11institutetext:
%the␣affiliations␣are␣given␣next;␣don’t␣give␣your␣e-mail␣address%unless␣you␣accept␣that␣it␣will␣be␣publishedhttp://www.citizendium.org/User:Tom_Morris
22institutetext: http://www.citizendium.org/User:Daniel_Mietchen
Correspondence: Daniel.Mietchen (at) uni-jena (dot) de

Tom Morris and Daniel Mietchen

# Collaborative Structuring of Knowledge
by Experts and the Public

Tom Morris 11 Daniel Mietchen 22

###### Abstract

There is much debate on how public participation and expertise can be brought
together in collaborative knowledge environments. One of the experiments
addressing the issue directly is Citizendium. In seeking to harvest the
strengths (and avoiding the major pitfalls) of both user-generated wiki
projects and traditional expert-approved reference works, it is a wiki to
which anybody can contribute using their real names, while those with specific
expertise are given a special role in assessing the quality of content. Upon
fulfillment of a set of criteria like factual and linguistic accuracy, lack of
bias, and readability by non-specialists, these entries are forked into two
versions: a stable (and thus citable) approved ”cluster” (an article with
subpages providing supplementary information) and a draft version, the latter
to allow for further development and updates. We provide an overview of how
Citizendium is structured and what it offers to the open knowledge
communities, particularly to those engaged in education and research. Special
attention will be paid to the structures and processes put in place to provide
for transparent governance, to encourage collaboration, to resolve disputes in
a civil manner and by taking into account expert opinions, and to facilitate
navigation of the site and contextualization of its contents.

###### Keywords:

open knowledge, open education, open science, open
governance, wikis, expertise, Citizendium, Semantic Web

##  1 Introduction

> Science is already a wiki if you look at it a certain way. It’s just a
> highly inefficient one – the incremental edits are made in papers instead of
> wikispace, and significant effort is expended to recapitulate existing
> knowledge in a paper in order to support the one to three new assertions
> made in any one paper.
>
> John Wilbanks [21]

There are many ways to structure knowledge, including collaborative
arrangements of digital documents. Only a limited number of the latter ones
have so far been employed on a larger scale. Amongst them are wikis – online
platforms which allow the aggregation, interlinking and updation of diverse
sets of knowledge in an Open Access manner, i.e. with no costs to the reader.

###  1.1 Wikis as an example of public knowledge environments online

As implied by the introductory quote, it is probably fair to say that turning
science (or any system of knowledge production, for that matter) into a wiki
(or a set of interlinked collaborative platforms) would make research,
teaching and outreach much more transparent, less prone to hype, and more
efficient. Just imagine you had a time slider and could watch the history of
research on general relativity, plate tectonics, self-replication, or cell
division unfold from the earliest ideas of their earliest proponents (and
opponents) onwards up to you, your colleagues, and those with whom you compete
for grants. So why don’t we do it?

Traditionally, given the scope of a particular journal, knowledge about
specialist terms (which may describe completely non-congruent concepts in
different fields), methodologies, notations, mainstream opinions, trends, or
major controversies could reasonably be expected to be widespread amongst the
audience, which reduced the need to redundantly say and then repeat the same
things all over again and again (in cross-disciplinary environments, there is
a higher demand for proper disambiguation of the various meanings of a term).
Nonetheless, redundancy is still quite visible in journal articles, especially
in the introduction, methods, and discussion sections and the abstracts, often
in a way characteristic of the authors (such that services like eTBLAST and
JANE can make qualified guesses on authors of a particular piece of text, with
good results if some of the authors have a lot of papers in the respective
database, mainly PubMed, and if they have not changed their individual
research scope too often in between).

A manuscript well-adapted to the scope of one particular journal is often not
very intelligible to someone outside its intended audience, which hampers
cross-fertilization with other research fields (we will get back to this
below). When using paper as the sole medium of communication there is not much
to be done about this limitation. Indeed, we have become so used to it that
some do not perceive it as a limitation at all. Similar thoughts apply to
manuscript formatting. However, the times when paper alone reigned over
scholarly communication have certainly passed, and wiki-like platforms provide
for simple and efficient means of storing information, updating it and
embedding it into a wider context.

Cross-field fertilization, for example, is crucial with respect to
interdisciplinary research projects, digital libraries and multi-journal (or
indeed cross-disciplinary) bibliographic search engines (e.g. Google Scholar),
since these dramatically increase the likelihood of, say, a biologist
stumbling upon a not primarily biological source relevant to her research
(think shape quantification or growth curves, for instance). What options do
we have to systematically integrate such cross-disciplinary hidden treasures
with the traditional intra-disciplinary background knowledge and with new
insights resulting from research?

The by now classical example of a wiki environment are the Wikipedias, a set
of interlinked wikis in multiple languages where basically anyone can edit any
page, regardless of subject matter expertise or command of the respective
language. As a consequence of this openness, the larger Wikipedias have a
serious problem with vandalism: take an article of your choice and look at its
history page for reverts - most of them will be about neutralizing subtle or
blunt forms of destructive edits that do nothing to improve the quality of the
articles, but may reduce it considerably. Few of these malicious edits persist
for long [14], but finding and fixing them takes time that could better be
spent on improving articles. This is less of an issue with more popular topics
for which large numbers of volunteers may be available to correct ”spammy”
entries but it is probably fair to assume that most researchers value their
time too much to spend it on repeatedly correcting information that had
already been correctly entered. Other problems with covering scientific topics
at the Wikipedias include the nebulous notability criteria which have to be
fulfilled to avoid an article being deleted, and the rejection of ”original
research” in the sense of not having been peer reviewed before publication.
Despite these problems, one scientific journal – RNA Biology – already
requires an introductory Wikipedia article for a subset of papers it is to
publish [16].

Peer review is indeed a central aspect of scholarly communication, as it paves
the way towards the reproducibility that forms one of the foundations of
modern science. Yet we know of no compelling reason to believe that it works
better before than after the content concerned has been made public (doing it
beforehand was just a practical decision in times when journal space was
measured in paper pages), while emerging movements like Open Notebook Science
– where claims are linked directly to the underlying data that are being made
public as they arise – represent an experiment in this direction whose initial
results look promising and call into question Wikipedia’s ”no original
research” as a valid principle for generating encyclopaedic content.

Although quite prominent at the moment, the Wikipedias are not the only wikis
around, and amongst the more scholarly inclined alternatives, there are even a
number of wiki-based journals, though usually with a very narrow scope and/or
a low number of articles. On the other hand, Scholarpedia (which has classical
peer review and an ISSN and may thus be counted as a wiki journal, too [17]),
OpenWetWare [12], Citizendium [2] and the Wikiversities [20] are cross-
disciplinary and structured (and of a size, for the moment) such that
vandalism and notability are not really a problem. With minor exceptions, real
names are required at the first three, and anybody can contribute to entries
about anything, particularly in their fields of expertise. None of these is
even close to providing the vast amount of context existing in the English
Wikipedia but the difference is much less dramatic if the latter were broken
down to scholarly useful content. Out of these four wikis, only OpenWetWare is
explicitly designed to harbour original research, while the others allow
different amounts thereof. Furthermore, a growing number of yet more
specialized scholarly wikis exist (e.g. WikiGenes [18], the Encyclopedia of
Earth [8], the Encyclopedia of Cosmos [7], the Dispersive PDE Wiki [6], or the
Polymath Wiki [13]), which can teach us about the usefulness of wikis within
specific academic fields.

##  2 The Citizendium model of wiki-based collaboration

Despite the above-mentioned tensions between public participation and
expertise in the collaborative structuring of knowledge, it is not
unreasonable to expect that these can be overcome by suitably designed public
knowledge environments, much like Citizen Science projects involve the public
in the generation of scientific data. One approach at such a design is
represented by Citizendium. The founder of Citizendium – Larry Sanger – is the
co-founder of Wikipedia. The two projects share the common goal of providing
free knowledge to the public, they are based on variants of the same software
platform, and they use the same Creative Commons-Attribution-Share Alike
license [4]. Yet they differ in a number of important ways, such that
Citizendium can be seen as composed of a Wikipedia core (stripped down in
terms of content, templates, categories and policies), with elements added
that are characteristic of the other wiki environments introduced above: A
review process leading to stable versions (as at Scholarpedia), an open
education environment (as at Wikiversity) and an open research environment (as
at OpenWetWare). Nonetheless, assuming that the reader is less familiar with
these three latter environments, we will follow previous commenters and frame
the discussion of Citizendium in terms of properties differentiating it from
Wikipedia, and specifically the latter’s English language branch [19].

###  2.1 Real names

The first of these is simply an insistence on real names. While unusual from a
Wikipedia perspective, this is custom in professional environments, including
traditional academic publishing and some of the above-mentioned wikis, e.g.
Scholarpedia and Encyclopedia of Earth. It certainly excludes a number of
legitimate contributors who prefer to remain anonymous but otherwise gives
participants accountability and allows to bring in external reputation to the
project.

###  2.2 Expert guidance

Figure 1: Screenshot of the main page of the [[Crystal Palace]] cluster while
logged in using a monobook skin that is the default at Wikipedia. It shows the
green cluster bar that indicates that the page has been approved and links to
all the supages. Also visible is the status indicator (green dots on the left
topped by green tick), mention of ”an editor” to distinguish the number of
editors involved (some pages can be approved by one rather than three
editors), links to the workgroups which have approval rights for the article
(in this case: the History and Architecture Workgroups), a prominent
disclaimer (unapproved articles have a much strong disclaimer), and links to
the ’unstable’ draft version of the article which any registered contributor
can update. Like traditional encyclopaedic environments, Citizendium does not
require every statement to be referenced, in the interest of narrative flow.

To compose and develop articles and to embed them in the multimedial context
of a digital knowledge environment, expert guidance is important. Of course,
many experts contribute to Wikipedia, and the Wikipedias in turn have long
started to actively seek out expert involvement, yet the possibility to see
their edits overturned by anonymous users that may lack even the most basic
education in that field keeps professionals away from spending their precious
time on such a project. The Citizendium approach of verifying expertise takes
a different approach – sometimes termed ”credentialism” – that rests on a
common sense belief that some people do know more than others: it is sometimes
the case that the thirteen-year-old kid in Nebraska does know more than the
physics professor. But most of the time, at least when matters of physics are
concerned, this is not the case. The role the experts have at Citizendium is
not, as frequently stated in external comments, that of a supreme leader who
is allowed to exercise his will on the populace. On the contrary, it is much
more about guiding. We use the analogy of a village elder wandering around the
busy marketplace [15] who can resolve disputes and whom people respect for
their mature judgement, expertise and sage advice. Wikipedia rejects
”credentialism” in much the same way that the Internet Engineering Task Force
(IETF) does. David Clark summarised the IETF process thusly [3]: ”We reject
kings, presidents and voting. We believe in rough consensus and running code.”
In an open source project, or an IETF standardisation project, one can decide
a great many of the disputes with reference to technical reality: the
compiler, the existing network protocols etc. If the code doesn’t compile,
think again. For rough consensus to happen under such circumstances, one needs
to get the people together who have some clear aim in mind: getting two
different servers to communicate with one another. The rough consensus
required for producing an encyclopaedia article is different – it should
attempt to put forward what is known, and people disagree on this to a higher
degree than computers do on whether a proper connection has been established.
It is difficult to get ”rough consensus, running code” when two parties are
working on completely different epistemological standards. At this point, one
needs the advice of the village elderly who will vet existing content and
provide feedback on how it can be expanded or otherwise improved. Upon
fulfillment of a set of criteria like factual and linguistic accuracy, lack of
bias, and readability by non-specialists, these vetted entries are forked into
two versions: a stable (and thus citable) approved ”cluster” (an article with
subpages providing supplementary information) and a draft version, the latter
to allow for further development and updates (cf. Fig. 1).

