Let $\mathsf{R_{\mathsf{ab}}}(\A,\B,\C)$ be a recollement of Grothendieck categories with enough projective objects and consider the associated recollement of categories of complexes $\mathsf{R_{\mathsf{ab}}}(\Ch(\A),\Ch(\B),\Ch(\C)),$ as in Remark <ref>.
In what follows, we make use of the standard projective and injective model structures on chain complexes that were recalled before Remark <ref>.

It is easy to see (cf. the discussion in the introductory section  <ref>) that we have a Quillen adjoint triple

\begin{equation}
\label{eq:Quillen_adj_tr_ch}
Q_{\mathrm{tr}}(\Ch(\B)_{\mathsf{proj/inj}}, \Ch(\C)_{\mathsf{proj/inj}}, \mathrm{id})
\end{equation}

as well as a Quillen adjoint triple

\begin{equation}
\label{eq:Quillen_adj_tr_ch2}
Q_{\mathrm{tr}}(\Ch(\A)_{\mathsf{proj/inj}}, \Ch(\B)_{\mathsf{proj/inj}}, \mathrm{id})
\end{equation}

where for both triples the morphisms $f,f',g,'g$ from Definition <ref> are identity morphisms. In particular, it makes sense to ask when $i\colon \Ch(\A)\to \Ch(\B)$ is a derived embedding (as in Definition <ref>).

The following are equivalent:

(i) The functor $i\colon \Ch(\A)\to \Ch(\B)$ is a derived embedding.

(ii) The functor $i\colon \A\to \B$ is a homological embedding.

We just need to check that the assumptions of <cit.> are satisfied. Indeed, since we are working with Grothendieck categories with enough projectives, we have in particular that $\Der(\B)$ is TR5 and TR5$^{*}$ and that $\Der(\A)$ (as well as $\Der(\B)$) is left-complete, see for instance <cit.>. Also the derived functor of $i$ preserves products and coproducts since there is an adjoint triple as in Theorem <ref>. Hence <cit.> applies.

Let $\mathsf{R_{\mathsf{ab}}}(\A,\B,\C)$ be a recollement of Grothendieck categories with enough projective objects. Then there exists a diagram of derived categories

\begin{equation}
\label{eq:der_dia}
\xymatrix@C=4pc{
\Der(\A<EMAIL_ADDRESS>|-{\, \textbf{R}i\cong\textbf{L}i\,}   &  \Der(\B<EMAIL_ADDRESS>^-{\textbf{R}p}<EMAIL_ADDRESS>_-{\textbf{L}q}  \ar[r] |-{\, \textbf{R}e\cong\textbf{L}e \,}&
\Der(\C<EMAIL_ADDRESS>_-{\textbf{L}l}<EMAIL_ADDRESS>^-{\textbf{R}r}
\end{equation}

which is a recollement if and only if the functor $i\colon \Ch(\A)\to \Ch(\B)$ is a derived embedding.

From the above remarks on the existence of the Quillen adjoint triples (<ref>, <ref>), together with Propositions <ref> and <ref> we deduce that the recollement $\mathsf{R_{\mathsf{ab}}}(\Ch(\A),\Ch(\B),\Ch(\C))$ satisfies Setup <ref>, hence from Theorem <ref> there exists a diagram as in (<ref>), which is a recollement if and only if conditions (i) and (ii) of Theorem <ref> hold. Hence, the proof will be over once we argue that condition (ii) of Theorem <ref> holds in the current situation.
From the proof of Theorem <ref>, this boils down to showing that the canonically induced functor $j'\colon \Der(\A)\rightarrow\mathrm{Ker}(\mathbf{R}e)$  is essentially surjective. But this follows from <cit.> (the fact that we can apply this last result in our context was explained in the proof of Proposition <ref>).

We consider a ring $R$ with an idempotent element $e$. Then we have the following  recollement of module categories$\colon$

\[
<EMAIL_ADDRESS>\Mod{R/ReR} \ar[rrr]^{i:=f_*} &&& \Mod{R} \ar[rrr]^{e \ } \ar
@/_1.5pc/[lll]_{q:=R/ReR\otimes_R-}  \ar
@/^1.5pc/[lll]^{p:=\Hom_R(R/ReR,-)} &&& \Mod{eRe}
\ar @/_1.5pc/[lll]_{l:=Re\otimes_{eRe}-} \ar
\]

where $f_*\colon \Mod{R/ReR}\to \Mod{R}$ is the restriction functor induced by the canonical morphism $f\colon R\to R/ReR$. Assume that $f_*$ is a homological embedding, equivalently, the morphism $f$ is a homological ring epimorphism <cit.>, equivalently, the ideal $ReR$ is stratifying <cit.>. In this case, by Cline-Parshall-Scott <cit.> it is known that the above recollement of module categories can be derived, that is, it induces a recollement $\mathsf{R_{\mathsf{tr}}}(\Der(R/ReR), \Der(R), \Der(eRe))$ of derived categories where all six functors are the derived functors of the underlying functors. Such a recollement situation was studied in <cit.> under the name of stratifying recollement, see also <cit.>.
Theorem <ref> implies this well-known result, stated in homotopical terms.

Let $R$ be a ring with an idempotent element $e$. The following statements are equivalent$\colon$

* The functor $i\colon \Ch(R/ReR)\to \Ch(R)$ is a derived embedding.

* There is a recollement of derived categories:

\[
\xymatrix@C=4pc{
\Der(R/ReR<EMAIL_ADDRESS>|-{\, \textbf{R}i\cong\textbf{L}i\,}   &
\Der(R<EMAIL_ADDRESS>^-{\textbf{R}p}<EMAIL_ADDRESS>_-{\textbf{L}q}  \ar[r] |-{\, \textbf{R}e\cong\textbf{L}e \,}&
\Der(eRe<EMAIL_ADDRESS>_-{\textbf{L}l}<EMAIL_ADDRESS>^-{\textbf{R}r} }
\]

AUTHOR = Angeleri Hügel, L.,
author = Koenig, S.,
author = Liu, Q.,
author = Yang, D.
TITLE = Jordan-Hölder theorems for derived module categories of
piecewise hereditary algebras,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 352,
YEAR = 2012,
PAGES = 361–381,
ISSN = 0021-8693,
MRCLASS = 16E35 (18E30),
MRNUMBER = 2862193,
MRREVIEWER = Grzegorz Bobiński,

URL = https://doi.org/10.1016/j.jalgebra.2011.09.041,

AUTHOR = Angeleri Hügel, L.,
author = Koenig, S.,
author = Liu, Q.,
author = Yang, D.
TITLE = On the uniqueness of stratifications of derived module
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 359,
YEAR = 2012,
PAGES = 120–137,
ISSN = 0021-8693,
MRCLASS = 16E35,
MRNUMBER = 2914629,
MRREVIEWER = Octavio Mendoza Hernández,

URL = https://doi.org/10.1016/j.jalgebra.2012.02.022,

AUTHOR = Angeleri Hügel, L.,
author = Koenig, S.,
author = Liu, Q.,
author = Yang, D.
TITLE = Ladders and simplicity of derived module categories,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 472,
YEAR = 2017,
PAGES = 15–66,
ISSN = 0021-8693,
MRCLASS = 16E35 (13C60 18E30 19A99),
MRNUMBER = 3584869,
MRREVIEWER = Octavio Mendoza Hernández,

URL = https://doi.org/10.1016/j.jalgebra.2016.10.023,

AUTHOR = Angeleri Hügel, L.,
author = Koenig, S.,
author = Liu, Q.,
author = Yang, D.
TITLE = Recollements and stratifying ideals,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 484,
YEAR = 2017,
PAGES = 47–65,
ISSN = 0021-8693,
MRCLASS = 16E35 (18E30),
MRNUMBER = 3656712,
MRREVIEWER = Yury Vladimirovich Volkov,

URL = https://doi.org/10.1016/j.jalgebra.2017.04.003,

AUTHOR = Beĭlinson, A. A.,
author = Bernstein, J.,
author = Deligne, P.,
TITLE = Faisceaux pervers,
BOOKTITLE = Analysis and topology on singular spaces, I (Luminy,
SERIES = Astérisque,
VOLUME = 100,
PAGES = 5–171,
PUBLISHER = Soc. Math. France, Paris,
YEAR = 1982,
MRCLASS = 32C38,
MRNUMBER = 751966,
MRREVIEWER = Zoghman Mebkhout,

AUTHOR = Bergh, P. A.,
author = Jorgensen, D. A.,
author = Moore, W. F.,
TITLE = Totally acyclic approximations,
JOURNAL = Appl. Categ. Structures,
FJOURNAL = Applied Categorical Structures. A Journal Devoted to
Applications of Categorical Methods in Algebra, Analysis,
Order, Topology and Computer Science,
VOLUME = 29,
YEAR = 2021,
NUMBER = 4,
PAGES = 729–745,

Author = Borceux, F.,
Title = Handbook of categorical algebra. 2: Categories and structures,
FJournal = Encyclopedia of Mathematics and Its Applications,
Journal = Encycl. Math. Appl.,
ISSN = 0953-4806,
Volume = 51,
ISBN = 978-0-521-06122-3,
Pages = xvii + 443,
Year = 2008,
Publisher = Cambridge: Cambridge University Press,

Author = Becker, H.,
Title = Models for singularity categories,
FJournal = Advances in Mathematics,
Journal = Adv. Math.,
ISSN = 0001-8708,
Volume = 254,
Pages = 187–232,
Year = 2014,
Publisher = Elsevier (Academic Press), San Diego, CA,

Author = Christensen, L. W.,
author = Foxby, H.-B.,
author = Holm, H.,
TITLE =	 Derived Category Methods in Commutative Algebra,
HOWPUBLISHED = Book in preparation,
YEAR= ,
NOTE = Available at ,

AUTHOR = Cline, E.,
author = Parshall, B.
author = Scott, L.
TITLE = Finite-dimensional algebras and highest weight categories,
JOURNAL = J. Reine Angew. Math.,
FJOURNAL = Journal für die Reine und Angewandte Mathematik. [Crelle's
VOLUME = 391,
YEAR = 1988,
PAGES = 85–99,
ISSN = 0075-4102,
MRCLASS = 18E30 (16A46 17B10 20G05 32C38),
MRNUMBER = 961165,
MRREVIEWER = Dieter Happel,

AUTHOR = Cline, E.,
author = Parshall, B.
author = Scott, L.
TITLE = Algebraic stratification in representation categories,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 117,
YEAR = 1988,
NUMBER = 2,
PAGES = 504–521,
ISSN = 0021-8693,
MRCLASS = 18E30 (16A46 17B10 17B56 20G05),
MRNUMBER = 957457,
MRREVIEWER = Dieter Happel,

URL = https://doi.org/10.1016/0021-8693(88)90123-8,

AUTHOR = Cline, E.,
author = Parshall, B.
author = Scott, L.
TITLE = Stratifying endomorphism algebras,
JOURNAL = Mem. Amer. Math. Soc.,
FJOURNAL = Memoirs of the American Mathematical Society,
VOLUME = 124,
YEAR = 1996,
NUMBER = 591,
PAGES = viii+119,
ISSN = 0065-9266,
MRCLASS = 16G10 (20C30),
MRNUMBER = 1350891,

URL = https://doi.org/10.1090/memo/0591,

AUTHOR = Dalezios, G.
author = Estrada, S.
author = Holm, H.,
TITLE = Quillen equivalences for stable categories,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 501,
YEAR = 2018,
PAGES = 130–149,
ISSN = 0021-8693,
MRCLASS = 18E10 (13D02 16E05 18E30 18G55),
MRNUMBER = MR3768129,
MRREVIEWER = Martin Frankland,
URL = https://doi.org/10.1016/j.jalgebra.2017.12.033,

AUTHOR = Dugger, D.
author = Shipley, B.,
TITLE = $K$-theory and derived equivalences,
JOURNAL = Duke Math. J.,
FJOURNAL = Duke Mathematical Journal,
VOLUME = 124,
YEAR = 2004,
NUMBER = 3,
PAGES = 587–617,
ISSN = 0012-7094,
MRCLASS = 19D99 (18E30 55U35),
MRNUMBER = 2085176,
MRREVIEWER = Daniel C. Isaksen,

