{\alpha}$ is defined, and ii) the
numerical prefactors $x_{\alpha,{\bf i}}$ which specify the linear combination on which
$\hat B^{\dagger}_{\alpha}$ is constructed. After fixing the $\hat B^{\dagger}_{\alpha}$
expression in (\ref{Keq13}), the equation (\ref{Keq12}) must be written for all
$\hat A_{p,{\bf j},\sigma}$ block operators, obtaining a coupled system of $16 N_c$
equations
which must be solved.

In performing this job one observes the following properties. a) The domain
${\cal{D}}_{B_{\alpha}}$ cannot be restricted (e.g. extended over $N_b < N_c$ cells). The
reason for this is that for restricted ${\cal{D}}_{B_{\alpha}}$, we always find at least
one
$\hat A_{p,{\bf j},\sigma}$ operator (holding some fixed indices $p=p', {\bf j}={\bf j}',
\sigma=\sigma'$) which touches the $\hat B^{\dagger}_{\alpha}$ operator only in a
single point
${\bf j}^{*}$. In this case, since only one kind of starting fermionic operator is acting
on each site and $\{\hat c_{{\bf j}^*,\sigma}, \hat c^{\dagger}_{{\bf j}^*,\sigma} \}=1$
holds, it results that $\{\hat A_{p',{\bf j}',\sigma'}, \hat B^{\dagger}_{\alpha} \} =0$
cannot be satisfied. Consequently, the $\hat B^{\dagger}_{\alpha}$ operators will be
extended operators (i.e. the domain ${\cal{D}}_{B_{\alpha}}$ will be extended along the
whole system). b) For similar reasons it is not possible to have in the expression of
$\hat B^{\dagger}_{\alpha}$ operators islands of $\hat c^{\dagger}_{{\bf i}',\sigma'}$
operators in the background of $\hat c^{\dagger}_{{\bf i},\sigma}$ operators where
$\sigma' \ne \sigma$. Consequently, for a given $\hat B^{\dagger}_{\alpha}$ in
(\ref{Keq13}) all $\sigma_{\bf i}$ indices become fixed, hence $\sigma_{\bf i}=\sigma$
holds.

Given by these properties, the $\alpha$ index in (\ref{Keq13}) specifies in fact a
fixed spin
index $\sigma$, and a label index $\ell$ which numbers the possible $\hat B^{\dagger}$
operators, hence $\alpha = (\ell,\sigma)$, and the expression of the $\hat B^{\dagger}$
operators from (\ref{Keq13}) becomes
\begin{eqnarray}
\hat B^{\dagger}_{\ell,\sigma} = \sum_{\bf i} x_{\ell,{\bf i}}
\hat c^{\dagger}_{{\bf i},\sigma},
\label{Keq14}
\end{eqnarray}
where the ${\bf i}$ sum extends over all sites of the system (in all sublattices).

\subsubsection{The calculation procedure}

For the effective calculation of the $x_{\ell,{\bf i}}$ numerical prefactors of the
extended operators as $\hat B^{\dagger}_{\ell,\sigma}$ present in (\ref{Keq14}),
special techniques
are needed. Such a technique has been developed and presented in Ref.\cite{TKG},
based on a recursive connection between coefficients in neighboring cells described
by a non-symmetric matrix $\tilde R$, the prefactors being deduced via the unity
eigenvalues of ${\tilde R}^m$, $m$ being an arbitrary positive integer. In order to
enlarge and develop the techniques applicable in treating extended operators, one
presents and uses here another alternative method. This is based on deducing the
$\hat B^{\dagger}_{\ell,\sigma}$ operators for a relatively small chain -- named test
chain -- containing $m$ cells, the test chain being closed in a ring by periodic boundary
conditions. Once these operators are deduced, the obtained result is used
in the description of a long chain containing $M = p \times m$
cells, the long chain being as well treated by periodic boundary conditions, $p$ being
an arbitrary positive integer.

In this method first one deduces the linearly independent
$\hat B^{\dagger}_{\ell,\sigma}$ operators for the test chain. This last contains
$m$ cells, which are in order numbered by the index $n_c=1,2,...,m$. The coefficients
$x_{\ell,{\bf i}}$ become in this way $x_{\ell,n_c,\nu}$ where $\nu$ denotes
the lattice site ${\bf j}+{\bf r}_{\nu}$, $\nu=1,2,...,8$ inside the $n_c$th cell
defined at the site ${\bf j}$ of the $S_1$ sublattice (see Fig.1). Consequently, in
this notation, by changing the sum over ${\bf i}$ in a sum over $n_c$ and $\nu$,
(\ref{Keq14}) becomes
\begin{eqnarray}
\hat B^{\dagger}_{\ell,\sigma} = \sum_{n_{c_{\bf j}}=1}^m \sum_{\nu=1}^8
x_{\ell,n_{c_{\bf j}},\nu} \hat c^{\dagger}_{{\bf j}+ {\bf a} n_{c_{\bf j}} +
{\bf r}_{\nu},\sigma},
\label{Keq15}
\end{eqnarray}
where ${\bf j}$ is an arbitrary site of the sublattice $S_1$.
Based on (\ref{Keq15}) and (\ref{Keq3}), the anti-commutation relation (\ref{Keq12})
provides the following system of equations for the prefactors of the
$\hat B^{\dagger}_{\ell,\sigma}$ operators