The respect for experts because of their knowledge of facts is only part of
the reasoning: the experts point out and correct factual mistakes, but they
also help to guide the structuring of content within an article and by means
of the subpages. The experts bring with them the experience and knowledge of
years of in-depth involvement with their subject matter, and the project is
designed to make best use of this precious resource, while still allowing
everyone to participate in the process. Of course, experts are likewise free
to bring in content, be it within their specialty or in other areas, where
others take over the guiding role. The Citizendium can also host ’Signed
Articles’, which are placed in a subpage alongside the main article. A Signed
Article is an article on the topic described by a recognised expert in the
field, but can express opinions and biases in a way that the main article
ought not to.

Figure 2: Top: Screenshot of the Related Articles subpage from the [[Biology]]
cluster (which is approved) while logged in. It shows the Parent topics and
the first section of the Subtopics – subdisciplines. For each related article,
there is a short definition or description of the topic, and a link to its
Related Articles subpage (hidden behind the [r]), as well as instructions on
mouseover and a Table of Content. Bottom: Related Articles subpage from the
[[Open Knowledge Conference]] cluster (while logged out) which has not yet
been converted to subpage style but can already be used for structuring
information related to the topic. In principle, on could also think of adding
[[Open Knowledge Conference 2010]] as a subtopic and using this article for
conference blogging. However, the current MediaWiki software cannot handle
parallel editing by multiple users, though tools like Etherpad [9] have shown
that it is feasible.

###  2.3 Contextualization

Citizendium attempts to structure knowledge in a different way. Each article
on Citizendium can make comprehensive use of Subpages, i.e. pages providing
additional information that are subordinate to an article’s page. Some of
these – e.g. the Sculptures subpage in Fig. 1 – are similar to but more
flexible than the supplementary online materials now being published routinely
along scholarly articles. Two subpages types are different, with keywords and
running title being the closest analogues from academic papers: All pages are
encouraged to have a short Definition subpage (around 30 words or 150
characters) which defines or describes the subject of the page. They are also
encouraged to have a comprehensive Related Articles subpage, which uses
templates to pull in the definitions from the pages that it links to (a
feature that relies on the absence of vandalism). If one looks at the Related
Articles subpage of [[Biology]] (cf. Fig. 2, top), one can see the parent
topics of biology (science), the subtopics - subdisciplines of biology like
zoology, genetics and biochemistry, articles on the history of biology and
techniques used by biologists - and finally other related topics, including
material on the life cycle, the various biochemical substances like DNA and
proteins, the components of the cell, and other specialised language. This
Related Articles page gives a pretty comprehensive contextual introduction to
what biology is all about, and is structured by the authors of the article in
a way that is consistent across the site (cf. Fig. 2, bottom). This goes
beyond Wikipedias categories, ”See also” sections and ad-hoc infoboxes.
Citizendium’s approach can be considered as an exploratory next step towards
linking encyclopaedic content with the Semantic Web.

Subpages (a further usage example is in (cf. Fig. 3) are one way in which
Citizendium is attempting to go beyond what is provided in either traditional
paper-based encyclopaedias or by Wikipedia: to engage with context, with
related forms of knowledge, and to emancipate knowledge from the page format
to which it was confined in the print era. Marx wrote that ”Philosophers have
hitherto only interpreted the world in various ways; the point is to change
it” [11]. Traditional encyclopaedias attempt to reflect the world, but we are
attempting to go further. The open science movement - which has formed around
the combination of providing open access to journal articles, making
scientific data more openly available in raw forms, using and sharing open
source software and experimenting with some of the new techniques appearing
from the community that is formed under the ’Web 2.0’ banner - is exploring
the edge of what is now possible for scientists to do to create new knowledge.
Some of the electronic engagements by academics has been for actual research
benefit, some has just been PR for universities - doing podcasts to sound
’relevant’. The Citizendium model, while a little bit more traditional than
some of the open science platforms, is willing to try a variety of new things.
Wikipedia has produced a pretty good first version of a collaboratively
written encyclopedia – the challenge is to see if we can go further and
produce a citizens’ compendium of structured and comprehensive knowledge and
update it as new evidence or insights arise.

Figure 3: Top: Screenshot of the main page of the [[English spellings]]
cluster while logged out. It shows the blue cluster bar that indicates that
the page has not been approved and links to all the supages. Also visible is
the status indicator (red dots on the left topped by grey dots, indicating the
status of a ’developing’ article), and a stronger disclaimer than on approved
pages. Below this standard header, a set of templates links to ’Catalog’
subpages that collect links, for each letter of the English alphabet, to
Alphabetical and Retroalphabetical lists of spellings, to lists of Common
misspellings as well as to an article on the specific Letter. Bottom: Close-up
of the Catalogs subpage hosting the retroalphabetical list of English
spellings for the letter T, again cross-linked with all the other subpages in
that cluster.

###  2.4 Open governance

Citizendium has an evolving, but hopefully soon-to-be clearly defined
governance process - a Charter is in the process of being drafted by an
elected group of writers that will allow for democratic governance and
oversight. The broad outline is this: we will have a democratically elected
Editorial Council which will deal with content policy and resolving disputes
regarding content, and we will also have a Management Committee, responsible
for anything not related to content. The Management Committee appoint
Constables who uphold community policy regarding behaviour. Disputes with the
Constables can be brought to an Ombudsman selected by the Editorial Council
and Management Committee. At the time of writing, the charter is still to be
ratified by the community. One of the reasons we have this is that although
there is a cost to having bureaucracy and democracy, the benefits of having an
open governance process outweigh the costs. We have a real problem when
governments of real-life communities are controlled by shadowy cabals who
invoke byzantine legal codes - all the same problems would seem to apply to
online communities. With a Wikipedia article, the debate seems to shift very
quickly from the truth value or relevance of the content itself into often
ritualized arguments about acronyms (AfDs, NPOV, CSD, ArbCom, OR etc.). There
is always a challenge in any knowledge-based community in attempting to
reconcile a fair and democratic process with a meritocratic respect for
expertise. There are no easy answers - if we go too far towards bureaucracy,
we risk creating a system where management is separated from the actual day-
to-day writing of the site, while if we attempt to let the site ’manage
itself’, we risk creating a rather conservative mob rule that doesn’t afford
due process to interested outsiders. A more traditional management structure,
combined with real names and civility, should help those outside of the online
community - the many experts in real life who work in universities, in
business and in public life - to participate on an equal footing. Hopefully,
if we get the governance decisions right, we can also not get in the way of
the people who engage on hobbyist terms with Citizendium.

###  2.5 Open education

An important part of the governance process is collaboration with partners
external to the Citizendium. One of our initiatives – called Eduzendium –
provides for educators in higher education to assign work on wiki articles as
part of a course. We have most recently had politics students from the
Illinois State University work on articles on pressure groups in American
public life, as well as medical students from Edinburgh, biologists from City
University of New York and the University of Colorado at Boulder, finance
students from Temple University and others. These courses reserve a batch of
articles for the duration of the course, and assign each article to one or
more students. The course instructor can reserve the articles for just the
group of students enrolled in the course, or invite the wider Citizendium
community to participate. Much of the formatting is achieved via course-
specific templates that can be generated semi-automatically by the instructor
and applied throughout the course pages, so that course participants can
concentrate on content.

##  3 Open questions

The project is still young, resulting in a number of challenges and
opportunities. In many fields, Citizendium does not meet our own standards –
we do not have a full range of expert editors. Larry Sanger once envisioned
that Citizendium could reach 100,000 articles by 2012. This would, on average,
require about 150 new articles a day to reach; the current level is around 15.
It is not obvious how the necessary shift from linear to exponential growth
can be achieved.

Motivating both editors and authors to take part in both writing and approving
of content remains a difficult challenge – most experts have very little time
to offer for projects that do not contribute to the metrics according to which
their performance is evaluated, and others shy away from contributing under
their real name and in the presence of experts. Another problem is that the
initial structure of the community, and the nature of its interaction with
Wikipedia, has led to a few articles on popular pseudoscientific topics which
are hard to handle from an editorial perspective because those willing to
invest their time on the topics are usually heavily biased in their approach,
and most of those capable of evidence-based comment prefer not to contribute
to these topics.

The project also needs to allow for more feedback by non-registered readers,
without harming the currently very collegial atmosphere that is to a large
extent due to the real-name policy and the respect for expertise. We may need
to explore how to codify our core policies and collaboration model as a
possible MediaWiki extension, from which other wikis could possibly benefit –
online, ”code is law” [10], as is currently being highlighted by sites like
Stack Overflow which have changed the social interactions of participants by
changing formal features of user experience and social structure. We need to
find financial backing and support. So far, the project has been run on a
basically volunteer-only basis, yet the envisioned growth and improvement of
English-language content and the possible start of branches in other languages
require a higher degree of professionalisation, for which the upcoming Charter
is meant as a basis.

##  4 Open perspectives

Citizendium is open for partnerships with other open science and online
knowledge communities and projects. Possible candidate projects would include,
for instance, AcaWiki [1] for references, OpenWetWare for primary research,
and Open Access journals [5] as possible content providers, and of course the
Wikipedias and other public wikis for exchange on matters of content
management, community development and user experience. The key strength we
think the Citizendium model brings is a greater focus on knowledge
contextualization: it will be interesting to see whether we can evolve the
social model for knowledge production to keep up with changes in the
technological possibilities. Many in the Citizendium community are looking
forward to working alongside both academics and those working in the Semantic
Web community to tie Citizendium into data projects. We feel that despite the
commoditization of Web 2.0 technologies, there is still plenty of
opportunities for reinventing and experimenting with new ways to render and
collaborate on knowledge production and to see if we can build a more stable,
sustainable and collegial atmosphere – with democratic and meritocratic
elements – for experts and the public to work together.

#### Acknowledgments.

The authors wish to thank Russell D. Jones, Howard C. Berkowitz, Steven
Mansour and Peter Schmitt for critical comments on earlier versions of this
draft as well as Claudia Koltzenburg, François Dongier and Charles van den
Heuvel for helpful discussions.

## References

  * [1] AcaWiki,
http://acawiki.org/
All URLs referenced in this article were functional as of March 31, 2010.

  * [2] Citizendium,
http://www.citizendium.org/

  * [3] Clark, D.: Plenary lecture, ”A Cloudy Crystal Ball – Visions of the Future”, Proc. 24th IETF: 539 (1992),
http://www.ietf.org/proceedings/prior29/IETF24.pdf

  * [4] Creative Commons-Attribution-Share Alike license 3.0,
http://creativecommons.org/licenses/by-sa/3.0/

  * [5] Directory of Open Access Journals,
http://www.doaj.org/

  * [6] Dispersive PDE Wiki,
http://tosio.math.utoronto.ca/wiki/

  * [7] Encyclopedia of Cosmos,
http://www.cosmosportal.org/

  * [8] Encyclopedia of Earth,
http://www.eoearth.org

  * [9] Etherpad source code,
http://code.google.com/p/etherpad/

  * [10] Lessig, Lawrence: Code and Other Laws of Cyberspace,
http://codev2.cc/

  * [11] Marx, Karl: Theses on Feuerbach,
http://www.marxists.org/archive/marx/works/1845/theses/theses.htm

  * [12] OpenWetWare,
http://www.openwetware.org/

  * [13] Polymath WIki,
http://michaelnielsen.org/polymath1/

  * [14] Priedhorsky R, Chen J, Lam STK, Panciera K, Terveen L, et al. (2007) Creating, destroying, and restoring value in wikipedia. In: GROUP ’07: Proceedings of the 2007 international ACM conference on Supporting group work. New York, NY, USA: ACM, pp. 259–268. http://doi.acm.org/10.1145/1316624.1316663.
  * [15] Raymond, Eric S.: The Cathedral and the Bazaar,
http://www.catb.org/~esr/writings/homesteading/

  * [16] RNA Biology, Guidelines for the RNA Families Track,
http://www.landesbioscience.com/journals/rnabiology/guidelines/

  * [17] Scholarpedia,
http://www.scholarpedia.org/

  * [18] WikiGenes,
http://www.wikigenes.org/

  * [19] English Wikipedia,
http://en.wikipedia.org

  * [20] Wikiversity,
http://www.wikiversity.org

  * [21] Wilbanks, J.: Publishing science on the web,
http://scienceblogs.com/commonknowledge/2009/07/publishing_science_on_the_web.php


# Constraint-induced breaking and restoration of ergodicity in spin-1 PXP
models

Bhaskar Mukherjee<EMAIL_ADDRESS>Wilczek Quantum Center,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Department of Physics and Astronomy, University of Pittsburgh,
Pittsburgh, PA 15260, USA  Zi Cai<EMAIL_ADDRESS>Wilczek Quantum Center,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Key Laboratory of Artificial Structures and Quantum Control,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Shanghai Research Center for Quantum Sciences, Shanghai 201315,
China  W. Vincent Liu<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, PA 15260, USA  Wilczek Quantum Center,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  T. D. Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China  Shanghai Research Center for Quantum Sciences, Shanghai 201315,
China  Shenzhen Institute for Quantum Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, China

###### Abstract

Eigenstate Thermalization Hypothesis (ETH) has played a pivotal role in
understanding ergodicity and its breaking in isolated quantum many-body
systems. Recent experiment on 51-atom Rydberg quantum simulator and subsequent
theoretical analysis have shown that hardcore kinetic constraint can lead to
weak ergodicity breaking. In this work, we demonstrate, using 1d spin-1 $PXP$
chains, that miscellaneous type of ergodicity can be realized by adjusting the
hardcore constraints between different components of nearest neighbor spins.
This includes ETH violation due to emergent shattering of Hilbert space into
exponentially many subsectors of various sizes, a novel form of non-
integrability with an extensive number of local conserved quantities and
strong ergodicity. We analyze these different forms of ergodicity and study
their impact on the non-equilibrium dynamics of a $\mathbb{Z}_{2}$ initial
state. We use forward scattering approximation (FSA) to understand the amount
of $\mathbb{Z}_{2}$-oscillation present in these models. Our work shows that
not only ergodicity breaking but an appropriate choice of constraints can lead
to restoration of ergodicity as well.