URL = https://doi.org/10.1215/S0012-7094-04-12435-2,

AUTHOR = Eshraghi, H.,
author = Hafezi, R.,
author = Salarian, Sh.,
TITLE = Total acyclicity for complexes of representations of quivers,
JOURNAL = Comm. Algebra,
FJOURNAL = Communications in Algebra,
VOLUME = 41,
YEAR = 2013,
NUMBER = 12,
PAGES = 4425–4441,

AUTHOR = Gao, N.
author = Koenig, S.
author = Psaroudakis, C.,
TITLE = Ladders of recollements of abelian categories,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 579,
YEAR = 2021,
PAGES = 256–302,
ISSN = 0021-8693,
MRCLASS = 16E10 (16E35 16E65 16G50 16S50 18E10 18G80),
MRNUMBER = 4241239,

AUTHOR = Garner, R.
author = Kȩdziorek, M.
author = Riehl, E.,
TITLE = Lifting accessible model structures,
JOURNAL = J. Topol.,
FJOURNAL = Journal of Topology,
VOLUME = 13,
YEAR = 2020,
NUMBER = 1,
PAGES = 59–76,
ISSN = 1753-8416,
MRCLASS = 18C35 (18G55 55U35),
MRNUMBER = 3999672,

AUTHOR = Geigle, W.
author = Lenzing, H.,
TITLE = Perpendicular categories with applications to representations and sheaves,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 144,
YEAR = 1991,
NUMBER = 2,
PAGES = 273–343,

AUTHOR = Gillespie, J.,
TITLE = Model structures on exact categories,
JOURNAL = J. Pure Appl. Algebra,
FJOURNAL = Journal of Pure and Applied Algebra,
VOLUME = 215,
YEAR = 2011,
NUMBER = 12,
PAGES = 2892–2902,
ISSN = 0022-4049,
CODEN = JPAAA2,
MRCLASS = 18E10 (18G35 55U15 55U35),
MRNUMBER = MR2811572,
MRREVIEWER = Timothy Porter,

AUTHOR = Gillespie, J.,
TITLE = Hereditary abelian model categories,
JOURNAL = Bull. Lond. Math. Soc.,
FJOURNAL = Bulletin of the London Mathematical Society,
VOLUME = 48,
YEAR = 2016,
NUMBER = 6,
PAGES = 895–922,

AUTHOR = Gillespie, J.,
TITLE = Canonical resolutions in hereditary abelian model categories,
JOURNAL = Pacific J. Math.,
FJOURNAL = Pacific Journal of Mathematics,
VOLUME = 313,
YEAR = 2021,
NUMBER = 2,
PAGES = 365–411,

AUTHOR = Göbel, R.,
Author = Trlifaj, J.,
TITLE = Approximations and endomorphism algebras of modules. Volume
SERIES = De Gruyter Expositions in Mathematics,
VOLUME = 41,
EDITION = extended,
NOTE = Approximations,
PUBLISHER = Walter de Gruyter GmbH & Co. KG, Berlin,
YEAR = 2012,
PAGES = xxviii+458

AUTHOR = Happel, Dieter,
TITLE = Triangulated categories in the representation theory of
finite-dimensional algebras,
SERIES = London Mathematical Society Lecture Note Series,
VOLUME = 119,
PUBLISHER = Cambridge University Press, Cambridge,
YEAR = 1988,
PAGES = x+208,
ISBN = 0-521-33922-7,
MRCLASS = 16A46 (16A48 16A62 16A64 18E30),
MRNUMBER = MR935124,
MRREVIEWER = Alfred G. Wiedemann,

AUTHOR = Holm, H.,
TITLE = Gorenstein homological dimensions,
JOURNAL = J. Pure Appl. Algebra,
FJOURNAL = Journal of Pure and Applied Algebra,
VOLUME = 189,
YEAR = 2004,
NUMBER = 1-3,
PAGES = 167–193,
ISSN = 0022-4049,
MRCLASS = 16E10 (16E05 16E30),
MRNUMBER = 2038564,
MRREVIEWER = Zhaoyong Huang,

AUTHOR = Holm, H.
author = Jørgensen, P.,
TITLE = Cotorsion pairs in categories of quiver representations,
JOURNAL = Kyoto J. Math,
FJOURNAL = ,
VOLUME = 59,
number = 3,
YEAR = 2019,
PAGES = 575–606,
ISSN = ,
CODEN = ,
MRCLASS = ,
MRNUMBER = ,
DOI = ,
URL = ,

AUTHOR = Hovey, M.,
TITLE = Cotorsion pairs, model category structures, and representation
JOURNAL = Math. Z.,
FJOURNAL = Mathematische Zeitschrift,
VOLUME = 241,
YEAR = 2002,
NUMBER = 3,
PAGES = 553–592,
ISSN = 0025-5874,
CODEN = MAZEAX,
MRCLASS = 55U35 (18E30 18G55),
MRNUMBER = MR1938704,

AUTHOR = Hovey, M.,
TITLE = Model categories,
SERIES = Mathematical Surveys and Monographs,
VOLUME = 63,
PUBLISHER = American Mathematical Society,
ADDRESS = Providence, RI,
YEAR = 1999,
PAGES = xii+209,
ISBN = 0-8218-1359-5,
MRCLASS = 55U35 (18D15 18G30 18G55),
MRNUMBER = MR1650134,
MRREVIEWER = Teimuraz Pirashvili,

Author = Joyal, A.,
Author =Tierney, M.,
Journal = Contemp. Math.,
Title = Quasi-categories vs Segal spaces,
BookTitle = In Categories in algebra, geometry and mathematical physics, Vol. 431., Amer. Math. Soc., Providence, RI.,
Pages = 277–326,
Year = 2007,

AUTHOR = Krause, H.,
TITLE = Highest weight categories and recollements,
JOURNAL = Ann. Inst. Fourier (Grenoble),
FJOURNAL = Université de Grenoble. Annales de l'Institut Fourier,
VOLUME = 67,
YEAR = 2017,
NUMBER = 6,
PAGES = 2679–2701,
ISSN = 0373-0956,
MRCLASS = 16G10 (16D90 16E65 18E30),
MRNUMBER = 3742477,
MRREVIEWER = Hongxing Chen,
URL = http://aif.cedram.org/item?id=AIF_2017__67_6_2679_0,

AUTHOR = Makkai, M.
author = Rosický, J.,
TITLE = Cellular categories,
JOURNAL = J. Pure Appl. Algebra,
FJOURNAL = Journal of Pure and Applied Algebra,
VOLUME = 218,
YEAR = 2014,
NUMBER = 9,
PAGES = 1652–1664,
ISSN = 0022-4049,
MRCLASS = 18C35 (18D05),
MRNUMBER = 3188863,
MRREVIEWER = Philippe Gaucher,

AUTHOR =	 Neeman, A.,
TITLE =	 Triangulated categories,
SERIES =	 Annals of Mathematics Studies,
VOLUME =	 148,
PUBLISHER =	 Princeton University Press,
ADDRESS =	 Princeton, NJ,
YEAR =	 2001,
PAGES =	 viii+449,
ISBN =	 0-691-08685-0; 0-691-08686-9,
MRCLASS =	 18E30 (55-02 55N20 55U35),
MRNUMBER =	 MR1812507,
MRREVIEWER =	 Stanisław Betley,

AUTHOR = Oppermann, S.,
author = Psaroudakis, C.,
author =Stai, T.,
TITLE = Change of rings and singularity categories,
JOURNAL = Adv. Math.,
FJOURNAL = Advances in Mathematics,
VOLUME = 350,
YEAR = 2019,
PAGES = 190–241,

AUTHOR = Parra, C. E.,
author = Vitória, J.,
TITLE = Properties of abelian categories via recollements,
JOURNAL = J. Pure and Appl. Algebra,
FJOURNAL = Journal of Pure and Applied Algebra,
VOLUME = 223,
YEAR = 2019,
PAGES = 3941–3963,
ISSN = ,
MRCLASS = ,
MRNUMBER = ,
MRREVIEWER = ,
URL = ,

AUTHOR = Psaroudakis, C.,
TITLE = Homological theory of recollements of abelian categories,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 398,
YEAR = 2014,
PAGES = 63–110,
ISSN = 0021-8693,
MRCLASS = 18E10 (16E10 18E30),
MRNUMBER = 3123754,
MRREVIEWER = Fernando Muro,
URL = https://doi.org/10.1016/j.jalgebra.2013.09.020,

AUTHOR = Psaroudakis, C.,
TITLE = A representation-theoretic approach to recollements of abelian
BOOKTITLE = Surveys in representation theory of algebras,
SERIES = Contemp. Math.,
VOLUME = 716,
PAGES = 67–154,
PUBLISHER = Amer. Math. Soc., [Providence], RI,
YEAR = 2018,
MRCLASS = 16E10 (16E65 16G50 16S50 18E30),
MRNUMBER = 3852400,

AUTHOR = Psaroudakis, C.,
author = Vitória, J.,
TITLE = Realisation functors in tilting theory,
JOURNAL = Math. Z.,
FJOURNAL = Mathematische Zeitschrift,
VOLUME = 288,
YEAR = 2018,
PAGES = 965–1028,

AUTHOR = Shulman, M.,
TITLE = Comparing composites of left and right derived functors,
JOURNAL = New York J. Math.,
FJOURNAL = New York Journal of Mathematics,
VOLUME = 17,
YEAR = 2011,
PAGES = 75–125,
MRCLASS = 55U35 (18A40 18D05 18G55),
MRNUMBER = 2781909,
MRREVIEWER = Kathryn P. Hess,
URL = http://nyjm.albany.edu:8000/j/2011/17_75.html,

AUTHOR = Stenström, B.,
TITLE = Rings of quotients,
NOTE = Die Grundlehren der Mathematischen Wissenschaften, Band 217,
An introduction to methods of ring theory,
PUBLISHER = Springer-Verlag, New York-Heidelberg,
YEAR = 1975,
PAGES = viii+309,
MRCLASS = 16-02,
MRNUMBER = 0389953,
MRREVIEWER = G. Michler,

AUTHOR = Štovíček, J.,
TITLE = Deconstructibility and the Hill lemma in Grothendieck
JOURNAL = Forum Math.,
FJOURNAL = Forum Mathematicum,
VOLUME = 25,
YEAR = 2013,
NUMBER = 1,
PAGES = 193–219,
ISSN = 0933-7741,
MRCLASS = 18E15 (16D70 18G35),
MRNUMBER = 3010854,
MRREVIEWER = Alex Martsinkovsky,

AUTHOR = Štovíček, J.,
TITLE = Exact model categories, approximation theory, and cohomology
of quasi-coherent sheaves,
BOOKTITLE = Advances in representation theory of algebras,
SERIES = EMS Ser. Congr. Rep.,
PAGES = 297–367,
PUBLISHER = Eur. Math. Soc., Zürich,
YEAR = 2013,
MRCLASS = 18E10 (18E30 18F20),
MRNUMBER = MR3220541,
MRREVIEWER = R. H. Street,

AUTHOR = Štovíček, J.,
Author = Positselski, L.,
Title = Derived, coderived, and contraderived categories of locally presentable abelian categories,
TITLE = Derived, coderived, and contraderived categories of locally
presentable abelian categories,
JOURNAL = J. Pure Appl. Algebra,
FJOURNAL = Journal of Pure and Applied Algebra,
VOLUME = 226,
YEAR = 2022,
NUMBER = 4,
PAGES = Paper No. 106883, 39,

AUTHOR = Verdier, J.-L.,
TITLE = Des catégories dérivées des catégories abéliennes,

JOURNAL = Astérisque,
FJOURNAL = Astérisque,
NUMBER = 239,
YEAR = 1996,
PAGES = xii+253 pp. (1997),
ISSN = 0303-1179,
MRCLASS = 18E30 (18-03 18E35),
MRNUMBER = 1453167,
MRREVIEWER = Amnon Neeman,

AUTHOR = Yang, G.
author = Liu, Z.,
TITLE = Cotorsion pairs and model structures on ${\rm Ch}(R)$,
JOURNAL = Proc. Edinb. Math. Soc. (2),
FJOURNAL = Proceedings of the Edinburgh Mathematical Society. Series II,
VOLUME = 54,
YEAR = 2011,
NUMBER = 3,
PAGES = 783–797,
ISSN = 0013-0915,
MRCLASS = 16D90 (18G35 18G55),
MRNUMBER = 2837480,

Author = Xiong, B. L.
AUTHOR = Zhang, P.,
TITLE = Gorenstein-projective modules over triangular matrix Artin
JOURNAL = J. Algebra Appl.,
FJOURNAL = Journal of Algebra and its Applications,
VOLUME = 11,
YEAR = 2012,
NUMBER = 4,
PAGES = 1250066, 14,
ISSN = 0219-4988,
MRCLASS = 16E65 (16G10),
MRNUMBER = 2959415,

AUTHOR = Zhang, P.,
TITLE = Gorenstein-projective modules and symmetric recollements,
JOURNAL = J. Algebra,
FJOURNAL = Journal of Algebra,
VOLUME = 388,
YEAR = 2013,
PAGES = 65–80,
ISSN = 0021-8693,
MRCLASS = 16D80,
MRNUMBER = 3061678,

Author = Williamson, J.,
Title = Algebraic models of change of groups functors in (co)free rational equivariant spectra,
FJournal = Journal of Pure and Applied Algebra,
Journal = J. Pure Appl. Algebra,
ISSN = 0022-4049,
Volume = 226,
Number = 11,
Pages = 53,
Note = Id/No 107108,
Year = 2022,


# CSS10: A Collection of Single Speaker Speech Datasets for 10 Languages

###### Abstract

We describe our development of CSS10, a collection of single speaker speech
datasets for ten languages. It is composed of short audio clips from LibriVox
audiobooks and their aligned texts. To validate its quality we train two
neural text-to-speech models on each dataset. Subsequently, we conduct Mean
Opinion Score tests on the synthesized speech samples. We make our datasets,
pre-trained models, and test resources publicly available. We hope they will
be used for future speech tasks.