\begin{eqnarray}
&&a_1 x_{\ell,1,1} + a_2 x_{\ell,1,2} + a_3 x_{\ell,1,3} + a_4 x_{\ell,1,4} = 0,
\nonumber\\
&&b_1 x_{\ell,1,1} + b_4 x_{\ell,1,4} + b_5 x_{\ell,1,5} + b_6 x_{\ell,1,6} = 0,
\nonumber\\
&&d_0 x_{\ell,2,1} + d_4 x_{\ell,1,4} + d_5 x_{\ell,1,5} + d_6 x_{\ell,2,6} = 0,
\nonumber\\
&&e_5 x_{\ell,1,5} + e_6 x_{\ell,2,6} + e_7 x_{\ell,1,7} + e_8 x_{\ell,1,8} = 0,
\nonumber\\
&&f_0 x_{\ell,2,1} + f_3 x_{\ell,1,3} + f_4 x_{\ell,1,4} + f_5 x_{\ell,1,5} = 0,
\nonumber\\
&&g_0 x_{\ell,2,1} + g_2 x_{\ell,2,2} + g_4 x_{\ell,1,4} + g_6 x_{\ell,2,6} = 0,
\nonumber\\
&&h_0 x_{\ell,2,1} + h_5 x_{\ell,2,5} + h_6 x_{\ell,2,6} + h_8 x_{\ell,1,8} = 0,
\nonumber\\
&&k_4 x_{\ell,1,4} + k_5 x_{\ell,1,5} + k_6 x_{\ell,1,6} + k_7 x_{\ell,1,7} = 0,
\nonumber\\
&&a_1 x_{\ell,2,1} + a_2 x_{\ell,2,2} + a_3 x_{\ell,2,3} + a_4 x_{\ell,2,4} = 0,
\nonumber\\
&&b_1 x_{\ell,2,1} + b_4 x_{\ell,2,4} + b_5 x_{\ell,2,5} + b_6 x_{\ell,2,6} = 0,
\nonumber\\
&&d_0 x_{\ell,3,1} + d_4 x_{\ell,2,4} + d_5 x_{\ell,2,5} + d_6 x_{\ell,3,6} = 0,
\nonumber\\
&&e_5 x_{\ell,2,5} + e_6 x_{\ell,3,6} + e_7 x_{\ell,2,7} + e_8 x_{\ell,2,8} = 0,
\nonumber\\
&&f_0 x_{\ell,3,1} + f_3 x_{\ell,2,3} + f_4 x_{\ell,2,4} + f_5 x_{\ell,2,5} = 0,
\nonumber\\
&&g_0 x_{\ell,3,1} + g_2 x_{\ell,3,2} + g_4 x_{\ell,2,4} + g_6 x_{\ell,3,6} = 0,
\nonumber\\
&&h_0 x_{\ell,3,1} + h_5 x_{\ell,3,5} + h_6 x_{\ell,3,6} + h_8 x_{\ell,2,8} = 0,
\nonumber\\
&&k_4 x_{\ell,2,4} + k_5 x_{\ell,2,5} + k_6 x_{\ell,2,6} + k_7 x_{\ell,2,7} = 0,
\nonumber\\
&&.................................................................
\nonumber
\end{eqnarray}
\begin{eqnarray}
&&a_1 x_{\ell,m,1} + a_2 x_{\ell,m,2} + a_3 x_{\ell,m,3} + a_4 x_{\ell,m,4} = 0,
\nonumber\\
&&b_1 x_{\ell,m,1} + b_4 x_{\ell,m,4} + b_5 x_{\ell,m,5} + b_6 x_{\ell,m,6} = 0,
\nonumber\\
&&d_0 x_{\ell,1,1} + d_4 x_{\ell,m,4} + d_5 x_{\ell,m,5} + d_6 x_{\ell,1,6} = 0,
\nonumber\\
&&e_5 x_{\ell,m,5} + e_6 x_{\ell,1,6} + e_7 x_{\ell,m,7} + e_8 x_{\ell,m,8} = 0,
\nonumber\\
&&f_0 x_{\ell,1,1} + f_3 x_{\ell,m,3} + f_4 x_{\ell,m,4} + f_5 x_{\ell,m,5} = 0,
\nonumber\\
&&g_0 x_{\ell,1,1} + g_2 x_{\ell,1,2} + g_4 x_{\ell,m,4} + g_6 x_{\ell,1,6} = 0,
\nonumber\\
&&h_0 x_{\ell,1,1} + h_5 x_{\ell,1,5} + h_6 x_{\ell,1,6} + h_8 x_{\ell,m,8} = 0,
\nonumber\\
&&k_4 x_{\ell,m,4} + k_5 x_{\ell,m,5} + k_6 x_{\ell,m,6} + k_7 x_{\ell,m,7} = 0,
\label{Keq16}
\end{eqnarray}
where the first eight equations from (\ref{Keq16}) represent (\ref{Keq12}) written with
the operators
from (\ref{Keq3}) defined in the first cell, the second eight equations from (\ref{Keq16})
represent
(\ref{Keq12}) written with the operators from (\ref{Keq3}) defined in the second cell,
and so on, the last eight equalities from (\ref{Keq16})
represent (\ref{Keq12}) written with the
operators from (\ref{Keq3}) defined in the last $m$th cell.
The coefficients $a_i,...,k_i$ are presented in (\ref{EquA12}) or (\ref{Keq7}).
One finds in this manner a system of $8m$ linear homogeneous equations for $8m$ unknown
variables, the $x_{\ell,n_c,\nu}$ prefactors, $\ell$ being fixed. Solutions are present
only if the determinant of the system (\ref{Keq16}) is zero. This condition specifies
different interconnection possibilities between $\hat H_0$ parameters (e.g. different
regions of the parameter space specified by the index $\gamma$) in a generic form
\begin{eqnarray}
F_{\gamma}(\{ t_{\alpha}\}, \{ \epsilon_{\alpha} \} ) =0,
\label{Keq17}
\end{eqnarray}
where solutions are possible to occur. In (\ref{Keq17}) the sets $\{ t_{\alpha} \}$ and
$\{\epsilon_{\alpha}\}$ are representing the set of hopping matrix elements, and on-site
one-particle potentials, respectively. The study of the physical meaning of the condition
(\ref{Keq17}) shows that since flat bands are not present in the bare band structure
(see Appendix B, Eq.(\ref{b7}) together with explications related to this equation),
(\ref{Keq17}) is not connected to the flat band notion, consequently, the ferromagnetism
which will be here deduced is not of flat band type. Instead, (\ref{Keq17}) specifies
regions in the parameter space where the density of states around the minimum system
energy value is high.