##  I Introduction

Eigenstate thermalization hypothesis (ETH) offers the most widely accepted
mechanism of thermalization of local observables in out of equilibrium closed
quantum many body systems rev1 ; deustch1 ; srednicki1 ; rigol1 . An ETH
satisfying system, prepared in an unentangled product state, gets strongly
entangled quickly under its own dynamics, losing all the information of the
initial state except the conserved quantities (e.g total energy). These
systems are usually strongly interacting in nature which makes the full
quantum system, though well isolated from external environment, suffer from
the presence of an indigenous heat bath. This makes the study of ETH-violating
systems not only of fundamental importance but also of technological
importance from the perspective of quantum information protection, quantum
state preparation and preservation of quantum coherence (which is the measure
of quantumness of a system) up to very long time.

Integrable systems sutherland , possessing an extensive number of conserved
quantities, have long been known to disobey ETH. A prototypical example is
transverse field Ising model (TFIM) which though appears an interacting system
in original spin language, becomes a free system via Jordan-Wigner
transformation sachdev . Many body localized (MBL) systems mblref  which are
mostly one-dimensional interacting quantum system with onsite disorder
potential, forms another class of ETH violating system and has been studied in
detail over the last decade. These systems are examples where we see strong
violation of ETH in the sense all eigenstates violates ETH.

Recently, anomalous oscillation from a density wave ($\mathbb{Z}_{2}$) state
observed in a quench dynamics experiment using a 51-atom Rydberg quantum
simulator expt  has revitalized the interest into the field of thermalization
and its violation. This phenomenon is understood by using spin-1/2 $PXP$ model
which hosts extensive number of ETH violating states with high
$\mathbb{Z}_{2}$-overlap, dubbed as quantum many body scars, in its
spectrumabanin1 . So far, a plethora of study olexei ; Shiraishi ; khemani1 ;
SU2  has not only revealed the full phenomenology of the scar states but they
have been found in a variety of models ranging from different spin models
iadecola1 ; moudgalya1 , Hubbard models etapairing , higher spin PXP
modelsTDVP ; Bull , Floquet systems Floquet , disordered systems onsagerscar ,
quantum Hall system Hall , in higher dimension higherdimension , via
confinement confined  etc. In many of these studies scar states are exactly
constructed either in matrix product state (MPS) form or by repeatedly acting
suitably designed creation operator on some mother state. These kind of
construction quite straightforwardly explains the ETH violating nature of the
scar states. But in spite of being the first experimentally realized model to
host quantum many body scar, a fully satisfactory explanation of scarring in
$PXP$ model is still a open problem. Except a few scar states for which exact
MPS representation was obtained olexei , numerics and semi-analytical
techniques like forward scattering approximation abanin1 ; abanin2 ; SU2 ,
single-mode approximation iadecola4  etc are the only option to study the
majority number of scar states in PXP model.

The spin-1 $PXP$ model TDVP ; Bull  is defined by the Hamiltonian :
$H=-\sum_{i}\mathcal{P}S^{x}_{i}\mathcal{P}$ in a chain of $L$ sites where the
local (per site) Hilbert space is spanned by the eigenstates of $S^{z}$
($|{m}\rangle\equiv|{-}\rangle,|{0}\rangle,|{+}\rangle$ for $m=-1,0,+1$) and
the operator $\mathcal{P}=\prod_{i}\mathcal{P}_{i,i+1}$ characterizes the
constrained Hilbert space. In traditional spin-1 $PXP$ TDVP  model at least
one of two consecutive spin must be in the $|{-}\rangle$ state which fixes the
form of the projector : $\mathcal{P}_{i,i+1}=P_{i}+P_{i+1}-P_{i}P_{i+1}$ with
$P_{i}=|{-}\rangle_{i}\langle{-}|_{i}$. This means,
$|{00}\rangle,|{+0}\rangle,|{0+}\rangle$ and $|{++}\rangle$ type of
configurations are not allowed in the constrained Hilbert space
($\mathbb{H}^{PXP}$). This opens up the question that what happen when
different set of constraints are used. In this work we show that the many body
spectrum of spin-1 $PXP$ model can get dramatically changed when certain
constraints are abolished. To this end we consider three different set of
constraints and construct three corresponding model Hamiltonians.

Three Models: Model-I, II & III are defined by the following Hamiltonians

$H^{\alpha}=\sum_{i=1}^{L}\mathcal{P}^{\alpha}S^{x}_{i}\mathcal{P}^{\alpha}$
(1)

where $\mathcal{P}^{\alpha}=\prod_{i}\mathcal{P}^{\alpha}_{i,i+1}$ for
$\alpha=I,II,III$ and
$\mathcal{P}^{I}_{i,i+1}=\mathcal{P}_{i,i+1}+[(|{+}\rangle\langle{+}|)_{i}\otimes(|{+}\rangle\langle{+}|)_{i+1}]$,
$\mathcal{P}^{II}_{i,i+1}=\mathbb{I}_{i}\otimes\mathbb{I}_{i+1}-[(|{0}\rangle\langle{0}|)_{i}\otimes(|{0}\rangle\langle{0}|)_{i+1}]$,
$\mathcal{P}^{III}_{i,i+1}=\mathcal{P}^{II}_{i,i+1}-[(|{+}\rangle\langle{+}|)_{i}\otimes(|{+}\rangle\langle{+}|)_{i+1}]$.
Note that, these three models are also in $PXP$ form, but to distinguish them
from traditional spin-1 $PXP$ model we use the model index ($\alpha$) in the
superscript.

In Fig.1 we show how these models can be obtained by imposing specific
constraints over a spin-1 free paramagnet ($H=\sum_{i}S^{x}_{i}$) or
abolishing specific constraints from traditional spin-1 $PXP$ model.

Figure 1: Schematic picture of construction of Model-I, II & III by imposing
(along the arrow direction) constraints on the Hilbert space of a spin-1 free
paramagnet and abolishing (opposite to the arrow direction) constraints from
traditional spin-1 $PXP$ model. The constraints (forbidden configurations on a
pair of adjacent sites) are shown on top of the arrows.

We find that if we allow $|{++}\rangle$ configurations on top of
$\mathbb{H}^{PXP}$, which we call Model-I, the spectrum gets shattered into
exponentially many emergent subsectors of different size including
$\mathbb{H}^{PXP}$ as one of the largest block. If we further allow
$|{+0}\rangle/|{0+}\rangle$ type configurations (Model-II), an extensive
number of local conserved quantities arises. This does not make Model-II
exactly solvable due to some degeneracies in the spectrum of the conserved
quantities. In fact, we find that these conserved quantities can be used
atmost to label different sectors of Model-II which are nothing but
disconnected patches of spin-1/2 $PXP$ model of different sizes. If we add
$|{++}\rangle$ type configurations on top of Model-II, all the scar states
disappear and the spectrum become strongly ergodic (Model-III). We demonstrate
the consequence of these different type of ergodicity in the non-equilibrium
dynamics of the $\mathbb{Z}_{2}$ state. Finally we use FSA to understand the
degree of $\mathbb{Z}_{2}$-oscillation as well as ergodic nature of these
three models.

The symmetries of these Hamiltonians include translation, inversion about
center bond/site and particle-hole symmetry characterized by the vanishing
anticommutator of the operator $\mathcal{C}=\prod_{i}(2(S^{z}_{i})^{2}-I_{i})$
with the Hamiltonians : $\\{H^{\alpha},\mathcal{C}\\}=0$ for
$\alpha=I,II,III$. The last symmetry guarantees that if there is an eigenstate
$\psi$ at energy $E$ then there will also be an eigenstate
($\mathcal{C}|{\psi}\rangle$) at $-E$. The Hilbert space dimension grows much
slowly than naive $3^{L}$ depending on the nature of constraints and choice of
boundary conditions (see Table.1). We utilize the first two symmetry and work
in zero momentum and inversion symmetric ($K=0,I=+1$) sector to access largest
possible system. The intertwining of the particle-hole and inversion symmetry
generates an exponentially large number of zero modes zeromode .

|  Model-I |  Model-II |  Model-III
---|---|---|---
Forbidden configurations |  $|{00}\rangle,|{+0}\rangle,|{0+}\rangle$ |  $|{00}\rangle$ |  $|{00}\rangle,|{++}\rangle$
$d_{L}^{OBC}$ |  $\approx 2.247^{L}$ |  $\frac{(1-\sqrt{3})^{L}(\sqrt{3}-2)+(1+\sqrt{3})^{L}(\sqrt{3}+2)}{2\sqrt{3}}$ $\approx 2.732^{L}$ |  $\frac{(1-\sqrt{2})^{L+1}+(1+\sqrt{2})^{L+1}}{2}$ $\approx 2.414^{L}$
Feature |  Emergent Hilbert space shattering |  Exponentially many local conserved quantities ; non-integrable |  Strongly Ergodic.
Table 1: Hardcore constraints, scaling of Hilbert space dimension in open
boundary condition (OBC)(see supp  for derivations) and main features of the
spectrum of Model-I, II & III.