Index Terms: text-to-speech, TTS, multi-lingual datasets, speech datasets,
deep learning

††footnotetext: † Both authors contributed equally.

##  1 Introduction

Text-to-speech (TTS) models aim at generating an audio sequence given some
input text. Recently there have been many TTS models based on neural networks.
Among them are WaveNet [1], Tacotron 1 & 2 [2][3], Char2Wav [4], DeepVoice 1,
2, & 3 [5][6][7], DCTTS [8], VoiceLoop [9], etc. All of them adopted an end-
to-end approach, taking as input a sequence of characters or phonemes and
returning the raw waveform or spectrogram.
All of those models have been introduced with results manufactured on their
internal datasets except DCTTS, which uses the public LJ Speech Dataset [10].
This status quo has made reproducing results in the papers difficult for
researchers outside. People have resorted to using less than ideal data and/or
creating their own datasets that try to match some of the properties of the
internal ones. What is even worse is that it is hard to compare different TTS
models, because there is no benchmark dataset.
Furthermore, the research is mostly focused on English. Except Deep Voice 2 &
3, which trained on the Chinese language, all the other models attempted to
model the text-to-audio mapping for the English language. The most likely
explanation is simply the dearth of freely available non-English datasets. We
strongly believe that languages other than English deserve to have attention
from researchers as well. This is why we construct multi-lingual speech
datasets and share them with the research community.
Our contribution is two-fold:

  * •

We construct single speaker speech datasets with aligned text for ten
different languages.

  * •

We train two famous TTS models on each dataset and evaluate them with Mean
Opinion Scores (MOS).

All of the resources mentioned above are available in our GitHub
repository1††footnotetext: 1 https://github.com/Kyubyong/CSS10.

##  2 Related work

Not surprisingly, for the English language, there are many public speech
datasets such as Blizzard [11], VCTK [12], LibriSpeech [13], TED-LIUM [14],
VoxForge [15], and Common Voice [16]. While these datasets are large—some
having more than 100 hours of audio—they are all from multiple speakers, which
make them unideal for the single speaker TTS task (namely, generating speech
from text in the voice of a single speaker) where more data from a single
speaker tends to help model performance [17]. One popular, newly-created
dataset is the LJ Speech dataset [10]. It consists of audio files segmented
from audiobooks in LibriVox recorded by a female volunteer. It has a large
cumulative audio length (20+ hrs) and has been verified to work with neural
TTS models. Another is the World English Bible (WEB) dataset [18], which was
sourced from bible recordings and text in the public domain. It shares the LJ
Speech dataset’s properties but is sampled at a relatively low rate of 12 kHz.
Both are publicly available.
For the Japanese language, there is a public dataset called the JSUT dataset
[19]. It is a single speaker dataset designed for speech synthesis that
includes around 10 hours of utterances with aligned text. One important
distinction between the JSUT dataset and the LJ Speech and WEB datasets is
that the JSUT dataset’s recording was carried out in a controlled environment
with specially designed scripts. One drawback of the JSUT dataset is that it
lacks a phonetic transcription of text. Additionally, there were no follow-up
experiments on the dataset.
For the German language, there is the Pavoque dataset [20]—a single speaker,
multi-style corpus of German speech. It has 12+ hours of audio clips each of
which is associated with phoneme-level annotations. However, it is more
suitable for speech style tasks rather than regular TTS tasks.
A dataset closer to our work is the Spoken WP Corpus Collection [21]. It has
hundreds of hours of audio with aligned text for three languages: Dutch,
German, and English. For each language, much of the audio comes from a single
speaker. However, the audio clips are generally large—on the order of
minutes—which makes using the dataset difficult for neural TTS models [22].
Perhaps the datasets closest to our work are the Tundra [22] and M-AILABS [23]
datasets, which have 14 and 9 languages, respectively, and are built from
audiobooks. The Tundra dataset uses a single speaker for each language, but
does so by using only one audiobook per language. The M-AILABS dataset, which
focuses on European languages, has nearly one-thousand hours of total audio
but uses multiple speakers in each language.

##  3 Datasets

We choose to use audiobooks from LibriVox [24], a website for free public
domain audiobooks, as the source of audio data for three major reasons. First,
audiobooks are inherently accompanied by text, which is essential for our
purpose. Second, although many readings are performances and hence use
abnormal intonation, readers tend to speak with a regular, constant speed.
Third, many audiobooks are performed by a single speaker.

###  3.1 Selection of audiobooks

As of now, there are audiobooks for 95 languages in LibriVox. We examine how
many hours of solo recordings each language has; many audiobook performers
record multiple audiobooks which makes finding large quantities of single
speaker data achievable. We exclude a language if it has less than 4 hours of
solo audio recordings, because we are not confident that training will
succeed2††footnotetext: 2This number was chosen based on our prior successes
of training DCTTS on an audiobook with 4 hours of data.. Then, we check for
text availability and for audio quality. If text is not available or if the
audio includes a noticeable amount of noise, the audiobook is excluded.
This process yields audiobooks for the following languages: Chinese (zh),
Dutch (nl), French (fr), Finnish (fi), German (de), Greek (el), Hungarian
(hu), Japanese (ja), Russian (ru), and Spanish (es)3††footnotetext: 3The
parenthetical expression next to each language is the language code from ISO
639-1.. All audio files are sampled at 22 kHz.

###  3.2 Audio processing

The audio from LibriVox usually comes in large files with lengthy audio clips
that do not suit the TTS task, so we fragment them into many small files. We
use the audio editor Audacity [25] to programmatically find split-points of
the audio anytime there is more than a 0.5-second duration of silence, except
for Spanish audiobooks, where we use a 0.25-second duration. Next, we adjust
the points such that neighboring clips are joined to have a duration around 10
seconds. We found these tricks improve computational efficiency. The
distribution of audio lengths for the Spanish dataset is shown in Figure 1. We
see about 85% of the samples have a duration between 5 and 11 seconds. All
other languages have distributions similar to this.

Figure 1: Histogram of audio durations for the Spanish dataset.

###  3.3 Text processing

We have experts align the text with each segmented audio clip to create an
$<$audio, text$>$ pair. At first we considered using a forced aligner such as
Gentle [26]. However, we forfeited the idea after we realized that it does not
guarantee the correct alignments and that it is language dependent.

####  3.3.1 Text normalization

Once we secure the $<$audio, text$>$ pairs, we request that our experts
normalize text; all abbreviations are expanded (e.g., Dra. $\rightarrow$
Doctora) and Arabic numerals are spelled out to match the context. Unlike Deep
Voice 3 or DCTTS, which ignore case, we decide to retain case, because it can
be a cue for new sentences or proper nouns or nouns. In addition, we remove
infrequent symbols other than punctuations such as the period, question mark,
exclamation point, colon, comma, and semi-colon.

####  3.3.2 Phonetic transcription

Because we take text as input, it is important to understand the writing
system of our target languages. Dutch, German, Finnish, French, Hungarian, and
Spanish are based on Latin alphabets, while Greek and Russian use Greek
letters and Cyrillic letters, respectively. Chinese uses Chinese characters,
and Japanese employs three different scripts (i.e., Hiragana, Katakana, and
Chinese characters). All the writing systems except Chinese characters are
phonetic. In other words, they are written as they are pronounced. On the
other hand, Chinese characters are ideographic, so they are not directly
associated with pronunciations. Thus, Chinese people typically use the
Romanization system for pronunciation, which is called pinyin, to input
Chinese characters in digital settings. Therefore, we convert original Chinese
text into pinyin (e.g., 思想言论举动丰采都没有什么大区别 $\rightarrow$ sī xiǎng yán lùn jǔ
dòng fēng cǎi dū méi yǒu shén me dài qū bié) using the Chinese segmenter Jieba
[27] and the open-source dictionary CC-CEDICT [28]. For Japanese, first we use
a morphological analyzer MeCab [29] to get the pronunciations of the text and
subsequently use romkan [30] to convert them into Roman letters. When MeCab
fails to return the pronunciations for words, we have a native speaker create
them manually.
Now, each of our datasets contains $<$audio, original text, processed text$>$
triplets.

##  4 Experiments

In order to validate our datasets, we train two neural TTS models—Tacotron [2]
and DCTTS [8]—on each dataset. We then synthesize audio samples and perform
Mean Opinion Score tests.

###  4.1 Models

Tacotron and DCTTS are well-known neural TTS models that have an attention-
based sequence-to-sequence architecture. Tacotron, which was introduced in
2017, is an impactful model that produced high quality results from an end-to-
end approach and has since served as an important benchmark. DCTTS is one of
the TTS models inspired by Tacotron. DCTTS is different from Tacotron in a few
ways. Tacotron’s computational backbone is the CBHG, a combination of
convolution layers, fully-connected layers, and recurrent layers, whereas
DCTTS only uses convolution layers. Tacotron is trained end-to-end whereas
DCTTS has two networks trained independently. DCTTS uses a few tricks that
help training such as guided attention and forcibly incremental attention.
One reason for choosing these models is that we have working implementations
of both [31][32] which have been tested on multiple datasets. We have found
that reproducing original work in neural TTS is non-trivial. Model performance
is dependent on many factors, such as data, model architecture,
hyperparameters, etc. The neural TTS field has yet to have the ossification of
helpful techniques for training that has seemingly already happened in the
neighboring field of neural computer vision. Thus, we find it paramount to use
models that have successful implementations in order to evaluate new datasets.

###  4.2 Training

We use Tesla P40 GPUs for training. For both models, we mostly use the given
hyperparameters in their respective papers. See our repository for the full
list of values. We use the same hyperparameters for all languages. The
original Tacotron paper uses more than 2 million training steps, which we find
impractical given our resources. In our preliminary tests, we found 400k steps
produced good results for DCTTS. For simplicity, we train both models for 400k
steps. It takes around ten days and three days to train Tacotron and DCTTS,
respectively.

###  4.3 Evaluation

####  4.3.1 Test sentences

To evaluate each model’s performance, 20 sentences per language are collected
from Tatoeba [33], a web database of sentences for multiple languages. These
sentences are carefully chosen to maximize the cover of letters in the
vocabulary so we can check the utterance quality of various phonemes. For
Chinese and Japanese, phonetic transcriptions, not the original text, are
considered. Some letters which are very rare in the language are left out. All
sentences are available in our repository.

####  4.3.2 Mean Opinion Scores

We leverage Amazon Mechanical Turk (MTurk) to gather workers to score the test
sentences. MTurk allows requesters to post Human Intelligence Tasks (HITs) for
a worker to complete. For each HIT, we ask the worker to listen to an audio
clip and score it.
We adopt the standard absolute category rating (ACR) test, where workers are
required to give integer scores between 1 and 5. We split evaluation into two
categories: speech naturalness and pronunciation accuracy. For speech
naturalness, we refer to the rubric for MOS scores in [34], which we include
in Table 1 for completeness. We design a new simple rubric for pronunciation
accuracy as shown in Table 2.