One given deduced $\hat B^{\dagger}$ operator from (\ref{Keq16}) receives a fixed
$\ell$ index,
for example $\ell=1$. It has hence the form
\begin{eqnarray}
\hat B^{\dagger}_{1,\sigma} = \sum_{n=1}^m \sum_{\nu=1}^8
x_{1,n,\nu} \hat c^{\dagger}_{{\bf j}+n{\bf a}+{\bf r}_{\nu},\sigma} \: ,
\label{Keq18}
\end{eqnarray}
where ${\bf j}$ is an arbitrary site of the sublattice $S_1$.
The obtained $x_{1,n,\nu}$ coefficients in (\ref{Keq18}) usually have the following
properties:
i) are  different in different cells,
e.g. for $n \ne n'$ one has $x_{1,n,\nu} \ne x_{1,n',\nu}$, and ii) rotating the
$\hat B^{\dagger}_{1,\sigma}$ operator by $\pi$ angle around the axis (the line)
of the chain, the prefactors attached to each site change their value, e.g.
\begin{eqnarray}
&&x_{1,n,1} \ne x_{1,n-1,5}, \quad x_{1,n,2} \ne x_{1,n-1,7}, \quad x_{1,n,3} \ne x_{1,n-1,8},
\quad x_{1,n,4} \ne x_{1,n,6},
\nonumber\\
&&x_{1,n,5} \ne x_{1,n,1}, \quad x_{1,n,6} \ne x_{1,n-1,4}, \quad
x_{1,n,7} \ne x_{1,n,2}, \quad x_{1,n,8} \ne x_{1,n,3}.
\label{Keq19}
\end{eqnarray}
Given by the properties i), ii), new linearly independent $\hat B^{\dagger}_{\ell,\sigma}$
operators are obtained as follows.

i) One translates all coefficients in (\ref{Keq18}) with one cell,
the procedure being possible to be repeated $m-1$ times. One obtains in this manner
\begin{eqnarray}
&&\hat B^{\dagger}_{2,\sigma} = \sum_{n=1}^m \sum_{\nu=1}^8
x_{1,n-1,\nu} \hat c^{\dagger}_{{\bf j}+n{\bf a}+{\bf r}_{\nu},\sigma},
\nonumber\\
&&\hat B^{\dagger}_{3,\sigma} = \sum_{n=1}^m \sum_{\nu=1}^8
x_{1,n-2,\nu} \hat c^{\dagger}_{{\bf j}+n{\bf a}+{\bf r}_{\nu},\sigma},
\nonumber\\
&&...........................
\nonumber\\
&&\hat B^{\dagger}_{m,\sigma} = \sum_{n=1}^m \sum_{\nu=1}^8
x_{1,n-m+1,\nu} \hat c^{\dagger}_{{\bf j}+n{\bf a}+{\bf r}_{\nu},\sigma}.
\label{Keq20}
\end{eqnarray}

ii) After this step one rotates by $\pi$ degree around the axis of the chain all operators
$\hat B^{\dagger}_{\ell,\sigma}$, $\ell=1,2,...,m$, obtaining $m$ new linearly independent
$\hat B^{\dagger}_{\ell,\sigma}$ operators for which $\ell=m+1,m+2,...,2m$ holds.
The expression of a
$\hat B^{\dagger}_{\ell',\sigma}$ operator obtained by rotating
$\hat B^{\dagger}_{\ell,\sigma}$ from (\ref{Keq15}) becomes
\begin{eqnarray}
\hat B^{\dagger}_{\ell',\sigma} = \sum_{n_{c_{\bf j}}=1}^m \sum_{\nu=1}^8
x'_{\ell',n_{c_{\bf j}},\nu} \hat c^{\dagger}_{{\bf j}+ {\bf a} n_{c_{\bf j}}+
{\bf r}_{\nu},\sigma},
\label{Keq21}
\end{eqnarray}
where ${\bf j}$ is an arbitrary site of $S_1$, and one has (see also (\ref{Keq19}))
\begin{eqnarray}
&&x'_{\ell',n,1}=x_{\ell,n-1,5}, \quad x'_{\ell',n,2}=x_{\ell,n-1,7}, \quad
x'_{\ell',n,3}=x_{\ell,n-1,8}, \quad x'_{\ell',n,4}=x_{\ell,n,6},
\nonumber\\
&&x'_{\ell',n,5}=x_{\ell,n,1}, \quad x'_{\ell',n,6}=x_{\ell,n-1,4},\quad
x'_{\ell',n,7}=x_{\ell,n,2}, \quad x'_{\ell',n,8}=x_{\ell,n,3}.
\label{Keq22}
\end{eqnarray}
The linear independence of the new operators obtained via (\ref{Keq20},\ref{Keq21})
must be always checked, and the presented procedure leads [for one given solution of
(\ref{Keq16})] to maximum $2m$ linearly independent $\hat B^{\dagger}_{\ell,\sigma}$
operators for a fixed spin index $\sigma$. The described procedure than must be repeated
for all independent solutions of (\ref{Keq16}) connected to the same parameter space
domain
(\ref{Keq17}). Explicit examples for the $m=6$ case (twelve hexagons connected in
six cells)
are presented in Appendix C.