The spectrum of Model-I posses a lot of degeneracies at nonzero (including
integers). We explore the connectivity of states in the Hilbert space of
Model-I and find that it (even each momentum and inversion symmetry sector) is
shattered into exponentially many emergent subsectors khemani2 ; pollman1 (see
Fig.2(a)). The lowest possible size (in $S^{z}$ basis) of such emergent blocks
is one, hence these are unentangled, zero energy eigenstates of $H^{I}$. These
inert states remain frozen under the dynamics generated by the Hamiltonian.
The number of such inert eigenstates ($I_{L}$) scales as $I_{L}\sim\phi^{L}$
in large $L$ limit where $\phi=\frac{\sqrt{5}-1}{2}$ is the Fibonacci number
and the proportionality constant depends on the choice of boundary condition
(see supp ). The eigenstates in low ($>1$) dimensional subsectors have very
small but nonzero entanglement, some of them also have interesting properties
like integer energy and magnetization. The minimally entangled states (in the
central region of the spectrum) are eigenstates of a subsector of size
$3\times 3$. These eigenstates (with energy $E=\pm 1,0$) are given by
$|{\psi_{E}}\rangle=\frac{1}{2}|{\psi_{-}}\rangle-\frac{e^{i\pi
E}}{2}|{\psi_{+}}\rangle+\frac{E}{\sqrt{2}}|{\psi_{0}}\rangle$ where
$|{\psi_{m}}\rangle=\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}[\otimes_{i=1}^{L-3}|{\color[rgb]{1,0,0}{+}}\rangle_{i}\otimes|{\color[rgb]{1,0,0}{-}\color[rgb]{0,0,1}{m}\color[rgb]{1,0,0}{-}}\rangle]$;
$m=\pm,0$ and $T$ is the translation operator. Red (blue) color is used to
denote inert (active) sites. The half chain entanglement entropy ($S_{L/2}$)
of these states are found to be
$S_{L/2}=\ln(\frac{2L}{L-4})+\frac{4}{L}\ln(\frac{L-4}{2})$ (see supp ) which
assumes area law behavior in thermodynamic limit (see inset of Fig.2(b)). We
also find that some special states with $E=\pm 2$ (belonging to subsectors of
size $6\times 6$) has logarithmic entanglement entropy ($S_{L/2}=\ln L-\ln 2$
; see supp ). The magnetization ($S_{z}=\sum_{i=1}^{L}S^{z}_{i}$) of such
eigenstates with energy $E=\pm n$ turns out to be $L-5n$ which are non-
negative at any system size as these type of states don’t appear for $L<5n$
(see supp ). This expectation value is much higher than the corresponding
thermal average ($\langle S_{z}\rangle_{\beta}=Tr[\rho_{\beta}S_{z}]$ with
$\rho_{\beta}=\exp(-\beta H^{I})/Tr[\exp(-\beta H^{I})]$) which is always
negative due to the excessiveness of $|{-}\rangle$ compared to $|{+}\rangle$
states in the constrained Hilbert space of Model-I. Thus such states clearly
violates ETH.

The number of $|{++}\rangle$ configurations
($N_{++}=\sum_{i}(|{++}\rangle\langle{++}|)_{i,i+1}$) turns out to be a
conserved quantity for Model-I which can be used to label (not uniquely)
different subsectors. The block with $N_{++}=0$ is nothing but the traditional
spin-1 $PXP$ model. Note that, this kinetic constraint (no two spin in
$|{+}\rangle$ state can sit next to each other) is emergent in nature as the
underlying Hamiltonian does not have it iadecola3 . The block with $N_{++}=1$
is the largest block in the largest possible system size ($L=16$) we have
explored numerically. Blocks with same $N_{++}$ ($>1$) can be further labeled
by $N_{+++}$ and so on. We note that there are many subsectors with $N_{++}>1$
and $N_{+++}=0$ whose states contains isolated $|{++}\rangle$ configurations,
separated by active sites. Such subsectors can be further labeled by number of
$|{++-++}\rangle$, $|{++--++}\rangle$ etc type of inert configurations. For
example, at $L=10$, there are 3 subsectors with $(N_{++}=2,N_{+++}=0)$ which
can be uniquely labeled by the quantum numbers ($N_{++-++},N_{++--++}$) with
values $(0,0)$, $(0,1)$ and $(1,0)$. For $L>10$, number of such subsectors is
$>3$, consequently unique labeling of them can not be achieved. In fact,
projectors on inert configurations of all system sizes $\leqslant L$ are
conserved quantities of a system of size $L$. Though the inert states are
unentangled, the projectors on them are nonlocal in nature. Moreover, their
number scales exponentially with system size and thus an unique set of
conserved quantities to distinguish different subsectors is lacking. That’s
why this is an emergent shattering of Hilbert space induced by the choice of
constraints. We note that ETH violation due to emergent fragmentation of
Hilbert space was first observed in fractonic circuit with local conservation
of charge and dipole moment khemani2  and in the corresponding Hamiltonian
system pollman1 . Subsequent works showed that Hilbert space fragmentation can
also happen from strict confinementiadecola2 .

Figure 2: Ergodicity property of Model-I, II & III. (a) Shattering of Hilbert
space in Model-I. Size of different emergent subsectors in $K=0,I=+1$ sector
for $L=16$. The traditional spin-1 $PXP$ model is the 2nd largest sector at
this system size. (b) $S_{L/2}$ of all the eigenstates in $K=0,I=+1$ sector of
Model-I for $L=16$. System size scaling of $S_{L/2}$ for the three lowest
entangled state is shown in the inset. (c) $S_{z}$ of all eigenstates in
$K=0,I=+1$ sector of Model-I for $L=16$. The three cluster of states is
characterized by the value of $N_{++}$. (d) $S_{L/2}$ of all the eigenstates
of Model-II which belong to the sectors where central $n$ ($n=2,\cdots L-2$)
sites are labeled by $O=+1$ and the rest are $O=-1$, $L=20$. (e) $S_{L/2}$ and
(f) $S_{z}$ of all the eigenstates in $K=0,I=+1$ sector of Model-III, $L=14$.
In Panel (b),(e) the Page value of $S_{L/2}$ is shown in red dashed line. For
panel (d) Page value ($S_{L/2}^{Page}=9.625$) is not shown. In Panel (c) and
(f) the canonical (Gibbs) ensemble prediction of $S_{z}$ is shown in black
dashed line.

In Model-II only $|{00}\rangle$ type configurations are not allowed. The
spectrum of Model-II also holds nonzero-energy-degeneracies inside the
$K=0,I=+1$ sector. This time we find an extensive ($=L$) number of local
quantities ($O_{i}$) which commutes with the Hamiltonian, i.e
$[O_{i},H^{II}]=0,\forall i$. We find that
$O_{i}=(|{+}\rangle\langle{-}|+|{0}\rangle\langle{0}|+|{-}\rangle\langle{+}|)_{i}=2(S^{x}_{i})^{2}-\mathbb{I}_{i}$
(see supp ). Interestingly, this does not make Model-II completely integrable
as each $O_{i}$ have degenerate eigenstates. The eigenvalues of $O_{i}$ are
$(+1,+1,-1)$ with corresponding eigenstates
$|{O^{+}_{1}}\rangle=\frac{1}{\sqrt{2}}(|{+}\rangle+|{-}\rangle),|{O^{+}_{2}}\rangle=|{0}\rangle,|{O^{-}}\rangle=\frac{1}{\sqrt{2}}(|{+}\rangle-|{-}\rangle)$.
Therefore all eigenstates of $H^{II}$ can’t be uniquely labeled by the
eigenvalues of all $O_{i}$’s as the number of such available quantum numbers
($2^{L}$) is far less than the total number of eigenstates ($\approx
2.7^{L}$). Thus the conserved quantities can only be used to block diagonalize
the Hamiltonian. The largest block is characterized by $O_{i}=+1,\forall i$.
It is easy to see that this sector is equivalent to the celebrated spin-1/2
$PXP$ model with the identification $|{\downarrow}\rangle=|{O^{+}_{1}}\rangle$
and $|{\uparrow}\rangle=|{O^{+}_{2}}\rangle$. Thus, in principle one can
expect to see all phenomenon observed in spin-1/2 $PXP$ model in this spin-1
model also. On the theoretical side, Lin-Motrunicholexei  type exact
eigenstates can also be constructed here, not only for the largest sector but
also for all sectors where island of sites with $O_{i}=+1$ are separated by
sites with $O_{i}=-1$ as these sectors are nothing but disconnected patches of
spin-1/2 $PXP$ model of different sizes. The smallest sector is of size 1 with
$O_{i}=-1$ at all sites, hence $H^{II}$ posses an unentangled, exact zero
energy eigenstates: $\prod_{i}|{O^{-}_{i}}\rangle$. We note that sectors where
at least two consecutive sites have $O_{i}=+1$ feels the hardcore interaction
whereas sectors where each $O_{i}=+1$ site is isolated by at least one
$O_{i}=-1$ from both side is basically non-interacting in nature. The number
of such non-interacting sectors scales as $\sim\phi^{L}$. Thus though many
sectors of this model is non-integrable in nature there exist exponentially
many eigenstates which violates $ETH$.

Model-III does not allow $|{00}\rangle$ and $|{++}\rangle$ type
configurations. This model neither have any conserved quantity (other than
translation, inversion and particle-hole) nor its spectrum have any emergent
shattering. Due to the high connectivity in Hilbert space, this model displays
strongly ergodic behavior (see Fig.2(e),(f)). In the next section we will
analyze it more using FSA and $\mathbb{Z}_{2}$-dynamics.

Figure 3: Schematic structure of a typical symmetry sector of Model-II and
projection of $\mathbb{Z}_{2}$ state in that sector.

$\mathbb{Z}_{2}$ dynamics and FSA : The difference in ergodicity of Model-I,
II & III can be probed by studying the dynamics of local observables (we
choose $O=(|{+}\rangle\langle{+}|)_{i}\otimes(|{+}\rangle\langle{+}|)_{i+1}$)
from the initial state
$|{\mathbb{Z}_{2}}\rangle=\otimes^{L/2}_{i=1}|{-}\rangle_{2i-1}|{+}\rangle_{2i}$.
It is worthy to point out here that all standard spin-s $PXP$ models studied
so far exhibits long lived coherent oscillation starting from the
$\mathbb{Z}_{2}$ stateTDVP . In our Model-I the $\mathbb{Z}_{2}$ state belong
to the emergent subsector $\mathbb{H}^{PXP}$ and so the corresponding dynamics
will be exactly same as that of traditional spin-1 $PXP$ model. On the other
hand, $|{\mathbb{Z}_{2}}\rangle$ state has uniform overlap with all $2^{L}$
symmetry sectors of Model-II as can be seen from the expression :
$|{\mathbb{Z}_{2}}\rangle=2^{-L/2}[\otimes^{L/2}_{i=1}(|{O^{+}_{1}}\rangle-|{O^{-}}\rangle)_{2i-1}(|{O^{+}_{1}}\rangle+|{O^{-}}\rangle)_{2i}]$.
Now how the state evolves inside a particular sector depends crucially on the
interacting nature of that sector. The system will have perfect revival and no
dephasing inside the fully non-interacting sectors whereas inside the
interacting sectors it will thermalize rapidly to infinite temperature (see
Fig.3). This is because the projection of the initial state in the interacting
sectors corresponds to the fully polarized down state
($\cdots\downarrow\downarrow\downarrow\cdots$) whereas the sectors themselves
are nothing but disconnected patches of spin-1/2 $PXP$ model of different
sizes. So the $\mathbb{Z}_{2}$-dynamics in Model-II is a mixture of maximally
thermal and maximally non-thermal effects, as a result neither it shows strong
coherent oscillation nor thermalizes quickly (see Fig.4(a)). We note that
though the number of fully non-interacting sectors ($\sim\phi^{L}$) is a
vanishingly small fraction ($\sim 0.81^{L}$) of total number of sectors in
thermodynamic limit, sectors with small interacting portions also increase
exponentially with system sizes. Therefore the nature of the dynamics in
thermodynamic limit is an interesting open question. Finally, we find that the
$\mathbb{Z}_{2}$-dynamics in Model-III thermalizes rapidly to infinite
temperature due to the strongly ergodic nature of the model. We show
entanglement dynamics of the three model in Fig.4 (b). The initial growth of
entanglement (which is related to the speed of information propagation) is
fastest in Model-III, slowest in Model-I and intermediate in Model-II. This is
consistent with the nature of the dynamics of local observable in these three
models.

Figure 4: (a) Dynamics of a local observable (see text) in the three models.
$L=12$ (b) Dynamics of $S_{L/2}$. $L=12$ (c) Behavior of total FSA errors as a
function of system size.