Table 1: Rubric of naturalness. Score | Quality | Definition
---|---|---
5 |  Excellent |  Imperceptible distortions (dist.)
4 |  Good |  Just perceptible but not annoying dist.
3 |  Fair |  Perceptible and slightly annoying dist.
2 |  Poor |  Annoying but not objectionable dist.
1 |  Bad |  Very annoying and objectionable dist.
Table 2: Rubric of pronunciation accuracy. Score | Quality | Definition
---|---|---
5 |  Excellent |  No mispronunciations
4 |  Good |  Few minor mispronunciations
3 |  Fair |  Many minor mispronunciations
2 |  Poor |  Few major mispronunciations
1 |  Bad |  Many major mispronunciations

With each HIT, we give the worker a reference sample from the audiobook and
tell the worker that the sample should score highly on both naturalness and
pronunciation; we then give the worker an audioclip with the corresponding
text and ask them score it.

For some languages, MTurk allows us to list qualifications for basic
proficiency. While we could opt for this requirement for some languages (e.g.,
French), we want consistency in the MOS procedure across all languages to make
the results more comparable. Thus, instead of relying on it, we require each
worker to listen to a reference sample chosen from the language’s dataset and
transcribe the text. We consider participants who have done this correctly to
be truthful and include their scores in our results.
We use the same method as crowdMOS [34] for computing confidence intervals
(C.I.). Because the number of available workers to complete HITs varies with
language, we allow languages to have varying numbers of total samples.

###  4.4 Results

We successfully trained models for all languages except Greek for Tacotron. We
believe it is due to the small data size of Greek ($\sim$4 hours of audio).
Although we successfully trained DCTTS on Greek, the samples were much worse
than those from other languages.
The MOS results for speech naturalness and pronunciation accuracy are shown in
Figures 2 and 3, respectively. For naturalness, DCTTS is statistically more
natural than Tacotron for German, French, and Spanish. For pronunciation
accuracy, we find that DCTTS and Tacotron are somewhat similar. The detailed
MOS scores are found in Table 3.

Table 3: MOS scores with 95% C.I. of Tacotron and DCTTS for all languages. Lang. | Dur. (hh:mm:ss) | # Workers | Speech Naturalness | Pronunciation Accuracy
---|---|---|---|---
|  |  | Tacotron | DCTTS | Tacotron | DCTTS
de | 16:08:01 | 75 | $2.82\pm 0.26$ | $3.59\pm 0.24$ | $4.31\pm 0.22$ | $4.25\pm 0.24$
el | 04:08:14 | 43 | N/A | $3.47\pm 0.41$ | N/A | $3.70\pm 0.39$
es | 23:49:49 | 78 | $3.45\pm 0.38$ | $4.31\pm 0.19$ | $4.31\pm 0.40$ | $4.72\pm 0.13$
fi | 10:32:03 | 62 | $3.69\pm 0.36$ | $4.15\pm 0.27$ | $4.16\pm 0.30$ | $4.43\pm 0.27$
fr | 19:09:03 | 75 | $2.51\pm 0.30$ | $3.50\pm 0.27$ | $3.48\pm 0.48$ | $4.24\pm 0.25$
hu | 10:00:25 | 39 | $3.60\pm 0.51$ | $4.21\pm 0.39$ | $4.27\pm 0.40$ | $4.51\pm 0.35$
ja | 14:55:36 | 79 | $3.39\pm 0.48$ | $4.01\pm 0.24$ | $3.61\pm 0.48$ | $4.20\pm 0.21$
nl | 14:06:40 | 97 | $3.06\pm 0.35$ | $3.63\pm 0.21$ | $3.82\pm 0.31$ | $3.93\pm 0.20$
ru | 21:22:10 | 73 | $2.55\pm 0.37$ | $3.43\pm 0.35$ | $3.64\pm 0.39$ | $4.02\pm 0.32$
zh | 06:27:04 | 122 | $3.41\pm 0.31$ | $4.10\pm 0.19$ | $4.19\pm 0.20$ | $4.46\pm 0.17$
Figure 2: MOS scores for speech naturalness with 95% C.I. Figure 3: MOS scores
for pronunciation accuracy with 95% C.I.

Although it’s common to use MOS as a performance metric for TTS models, we
recognize that it may not be appropriate to take the mean of Likert scores,
because each score belongs to a category and the semantic meaning of these
categories need not be evenly distributed along a number line as we (and
others) have implied in our rubrics. Thus, we also present the distributions
of scores for each model-language pair; the distributions for naturalness and
pronunciation accuracy are shown in Tables 4 and 5, respectively. We see that
for both speech naturalness and pronunciation accuracy, DCTTS has
distributions more skewed towards higher scores than Tacotron, but the effect
is more marked for naturalness.

Figure 4: Percentages of naturalness scores shown by model and language.
Figure 5: Percentages of pronunciation accuracy scores shown by model and
language.

In general, when listening to samples, both models capture the original voice,
but the samples of DCTTS sound better than those of Tacotron. We found DCTTS
produced fairly clean speech, while Tacotron consistently had noise in the
outputs.
Both models exhibited mumbling at the end of a few generated samples. Although
this was not expected–the models should have learned trailing silences–we were
able to remove mumbling via trimming.
In Japanese, we found utterances that were not relevant to the input text were
generated at the end in a few samples. We suspect it is derived from the
imperfection of the automatic phonetic transcription in the training data.

##  5 Conclusions

We discussed how we built CSS10, a collection of single-speaker speech
datasets for 10 languages, and how we used it for a TTS task. Despite the fact
that there were differences in model performance depending on the language, we
were able to train the models successfully on our datasets. We release all the
resources for this project including source code, datasets, pre-trained
models, and evaluation data to the public.

##  6 Future work

We hope CSS10 and our experiments serve as benchmarks for future non-English
TTS research. We found that some automatic phonetic transcriptions for
Japanese and Chinese contain errors. If these errors are corrected, perhaps
the model performance will improve. Additionally, we plan to add a Korean
dataset. Because there are not enough audiobooks available for Korean in
LibriVox, we are willing to produce recordings ourselves. When they are ready,
we will release them with our other languages. Although we validated our
datasets on TTS models in this work, CSS10 can be used for other speech tasks
such as multi-lingual speech recognition.

##  7 Acknowledgments

We thank the LibriVox creator for his platform which allows public access to
audiobooks, the performers who recorded their readings of audiobooks, and
Kakao Brain for funding this work.

## References

  * [1] A. Van Den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, “Wavenet: A generative model for raw audio,” _arXiv preprint arXiv:1609.03499_ , 2016.
  * [2] Y. Wang, R. Skerry-Ryan, D. Stanton, Y. Wu, R. J. Weiss, N. Jaitly, Z. Yang, Y. Xiao, Z. Chen, S. Bengio _et al._ , “Tacotron: A fully end-to-end text-to-speech synthesis model,” _arXiv preprint arXiv:1703.10135_ , 2017\.
  * [3] J. Shen, R. Pang, R. J. Weiss, M. Schuster, N. Jaitly, Z. Yang, Z. Chen, Y. Zhang, Y. Wang, R. Skerry-Ryan _et al._ , “Natural tts synthesis by conditioning wavenet on mel spectrogram predictions,” _arXiv preprint arXiv:1712.05884_ , 2017.
  * [4] J. Sotelo, S. Mehri, K. Kumar, J. F. Santos, K. Kastner, A. Courville, and Y. Bengio, “Char2wav: End-to-end speech synthesis,” 2017.
  * [5] S. O. Arik, M. Chrzanowski, A. Coates, G. Diamos, A. Gibiansky, Y. Kang, X. Li, J. Miller, J. Raiman, S. Sengupta _et al._ , “Deep voice: Real-time neural text-to-speech,” _arXiv preprint arXiv:1702.07825_ , 2017.
  * [6] S. O. Arık, G. Diamos, A. Gibiansky, J. Miller, K. Peng, W. Ping, J. Raiman, and Y. Zhou, “Deep voice 2: Multi-speaker neural text-to-speech,” _arXiv preprint arXiv:1705.08947_ , 2017.
  * [7] W. Ping, K. Peng, A. Gibiansky, S. O. Arik, A. Kannan, S. Narang, J. Raiman, and J. Miller, “Deep voice 3: 2000-speaker neural text-to-speech,” _arXiv preprint arXiv:1710.07654_ , 2017.
  * [8] H. Tachibana, K. Uenoyama, and S. Aihara, “Efficiently trainable text-to-speech system based on deep convolutional networks with guided attention,” _arXiv preprint arXiv:1710.08969_ , 2017.
  * [9] Y. Taigman, L. Wolf, A. Polyak, and E. Nachmani, “Voice synthesis for in-the-wild speakers via a phonological loop,” _arXiv preprint arXiv:1707.06588_ , 2017.
  * [10] K. Ito, “The lj speech dataset,” https://keithito.com/LJ-Speech-Dataset/, 2017.
  * [11] “Blizzard challenge 2018,” https://www.synsig.org/index.php/Blizzard_Challenge_2018, 2018.
  * [12] J. Yamagishi, T. Nose, H. Zen, Z. H. Ling, T. Toda, K. Tokuda, S. King, and S. Renals, “Robust speaker-adaptive hmm-based text-to-speech synthesis,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 17, no. 6, pp. 1208–1230, Aug 2009.
  * [13] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Librispeech: an asr corpus based on public domain audio books,” http://www.openslr.org/12/, 2015.
  * [14] A. Rousseau, P. Deléglise, and Y. Estève, “Ted-lium: an automatic speech recognition dedicated corpus,” http://www-lium.univ-lemans.fr/en/content/ted-lium-corpus, 2014.
  * [15] “Voxforge,” http://www.voxforge.org/, 2006.
  * [16] Mozilla, “Common voice,” https://voice.mozilla.org/, 2017.
  * [17] P. Baljekar, “Speech synthesis from found data,” 2018.
  * [18] K. Park, “The world english bible,” https://www.kaggle.com/bryanpark/the-world-english-bible-speech-dataset, 2017\.
  * [19] R. Sonobe, S. Takamichi, and H. Saruwatari, “Jsut corpus: Free large-scale japanese speech corpus for end-to-end speech synthesis,” http://arne.chark.eu/static/spoken-wp-corpus-collection.pdf,https://nats.gitlab.io/swc/, 2017\.
  * [20] “Pavoque corpus of expressive speech,” https://github.com/marytts/pavoque-data, 2009.
  * [21] T. Baumann, A. Kohn, and F. Hennig, “The spoken wikipedia corpus collection,” http://arne.chark.eu/static/spoken-wp-corpus-collection.pdf,https://nats.gitlab.io/swc/, 2016\.
  * [22] A. Stan, O. Watts, Y. Mamiya, M. Giurgiu, R. A. J. Clark, J. Yamagishi, and S. King, “Tundra: A multilingual corpus of found data for tts research created with light supervision,” 08 2013.
  * [23] I. Solak, “The m-ailabs speech dataset,” https://www.caito.de/2019/01/the-m-ailabs-speech-dataset/, January 2019\.
  * [24] “Librivox,” https://librivox.org/, 2018.
  * [25] “Audacity,” http://audacity.sourceforge.net/, 2018.
  * [26] R. Ochshorn and M. Hawkins, “Gentle,” https://github.com/lowerquality/gentle, 2017.
  * [27] J. Sun, “Jieba,” https://github.com/fxsjy/jieba, 2017.
  * [28] P. Denisowski, “Cc-cedict,” https://cc-cedict.org/editor, 2018.
  * [29] “Mecab: Yet another part-of-speech and morphological analyzer,” http://taku910.github.io/mecab, 2006.
  * [30] M. Yao, “python-romkan,” https://www.soimort.org/python-romkan/, 2013.
  * [31] K. Park and T. Mulc, “A (heavily documented) tensorflow implementation of tacotron: A fully end-to-end text-to-speech synthesis model,” https://github.com/Kyubyong/tacotron, 2018.
  * [32] K. Park, “A tensorflow implementation of dc-tts: yet another text-to-speech model,” https://github.com/Kyubyong/dc_tts, 2018.
  * [33] Sysko, “Tatoeba,” https://tatoeba.org, 03 2013.
  * [34] F. P. Ribeiro, D. Florencio, C. Zhang, and M. Seltzer, “Crowdmos: An approach for crowdsourcing mean opinion score studies,” in _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , 05 2011, pp. 2416–2419.