The deduced $\hat B^{\dagger}_{\ell,\sigma}$ operators
on the test chain containing $m$ cells,
can be used for
the description of a chain containing $M=p \times m$ cells as well, as stated before, $p$
being an arbitrary integer number. For this, a given
$\hat B^{\dagger}_{\ell,\sigma}$ operator (maintaining $\ell$ and $\sigma$ fixed)
is periodically repeated $p$ times to cover completely the
chain containing $M$ cells (treated with periodic boundary
conditions). In order to increase the number of linearly independent
$\hat B^{\dagger}$ operators relating the same chain containing $M$ cells, one must
collect
all solutions describing the same parameter space region (\ref{Keq17}) with different
$m'$ values such to reobtain the same fixed $M$ via $M=p' \times m'$, where both
$p'$ and $m'$ are integers.

\subsection{The ground state wave function}

\subsubsection{The expression of the ground state}

Collecting together all linearly independent $\hat B^{\dagger}_{\ell,\sigma}$ operators
deduced in the previous subsection and denoting their number by $N_B$,  one has for
$|\Psi(N_B)\rangle = \prod_{\sigma} \prod_{\ell=1}^{N_B} \hat B^{\dagger}_{\ell,\sigma}
|0\rangle$ the property
\begin{eqnarray}
\hat H_A |\Psi(N_B)\rangle = 0.
\label{Keq23}
\end{eqnarray}
The equality (\ref{Keq23}) is a direct consequence of (\ref{Keq12}) [see also the
explications presented just above (\ref{Keq12})].
Since the kernel $ker(\hat H_A)$ is intimately connected to (\ref{Keq12}), it
results that
in the studied conditions, the wave vector $|\Psi(N_B)\rangle$ is unique in satisfying
(\ref{Keq23}).

The ground state is obtained from $|\Psi(N_B)\rangle$
based on (\ref{Keq10a})
by taking from it the contributions
which are present also in $ker(\hat H_U)$, i.e. does not contain double
occupancy. Since all $\hat B^{\dagger}_{\ell,\sigma}$ operators are extended along a
confined space region defined by the studied chain, different
$\hat B^{\dagger}_{\ell,\sigma}$ operators holding different $\ell$ indices intersect
each other on several sites of the system. Consequently, the double occupancy can be
avoided
only if one fixes to the same value all spin indices of all
$\hat B^{\dagger}_{\ell,\sigma}$
operators. This property remains also true if one decreases the number of the
$\hat B^{\dagger}$ operators present in the wave vector. Hence,
the ground state wave function at $N \leq N_B$ becomes
\begin{eqnarray}
|\Psi_g(N \leq N_B)\rangle = \prod_{\ell=1}^{N} \hat B^{\dagger}_{\ell,\sigma} |0\rangle,
\label{Keq24}
\end{eqnarray}
where $\sigma$ is fixed. The corresponding ground state energy according to (\ref{Keq4})
is
$E_g=K N$, where the $K$ constant is given in (\ref{Keq8}). The ground state nature arises
since because of the missing double occupancy one has
$\hat H_U |\Psi_g(N \leq N_B)\rangle =0$, while
because of (\ref{Keq23}), $\hat H_A |\Psi_g(N \leq N_B)\rangle =0$. Hence indeed
$(\hat H_A + \hat H_U) |\Psi_g(N \leq N_B)\rangle = \hat H'|\Psi_g(N \leq N_B)\rangle =0$
is satisfied.

One further note that the described ferromagnetic ground state emerges in the relatively
low concentration limit. For example, using the explicit solution presented in Appendix C
in the case of a system with $N_c=36$ cells (containing 72 hexagons), the number of
electrons per cell $N/N_c \leq 0.69$ holds, where $max(N)=N_B$ is satisfied.

\subsubsection{Physical properties of the ground state}

The deduced ground state (\ref{Keq24}) is a fully polarized ferromagnet. Besides, it is a
conducting state since the operators building up the ground state wave vector are
extended,
and $\delta \mu = \mu_{+}-\mu_{-}= (E_g(N+1)-E_g(N))-(E_g(N)-E_g(N-1))= 0$ holds for
$N < N_B$. All itinerant electrons being spin-polarized, such systems are potentially
applicable in spintronics devices.

The obtained ferromagnetism is not of flat band type, since as shown in Appendix B,
flat bands are not present in the system. The spin polarization occurs since
extended operators confined
in the restricted space region of the chain touch each other, hence must correlate their
spin index in order to reduce the double occupancy. A such type of ferromagnetism is
created by the common effect of correlations and confinement, and was observed in
other hexagon chains as well \cite{TKG}. For the appearance of this phase, besides the
confined geometry, the following aspects should be present: a) The operators
$\hat B^{\dagger}$ building up the ground state wave vector must be extended. Such a
property is not {\it a priori} a chain characteristics, and several chain examples are
known where it does not occur \cite{GKV1,GKV2,Hon}.
Regarding this property one mentions that
always when a such type of behavior was observed for chain structures not possessing
flat bands, {\it external hoppings} (i.e. hopping matrix elements connecting different
sites from different neighboring cells -- see $t'_1$ in Fig.2) were present.
Consequently,
when the hopping terms allow the movement along the whole chain (e.g. cage regions where
the hopping amplitudes are missing along a chain section perpendicular to the line
of the chain are not present),
this seems to be a sufficient condition preserving the extended nature of
$\hat B^{\dagger}$ operators. b) The density of states at and around the minimum
one-particle energy should be high. This preserves the presence of the fully saturated
ferromagnetic state at arbitrary repulsive $U$ in the low concentration limit and finite
chain length in the presented case. The mechanism however surely works (probably with
lower
$U$ bound) in the thermodynamic limit as well, but the effective proof  of this statement
remains a challenging problem for further investigations.