We now corroborate our dynamics results using FSA which has been very
successful in capturing the scar states in large $PXP$ systemsabanin1 ;
abanin2 . We begin by decomposing $H^{\alpha}$ into two parts :
$H^{\alpha}=H^{\alpha}_{+}+H^{\alpha}_{-}$ such that $H^{\alpha}_{-}$
annihilates the initial state $|{v_{0}}\rangle=|{\mathbb{Z}_{2}}\rangle$ and
$H^{\alpha}_{+}=(H^{\alpha}_{-})^{\dagger}$. The repeated application of
$H^{\alpha}_{+}$ on $|{\mathbb{Z}_{2}}\rangle$ generates the FSA vectors
:$|{v_{n}}\rangle=\frac{1}{\beta_{n}}H^{\alpha}_{+}|{v_{n-1}}\rangle$. For the
spin-1 models in this paper, $n$ runs from 0 to $2L$ after which the state
gets annihilated. The FSA errors quantified by
$\delta^{\alpha}_{n}=||H^{\alpha}_{-}|{v_{n}}\rangle-\beta_{n}|{v_{n-1}}\rangle||,n=1,\cdots
2L$ are a measure of damping force felt by the dynamical system due to many
body interaction effects. The system exhibits undamped oscillations when all
the $\delta_{n}$ are zero which can be seen in a free paramagnet (no
constraints) or by adding suitable perturbation of appropriate strength with
the constrained model SU2 . Though, in general, FSA errors are non-zero in
$PXP$ type constrained systems considered here, it could be zero in first few
steps depending on the nature of the constraints. For example, we find that in
our Model-I, first four FSA steps are exact (i.e error free) and error arises
at fifth ($n_{f}$) step. For Model-II and III error arises at 3rd and 2nd FSA
step respectively. This errors cause dephasing ; higher the errors, higher is
the rate of dephasing. Here we give analytical expression (See supp ) of first
nonzero FSA error ($\delta^{\alpha}_{n_{f}}$) for the three models

$\displaystyle\delta^{I}_{n_{f}(=5)}$ $\displaystyle=$
$\displaystyle\frac{12(L^{3}-6L^{2}+11L-18)}{(L-1)(L-2)(L-3)(5L^{4}-50L^{3}+175L^{2}-250L+144)}$
$\displaystyle\delta^{II}_{n_{f}(=3)}$ $\displaystyle=$
$\displaystyle\frac{50(2L-9)}{(2L-5)(6L^{2}-45L+95)}$
$\displaystyle\delta^{III}_{n_{f}(=2)}$ $\displaystyle=$
$\displaystyle\frac{1}{4(4L-11)}$

In brief, $\delta_{n_{f}}\sim L^{-(n_{f}-1)}$ at large $L$, from which it is
easy to see that
$\delta^{I}_{n_{f}}<\delta^{II}_{n_{f}}<\delta^{III}_{n_{f}}$. We numerically
calculate the FSA errors in higher steps ($n>n_{f}$) and plot the behavior of
total FSA error ($\delta_{tot}=\sum_{n=1}^{2L}\delta_{n}$) in Fig.4. One can
again see that $\delta^{I}_{tot}<\delta^{II}_{tot}<\delta^{III}_{tot}$. This
gives a qualitative understanding of the hierarchy of entanglement growth,
amount of oscillation present in the dynamics of local observable and
ergodicity of the models.

Conclusion & Discussion :  In summary we have studied the change in ergodic
properties of spin-1 $PXP$ model for three specific choices of constraints. We
find that whereas certain set of constraints can shatter the Hilbert space
into exponentially many emergent subsectors, thus leading to violation of ETH,
some other set of constraints can destroy all the anomalous states and make
the spectrum strongly ergodic. Our choice of constraints (only between the
excited states $|{0}\rangle$ and $|{+}\rangle$ in nearest neighbor (n.n)
sites) is motivated by the experimental realization of spin-1/2 $PXP$ model in
Rydberg atom systems where strong repulsive interaction between n.n atoms is
turned on only when they are simultaneously in the excited (Rydberg)
statesexpt . A detail study of all such set of constraints is beyond the scope
of the current workBull . But we have checked that there are other set of
constraints which belong to the first category (i.e in the class of Model-I).
For example, if only $|{+0}\rangle$ and $|{0+}\rangle$ type of constraints are
not allowed, then also the constrained Hilbert space ($d_{L}^{OBC}\sim
2.414^{L}$) gets shattered into many blocks of different sizes. In fact the
special eigenstates of Model-I (with integer energies) are also the
eigenstates of this Model. Note that this model differs from Model-I by only
the presence of $|{00}\rangle$ type of configurations which means the inert
sector or the special eigenstates of Model-I are robust against this change of
constraints. On the experimental side, we note that non-ergodic quantum
dynamics due to emergent kinetic constraint and Hilbert space fragmentation is
recently observed in tilted Fermi-Hubbard model at large tilt potentialexpt2 .
Secondly, we believe that the Model-II is a minimal interacting model where
non-integrability and extensive number of local conserved quantities coexist.
It will be interesting to explore the existence of such models in other type
of systems. Finally, the strong ergodic nature of Model-III is also remarkable
because we find that quantum scars exist and ETH violation happens even when
only $|{++}\rangle$ type of configurations are forbidden. We leave the study
of detailed mechanism behind this constraint induced strong ergodicity and the
exploration of class of constraints which leads to the same behavior as a
future problem.

###### Acknowledgements.

Acknowledgments :  This work is supported by the Shanghai Municipal Science
and Technology Major Project (Grant No. 2019SHZDZX01) [B.M., Z.C., W.V.L.],
National Key Research and Development Program of China (Grants No.
2016YFA0302001 and 2020YFA0309000) and NSFC of China (Grants No. 11674221 and
No. 11574200) [Z.C] and by the AFOSR Grant No. FA9550-16-1-0006 and the MURI-
ARO Grant No. W911NF17-1-0323 through UCSB [W.V. L.].

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##  Appendix A Supplemental material for “Constraint-induced breaking and
restoration of ergodicity in spin-1 PXP models”

##  Appendix B Growth of Hilbert space dimension for Model-I, II & III

The constraints leads to a slower growth of Hilbert space dimension compared
to naive $3^{L}$. Atfirst we demonstrate it for open boundary condition (OBC).
We start by Model-I for which $|{+0}\rangle,|{0+}\rangle,|{00}\rangle$ type of
configurations are not allowed. Any state in a system of size $L$ may end by
$|{-}\rangle,|{0}\rangle$ or $|{+}\rangle$. All states in a $L$ site system
which are ending by $|{-}\rangle$ can be obtained by simply appending a
$|{-}\rangle$ to all states in a $L-1$ site system. The states which are
ending by $|{0}\rangle$ in a $L$ site system can be obtained by appending
$|{-0}\rangle$ to all states in a $L-2$ site system whereas $|{+}\rangle$ can
only be appended if the last site is not in the state $|{0}\rangle$. This
leads to the following recurrence relation of total number of states ($d_{L}$)
in a system of size $L$

$\displaystyle d_{L}$ $\displaystyle=$ $\displaystyle
d_{L}^{-}+d_{L}^{0}+d_{L}^{+}$ (3) $\displaystyle=$ $\displaystyle
d_{L-1}+d_{L-2}+(d_{L-1}-d_{L-3})$ $\displaystyle=$ $\displaystyle
2d_{L-1}+d_{L-2}-d_{L-3}$

which can be cast in the following matrix form

$\left(\begin{array}[]{c}d_{L}\\\ d_{L-1}\\\ d_{L-2}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}2&1&-1\\\ 1&0&0\\\ 0&1&0\\\
\end{array}\right)\left(\begin{array}[]{c}d_{L-1}\\\ d_{L-2}\\\ d_{L-3}\\\
\end{array}\right)$ (4)

The eigenvalues of this matrix are $\approx(2.247,-0.802,0.555)$. It is
difficult to get an exact expression of $d_{L}$ for Model-I but the leading
behavior in the large $L$ limit will be controlled by the largest eigenvalue
2.247 (as this is the only one with magnitude $>1$).

Model-II does not allow $|{00}\rangle$ type configurations only. So,
$|{0}\rangle$ can only be appended if the last site is either in state
$|{+}\rangle$ or $|{-}\rangle$. The recurrence relation for $d_{L}$ is given
by

$\displaystyle d_{L}$ $\displaystyle=$ $\displaystyle
d_{L}^{-}+d_{L}^{0}+d_{L}^{+}$ (5) $\displaystyle=$ $\displaystyle
d_{L-1}+(d^{+}_{L-1}+d^{-}_{L-1})+d_{L-1}$ $\displaystyle=$ $\displaystyle
2d_{L-1}+2d_{L-2}$

which can be cast in the following matrix form

$\left(\begin{array}[]{c}d_{L}\\\ d_{L-1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}2&2\\\ 1&0\\\
\end{array}\right)\left(\begin{array}[]{c}d_{L-1}\\\ d_{L-2}\\\
\end{array}\right)=\left(\begin{array}[]{cc}2&2\\\ 1&0\\\
\end{array}\right)^{L-2}\left(\begin{array}[]{c}d_{2}\\\ d_{1}\\\
\end{array}\right)$ (6)

using $d_{1}=3$ and $d_{2}=8$ one get the expression of $d_{L}$ as a function
of $L$ (see Table.-1 in main text).

In Model-III $|{+0}\rangle/|{0+}\rangle$ type of configurations are not
allowed. States which are ending by $|{0}\rangle(|{+}\rangle)$ in a $L$ site
system can be obtained by appending $|{0}\rangle(|{+}\rangle)$ to all states
of a $L-1$ site system which are not ending by $|{+}\rangle(|{0}\rangle)$.
Therefore,

$\displaystyle d_{L}$ $\displaystyle=$ $\displaystyle
d_{L}^{-}+d_{L}^{0}+d_{L}^{+}$ (7) $\displaystyle=$ $\displaystyle
d_{L-1}+(d_{L-1}-d_{L-1}^{+})+(d_{L-1}-d_{L-1}^{0})$ $\displaystyle=$
$\displaystyle 3d_{L-1}-(d^{0}_{L-1}+d_{L-1}^{+})$ $\displaystyle=$
$\displaystyle 2d_{L-1}+d_{L-2}$

this recurrence relation can be represented in the following matrix form

$\left(\begin{array}[]{c}d_{L}\\\ d_{L-1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}2&1\\\ 1&0\\\
\end{array}\right)\left(\begin{array}[]{c}d_{L-1}\\\ d_{L-2}\\\
\end{array}\right)=\left(\begin{array}[]{cc}2&1\\\ 1&0\\\
\end{array}\right)^{L-2}\left(\begin{array}[]{c}d_{2}\\\ d_{1}\\\
\end{array}\right)$ (8)

the exact expression of $d_{L}$ for Model-I can be calculated using $d_{1}=3$
and $d_{2}=7$ (see Table-1 in main text).

Hilbert space dimension in $PBC$ will be somewhat smaller than the
corresponding number in $OBC$ as some of the configurations( which does not
satisfies the constraints between the two end spins) will be eliminated.

##  Appendix C Special states of Model-I and their properties

###  C.1 Inert states

There are exponentially many states inside the constrained Hilbert space of
Model-I that are not connected by the Hamiltonian with other states and hence
inert khemani1 ; pollman1 . Inert states are unentangled, zero-energy
eigenstates of $H^{I}$. We first consider the most obvious one :
$|{+++\cdots}\rangle$, the product of $|{S_{z}=1}\rangle$ state at all sites.
Starting from this state, one can construct other inert states by inserting
one or more $|{-}\rangle$ in the sea of $|{+}\rangle$ states. There can not be
three consecutive $|{-}\rangle$ state and isolated (i.e surrounded by at least
one $|{-}\rangle$ from both side) $|{+}\rangle$ state, as the presence of such
configurations makes the state active. Note that, due to the same reason,
presence of any $|{0}\rangle$ state is also not allowed. We find that the
exact number of inert states ($I_{L}$) for system size $L$ follows the
recurrence relation $I_{L}=I_{L-1}+I_{L-3}+I_{L-4}$ which can be cast in the
following matrix form

$\left(\begin{array}[]{c}I_{L}\\\ I_{L-1}\\\ I_{L-2}\\\
I_{L-3}\end{array}\right)=\left(\begin{array}[]{cccc}1&0&1&1\\\ 1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ \end{array}\right)\left(\begin{array}[]{c}I_{L-1}\\\
I_{L-2}\\\ I_{L-3}\\\ I_{L-4}\end{array}\right)$ (9)

Interestingly this holds for both $PBC$ and $OBC$. Therefore, raising this
matrix to appropriate power and using suitable boundary conditions we get

$\displaystyle I_{L}^{OBC}$ $\displaystyle=$
$\displaystyle\frac{1}{10}((3+\sqrt{5})\phi^{L}-(3-\sqrt{5})(\frac{-1}{\phi})^{L}+4(\cos(\frac{\pi
L}{2})-2\sin(\frac{\pi L}{2})))$ $\displaystyle I_{L}^{PBC}$ $\displaystyle=$
$\displaystyle 2\cos(\frac{\pi L}{2})+\phi^{L}+\phi^{-L}$ (10)

where $\phi(=\frac{1+\sqrt{5}}{2})$ is the Fibonacci number. It is easy to see
that for large systems $I_{L}\sim c\phi^{L}$ where $c=1(0.523)$ for $PBC(OBC)$
which means there is nearly half amount of inert states in $OBC$ compared to
$PBC$.