# Tunable mode entanglement: topological Su-Schrieffer-Heeger (SSH) chain with
an embedded Aharonov-Bohm quantum ring

A. P. Garrido  Departamento de Física, Facultad de Ciencias, Universidad
Católica del Norte, Angamos 0610, Antofagasta, Chile  Departamento de Física,
Universidad Técnica Federico Santa María, Valparaíso, Chile  A. R. Plastino
CeBio y Departamento de Ciencias Básicas, Universidad Nacional del Noroeste de
la Província de Buenos Aires, UNNOBA, Conicet, Roque Saenz Peña 456, Junin,
Argentina  M. L. Ladrón de Guevara  Departamento de Física, Facultad de
Ciencias, Universidad Católica del Norte, Angamos 0610, Antofagasta, Chile  V.
M. Apel<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias,
Universidad Católica del Norte, Angamos 0610, Antofagasta, Chile

###### Abstract

We investigate a variant of the SSH model consisting of an SSH chain with an
embedded Aharonov-Bohm quantum ring. The embedded ring gives rise to domain
wall states whose energy levels are in the band gap. The dependence of some of
the states on the ring’s magnetic flux provides a mechanism to control the
entanglement between the two SSH edge fermion modes. The concurrence between
the edge modes depends periodically on the magnetic flux. In the deep
topological zone, for appropriate values of the flux, the concurrence reaches
the extreme values of $0$ and $1$, corresponding to vanishing entanglement and
to maximum entanglement. Therefore, the QR-SSH system offers a tuning
mechanism that can be used either as an entanglement modulator or as an
entanglement switch.

###### pacs:

03.67.Bg, 03.65.Ud, 05.30.Fk, 73.21.Cd

##  I Introduction

The SSH model, which describes the polyacetylene molecule, constitutes a
paradigmatic example of a model quantum system admitting non-trivial
topological properties Su _et al._ (1979). The quintessential feature of this
kind of systems is that they can hold gapless states at the edges that are
immune to weak or moderated disorder. As an example we can mention the quantum
Hall bridge which, in the appropriate regime, while possessing robust metallic
states at its edges is insulator at the bulk. The conducting states occur
within the so called topological phase. In contrast to the topological phases,
there exist also trivial phases where such states do not arise. Bearing in
mind that the edges of a system have one dimension less than the bulk
dimension, it is plain that for $1$D topological systems the edges states are
zero dimensional, and therefore do not conduct. The topological states are
then known as edge zero modes. The different phases in a topological system
can be explained in terms of topological properties of a gapped Hamiltonian.
It is said that two Hamiltonians belong to the same topological class if, in
the parameter space where they live, there is a continuous path along which it
is possible to transform one Hamiltonian into the other without closing the
gap. There is a subcategory within the topological systems called symmetry
protected topological (SPT) systems, in which certain symmetry properties of
the Hamiltonian are preserved during the transformation Chiu _et al._ (2016).
The SSH model falls into such category.

The $1$D SSH system is governed by a one dimensional tight-binding Hamiltonian
with periodically alternating hoppings between neighbor sites Su _et al._
(1979, 1980), that describes the transport of spinless fermions in a dimerized
one-dimensional lattice. The difference between the values of the alternate
hoppings is usually expressed in terms of an asymmetry parameter $\lambda$,
adopting values in the range $[-1,1]$. This parameter determines the two
phases of the system: the trivial phase corresponds to $\lambda<0$ and the
topological one to $\lambda>0$.

Although originally proposed in the context of solid state physics, SSH-like
models have recently been the focus of considerable attention in various other
fields, that include quantum optics and quantum information theory. The
increasing attention is due, among other things, to the appealing variety of
non-trivial phenomena that can take place in models based on SSH chains. For
instance, it has been pointed out that the non-local nature of topological
phases should be reflected in purely quantum mechanical correlations, such as
quantum entanglement Fidkowski (2010); Turner _et al._ (2011); Azses _et al._
(2020); Lima (2019); Micallo _et al._ (2020); Cornfeld _et al._ (2019); Zeng
_et al._ (2019). Regarding this point, Cho and Kim Cho and Kim (2017)
discovered that mode entanglement between two first neighbor sites of an SSH
chain exhibits critical behaviour at the phase transition point ($\lambda=0$).
Monkman Monkman and Sirker (2020) showed that the topological edge states in a
three leg SSH ladder can be a source of genuine non-bipartite many-body
entanglement that can be transferred to a quantum register. These studies are
related to another interesting phenomenon that deserves investigation: the
appearance of domain walls in the SSH chains. In this regard, Huda et al.
recently reported the experimental realization of dimer and trimer topological
chains with domain wall structures precisely controlled by atomic manipulation
Huda _et al._ (2020).

In the present work we investigate an SSH-based model, paying special
attention to two of the phenomena mentioned above: quantum entanglement and
the existence of domain walls. We consider the SSH-based model schematically
depicted in Fig. 1, which consists of an SSH chain with an embedded Aharonov-
Bohm quantum ring (QR). The motivation for including the ring is to explore
how the concomitant magnetic flux affects the system’s topological states, and
the mode entanglement features of the system’s ground state. In addition to
the topological edge modes, we have found that the ring embedded in the SSH
chain configures a domain wall where two different kind of topological states
are supported, according to the value adopted by the chain’s asymmetry
parameter. One of the states has zero energy, independently of the magnetic
flux, while the other, which is within the gap, has an energy tunable by the
flux. The dependence of the topological state on the ring’s magnetic flux
provides a mechanism to control the mode entanglement between the edge modes.
In the deep topological zone ($\lambda\approx 1$), as the flux is varied, the
concurrence has a discontinuous jump from $0$ to its maximal value $1$. In
addition, there are critical values of both the flux and the asymmetry
parameter for which sudden births or deaths of entanglement are observed.

The article is organized as follows. The model to be investigated in the
present work is described in section II. Some necessary background material on
the phenomenon of entanglement between fermion modes, including relevant
technical details concerning reduced density operators in the context of
identical fermions, are reviewed in section III. Our main results are
presented in section IV, where the system’s energy spectrum and topological
states are determined, and the states’ entanglement features are investigated.
Finally, some conclusions are drawn in section V.

##  II The Model

The system under consideration consists of a SSH chain with an embedded
Aharonov-Bohm ring in the form described in Fig. 1.

Figure 1:  A Su-Schrieffer-Heeger (SSH) chain with an embedded Aharonov-Bohm
quantum ring (QR). The SSH chain comprises two branches. Each of them is
composed of cells containing two sites: one site of type $\alpha$ and one site
of type $\beta$.

Each branch of the SSH chain is composed of cells containing two sites: one of
type $\alpha$ and one of type $\beta$. The intra-cell and inter-cell hoppings
are $v$ and $w$, respectively. We denote by $t_{R}$ the hoping between the
sites in the ring, and by $t^{\prime}$ the chain-ring hoping. The Hamiltonian
of the system can be written as

$H=H_{SSH}+H_{QR}+H_{QR-SSH}.$ (1)

The term $H_{SSH}=H_{SSH}^{L}+H_{SSH}^{R}$ describes the Hamiltonian of the
left and right branches of the SSH chain, with

$\displaystyle H_{SSH}^{L(R)}$ $\displaystyle=$ $\displaystyle\sum_{i\in
L(R)}(vc_{\beta,i}^{\dagger}c_{\alpha,i}+\mbox{h.c.})$ (2)
$\displaystyle+\sum_{i\in
L(R)}(wc_{\alpha,i+1}^{\dagger}c_{\beta,i}+\mbox{h.c.}),$

where the index $i$ corresponds to the $i$-th pair in the left (right) chain
and $c_{\alpha(\beta)}$ denotes the spinless fermion operator in a
$\alpha(\beta)$-type site. The intra and inter cell hopings, $v$ and $w$, can
be expressed in terms of the asymmetry parameter $\lambda$ ($-1\leq\lambda\leq
1$) as

$v=-t(1-\lambda)\;\mbox{ and }\;w=-t(1+\lambda),$ (3)

where $t$ has energy dimensions. $H_{QR}$ is the Hamiltonian of the quantum
ring

$H_{QR}=\sum_{j=1}^{n}t_{R}(e^{i\varphi}d_{j}^{\dagger}d_{j+1}+\mbox{h.c.}),$
(4)

where $n$ is the number of sites in the ring, $t_{R}$ the hoping between
sites, $d_{j}$ is the spinless fermion operator on site $j$ of the ring, and
$\varphi=2\pi\phi/\phi_{0}$ is the Aaharonov-Bohm phase, which is expressed in
terms of the ring’s magnetic flux $\phi$ and the flux quantum $\phi_{0}=h/e$.
Finally, $H_{R-SSH}$ is the Hamiltonian describing the coupling between the
chains and quantum ring

$H_{R-SSH}=t^{\prime}(d_{L}^{\dagger}c_{\alpha,N_{SL}}+c^{\dagger}_{\alpha,1_{R}}d_{R})+\mbox{h.c.},$
(5)

where $t^{\prime}$ is the chain-ring hoping and $d_{L(R)}$ destroys an
electron in the site in the ring connected to the left(right) SSH chain.

##  III Reduced density matrices in the modes representation and entanglement
between fermionic modes

In this section we discuss some background material on reduced density
matrices within the mode representation of fermionic systems, and on the
quantitative evaluation of entanglement between fermionic modes.

###  III.1 Reduced density matrices in the fermionic modes representation

In order to calculate the amount of entanglement between the components of a
subsystem of a quantum system, it is necessary first to obtain the subsystem’s
reduced (also known as marginal) density matrix. For instance, to calculate
the entanglement between two particular modes of a fermionic system, one needs
first to calculate the reduced density matrix that jointly describes the two
modes. When considering entanglement between modes in systems of identical
particles, the natural framework to use is the occupation number
representation Balachandran _et al._ (2013); Friis _et al._ (2013). In the
Appendix we describe the procedure to calculate the reduced density matrix in
the occupation number (or fermionic modes) representation, which in addition
will allow us to interpret our results in the context of many identical
particles. In the modes representation for identical fermions, the natural
basis of the many-body state-space (Fock space) consists of states
characterized by the occupation numbers $n_{s}$ of each site $s$ ($1\leq s\leq
N_{L}$), where $N_{L}$ is the total number of sites in the system. The vacuum
state $|\emptyset\rangle$ is the state with $n_{s}=0$ for all $s$. The basis
of the Fock space contains the vacuum and states of the form
$|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle$, which are obtained by the action of
the creation operators $c_{i}^{{\dagger}}$ on $|\emptyset\rangle$,

$|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle=\prod_{l=1}^{r}c^{{\dagger}}_{i_{l}}|\emptyset\rangle,$
(6a) where $r$ is the number of fermions, the Fermi annihilation and creation
operators obey $\\{c_{i},c^{\dagger}_{j}\\}=\delta_{ij}$, and we have adopted
the convention,
$\prod_{l=1}^{r}c^{{\dagger}}_{i_{l}}=c^{{\dagger}}_{i_{r}}\dots
c^{{\dagger}}_{i_{2}}c^{{\dagger}}_{i_{1}},\quad i_{1}<i_{2}<\dots<i_{r}.$
(6b)

We have denoted by $I$ the multi-index $(i_{1},i_{2},\dots,i_{N_{L}})$. In the
state of Eq.(6a), the label $I$ indicates the occupied sites of the system,
i.e., if $s$ ($1\leq s\leq N_{L}$) is equal to some index of $I$, then
$n_{s}=1$, otherwise, $n_{s}=0$. Notice that, as follows from Eq. (6b), the
indices in $I$ are defined in increasing order.

Given the anticommutative character of the fermionic operators, it is
convenient to use the generalized Kronecker delta symbol, $\delta_{I}^{J}$,
defined as,

$\delta^{J}_{I}=\begin{cases}1\textrm{ }\textrm{ }\textrm{ if }J\textrm{ is an
even permutation of }I\\\ -1\textrm{ if }J\textrm{ is an odd permutation of
}I\\\ 0\textrm{ }\textrm{ }\textrm{ if }J\textrm{ is not permutation of
}I\end{cases}$

where $I=(i_{1},\dots,i_{f})$ and $J=(j_{1},\dots,j_{f})$ are two multi-
indices. In this definition, the indices in $I$ or $J$ are not necessarily
ordered.