One further notes that the described ferromagnetic phase emerges for a sequence of
different possible interdependences between hopping amplitudes
[see the different $A=(t_1-4t'_2)/(t_1+ 2 t'_2)$ values presented after Eq.(\ref{c5})].
Since it is known that carbon-based nanoribbons are sensitive to uniaxial or
shear strains
\cite{st1}, and/or external pressure \cite{st2}, the needed interdependences between
$\hat H_0$ parameters can be achieved for example by strain effects
(see also Ref.\cite{st3}).

\section{Summary and conclusions}

An armchair hexagon chain (i.e. polyphenanthrene)
as a representative of a thin armchair ribbon has been
described in exact terms by the use of a technique based on positive semidefinite
operator properties. The deduced ground state for mesoscopic samples, in the small
concentration limit was shown to be ferromagnetic and metallic. This appears in
conditions in which flat
bands are not present in the bare band structure of the system, the spin polarization
being provided by a common effect of correlations and confinement. Since increasing
the ribbon width there are not present physical reasons to cancel in exact terms
this behavior, at least for a restricted domain of hopping amplitudes we
expect similar effect to occur for thicker ribbons as well.

\acknowledgements
For Z.G. financial support provided by the Alexander von Humboldt Foundation.

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India's Nuclear Doctrine
Pinto Uncategorized October 23, 2020 | 0
India has once again said that it is, committed to know first use of strategic weapons under its Nuclear Doctrine
Speaking at the conference at the 75th Session of the first committee of United Nations General Assembly, India said that its nuclear weapons are meant to maintain a posture of minimum deterrence.
Earlier this month foreign secretary Harsh Vardhan said "Nuclear disarmament can be achieved through an agreed multi-lateral framework and India will push for it."
India's nuclear Doctrine came out in 2003 and its 17-year now and it has been quite consistent. It hasn't changed at all despite of defeat and debate whether it has to be changed or not and the debate continues.
The Government has considered all the pros and cons and feels that no first use doctrine which is really a breathe-in our deterrence capabilities, the credible minimum deterrent and second Strike is very much intact. It is good that it is re-altered by the United Nations and clarified at Lok Sabha when there were many discussions whether India is considering a revision.
So this is a strong signal that India is quite confident of its credible minimum deterrent and India is in a position to deal with threats.
Nuclear environment and security environment in the world has changed drastically .
Transition of being a nuclear weapons state was meant to be with threats like black mail.
Nuclear doctrine is a defensive doctrine but it assures credibility of retaliation
In nuclear age the only stability will come from defensive posture.
Anything, which is aggressive, leads to counter relations
India as a responsible stakeholder in the global nuclear architecture it is always been the case that India would represent our case with modicum of an Sovereignty and modicum of Stability.
Nuclear doctrine had provision for reassessment.
India by its structure and faith in nuclear disarmament should continue to press for global discourse on a nuclear weapon convention .
In fear of extent of nuclear weapons it should be removed rather than worshipping it. Weapons should be eliminated and we should not believe in aggression
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MCB Factors
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Good corporate governance practices represent a key pillar of the way in which MCB Group operates and conducts business. Indeed, the Board of MCB Factors Ltd is committed to upholding the highest standards of corporate governance, by promoting principles of integrity, accountability and transparency across the organisation, with the aim of maximising the long-term value creation for its stakeholders. Adherence to advocated norms is viewed as being crucial to effectively cope with the challenges posed by the increasingly exigent operating environment.
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Profile of DirectorsPDF 42KB
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The Directors have approved the following Statement of Accountabilities:
The Board assumes the responsibility for leading and controlling the Company and meeting all legal and regulatory requirements. Directors are aware of their legal duties.
The Board is accountable for the performance and affairs of the Company and for achieving sustainable growth.
The Board is responsible for ensuring that the Company adheres to high standards of ethical behavior and acts in the best interest of shareholders.
The Board has the responsibility of reviewing and approving the results announcements of the Company.
Additionally, the Board Charter and the Position Statements, which have been approved by the Board, provides for a clear definition of the roles and responsibilities of the Chairperson, the Directors and the Company Secretary.
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The Board has reviewed and approved the Board Charter, the Organisational Chart, the Job positions and the Statement of Accountabilities of the Company. These documents are monitored regularly and are updated in a timely manner further to any changes in laws, requirements from regulatory bodies or decisions taken by the Board.
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text:
class ResumeController < ApplicationController
  def index
    if request.xhr?
      render partial: 'resume'
    end
  end
end

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Hastings College of the Law - J.D.
Evan Field is the sole proprietor of his own law practice, focused on estate planning and decedents' estates and trusts, with substantial experience in employment law, real estate law, mortgage banking, and litigation. Carl teaches the Legal Environment of Business at Sonoma State University.