###  C.2 Special states with integer energies

Special states with energy $E=\pm 1$ are given by

$|{\psi_{\pm
1}}\rangle=\frac{1}{2}|{\psi_{-}}\rangle+\frac{1}{2}|{\psi_{+}}\rangle\pm\frac{1}{\sqrt{2}}|{\psi_{0}}\rangle$
(11)

where

$\displaystyle|{\psi_{-}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}|{\underset{L-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}-\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{+}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}|{\underset{L-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}+\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{0}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}|{\underset{L-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}0\
\color[rgb]{1,0,0}-}\rangle$ (12)

Next, eigenstates with energy $E=\pm 2$ are given by

$|{\psi_{\pm
2}}\rangle=\pm\frac{1}{4}|{\psi_{--}}\rangle\pm\frac{1}{\sqrt{8}}|{\psi_{+-}}\rangle\pm\frac{1}{4}|{\psi_{++}}\rangle+\frac{1}{2}|{\psi_{0-}}\rangle+\frac{1}{2}|{\psi_{+0}}\rangle\pm\frac{1}{2}|{\psi_{00}}\rangle$
(13)

where

$\displaystyle|{\psi_{--}}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{L}}\sum_{n=1}^{L/2}T^{n}|{\underset{\frac{L}{2}-3}{\underbrace{{\color[rgb]{1,0,0}++\cdots++}}}{\color[rgb]{1,0,0}-}{\color[rgb]{0,0,1}-}{\color[rgb]{1,0,0}-}\underset{\frac{L}{2}-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}-\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{+-}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}|{\underset{\frac{L}{2}-3}{\underbrace{{\color[rgb]{1,0,0}++\cdots++}}}{\color[rgb]{1,0,0}-}{\color[rgb]{0,0,1}+}{\color[rgb]{1,0,0}-}\underset{\frac{L}{2}-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}-\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{++}}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{L}}\sum_{n=1}^{L/2}T^{n}|{\underset{\frac{L}{2}-3}{\underbrace{{\color[rgb]{1,0,0}++\cdots++}}}{\color[rgb]{1,0,0}-}{\color[rgb]{0,0,1}+}{\color[rgb]{1,0,0}-}\underset{\frac{L}{2}-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}+\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{0-}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}|{\underset{\frac{L}{2}-3}{\underbrace{{\color[rgb]{1,0,0}++\cdots++}}}{\color[rgb]{1,0,0}-}{\color[rgb]{0,0,1}0}{\color[rgb]{1,0,0}-}\underset{\frac{L}{2}-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}-\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{+0}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{n=1}^{L}T^{n}|{\underset{\frac{L}{2}-3}{\underbrace{{\color[rgb]{1,0,0}++\cdots++}}}{\color[rgb]{1,0,0}-}{\color[rgb]{0,0,1}+}{\color[rgb]{1,0,0}-}\underset{\frac{L}{2}-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}0\color[rgb]{1,0,0}-}\rangle$
$\displaystyle|{\psi_{00}}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{L}}\sum_{n=1}^{L/2}T^{n}|{\underset{\frac{L}{2}-3}{\underbrace{{\color[rgb]{1,0,0}++\cdots++}}}{\color[rgb]{1,0,0}-}{\color[rgb]{0,0,1}0}{\color[rgb]{1,0,0}-}\underset{\frac{L}{2}-3}{\underbrace{\color[rgb]{1,0,0}++\cdots++}}\color[rgb]{1,0,0}-\color[rgb]{0,0,1}0\color[rgb]{1,0,0}-}\rangle$
(14)

Red (blue) colored sites are inert (active). Note that there is a structural
resemblance between the states in Eq.(11),(12) and in Eq.(13),(14). The later
type of states (in a system of size $L$) is obtained by a spatial addition of
different combinations of the former type of states (in a system of size
$L/2$). This also explains the additivity of their energies.

####  C.2.1 Proof of eigenstates

Here we prove that the special states in Eq. (11), (13) are exact eigenstates
of Model-I. We find that the following relations hold for the states in Eq.
(12) and Model-I.

$\displaystyle H^{I}|{\psi_{-}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|{\psi_{0}}\rangle$ $\displaystyle
H^{I}|{\psi_{+}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|{\psi_{0}}\rangle$ $\displaystyle
H^{I}|{\psi_{0}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|{\psi_{+}}\rangle+|{\psi_{-}}\rangle)$ (15)

These relations together gives $H^{I}|{\psi_{\pm 1}}\rangle=\pm|{\psi_{\pm
1}}\rangle$

Similarly for the states in Eq. (14), we find

$\displaystyle H^{I}|{\psi_{--}}\rangle$ $\displaystyle=$
$\displaystyle|{\psi_{0-}}\rangle$ $\displaystyle H^{I}|{\psi_{+-}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|{\psi_{0-}}\rangle+|{\psi_{+0}}\rangle)$
$\displaystyle H^{I}|{\psi_{++}}\rangle$ $\displaystyle=$
$\displaystyle|{\psi_{+0}}\rangle$ $\displaystyle H^{I}|{\psi_{0-}}\rangle$
$\displaystyle=$
$\displaystyle|{\psi_{--}}\rangle+|{\psi_{00}}\rangle+\frac{1}{\sqrt{2}}|{\psi_{+-}}\rangle$
$\displaystyle H^{I}|{\psi_{+0}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|{\psi_{+-}}\rangle+|{\psi_{++}}\rangle+|{\psi_{00}}\rangle$
$\displaystyle H^{I}|{\psi_{00}}\rangle$ $\displaystyle=$
$\displaystyle|{\psi_{0-}}\rangle+|{\psi_{+0}}\rangle$

These relations together gives $H^{I}|{\psi_{\pm 2}}\rangle=\pm 2|{\psi_{\pm
2}}\rangle$.

####  C.2.2 Magnetization

Here we show that the special eigenstates have integer magnetization
($S_{z}=\sum_{i}S^{z}_{i}$).

For states in Eq.(11)

$\langle{\psi_{\pm 1}}|S_{z}|{\psi_{\pm
1}}\rangle=\frac{L-6}{4}+\frac{L-4}{4}+\frac{L-5}{2}=L-5$ (17)

For states in Eq.(13)

$\langle{\psi_{\pm 2}}|S_{z}|{\psi_{\pm
2}}\rangle=\frac{L-12}{16}+\frac{L-10}{8}+\frac{L-8}{16}+\frac{L-11}{4}+\frac{L-9}{4}+\frac{L-10}{4}=L-10$
(18)

Similarly for states with energy $E=\pm n$, magnetization will be $L-5n$. We
note that there are lots of eigenstates at integer energies in Model-I. The
specialty of integer energy eigenstates (e.g in (11),(13)) studied in this
work is that they have minimum entanglement entropy and maximum magnetization
in the corresponding manifold of states.

####  C.2.3 Entanglement entropies

The entanglement of a state can be quantified using various schemes. We use
Von-Neumann formula which works in the following way : first divide the full
system $AB$ into two parts, $A$ and $B$. Then the entropy of entanglement of
part $A$ with part $B$ is given by $S_{A}=Tr[\rho_{A}\ln(\rho_{A})]$ where
$\rho_{A}=Tr_{B}[\rho_{AB}]$ is the reduced density matrix of part $A$ and
$\rho_{AB}$ is the density matrix corresponding to the state of the full
system. The size of $A$ and $B$ can be anywhere in between 1 and $L-1$ with
the constraint $L_{A}+L_{B}=1$. As we use $PBC$ for all our calculation, the
full system $AB$ will be in $PBC$ we have to treat the individual part $A$ and
$B$ using $OBC$. One needs to be careful while taking the partial trace of the
full density matrix (to ensure $Tr[\rho_{A}]=1$) as due to the constraints in
Hilbert space, the state of $AB$ is not tensor product of states in $A$ and
$B$ (see Fig.5).

We first derive the single site reduced density matrices ($\rho_{1}$) for the
special states in Eq.(11). The local HSD per site is 3, so $\rho_{1}$ is a
$3\times 3$ matrix, whose diagonal elements for the states in Eq.(11) are

$\displaystyle\rho_{1}^{++}$ $\displaystyle=$
$\displaystyle\frac{1}{4}.\frac{1}{L}.(L-3)+\frac{1}{4}.\frac{1}{L}(L-2)+\frac{1}{2}.\frac{1}{L}.(L-3)$
$\displaystyle=$ $\displaystyle 1-\frac{11}{4L}$ $\displaystyle\rho_{1}^{00}$
$\displaystyle=$ $\displaystyle\frac{1}{2}.\frac{1}{L}$
$\displaystyle\rho_{1}^{--}$ $\displaystyle=$
$\displaystyle\frac{1}{4}.\frac{1}{L}.3+\frac{1}{4}.\frac{1}{L}.2+\frac{1}{2}.\frac{1}{L}.2=\frac{9}{4L}$
(19)

The off-diagonal elements are

$\displaystyle\rho_{1}^{+0}$ $\displaystyle=$
$\displaystyle\rho_{1}^{0+}=\frac{1}{2}.\frac{1}{\sqrt{2}}.\frac{1}{L}$
$\displaystyle\rho_{1}^{+-}$ $\displaystyle=$
$\displaystyle\rho_{1}^{-+}=\frac{1}{2}.\frac{1}{2}.\frac{1}{L}$
$\displaystyle\rho_{1}^{0-}$ $\displaystyle=$
$\displaystyle\rho_{1}^{-0}=\frac{1}{2}.\frac{1}{\sqrt{2}}.\frac{1}{L}$ (20)

note that all matrix elements except one diagonal element die out in the
thermodynamic limit which means $\displaystyle{\lim_{L\to\infty}S_{1}=0}$.
Therefore, in the thermodynamic limit, any site is unentangled with the rest
of the system. This holds for the state in Eq.(13) also.

Now we concentrate on half chain entanglement entropy. Though the dimension of
the corresponding reduced density matrices ($\rho_{L/2}$) scales as $\approx
2.249^{L/2}$, for the states in Eq.(11) we find only six eigenvalues are
nonzero at any system size $L$, which are $\frac{L-4}{2L}$ (multiplicity 2)
and $\frac{1}{L}$ (multiplicity 4). This gives
$S_{L/2}=-2.\frac{L-4}{2L}\ln(\frac{L-4}{2L})-4.\frac{1}{L}\ln(\frac{1}{L})$
which has been plotted in Fig. 5. The decrease of $S_{L/2}$ with $L$ and its
saturation to the area law value ($\ln 2$) in asymptotically large system size
is an artifact of the non-tensorproduct structure of the constrained Hilbert
space. For the states in Eq.(13) we find the number of nonzero eigenvalues to
be $L/2$ which are all equal with magnitude $\frac{2}{L}$. This gives
$S_{L/2}=\frac{L}{2}.\frac{2}{L}.\ln(\frac{2}{L})=\ln(L)-\ln(2)$.