Let $A|B$ be a partition of a lattice $L$, i.e., $L=A\cup B$, $A\cap
B=\emptyset$, with $A=\\{\alpha_{1},\dots,\alpha_{N_{A}}\\}$ and
$B=\\{\beta_{1},\dots,\beta_{N_{B}}\\}$, where $N_{A}+N_{B}=N_{L}$. As
described in details in the Appendix, the reduced density matrix of the
subpart $A$ in the fermionic modes representation is given by

$\rho_{A}=\sum_{J,J^{\prime}}[\rho_{A}]_{J,J^{\prime}}|J;n_{\alpha_{1}},\dots,n_{N_{A}}\rangle\langle
J^{\prime};n_{\alpha_{1}},\dots,n_{N_{A}}|,$ (7a)
$[\rho_{A}]_{J,J^{\prime}}=\sum_{I,I^{\prime},K}\delta_{JK}^{I}\delta_{J^{\prime}K}^{I^{\prime}}[\rho]_{I,I^{\prime}},$
(7b)

where the labels $J$ or $J^{\prime}$ denote the sites occupied in the subpart
A, $K$ denotes the sites occupied in subpart $B$, and
$JK=(j_{1}<\dots<j_{p},k_{1}<\dots<k_{q})$, $p+q=r$, where $r$ is the number
of occupied sites in all the system. Notice that the indices in $JK$ are not
necessarily ordered in increasing order, in spite that $J$ and $K$ are
ordered.

An alternative procedure to obtain the reduced density matrix is based on the
analytical procedure discussed by Peschel Peschel (2003) and by Cheung and
Henley Cheong and Henley (2004), which is consistent with what was defined in
Eqs. (7b). To calculate the reduced density matrix $\rho_{A}$ it is necessary
to evaluate the correlation matrix $G$ defined by the elements $\langle
c^{{\dagger}}_{i}c_{j}\rangle$, where $i,j$ are the sites that belong to the
subpart $A$. Once the correlation matrix $G$ is calculated, the reduced
density matrix $\rho_{A}$ can be calculated using the formula,

$\displaystyle\rho_{A}$ $\displaystyle=$
$\displaystyle\det(I-G)\exp{\left\\{\sum_{ij}\left[\log
G(I-G)^{-1}\right]_{ij}c_{i}^{\dagger}c_{j}\right\\}},$ (8) $\displaystyle
i,j$ $\displaystyle\in A.$

Although Eqs. (7b) and (8) are equivalent, the latter is easier to implement
in the calculation of reduced density matrices, and for this reason we have
used this to obtain our results. As for Eq. (7b), its usefulness lies on the
fact that it illustrates in a straightforward way the meaning of the reduced
density matrix, and it highlights some technical aspect its computation when
working in the modes representation.

###  III.2 Entanglement between modes in fermionic systems

Entanglement is a key piece of quantum mechanical theory. This notion is
central for understanding the physical properties of composite quantum
systems, including systems consisting of identical particles. Considerable
effort has been devoted to analyze and clarify the concept of entanglement in
systems of identical particles Amico _et al._ (2008); Lo Franco and Compagno
(2016); Shi (2003); Zanardi (2002); Wiseman and Vaccaro (2003); Friis _et al._
(2013); Zander and Plastino (2010); Gigena and Rossignoli (2015); Majtey _et
al._ (2016), and to apply it to the study of specific physical systems Bouvrie
_et al._ (2012); Simonović and Nazmitdinov (2015); Iemini _et al._ (2015);
Melnikov and Fedichkin (2016, 2018); Di Tullio _et al._ (2018, 2019); Bouvrie
_et al._ (2019); Bussandri and Majtey (2020); Christov (2021); Ferreira _et
al._ (2022); Capizzi _et al._ (2022); Fraenkel and Goldstein (2022). In the
present work, as already mentioned, we shall focus on the entanglement between
modes. When addressing entanglement from the modes’ perspective each mode can
be distinguished from the others, because each of them is associated to a
specific lattice-site. Since the modes are spatially localized, the
entanglement between modes considered here can be regarded as spatial
entanglement. For the sake of clarity and completeness, it is worth to briefly
discuss the notion of entanglement between modes. Following the Wiseman-
Vaccaro argument Wiseman and Vaccaro (2003) let us consider, as a simple
illustration of entanglement between modes, a four site system with only one
electron in a superposition state

$\frac{1}{\sqrt{2}}(|\varphi_{1}\rangle+|\varphi_{2}\rangle).$ (9)

This state doesn’t seem to be entangled, since it is a one particle state. In
the occupation number representation, however, this state is given by

$\frac{1}{\sqrt{2}}(|1000\rangle+|0100\rangle),$ (10)

which is entangled. In fact, from the perspective of fermionic modes, (10) is
a Bell state. The reverse situation can also occur. That is, a fermion state
that in the particle representation looks like a Bell state, but in the mode
representation is a separable state. For example, in our two spinless electron
system in a four-site lattice, the state

$\frac{1}{\sqrt{2}}(|\varphi_{1}\rangle\otimes|\varphi_{3}\rangle-|\varphi_{3}\rangle\otimes|\varphi_{1}\rangle)$
(11)

looks non-separable. From the fermionic mode perspective, however, this state
is expressed as,

$|1010\rangle$ (12)

which is a separable state. These simple examples correspond to pure states,
but the concept of entanglement between fermionic modes can be
straightforwardly extended to mixed states. The amount of entanglement between
fermionic modes, both for pure and mixed states, can be calculated using
standard procedures analogous to those used for systems consisting of
distinguishable qubits. In this work, in order to calculate the entanglement
between two modes jointly described by a mixed state $\rho$, we shall use the
concurrence $\mathcal{C(\rho)}$. To calculate $C(\rho)$, previously we require
to specify an object called spin-flip superoperator $\tilde{\rho}$,

$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y}),$
(13)

where $\rho$ is represented in the basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, $*$ denotes complex
conjugation, and $\sigma_{y}$ is the Pauli matrix
$\left(\begin{array}[]{cc}0&-i\\\ i&0\\\ \end{array}\right)$,
($\sigma_{y}|0\rangle=i|1\rangle$ and $\sigma_{y}|1\rangle=-i|0\rangle$). The
concurrence $\mathcal{C}$ is evaluated using Wootter’s Wootters (1998)
celebrated expression,

$\mathcal{C}=\max\left\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\right\\},$
(14)

where $\lambda_{i}$ ($i=1,2,3,4$) are the eigenvalues, in decreasing order, of
the operator $R=\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$. The concurrence
$\mathcal{C}$ can take values between $0$ and $1$, where $\mathcal{C}=0$
indicates absence of entanglement and $\mathcal{C}=1$ corresponds to a
maximally entangled state.

##  IV Topological States and Entanglement Between Modes

In the numerical simulations performed in this work, it is observed that the
properties of the system remain qualitatively invariant when varying some of
the model’s parameters, such as the hoppings $t_{R}$ and $t^{\prime}$. In what
follows, it is assumed that the two SSH branches have the same number of
sites, and that the ring has five sites. Also, the values of all the
parameters appearing in the Hamiltonian are express in units of $t$. Our main
results are presented in the following order: In subsection IV.1 we analyze
the effect of the quantum ring on the energy levels of the system determining,
in particular, the levels’ dependence on the ring’s magnetic flux. In section
IV.2 we study the amount of entanglement between fermionic modes, as measured
by the concurrence evaluated on the marginal density matrix associated with a
given pair of modes.

###  IV.1 Energy Spectrum and topological states

The energy spectrum provides essential information about the system’s physical
properties. In particular, it sheds lights on the presence of topological
states. For example, the spectrum of the standard SSH model presents a quasi-
degenerate couple of levels around $E=0$, characterized by their robustness in
the region of $\lambda>0$. The presence of such levels is associated with the
existence of topological states. In the following analysis, as usually done
when discussing the standard SSH chain, we refer to the region with
$\lambda>0$ as the “topological zone”, and to the region with $\lambda<0$ as
the “trivial zone” Asbóth _et al._ (2016). In our model, however, the
existence of topological states is not restricted to $\lambda>0$. Indeed, in
the QR and its neighboring sites, topological states are observed for both
$\lambda>0$ and $\lambda<0$. Fig. 2 depicts the dependence of the system’s
energy levels on the asymmetry parameter $\lambda$. Fig. 2a corresponds to a
standard SSH chain, while Figures 2b, 2c, and 2d correspond to the SSH-QR
system with different values of the ring’s magnetic flux.

Figure 2: Energy levels as a function of asymmetry parameter $\lambda$. (a)
corresponds to standard SSH chain without QR with $N_{T}=90$ sites. (b), (c)
and (d) correspond to SSH-QR system $N_{T}=97$ sites and $t_{R}=0.5t$. The
corresponding magnetic flux are (b) $\Phi/\Phi_{0}=0$, (c)
$\Phi/\Phi_{0}=0.25$ and (d) $\Phi/\Phi_{0}=0.5$

In the panels of Fig. 2 we specify the degeneration degree $g$ of the level
$E=0$ in each zone. As shown in all the figures, there exist robust zero
energy levels in the topological zone, but one such state is also observed in
the trivial zone. We find that the standard topological states at the free
edges of the chains arise for any value of the magnetic flux (a two-fold
degenerate state appearing in the region $\lambda>0$). In the particular case
of $\Phi/\Phi_{0}=0.25$ the degeneration in the topological zone is triple; to
the standard edge states it is added a robust state localized at the ring and
its neighbor sites. The robust level around $E=0$ for $\lambda<0$ indicates
the presence of a non-expected topological state in this region, which
corresponds to a state located in some sites of the ring and its neighborhood.

In Fig. 3, it is plotted the probability that the electron is found at the $n$
site, where the features mentioned above are revealed. In Fig. 3(a) we observe
the standard topological edge states, which occur for $\lambda>0$ and any
value of the magnetic flux. Figs. 3(b) and (c) give account of the two kinds
of topological states confined in the QR region. Fig. 3(b) corresponds to the
probability of the non degenerate eigenstate with energy $E=0$, for
$\lambda<0$, which does not depend on the magnetic flux. The electron’s
probability distribution localizes just in one site of the ring and in some
neighboring sites. This localization in one site of the ring explains why the
energy of this state is insensitive to the magnetic flux. Let us recall that,
for the Aharonov-Bohm effect to take place, it is necessary that the electric
charge can circulate along a closed circuit around the magnetic field. For the
state under consideration, the probability that the electron transits along
one of the ring’s arms is zero, and, consequently, the magnetic field does not
produce any observable effect in the system. In particular, the energy does
not depend on the flux. In the topological zone ($\lambda>0$), in addition to
the two topological edge states, a third topological state arises, which has
$E=0$ for $\Phi/\Phi_{0}=l+1/4$ ($l$ integer). The concomitant probability
distribution is uniform in the ring, as transpires from Fig. 3(c).

Figure 3: Colour online. Panels (a), (b) and (c) show the probability as a
function of the site position $n$ for three different of system’s eigenstates
with energy $E/t\approx 0$ ($|E/t|\ll 1$). The panel (a) shows the topological
edge states for $\lambda=0.9$ and an arbitrary value of $\Phi$. Panel (b)
corresponds to the state with energy $E=0$ in the trivial zone, for
$\lambda=-0.9$ and an arbitrary value of $\Phi$. Panel (c) corresponds to an
additional state in the topological zone with $\Phi/\Phi_{0}=(l+1/4)$
($l\in\mathds{Z}$) and $\lambda=0.9$. In panel (d) we specify the
correspondence of the values of $n$ their respective position in the system.

In contrast with the other topological QR state, in this case the Aharonov-
Bohm effect takes place, since the electron can circulate through both the
ring’s arms. Fig. 4 shows the energy as a function of the magnetic flux, for
the two topological QR states. We can see that the energy of the state with
uniform probability in the QR ($\lambda>0$) oscillates with the magnetic flux,
while for the case of the state localized in only one QR site ($\lambda<0$),
the energy is constant and equal to $0$. This means that the former state
supports persistent currents ($I_{P}=-dE/d\Phi$), while the latter does not.
These topological states, which are localized in the ring and its
surroundings, configure domain walls. It is worth emphasizing that the
topological states having a uniform probability distribution in the ring do
not necessarily have vanishing energy. In fact, the energy can be controlled
by the intensity of the magnetic flux.