text:
Minnesota History
MinnPost's Minnesota History articles are produced in partnership with the Minnesota Historical Society and its MNopedia project, which is made possible by the Legacy Amendment's Arts and Cultural Heritage Fund.
In 1977, boss tells Willmar 8 'We're not all equal, you know'; strike ensues
Things didn't work out so well for the Willmar 8. But for the women's movement, the 1977-1979 strike was a resounding success.
By John Fitzgerald
For nearly two years the Willmar 8 picketed in front of downtown Willmar's Citizens National Bank.
Courtesy of the Minnesota Historical Society
Things didn't work out so well for the Willmar 8. For nearly two years they picketed in front of downtown Willmar's Citizens National Bank. They never got their pay increases; the NCLB said it was an "economic" strike, so they never got strike-related compensation; and after the strike, seven of them never really got their jobs back.
But for the women's movement, the 1977-1979 strike was a resounding success. It was a huge chink in the armor of the institutional sexism women faced in the workplace. Women were becoming more common in the workplace and they demanded equal treatment and wages. It's a battle that is still being fought today as women employees make only 77 cents to the male employee's dollar nationally.
Despite the Equal Pay Act of 1963, which guarantees women the right to equal pay, the problem was still common and sexism still casual in 1977 when trouble boiled over in Willmar. According to a story written by Asa Wilson in a 2006 edition of the Workday Minnesota newspaper, eight women — Doris Boshart, Sylvia Erickson, Jane Harguth, Teren Novotny, Shirley Solyntjes, Glennis Ter Wisscha, Sandi Treml and Irene Wallin – grew tired of making nearly $300 per week less than their male counterparts.
"The starting salary for men averaged $700 per month, the same amount Boshart made after 10 years at the bank," Wilson recounted. They were also expected to work overtime without pay. The bank's only female officer earned $4,000 less than her male counterparts, whom she supervised.
The issue came to a head in April 1977, when the women were told to train a young male employee, who was hired at a better wage, to become their boss, Wilson recounts. They went to see Bank President Leo Pirsch and demanded an end to the discrimination.
Pirsch told the women, "We're not all equal, you know." Men need more money because they have to pay for dates, he said.
Complaint filed with EEOC
Wilson's narrative picks up the timeline of frustration: "The women filed a gender discrimination complaint with the Equal Employment Opportunity Commission and an unfair labor practice charge with the National Labor Relations Board. In May 1977, they formed Minnesota's first bank union.
"In June 1977, the EEOC ruled there was 'reasonable cause to believe' there had been gender discrimination at the bank, and the board of directors finally agreed to negotiate." But "Ter Wisscha calls the ensuing negotiations 'an absolute effort in futility."
So, on Dec. 16, 1977, the eight female bank employees went on strike, picketing the bank in a wind chill of -70 degrees below zero.
It was a strike many of the Willmar 8 thought was unnecessary. In a report written for the American Federation of State, County and Municipal Employees Council 5 Local 404 in St. Peter, members of the Willmar 8 said they thought it could have been averted. But, Novotny says, "They weren't going to let a bunch of female employees tell them what to do."
Instead, they became a cause célèbre for the Women's Movement. They received thousands of cards. They appeared in national magazines. Phil Donahue featured them on his talk show, as did NBC's "Today Show." Actress Lee Grant made a behind-the-scenes documentary. NBC made a TV movie.
Willmar community was split
But the 14,000 people in Willmar were split. Five female employees remained with the bank. Some businesses blacklisted the Willmar 8 for years. The strikers made huge personal and financial sacrifices. Their children lost friends. One striker's marriage fell apart. The lawyer who took the Willmar 8's case, John Mack, lost his position as county chair of the GOP but stayed with the case.
Support did come. They received thousands of letters each week, many with donations. Many of the townspeople were sympathetic to their cause, honking and waving as they drove by.
The National Organization for Women sent supporters to join them on the picket line. At first, the Willmar 8 were wary of the NOW supporters. Feminism "was somebody else's fight. They were the bra-burners and the radicals." Ter Wisscha said. Then a NOW member told Ter Wisscha that feminism did not mean preventing women from staying home, but instead meant giving them choices.
"It was like a light bulb came on in my head. It was like, wow, I get it! I understand," Ter Wisscha told Wilson.
On another February day, more than 250 people joined the United Automobile Workers at a support rally for the Willmar 8. "They were friends, they were supporters, and they did this without being asked," Ter Wisscha said. It was not the last visit by UAW supporters, she said, and they always received a police escort out of town.
Summer of '78: Suit, demands dropped
The strike fund dried up in the summer of 1978 and the Willmar 8 dropped their discrimination lawsuit against the bank in exchange for a tiny financial settlement. In September, the strikers dropped their demands and offered to return to work without a contract, even though the bank had filled their jobs and told the women they could return only as openings became available.
Boshart was the only one immediately called back to work. She was demoted from head bookkeeper to teller. The bank, which saw a severe drop-off in deposits, was sold, then sold again. In 1980, four of the women returned to work, though only Boshart stayed more than a few months.
Except for the small settlement in 1978, the Willmar 8 received no financial gain from the strike. The NLRB ruled in 1979 that the bank was guilty of unfair labor practices, but those practices did not cause the strike, therefore the strike was "economic" and no back pay was due.
But the women realized that even though they lost the battle, they were big players in the war against discrimination. Novotny remembers her mother-in-law noticing how women were treated with increased respect at the bank where she worked.
The Willmar 8 also remains in the classroom. Each year, Boshart receives letters from college students thanking her for her role in the struggle for women's rights. Ter Wisscha also gets letters and phone calls from students learning about the strike. "That's what fuels my belief that we're not done winning yet," she said. "People are still asking the questions. People still want to try to understand."
At Willmar Junior High School, history teacher Suzanne Nelson teaches her students about the Willmar 8 in the local history course to show them how "a small group of people can make a lot of difference."
John Fitzgerald is a longtime journalist who has worked at newspapers across the country.
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In 1977, boss tells Willmar 8 'We're not all equal, you know'; strike ensues

By John Fitzgerald

June 28, 2013

Things didn't work out so well for the Willmar 8. For nearly two years they picketed in front of downtown Willmar's Citizens National Bank. They never got their pay increases; the NCLB said it was an "economic" strike, so they never got strike-related compensation; and after the strike, seven of them never really got their jobs back.