Figure 5: left : Partition of a constrained system into two parts. right :
$S_{L/2}$ as a function of $L$ for the states in (11)

##  Appendix D Conserved quantities of Model-II

Here we show that the operators $O_{i}$ commutes with the Hamiltonian
$H^{II}$. We expand the relevant portion of $H^{II}$

$\displaystyle\mathcal{P}^{II}_{i-1,i}S^{x}_{i}\mathcal{P}^{II}_{i,i+1}$
$\displaystyle=$ $\displaystyle\mathbb{I}_{i-1}\otimes
S^{x}_{i}\otimes\mathbb{I}_{i+1}-\frac{1}{\sqrt{2}}\mathbb{I}_{i-1}\otimes(|{+}\rangle\langle{0}|+|{-}\rangle\langle{0}|)_{i}\otimes(|{0}\rangle\langle{0}|)_{i+1}-\frac{1}{\sqrt{2}}(|{0}\rangle\langle{0}|)_{i-1}\otimes(|{0}\rangle\langle{+}|+|{0}\rangle\langle{-}|)_{i}\otimes\mathbb{I}_{i+1}$
(21)

where we have used that
$S^{x}_{i}=\frac{1}{\sqrt{2}}(|{+}\rangle\langle{0}|+|{0}\rangle\langle{+}|+|{0}\rangle\langle{-}|+|{-}\rangle\langle{0}|)_{i}$
and
$\mathcal{P}^{II}_{i,i+1}=\mathbb{I}_{i}\otimes\mathbb{I}_{i+1}-(|{0}\rangle\langle{0}|)_{i}\otimes(|{0}\rangle\langle{0}|)_{i+1}$.
It is easy to see that each part of Eq.(21) individually commutes with
$O_{i}(=|{+}\rangle\langle{-}|+|{0}\rangle\langle{0}|+|{-}\rangle\langle{+}|)$.
Hence, $[H^{II},O_{i}]=0,\forall i$.

##  Appendix E Forward Scattering Approximation

Forward scattering approximation (FSA) has been an important tool to analyze
the scar induced oscillation since its first usage in spin-1/2 PXP
modelabanin1 . Here we will apply FSA in detail in our spin-1 models Bull .
The essential idea is to first break the model Hamiltonians in $H^{+}$ and
$H^{-}$ (one is conjugate transpose to other) such that one (lets say $H^{-}$)
annihilate the state $|{\mathbb{Z}_{2}}\rangle$ and the other ($H^{+}$)
annihilate $|{\bar{\mathbb{Z}}_{2}}\rangle$. One state
($|{\bar{\mathbb{Z}}_{2}}\rangle$ / $|{\mathbb{Z}_{2}}\rangle$ ) can be
obtained from the other ($|{\mathbb{Z}_{2}}\rangle$ /
$|{\bar{\mathbb{Z}}_{2}}\rangle$) by repeated action ($2L$ times) of ($H^{+}$
/ $H^{-}$). The oscillatory dynamics then can be visualized as a coherent
forward and backward scattering in between these two states.

We define the n’th FSA vector ($|{v_{n}}\rangle)$) by
$|{v_{n}}\rangle=\frac{1}{\beta_{n}}H^{+}|{v_{n-1}}\rangle$ where
$|{v_{0}}\rangle=|{\mathbb{Z}_{2}}\rangle$ and $\beta_{n}$ is the
normalization constant. Due to the choice of the initial state and structure
of the Hamiltonian, FSA vectors form a closed orthonormal subspace of
dimension $2L+1$. Representation of the Hamiltonian in this subspace forms a
tridiagonal matrix :
$H_{FSA}=\beta_{n}\sum_{n=1}^{2L}|{v_{n}}\rangle\langle{v_{n+1}}|+h.c$. Thus,
if one is interested only in the scar subspace, an enormous simplification can
be achieved, namely, one need to diagonalize a matrix whose dimension scales
only linearly with system size. This enables to deal with larger system size.
In Fig. 6 we have compared the $\mathbb{Z}_{2}$-overlap of the eigenstates of
$H_{FSA}$ and fidelity dynamics of the $\mathbb{Z}_{2}$ state generated by
$H_{FSA}$ with the same quantities obtained by the full Hamiltonian ($H^{I}$)
for Model-I.

Figure 6: Comparison of FSA with exact numerics for Model-I (a)
$\mathbb{Z}_{2}$-overlap (b) Fidelity dynamics of the $\mathbb{Z}_{2}$ state.
L=10.

To calculate other quantities (e.g. some observables, entanglement entropy
etc) one needs to store the FSA vectors which again consumes exponential
memory (though this scales as $\sim Le^{L}$ but not as $\sim e^{2L}$.). The
mismatch of the FSA results with exact numerics in Fig. 6 is due to the fact
that the FSA vectors does not form a complete set and hence can’t span the
Hilbert space, that’s why the leakage of the $\mathbb{Z}_{2}$ dynamics out of
the scar (/FSA) manifold is inevitable. This causes the revival amplitude to
decrease and dephase the dynamics. This can be quantified using the FSA errors
: $\delta_{n}=||H^{-}|{v_{n}}\rangle-\beta_{n}|{v_{n-1}}\rangle||$. If the
action of $H^{-}$ on a FSA vector completely undo the action of $H^{+}$ on the
same vector, $\delta$ is zero and the corresponding FSA step is exact. In an
ideal paramagnet (described by $H=\sum_{i}S^{x}_{i}$) all FSA steps are exact
but due to the constraints induced by the projectors, FSA errors are nonzero
in $PXP$ models. Interestingly suitable term can be added to the bare $PXP$
model which can reduceFloquet  the FSA errors even to zeroSU2 . In this
situation the $\mathbb{Z}_{2}$ dynamics remains totally confined in the scar
manifold which results in an undamped oscillation up to very long time. This
suggests that the FSA errors produces some kind of frictional force in the
system and hence the amplitude of oscillation (/degree of fidelity revival)
should be inversely proportional to the total (summed over all the FSA steps)
amount of FSA errors present in the system. Motivated by these arguments, we
plan to perform a detail analysis of the FSA errors for Model-I–III.

###  E.1 FSA for Model-I

In Model-I $|{00}\rangle$,$|{+0}\rangle$,$|{0+}\rangle$ type of configurations
are not allowed. We start with

$H^{+}_{I}|{v_{0}}\rangle=\frac{1}{\sqrt{2}}\sum_{i\
even}|{\cdots\mathop{0}_{i}\cdots}\rangle$ (22)

where $\cdots$ represents sea of repeated $+-/-+$ configurations (with a
suitably added $+$ or $-$ at the end). So,
$\beta_{1}=\sqrt{||H^{+}_{I}|{v_{0}}\rangle||}=\sqrt{L}/2$.

$|{v_{1}}\rangle=\frac{1}{\beta_{1}}H^{+}_{I}|{v_{0}}\rangle=\sqrt{\frac{2}{L}}\sum_{i\
even}|{\cdots\mathop{0}_{i}\cdots}\rangle$ (23)

One can easily check : $H^{-}_{I}|{v_{1}}\rangle=\beta_{1}|{v_{0}}\rangle$.
Therefore, $\delta^{I}_{1}=0$.
Next,

$\displaystyle H^{+}_{I}|{v_{1}}\rangle=\frac{1}{\sqrt{L}}(\sum_{i\
even}|{\cdots-\mathop{-}_{i}-\cdots}\rangle+2\sum_{\begin{subarray}{c}i,j\
even\\\ i\neq
j\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots}\rangle)$
(24)

There are $\frac{L}{4}(\frac{L}{2}-1)$ distinct 2nd type of states. The 2nd
summation should be understood as a sum over only this many number of states
(that’s why we bring a factor 2 before it). All such summation notations in
this paper avoid double counting. We get,
$\beta_{2}=\sqrt{||H^{+}_{I}|{v_{1}}\rangle||}=\sqrt{\frac{1}{L}.\frac{L}{2}+\frac{4}{L}.\frac{L}{4}.(\frac{L}{2}-1)}=\frac{\sqrt{L-1}}{\sqrt{2}}$.
and $|{v_{2}}\rangle=\frac{1}{\beta_{2}}H^{+}_{I}|{v_{1}}\rangle$. It is easy
to see that $H^{-}_{I}|{v_{2}}\rangle=\beta_{2}|{v_{1}}\rangle$. $\therefore$
$\delta^{I}_{2}=0$.

Next we find

$\displaystyle
H^{+}_{I}|{v_{2}}\rangle=\sqrt{\frac{1}{L(L-1)}}(3\sum_{\begin{subarray}{c}i,j\
even\\\ i\neq
j\end{subarray}}|{\cdots\mathop{0}_{i}\cdots-\mathop{-}_{j}-\cdots}\rangle+6\sum_{\begin{subarray}{c}i,j,k\
even\\\ i\neq j\neq
k\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots\mathop{0}_{k}\cdots}\rangle)$
(25)

the number of distinct 2nd type of states is
$\frac{1}{3!}\frac{L}{2}(\frac{L}{2}-1)(\frac{L}{2}-2)=\frac{L(L-2)(L-4)}{48}$.
Therefore,
$\beta_{3}^{2}=\frac{1}{L(L-1)}(9.\frac{L}{2}.(\frac{L}{2}-1)+36.\frac{L(L-2)(L-4)}{48})=\frac{3(L-2)}{4}$.
$|{v_{3}}\rangle=\frac{1}{\beta_{3}}H^{+}_{I}|{v_{2}}\rangle$ and we again
find $H^{-}_{I}|{v_{3}}\rangle=\beta_{3}|{v_{2}}\rangle$ which means
$\delta^{3}_{I}=0$.

Next we find

$\displaystyle
H^{+}_{I}|{v_{3}}\rangle=\frac{\sqrt{2}}{\sqrt{3L(L-1)(L-2)}}(3\sum_{\begin{subarray}{c}i,j\
even\\\ i\neq
j\end{subarray}}|{\cdots-\mathop{-}_{i}-\cdots}\rangle+12\sum_{\begin{subarray}{c}i,j,k\
even\\\ i\neq j\neq
k\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots-\mathop{-}_{k}-\cdots}\rangle+$
$\displaystyle 24\sum_{\begin{subarray}{c}i,j,k,l\ even\\\ i\neq j\neq k\neq
l\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots\mathop{0}_{k}\cdots\mathop{0}_{l}\cdots}\rangle)$
(26)

Therefore,
$\beta_{4}^{2}=||H^{+}_{I}|{v_{3}}\rangle||=\frac{2}{3L(L-1)(L-2)}(36.\frac{L}{4}(\frac{L}{2}-1)+144.\frac{L(L-2)(L-4)}{16}+576.\frac{L(L-2)(L-4)(L-6)}{384})=L-3$
where we have used the fact that the number of distinct 2nd and 3rd type of
states in Eq.(26) are $\frac{L(L-2)(L-4)}{16}$ and
$\frac{L(L-2)(L-4)(L-6)}{384}$ respectively.
$|{v_{4}}\rangle=\frac{1}{\beta_{4}}H^{+}_{I}|{v_{3}}\rangle$ and one can
again check $\delta^{I}_{4}=0$.Note that the first part of $H^{+}_{I}$ (i.e
$\sum_{i\ odd}\sigma^{+}_{i}$) has null effect till now.