Figure 4: Colour online. Energy of the QR confined states as a function of
magnetic flux. Black corresponds to QR confined state in the topological zone
($\lambda=0.4$), and constant energy level (red line) corresponds to localized
QR state in the trivial zone ($\lambda<0$). The QR hoping is $t_{R}=0.5t$

###  IV.2 Entanglement

In this section we investigate the amount of entanglement exhibited by
particular couples of modes in the system, and analyze how entanglement is
related to the topological phases. Specifically, we explore the concurrence
between adjacent modes, as well as between the two edge modes. For relatively
large systems (say, for systems with a total number of sites $N_{T}$ similar
or greater than $20$) the numerical simulations are too costly in terms of
computational time. Therefore, we have considered $N_{T}=19$ total sites: five
sites in the quantum ring and $7$ sites in each branch of the SSH chain. In
spite of the size limitations, the results reported here strongly suggest that
the observed effects can be extrapolated to systems of much larger size.
Moreover, it is to be expected that some of the effects should be enhanced
when considering larger systems. In all our computations we assumed that the
system is near half-filling. That is, we assumed that the number of electrons
is equal to $(N_{T}-1)/2$.
Concurrence between nearest neighbors in the SSH chains. Here, we focus on the
concurrence (see Eq. (14)) between nearest neighbors in the SSH chains. The
reduced density matrix corresponding to a pair of neighbour sites is obtained
from Eq. (8). As shown in Fig. 5, the concurrence between nearest neighbor
sites of the chains can be different from zero. This is in agreement with the
results reported by Cho and Kim Cho and Kim (2017) for the standard SSH chain.

Figure 5: Colour online. Concurrence between first neighbors as a function of
$\lambda$ and $t_{R}=2t$. Panel a) corresponds to intra-cell sites, and panel
(b) corresponds to inter-cell sites.

The panels (a) and (b) depict, for different values of the magnetic flux, the
dependence on the parameter $\lambda$ of the concurrence $\mathcal{C}$ between
nearest neighbors of the SSH chains. Panel (a) corresponds to next neighbours
belonging to the same cell ($\alpha_{n},\beta_{n}$), and panel (b) to next
neighbours of adjacent cells ($\beta_{n},\alpha_{n+1}$). The qualitative
behaviour of the concurrences are independent of the cell position. Notice
that
$\mathcal{C}_{\alpha_{n},\beta_{n}}(\lambda)\sim\mathcal{C}_{\beta_{n},\alpha_{n+1}}(-\lambda)$,
which follows from the fact that the hoppings intra and inter-cell satisfy the
relation $v(\lambda)=w(-\lambda)$. In addition, the concurrence
$\mathcal{C}_{\alpha_{n},\beta_{n}}$ ($\mathcal{C}_{\beta_{n},\alpha_{n+1}}$)
reaches its maximum $\mathcal{C}=1$ in the trivial (topological) zone, for
$\lambda=-1$ ($\lambda=1$). To understand this behaviour, let us consider the
limits $\lambda=\pm 1$. In the limit $\lambda=-1$ one has $w=0$ and $v=-2t$,
and each branch of the SSH chain decomposes in decoupled bi-atomic molecules
consisting of the sites ($\alpha_{n},\beta_{n}$), that are in the bonding
states

$\displaystyle|\Psi_{n}(\lambda=-1)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|\alpha_{n}\rangle+|\beta_{n+1}\rangle)$ (15)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|1_{n}0_{n}\rangle+|0_{n}1_{n}\rangle).$

Analogously, for $\lambda=1$ one has $v=0$ and $w=-2t$, and the SSH branches
decompose in decoupled molecules consisting of the sites
($\alpha_{n},\beta_{n}$), in the bonding states

$\displaystyle|\Psi_{n}(\lambda=1)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|\beta_{n}\rangle+|\alpha_{n+1}\rangle)$ (16)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|1_{n}0_{n+1}\rangle+|0_{n}1_{n+1}\rangle).$

It is plain that both (15) and (16) are formally maximally entangled Bell-like
states of two qubits. When the value of $\lambda$ departs from $\pm 1$, each
molecule interacts and becomes entangled with its neighbors, and the
entanglement between next-neighbour sites decreases. It is worth noticing that
there are some critical values of $\lambda$ for which entanglement between
next-neighbour sites is completely suppressed Cho and Kim (2017), in a way
reminiscent of the sudden death of entanglement predicted and observed in the
context of quantum optics Almeida _et al._ (2007).

Concurrence between edges sites. Here we present the central result of our
work, the existence of entanglement between the edge sites of the system. The
behaviour of the concurrence $\mathcal{C}_{L,R}$ between the edge sites of the
system is shown in Fig. 6, as a function of the parameter $\lambda$, for four
different values of the magnetic flux ($\Phi/\Phi_{0}=0$, $0.2$, $0.25$ and
$0.5$ ).

Figure 6: Colour online. Edge-edge concurrence as a function of $\lambda$ for
four values of magnetic fluxes and $t_{R}=2t$. The insets show the concurrence
as a function of magnetic flux for two different values of $\lambda$.

In all cases the concurrence $\mathcal{C}_{L,R}$ vanishes in the trivial zone.
For $\Phi/\Phi_{0}=0.5$ the concurrence also vanishes in the topological zone,
while for the remaining values of the ring’s magnetic flux ($\Phi/\Phi_{0}=0$,
$0.2$ and $0.25$) it departs from zero at a critical positive value
$\lambda_{\Phi}$ of the asymmetry parameter, and then increases monotonously
with $\lambda$ in the interval $(\lambda_{\Phi},1)$. In the limit $\lambda\to
1^{-}$ $\mathcal{C}_{L,R}\to 1$ for $\Phi/\Phi_{0}=0$ and $0.2$, and falls
abruptly to zero at $\lambda=1$, when $v=0$ and the edge sites become
disconnected from the rest of the SSH chains. For $\Phi/\Phi_{0}=0.25$ the
concurrence behaves in a similar way, but its maximum value, reached in the
limit $\lambda\to 1^{-}$, is $1/2$. In the inset of the Fig 6, we also show
the dependence of the concurrence $\mathcal{C}_{L,R}$ on the magnetic flux
$\Phi$ for two fixed values of asymmetry parameter, $\lambda=0.8$ and
$\lambda=0.2$. We observe that the concurrence between the edge sites is a
symmetric and periodic function of the ring’s magnetic flux $\Phi$, with
period equal to $\Phi_{0}$. In the deep topological zone ($\lambda\approx 1$)
the concurrence is a discontinuous function of the flux, with
$\mathcal{C}_{L,R}=1$ for $|\Phi/\Phi_{0}|<0.25$ and $\mathcal{C}_{L,R}=0$ for
$0.25<|\Phi/\Phi_{0}|<0.5$.

The entanglement between the edge modes is closely related to the occupation
of the quasi-degenerate levels associated to the topological edge states of
the SSH chains. These edge states have energies $E_{+}\gtrsim 0$ and
$E_{-}\lesssim 0$ Asbóth _et al._ (2016). More precisely,

$E_{\pm}\propto\pm|ve^{-N_{T}\xi}|,$ (17)

where $\xi=(\log|w|-\log|v|)^{-1}$ is the localization length of the states
$|E_{\pm}\rangle$. In the limit $v\to 0$, these eigenstates tend to the
superpositions of localized edge states given by

$|E_{\pm}\rangle=\frac{1}{\sqrt{2}}\left(e^{-i\varphi/2}|1_{L},0_{R}\rangle\pm
e^{i\varphi/2}|0_{L},1_{R}\rangle\right),$ (18)

which are maximally entangled states.
In half-filling, for $|\Phi/\Phi_{0}|<0.25$ and $\lambda=1^{-}$, the state
$|E_{-}\rangle$ is occupied and the state $|E_{+}\rangle$ is empty.
Consequently, there is only one electron shared by the edge sites, and the
concurrence $\mathcal{C}_{L,R}$ reaches the maximum value $1$. When the
magnetic flux increases, the energy of one of the states of the QR drops. For
$\Phi/\Phi_{0}=0.25$ the energy of the ring’s state vanishes, the state
hybridizes with the edge states zero modes, and the concurrence
$\mathcal{C}_{L,R}$ falls to $1/2$. When $0.25<|\Phi/\Phi_{0}|<0.5$, the
energy of the same state of the ring drops below zero and absorbs the edge
electron. Consequently, both edge levels get empty, and the edge-edge
concurrence falls to zero.

##  V Conclusions

In this work we investigated the topological states of a Su-Schrieffer-Heeger
(SSH) chain with an embedded Aharonov-Bohm quantum ring. We payed special
attention to the states’ entanglement-related features, focusing on the
entanglement between fermionic modes of the system. We have found that, in
addition to the topological edge states exhibited by the standard SSH chain,
our present system also admits two kinds of topological states in the bulk. On
the one hand, a topological state with a vanishing energy not depending on the
magnetic flux, arises in the region usually known as the trivial zone. On the
other hand, bulk topological states with energies depending on the flux,
emerge in the standard topological zone. In both cases the ring plays the role
of a domain wall, where it is known that another zero energy topological state
also occurs Asbóth _et al._ (2016).

Part of this work was devoted to the analysis of the quantum mode entanglement
exhibited by pairs of sites in the system. We found that the effects of the
magnetic flux on the entanglement between adjacent sites are small and,
consequently, when considering adjacent sites, the behavior of our system is
consistent with the one observed in a single SSH chain Cho and Kim (2017). On
the other hand, the ring’s magnetic flux has a strong effect on the of edge-
edge entanglement. This effect is determined by the state of occupation of the
zero energy eigenstates located at the edges. The concurrence between edges
depends on the magnetic flux $\Phi$. When the chain’s asymmetry parameter
$\lambda$ approaches the limit $\lambda\rightarrow 1^{-}$, the oscillations of
the concurrence, as a function of $\Phi$, tend to reach the extreme values $0$
and $1$. In the limit $\lambda\rightarrow 1^{-}$, the dependence of the
concurrence on the flux has the shape of a “square wave”, with the concurrence
changing abruptly between $0$ and $1$ when $\Phi/\Phi_{0}$ approaches the
values $l+1/4$ ($l$ integer). The dependence of entanglement on the ring’s
magnetic flux provides a mechanism that can be used to design an entanglement
switch between the system’s edges.

The present work suggests various possible directions for further research.
Here we have focused on the entanglement features of a Su-Schrieffer-Heeger
chain with an embedded Aharonov-Bohm ring, related to the eigenstates of the
system. It would be interesting to investigate explicit dynamical entanglement
features of this system, corresponding to time-dependent states, and compare
them to the entanglement dynamics exhibited by other fermionic systems, such
as those considered in Melnikov and Fedichkin (2016, 2018). It would also be
worth to explore possible connections between our developments, and very
recent works on the topology and/or the entanglement in junctions of free
fermionic chains akin to the ones consider by us, such as Capizzi _et al._
(2022) (where the junction is conformally-invariant), Fraenkel and Goldstein
(2022) (where the junction is a scatterer), or Ares _et al._ (2022) (where the
junction is a hopping defect). Any advances along these or related lines will
certainly be welcome.

APPENDIX: Many Fermion State, Convention, and Basic Issues

Here, in order to make our work self-contained, we briefly describe the
procedure for calculating the reduced density matrix in the occupation number
representation. In this representation, each state of the many fermion basis
is labeled by a sequence of zeros and ones. The zeros (ones) of a given
sequence corresponds to the empty (occupied) one fermion states. It is well-
known that in this formalism there is an implicit convention related to the
anticommutative character of fermionic many-body states. This convention plays
a relevant role in the computation of reduced density matrices.

Let us consider a lattice system with $r$ electrons distributed in $N_{L}$
sites. In the occupation number representation, each lattice site can be
considered as a physical object with two available states, empty ($0$) or
occupied ($1$). A state of the system in this representation can be described
as,

$|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle,$ (19)

where $n_{s}=1$ if site $s$ is occupied and $n_{s}=0$ otherwise, and the
ordered multi-index $I=(i_{1}<i_{2}<\dots<i_{r})$, where $1\leq i_{j}\leq
N_{L}$ indicates explicitly which sites of the lattice $L$ are occupied. In
this description, each site of the lattice defines a mode $s$, and the value
of $n_{s}$ specifies its state. In the standard notation the label $I$ is not
present in the ket $|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle$, but here we
include it for convenience. We also assume that each multi-index appearing in
what follows is in increasing order, unless otherwise indicated.