But for the women's movement, the 1977-1979 strike was a resounding success. It was a huge chink in the armor of the institutional sexism women faced in the workplace. Women were becoming more common in the workplace and they demanded equal treatment and wages. It's a battle that is still being fought today as women employees make only 77 cents to the male employee's dollar nationally.

Despite the Equal Pay Act of 1963, which guarantees women the right to equal pay, the problem was still common and sexism still casual in 1977 when trouble boiled over in Willmar. According to a story written by Asa Wilson in a 2006 edition of the Workday Minnesota newspaper, eight women — Doris Boshart, Sylvia Erickson, Jane Harguth, Teren Novotny, Shirley Solyntjes, Glennis Ter Wisscha, Sandi Treml and Irene Wallin – grew tired of making nearly $300 per week less than their male counterparts.

"The starting salary for men averaged $700 per month, the same amount Boshart made after 10 years at the bank," Wilson recounted. They were also expected to work overtime without pay. The bank's only female officer earned $4,000 less than her male counterparts, whom she supervised.

The issue came to a head in April 1977, when the women were told to train a young male employee, who was hired at a better wage, to become their boss, Wilson recounts. They went to see Bank President Leo Pirsch and demanded an end to the discrimination.

Pirsch told the women, "We're not all equal, you know." Men need more money because they have to pay for dates, he said.

Complaint filed with EEOC

Wilson's narrative picks up the timeline of frustration: "The women filed a gender discrimination complaint with the Equal Employment Opportunity Commission and an unfair labor practice charge with the National Labor Relations Board. In May 1977, they formed Minnesota's first bank union.

"In June 1977, the EEOC ruled there was 'reasonable cause to believe' there had been gender discrimination at the bank, and the board of directors finally agreed to negotiate." But "Ter Wisscha calls the ensuing negotiations 'an absolute effort in futility."

So, on Dec. 16, 1977, the eight female bank employees went on strike, picketing the bank in a wind chill of -70 degrees below zero.

It was a strike many of the Willmar 8 thought was unnecessary. In a report written for the American Federation of State, County and Municipal Employees Council 5 Local 404 in St. Peter, members of the Willmar 8 said they thought it could have been averted. But, Novotny says, "They weren't going to let a bunch of female employees tell them what to do."

Instead, they became a cause célèbre for the Women's Movement. They received thousands of cards. They appeared in national magazines. Phil Donahue featured them on his talk show, as did NBC's "Today Show." Actress Lee Grant made a behind-the-scenes documentary. NBC made a TV movie.

Willmar community was split

But the 14,000 people in Willmar were split. Five female employees remained with the bank. Some businesses blacklisted the Willmar 8 for years. The strikers made huge personal and financial sacrifices. Their children lost friends. One striker's marriage fell apart. The lawyer who took the Willmar 8's case, John Mack, lost his position as county chair of the GOP but stayed with the case.

Support did come. They received thousands of letters each week, many with donations. Many of the townspeople were sympathetic to their cause, honking and waving as they drove by.

The National Organization for Women sent supporters to join them on the picket line. At first, the Willmar 8 were wary of the NOW supporters. Feminism "was somebody else's fight. They were the bra-burners and the radicals." Ter Wisscha said. Then a NOW member told Ter Wisscha that feminism did not mean preventing women from staying home, but instead meant giving them choices.

"It was like a light bulb came on in my head. It was like, wow, I get it! I understand," Ter Wisscha told Wilson.

On another February day, more than 250 people joined the United Automobile Workers at a support rally for the Willmar 8. "They were friends, they were supporters, and they did this without being asked," Ter Wisscha said. It was not the last visit by UAW supporters, she said, and they always received a police escort out of town.

Summer of '78: Suit, demands dropped

The strike fund dried up in the summer of 1978 and the Willmar 8 dropped their discrimination lawsuit against the bank in exchange for a tiny financial settlement. In September, the strikers dropped their demands and offered to return to work without a contract, even though the bank had filled their jobs and told the women they could return only as openings became available.

Boshart was the only one immediately called back to work. She was demoted from head bookkeeper to teller. The bank, which saw a severe drop-off in deposits, was sold, then sold again. In 1980, four of the women returned to work, though only Boshart stayed more than a few months.

Except for the small settlement in 1978, the Willmar 8 received no financial gain from the strike. The NLRB ruled in 1979 that the bank was guilty of unfair labor practices, but those practices did not cause the strike, therefore the strike was "economic" and no back pay was due.

But the women realized that even though they lost the battle, they were big players in the war against discrimination. Novotny remembers her mother-in-law noticing how women were treated with increased respect at the bank where she worked.

The Willmar 8 also remains in the classroom. Each year, Boshart receives letters from college students thanking her for her role in the struggle for women's rights. Ter Wisscha also gets letters and phone calls from students learning about the strike. "That's what fuels my belief that we're not done winning yet," she said. "People are still asking the questions. People still want to try to understand."

At Willmar Junior High School, history teacher Suzanne Nelson teaches her students about the Willmar 8 in the local history course to show them how "a small group of people can make a lot of difference."

This article first appeared on MinnPost and is republished here under a Creative Commons license.