Finally we arrive at the 5th step where non-zero FSA error arises for the
first time. Calculations of FSA vectors become cumbersome from this step
onward as both part of $H^{+}_{I}$ will now have non-zero actions. After
regrouping all similar type of states we write a consolidated expression of
the action of $H^{+}_{I}$ on $|{v_{4}}\rangle$

$\displaystyle
H^{+}_{I}|{v_{4}}\rangle=\frac{1}{\sqrt{3L(L-1)(L-2)(L-3)}}(30\sum_{\begin{subarray}{c}i,j,k\
even\\\ i\neq j\neq
k\end{subarray}}|{\cdots\mathop{0}_{i}\cdots-\mathop{-}_{j}-\cdots-\mathop{-}_{k}-\cdots}\rangle+6\sum_{i\
odd}|{\cdots--\mathop{0}_{i}--\cdots}\rangle+$ $\displaystyle
60\sum_{\begin{subarray}{c}i,j,k,l\ even\\\ i\neq j\neq k\neq
l\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots\mathop{0}_{k}\cdots-\mathop{-}_{l}-\cdots}\rangle+120\sum_{\begin{subarray}{c}i,j,k,l,m\
even\\\ i\neq j\neq k\neq l\neq
m\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots\mathop{0}_{k}\cdots\mathop{0}_{l}\cdots\mathop{0}_{m}\cdots}\rangle)$
(27)

Counting the respective number of different states in Eq.(27) we get

$\displaystyle\beta_{5}^{2}$ $\displaystyle=$
$\displaystyle||H^{+}_{I}|{v_{4}}\rangle||$ (28) $\displaystyle=$
$\displaystyle\frac{1}{3L(L-1)(L-2)(L-3)}[30^{2}.\frac{L(L-2)(L-4)}{16}+36.\frac{L}{2}+60^{2}.\frac{L(L-2)(L-4)(L-6)}{96}$
$\displaystyle+120^{2}.\frac{L(L-2)(L-4)(L-6)(L-8)}{3840}]$ $\displaystyle=$
$\displaystyle\frac{5L^{4}-50L^{3}+175L^{2}-250L+144}{4(L-1)(L-2)(L-3)}$

The action of $H^{-}_{I}$ on
$|{v_{5}}\rangle(=\frac{1}{\beta_{5}}H^{+}_{I}|{v_{4}}\rangle)$ gives after
grouping same type of states together

$\displaystyle H^{-}_{I}|{v_{5}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\beta_{5}\sqrt{6L(L-1)(L-2)(L-3)}}[15(L-4)\sum_{\begin{subarray}{c}i,j\
even\\\ i\neq
j\end{subarray}}|{\cdots-\mathop{-}_{i}-+-\cdots-\mathop{-}_{j}-\cdots}\rangle+$
(29) $\displaystyle(15L-54)\sum_{i\
odd}|{\cdots--\mathop{-}_{i}--\cdots}\rangle+30(L-4)\sum_{\begin{subarray}{c}i,j,k\
even\\\ i\neq j\neq
k\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots-\mathop{-}_{k}-\cdots}\rangle+$
$\displaystyle 60(L-4)\sum_{\begin{subarray}{c}i,j,k,l\ even\\\ i\neq j\neq
k\neq
l\end{subarray}}|{\cdots\mathop{0}_{i}\cdots\mathop{0}_{j}\cdots\mathop{0}_{k}\cdots\mathop{0}_{l}\cdots}\rangle]$

note that the 1st and 2nd type of states in Eq.(29) had same strength in
$|{v_{4}}\rangle$ which is sufficient to see that $H^{-}_{I}|{v_{5}}\rangle$
is not proportional to $|{v_{4}}\rangle$. Thus finally error arises in 5th FSA
step which can be easily calculated by evaluating the norm :
$\delta^{I}_{5}=||H^{-}_{I}|{v_{5}}\rangle-\beta_{5}|{v_{4}}\rangle||=\frac{12(L^{3}-6L^{2}+11L-18)}{(L-1)(L-2)(L-3)(5L^{4}-50L^{3}+175L^{2}-250L+144)}$.

###  E.2 FSA for Model-II

Only $|{00}\rangle$ type of configurations are not allowed in Model-II. We
start by

$\displaystyle H^{+}_{II}|{\mathbb{Z}_{2}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\sum_{i\
even}|{\cdots-\mathop{0}_{i}-\cdots}\rangle+\sum_{i\
odd}|{\cdots-+\mathop{0}_{i}+-\cdots}\rangle)$ (30)

So $\beta_{1}=\sqrt{||H^{+}_{II}|{\mathbb{Z}_{2}}\rangle||}=\sqrt{L/2}$ and
$|{v_{1}}\rangle=\frac{1}{\beta_{1}}H^{+}_{II}|{\mathbb{Z}_{2}}\rangle$. It is
easy to see that $H^{-}_{II}|{v_{1}}\rangle=\beta_{1}|{\mathbb{Z}_{2}}\rangle$
and hence $\delta^{II}_{1}=0$. Next,

$\displaystyle H^{+}_{II}|{v_{1}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2L}}[\sum_{i\
even}|{\cdots-\mathop{-}_{i}-\cdots}\rangle+2\sum_{\begin{subarray}{c}i,j\
even\\\ i\neq
j\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots-\mathop{0}_{j}-\cdots}\rangle+2\sum_{\begin{subarray}{c}i\
even\\\ j\
odd\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots+\mathop{0}_{j}+\cdots}\rangle+$
(31) $\displaystyle 2\sum_{\begin{subarray}{c}i,j\ odd\\\ i\neq
j\end{subarray}}|{\cdots+\mathop{0}_{i}+\cdots+\mathop{0}_{j}+\cdots}\rangle+\sum_{i\
odd}|{\cdots+\mathop{+}_{i}+\cdots}\rangle]$

We get $\beta^{2}_{2}=||H^{+}_{II}|{v_{1}}\rangle||=\frac{2L-5}{2}$ and
$|{v_{2}}\rangle=\frac{1}{\beta_{2}}H^{+}_{II}|{v_{1}}\rangle$. It is easy to
check that $H^{-}_{II}|{v_{2}}\rangle=\beta_{2}|{v_{1}}\rangle$ which means
$\delta^{II}_{2}=0$. Rarity of constraints produces a large number of states
in the next step, we write the consolidated expression

$\displaystyle H^{+}_{II}|{v_{2}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2L(2L-5)}}[3\sum_{\begin{subarray}{c}i\ even\\\
j\
odd\end{subarray}}|{\cdots-\mathop{-}_{i}-\cdots+\mathop{0}_{j}+\cdots}\rangle+\sum_{i\
even}(|{\cdots-\mathop{-}_{i}0\cdots}\rangle+|{\cdots
0\mathop{-}_{i}-\cdots}\rangle)+$ (32) $\displaystyle 3\sum_{i,j\
even}|{\cdots-\mathop{-}_{i}-\cdots-\mathop{0}_{j}-\cdots}\rangle+3\sum_{i,j\
odd}|{\cdots+\mathop{+}_{i}+\cdots+\mathop{0}_{j}+\cdots}\rangle+\sum_{i\
even}(|{\cdots+\mathop{+}_{i}0\cdots}\rangle+|{\cdots
0\mathop{+}_{i}+\cdots}\rangle)$ $\displaystyle 3\sum_{\begin{subarray}{c}i\
even\\\ j\
odd\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots+\mathop{+}_{j}+\cdots}\rangle+6\sum_{\begin{subarray}{c}i,j,k\
even\\\ i\neq j\neq
k\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots-\mathop{0}_{j}-\cdots-\mathop{0}_{k}-\cdots}\rangle+$
$\displaystyle 6\sum_{\begin{subarray}{c}i,j,k\ odd\\\ i\neq j\neq
k\end{subarray}}|{\cdots+\mathop{0}_{i}+\cdots+\mathop{0}_{j}+\cdots+\mathop{0}_{k}+\cdots}\rangle+6\sum_{\begin{subarray}{c}i,j\
even\\\ k\
odd\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots-\mathop{0}_{j}-\cdots+\mathop{0}_{k}+\cdots}\rangle$
$\displaystyle 6\sum_{\begin{subarray}{c}i,j\ odd\\\ k\
even\end{subarray}}|{\cdots+\mathop{0}_{i}+\cdots+\mathop{0}_{j}+\cdots-\mathop{0}_{k}-\cdots}\rangle]$

by carefully counting the number of each type of states in Eq.(32) we find

$\beta_{3}^{2}=\frac{1}{2L(2L-5)}[2.(9.\frac{L}{2}.(\frac{L}{2}-2)+L+9.\frac{L}{2}.(\frac{L}{2}-2))+2.(36.\frac{L(L-2)(L-4)}{48}+36.\frac{L(L-4)(L-6)}{16})]=\frac{6L^{2}-45L+95}{4L-10}$
(33)

and $|{v_{3}}\rangle=\frac{1}{\beta_{3}}H^{+}_{II}|{v_{2}}\rangle$. We now
calculate $H^{-}_{II}|{v_{3}}\rangle$

$\displaystyle H^{-}_{II}|{v_{3}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2L(2L-5)}}.\frac{1}{\beta_{3}}[(3.(\frac{L}{2}-2)+2+3.(\frac{L}{2}-1))(\sum_{i\
even}|{\cdots-\mathop{-}_{i}-\cdots}\rangle+\sum_{i\
odd}|{\cdots+\mathop{+}_{i}+\cdots}\rangle)+$ (34)
$\displaystyle(6+6.\frac{L-6}{2}+6.\frac{L-6}{2})\sum_{\begin{subarray}{c}i\
even\\\ j\
odd\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots+\mathop{0}_{j}+\cdots}\rangle+$
$\displaystyle(6+6.\frac{L-4}{2}+6.\frac{L-8}{2})(\sum_{i,j\
even}|{\cdots-\mathop{0}_{i}-\cdots+-+\cdots-\mathop{0}_{j}-\cdots}\rangle+\sum_{i,j\
odd}|{\cdots+\mathop{0}_{i}+\cdots+-+\cdots+\mathop{0}_{j}+\cdots}\rangle)$
$\displaystyle(6+6.\frac{L-4}{2}+6.\frac{L-6}{2})\sum_{i\
even}(|{\cdots-\mathop{0}_{i}-0-\cdots}\rangle+|{\cdots+\mathop{0}_{i}+0+\cdots}\rangle)]$

we then find that
$\delta^{II}_{3}=||H^{-}_{II}|{v_{3}}\rangle-\beta_{3}|{v_{2}}\rangle||=\frac{50(2L-9)}{(2L-5)(6L^{2}-45L+95)}$.

###  E.3 FSA for Model-III

In Model-III the $|{00}\rangle$ and $|{++}\rangle$ type of configurations are
forbidden but $|{+}\rangle$ and $|{0}\rangle$ can sit next to each other. We
start by

$\displaystyle H^{+}_{III}|{\mathbb{Z}_{2}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\sum_{i\
even}|{\cdots-\mathop{0}_{i}-\cdots}\rangle+\sum_{i\
odd}|{\cdots-+\mathop{0}_{i}+-\cdots}\rangle)$ (35)

So $\beta_{1}=\sqrt{||H^{+}_{III}|{\mathbb{Z}_{2}}\rangle||}=\sqrt{L/2}$ and
$|{v_{1}}\rangle=\frac{1}{\beta_{1}}H^{+}_{III}|{\mathbb{Z}_{2}}\rangle$. We
checked that $H^{-}_{III}|{v_{1}}\rangle=\beta_{1}|{\mathbb{Z}_{2}}\rangle$
and hence $\delta^{III}_{1}=0$.

Next

$\displaystyle H^{+}_{III}|{v_{1}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2L}}(\sum_{i\
even}|{\cdots-\mathop{-}_{i}-\cdots}\rangle+2\sum_{\begin{subarray}{c}i,j\
even\\\ i\neq
j\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots-\mathop{0}_{j}-\cdots}\rangle+2\sum_{\begin{subarray}{c}i\
even\\\ j\
odd\end{subarray}}|{\cdots-\mathop{0}_{i}-\cdots+\mathop{0}_{j}+\cdots}\rangle+$
(36) $\displaystyle 2\sum_{\begin{subarray}{c}i,j\ odd\\\ i\neq
j\end{subarray}}|{\cdots+\mathop{0}_{i}+\cdots+\mathop{0}_{j}+\cdots}\rangle)$

counting distinct number of 4 different type of states in Eq.(36), we get
$\beta_{2}^{2}=\frac{1}{2L}[\frac{L}{2}+4.\frac{L}{4}(\frac{L}{2}-1)+4.\frac{L}{2}(\frac{L}{2}-2)+4.\frac{L}{4}(\frac{L}{2}-1)]=L-\frac{11}{4}$.
$|{v_{2}}\rangle=\frac{1}{\beta_{2}}H^{+}_{III}|{v_{1}}\rangle$.

Next we find

$\displaystyle H^{-}_{III}|{v_{2}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L(4L-11)}}[(2L-5)\sum_{i\
even}|{\cdots-\mathop{0}_{i}-\cdots}\rangle+(2L-6)\sum_{i\
odd}|{\cdots+\mathop{0}_{i}+\cdots}\rangle]$ (37)

clearly $H^{-}_{III}|{v_{2}}\rangle$ is not equal to $|{v_{1}}\rangle$
multiplied by $\beta_{2}$ and this produces error in the 2nd FSA step. The
norm of the difference is given by

$\displaystyle\delta^{III}_{2}=||H^{-}_{III}|{v_{2}}\rangle-\beta_{2}|{v_{1}}\rangle||=\frac{1}{4(4L-11)}$
(38)