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BoI boss signals further payment break extension beyond six months unlikely
The chief executive of Bank of Ireland has said she does not currently see repayment breaks for borrowers who are struggling financially because of the Covid-19 crisis being extended beyond the six months agreed by the main banks here.
Francesca McDonagh said the sector had been proactive in agreeing payment breaks initially for three months and then extending it to six.
But asked by RTE News if she saw the six months being extended further, Ms McDonagh replied, "not where I am sitting right now."
She said "leaning in" and supporting customers, whether they be in households or in businesses, is the bank's priority.
"We need to do that in a sustainable way and I believe the payment breaks give the space for homes and business owners," she said.
"But the current plan is for those to be up to a maximum six months and then we will be there to help the Irish economy get back on its feet."
Speaking following publication of the bank's first quarter results today, Ms McDonagh said the bank is continuing to lend to customers who need it where it is appropriate and in their best interests.
"I'm not aware of any single case today where a customer has lost their deposit as a result of a change in a Bank of Ireland mortgage," she said.
"But we must be careful when we support customers that if their circumstances have changed significantly and their income is very different it is not in anyone's interest for them to draw down a mortgage that they cannot afford to pay for the first month onwards," she added.
The bank has set aside €250m as an impairment charge to deal with the fallout from Covid-19.
But Ms McDonagh said it was difficult to be specific at this point about about the future extent of bad loans arising from the challenges, saying it varies by sector.
"Different sectors are experiencing Covid-19 in different ways and we also see in what we call the Horizon 2, the gradual easing of restrictions coming at a different pace in different sectors," she said.
She said that sectors such as travel, hospitality or non-food retail are being very much impacted.
"Our focus is on supporting our customers and that is why we have put in place mortgage payment breaks. This is an example of where we are leaning into the national effort to support businesses through this period of uncertainty," the bank boss said.
Francesca McDonagh said the losses recorded by the bank in the first quarter were the result of the coronavirus, with otherwise strong lending growth and reduction in costs.
"But the results today reflect the realities of Covid-19 and that includes a major part of the reason why we would have recorded a loss in the first quarter," she said.
"We decided to take a Covid-19 management overlay for a provision for a potential credit loss for the future of €250m. That's not specific against any deterioration we've seen in our loan book, but it is a recognition that the economic outlook for Ireland this year is of a quite significant contraction and an increase in unemployment."
"We expect to see that recover towards the end of this year and into 2021. But we wanted to take that prudent management overlay charge given the crisis we are all in," she added.
Despite the challenges facing the bank, Ms McDonagh indicated that she does not anticipate further job losses or permanent branch closures.
She said the bank had already been on a cost cutting trajectory for the last two and a half years.
"Our colleagues right now are totally focused on supporting our customers and communities," she said.
"We have 70% working from home and 30% are on site in the frontline doing exceptional jobs that I am hugely appreciative of. When we think about our costs we think about efficiency, half of that is related to people and half of that is non-people costs," she said.
"In terms of branches, the branch network is a really important part of our offering," she added.
Ms McDonagh also said she is confident that the bank is well capitalised for whatever lies ahead.
"I'm incredibly confident and comfortable about the outlook. We've taken the impairment charge today as a prudent and conservative response to the economy," she stated.
"But our underlying capital is in excess of what we are required to hold by the regulator and even in a range of scenarios and further more pessimistic outlooks in the economy if that happens, and there is no one who can be certain about how this is going to play out, our capital ratio does remain above even our previous minimum requirement," she added.
Bank of Ireland swings to Q1 loss due to Covid-19 impairment charges
Earlier, Bank of Ireland reported an underlying loss before tax of €235m for the three months to the end of March as the coronavirus had a "material impact" on the bank's results.
This compares to underlying profits of €143m the same time last year.
In an interim management statement, the bank said the economic outlook for its core markets in Ireland and the UK has deteriorated, with reduced levels of activity across its businesses.
Bank of Ireland said today it had taken an impairment charge of €266m, including Covid-19 management overlay of €250m, which it said mainly reflected the initial impact of the deteriorating economic environment.
Its net interest income for the three months to March came to €540m, which the bank said was in line with its expectations.
The lender reported a pre-tax loss of €214m for the three month period, down from a pre-tax profit of 4123m.
"The economic effects will have a material impact on the group's 2020 financial performance," Bank of Ireland's group chief executive Francesca McDonagh said today.
"The full impact remains uncertain and will be driven by the duration of Covid-19 restrictions and the successful reopening of the Irish and UK economies," Ms McDonagh added.
Bank of Ireland said today that Covid-19 hit its markets "fast and hard", but said it took immediate steps to help it customers as it launched a range of supports for personal and business customers impacted by the outbreak of the virus.
A total of 86,000 payment breaks have been approved for customers in Ireland and the UK since the middle of March.
Bank of Ireland said it has strong capital, funding and liquidity to provide this support for customers
The bank said today that in a range of scenarios its fully loaded core Tier 1 capital ratio would remain above its previous minimum regulatory capital requirement of 11.45%.
Its CET1 ratio – a key measure of financial strength – fell to 13.5% at the end of March, from 13.8% at the end of 2019.
Bank of Ireland, the country's largest bank by assets, said it expected more impairment charges and loan losses during 2020, with new lending potentially dropping to 50% to 70% of 2019 volumes.
The bank said that in a range of scenarios its fully loaded core Tier 1 capital ratio would remain above its previous minimum regulatory capital requirement of 11.45%.
Bank of Ireland, 14% owned by the state, reduced costs by 3% year-on-year and said it was maintaining a target introduced in February for more aggressive cost cuts by 2021.
The lender said today that 70% of its staff are working from home due to the Covid-19 outbreak, while it has also temporarily closed smaller branches due to reduced footfall.
It has also introduced an allowance scheme for staff working in front line and on site locations and is supporting staff who are needed to provide childcare or family support.
Bank of Ireland shares have halved in value since the first coronavirus case was reported here at the end of February. They were down 14.9% in Dublin trade today.