In the second quantization formalism, each mode $i$ is associated to creation
and annihilation operators, $c_{i}^{\dagger}$ and $c_{i}$, respectively. The
operator $c^{{\dagger}}_{i}$ ($c_{i}$) creates (annihilates) a fermion in a
site $i$ when this site is empty (occupied), otherwise it returns a null
vector. These operators obey the anticommutation relations

$\\{c_{i},c_{j}\\}=\\{c_{i}^{{\dagger}},c_{j}^{{\dagger}}\\}=0,\;\\{c_{i},c_{j}^{{\dagger}}\\}=\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta and the curly brackets mean
$\\{\hat{O}_{1},\hat{O}_{2}\\}=\hat{O}_{1}\hat{O}_{2}+\hat{O}_{2}\hat{O}_{1}$
for any pair of operators $\hat{O}_{1}$ and $\hat{O}_{2}$.

The vacuum state $|\emptyset\rangle$ corresponds to a state with $n_{s}=0$ for
all $1\leq s\leq N_{L}$. The state $|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle$ is
obtained by the action of the creation operators $c_{i}^{{\dagger}}$ on the
vacuum state $|\emptyset\rangle$ as follows,

$|I;n_{1},n_{2},\dots,n_{N_{C}}\rangle=\prod_{l=1}^{r}c^{{\dagger}}_{i_{l}}|\emptyset\rangle,$
(20a) where we adopt the convention,
$\prod_{l=1}^{r}c^{{\dagger}}_{i_{l}}=c^{{\dagger}}_{i_{r}}\dots
c^{{\dagger}}_{i_{2}}c^{{\dagger}}_{i_{1}},\quad i_{1}<i_{2}<\dots<i_{r}.$
(20b)

Note that in the definition of the state in the occupation number
representation via Eqs. (20b) it was implicitly adopted a convention of sign.
If it is carried out an odd permutation of the order in which the creation
operators are applied, a change of sign in the resulting state must be
included. Whatever is the sign convention chosen, it must be applied
systematically when calculating quantum correlations between fermion modes.

If $J=(j_{1},\dots,j_{r})$ is a permutation of $I=(i_{1},\dots,i_{r})$ then,

$|J;n_{1},\dots
n_{N_{L}}\rangle=\prod_{l=1}^{r}c^{{\dagger}}_{j_{l}}|\emptyset\rangle=\delta^{J}_{I}|I;n_{1},\dots
n_{N_{L}}\rangle,$

where $\delta_{I}^{J}$ stands for the generalized Kronecker delta symbol (see
(III.1)) and $I$ and $J$ are not necessarily in increasing order.
Let $A|B$ be a partition of the lattice $L$, i.e., $L=A\cup B$ and $A\cap
B=\emptyset$, given by,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\\{\alpha_{1},\dots,\alpha_{N_{A}}\\}$ (21a) $\displaystyle B$
$\displaystyle=$ $\displaystyle\\{\beta_{1},\dots,\beta_{N_{B}}\\}$ (21b)

where $N_{A}+N_{B}=N_{L}$. Let us consider a state of $p$ fermions distributed
among the sites represented by ${J}=(j_{1}<j_{2}<\dots<j_{p})$ of subpart $A$,
and another of $q$ fermions distributed among $K=(k_{1}<k_{2}<\dots<k_{q})$
sites of subset $B$, with $p+q=r$, the number of fermions in the whole
lattice. These states can be written as,

$\displaystyle|J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}\rangle$
$\displaystyle=$
$\displaystyle\prod_{l=1}^{p}c^{{\dagger}}_{j_{l}}|\emptyset\rangle,$ (22a)
$\displaystyle|K;n_{\beta_{1}},\dots,n_{\beta_{N_{B}}}\rangle$
$\displaystyle=$
$\displaystyle\prod_{l=1}^{q}c^{{\dagger}}_{k_{l}}|\emptyset\rangle,$ (22b)

The wedge product between $|J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}\rangle$
and $|K;n_{\beta_{1}},\dots,n_{\beta_{N_{B}}}\rangle$ is defined as,

$|J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}\rangle\wedge|K;n_{\beta_{1}},\dots,n_{\beta_{N_{B}}}\rangle=\prod_{l=1}^{p}c^{{\dagger}}_{j_{l}}\prod_{l=1}^{q}c^{{\dagger}}_{k_{l}}|\emptyset\rangle.$
(23)

This product will allow us later to obtain the reduced density matrix of the
subparts $A$ or $B$ in the fermionic modes representation.

Given a bipartite system $A|B$ in a state described by a density matrix
$\rho$, the reduced density matrix of the subpart $A$ is obtained by tracing
out the degrees of freedom of the subpart $B$,

$\rho_{A}=\mathrm{Tr}_{B}\rho.$ (24)

Physically, the reduced density operator $\rho_{A}$ ($\rho_{B}$) stores the
available information of the state $\rho$ located in subpart $A$ ($B$)
ignoring any information associated to subpart $B$ ($A$) Friis _et al._
(2013). The reduced density matrix $\rho_{A}$ has to obey the consistency
condition,

$\mathrm{Tr}_{A}(\rho_{A}O_{A})=\mathrm{Tr}(\rho O_{A}),$ (25)

where $O_{A}$ an observable acting over subpart $A$. Besides, the trace and
partial trace operations have to comply with linearity,

$\displaystyle\mathrm{Tr}(\mathrm{a}_{1}\rho^{(1)}+\mathrm{a}_{2}\rho^{(2)})$
$\displaystyle=$
$\displaystyle\mathrm{a}_{1}\mathrm{Tr}(\rho^{(1)})+\mathrm{a}_{2}\mathrm{Tr}(\rho^{(2)}),$
$\displaystyle\mathrm{Tr}_{B}(\mathrm{a}_{1}\rho^{(1)}+\mathrm{a}_{2}\rho^{(2)})$
$\displaystyle=$
$\displaystyle\mathrm{a}_{1}\mathrm{Tr}_{B}(\rho^{(1)})+\mathrm{a}_{2}\mathrm{Tr}_{B}(\rho^{(2)}),$

where $\mathrm{a}_{i}$ ($i=1,2$) are real numbers. The operators $\rho^{(i)}$
($i=1,2$) are density matrices of the whole system.

The antisymmetric character of fermionic many-body states, in addition to the
requirements imposed by (25) and (26), completely determine the expressions
for the trace $\mathrm{Tr}(\,)$ and the partial trace $\mathrm{Tr}_{A}(\,)$
($\mathrm{Tr}_{B}(\,)$) operations. We shall focus on $\mathrm{Tr}_{B}(\,)$,
leaving aside $\mathrm{Tr}_{A}(\,)$, since it obviously has the same form as
the former.

Returning to the our lattice $L$ and its partition $A|B$, let us suppose that
$I=(i_{1}<i_{2}<\dots<i_{p+q})$ results from a permutation $\sigma$ of the set
of numbers $JK=(j_{1}<j_{2}<\dots<j_{p},k_{1}<k_{2}<\dots<k_{q})$ (note that
in general $JK$ is not in increasing order). Hence

$\displaystyle|J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}\rangle\wedge|K;n_{\beta_{1}},\dots,n_{\beta_{N_{B}}}\rangle$
$\displaystyle=\delta_{JK}^{I}|I;n_{1},\dots,n_{N_{L}}\rangle.$ (27)

Analogously, the wedge product between the bras $\langle
J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}|$ and $\langle
K;n_{\beta_{1}},\dots,n_{\beta_{N_{B}}}|$ results in,

$\displaystyle\langle J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}|\wedge\langle
K;n_{\beta_{1}},\dots,n_{\beta_{N_{B}}}|$
$\displaystyle=\delta_{JK}^{I}\langle I;n_{1},\dots,n_{N_{L}}|.$ (28)

Now we are in position to include into the trace $\mathrm{Tr}_{B}(\,)$ of
operator $|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle\langle
I^{\prime};n_{1},n_{2},\dots,n_{N_{L}}|$ the underlying many fermion
antisymmetric property,

$\displaystyle\mathrm{Tr}_{B}\big{(}|I;n_{1},n_{2},\dots,n_{N_{L}}\rangle\langle
I^{\prime};n_{1},n_{2},\dots,n_{N_{L}}|\big{)}=$
$\displaystyle\delta_{JK}^{I}\delta_{J^{\prime}K^{\prime}}^{I^{\prime}}\delta_{K,K^{\prime}}|J;n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}\rangle\langle
J^{\prime};n_{\alpha_{1}},\dots,n_{\alpha_{N_{A}}}|$ (29)

where as usual $\delta_{K,K^{\prime}}=1$ if $K=K^{\prime}$ and
$\delta_{K,K^{\prime}}=0$ otherwise. In the above equations we have used
$I,J,K$ ($I^{\prime},J^{\prime},K^{\prime}$) to label kets (bras). With this
procedure, the degrees of freedom associated to the fermion modes belonging to
subpart $B$ are eliminated, obtaining the reduced operator of the subpart $A$
in the fermionic modes context.
Now, let us consider the density operator,

$\rho=\sum_{I,I^{\prime}}[\rho]_{I,I^{\prime}}|I;n_{1},\dots,n_{N_{L}}\rangle\langle
I^{\prime};n_{1},\dots,n_{N_{L}}|,$ (30)

where $[\rho]_{I,I^{\prime}}$ are complex numbers, which by definition satisfy
the following properties,

$\displaystyle[\rho]_{I,I^{\prime}}=[\rho]^{*}_{I^{\prime},I}$ (31)
$\displaystyle\mathrm{Tr}(\rho)=\sum_{I}[\rho]_{I,I}=1$ (32)

where the symbol ”$*$” denotes complex conjugation. Now, to calculate the
reduced fermion density operator $\rho_{A}$ we apply Eq. (LABEL:linearidad-
reducida), i.e., the linearity of partial trace condition,

$\displaystyle\rho_{A}$ $\displaystyle=$ $\displaystyle\mathrm{Tr}_{B}(\rho)$
$\displaystyle=$
$\displaystyle\sum_{I,I^{\prime}}[\rho]_{II^{\prime}}\mathrm{Tr}_{B}\big{(}|I;n_{1},\dots,n_{N_{L}}\rangle\langle
I^{\prime};n_{1},\dots,n_{N_{L}}|\big{)},$

what results in,

$\rho_{A}=\sum_{J,J^{\prime}}[\rho_{A}]_{J,J^{\prime}}|J;n_{\alpha_{1}},\dots,n_{N_{A}}\rangle\langle
J^{\prime};n_{\alpha_{1}},\dots,n_{N_{A}}|.$ (34)

From Eqs. (29), (LABEL:reduced-rho-A1) and (34) it follows that the
coefficients $[\rho_{A}]_{J,J^{\prime}}$ of reduced density matrix can be
written as,

$[\rho_{A}]_{J,J^{\prime}}=\sum_{I,I^{\prime},K}\delta_{JK}^{I}\delta_{J^{\prime}K}^{I^{\prime}}[\rho]_{I,I^{\prime}}.$
(35)

## References

  * Su _et al._ (1979) W. P. Su, J. R. Schrieffer and A. J. Heeger, _Phys. Rev. Lett._ 1979, _42_ , 1698.
  * Chiu _et al._ (2016) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder and S. Ryu, _Rev. Mod. Phys._ 2016, _88_ , 035005.
  * Su _et al._ (1980) W. P. Su, J. R. Schrieffer and A. J. Heeger, _Phys. Rev. B_ 1980, _22_ , 2099.
  * Fidkowski (2010) L. Fidkowski, _Phys. Rev. Lett._ 2010, _104_ , 130502.
  * Turner _et al._ (2011) A. M. Turner, F. Pollmann and E. Berg, _Phys. Rev. B_ 2011, _83_ , 075102\.
  * Azses _et al._ (2020) D. Azses, R. Haenel, Y. Naveh, R. Raussendorf, E. Sela and E. G. Dalla Torre, _Phys. Rev. Lett._ 2020, _125_ , 120502.
  * Lima (2019) L. S. Lima, _Int. J. Mod. Phys. B_ 2019, _33_ , 1950284.
  * Micallo _et al._ (2020) T. Micallo, V. Vitale, M. Dalmonte and P. Fromholz, _SciPost Phys. Core_ 2020, _3_ , 12.
  * Cornfeld _et al._ (2019) E. Cornfeld, E. Sela and M. Goldstein, _Phys. Rev. A_ 2019, _99_ , 062309\.
  * Zeng _et al._ (2019) B. Zeng, X. Chen, D.-L. Zhou and X.-G. Wen, _Quantum Information Meets Quantum Matter_ , Springer, New York 2019.
  * Cho and Kim (2017) J. Cho and K. W. Kim, _Sci. Rep._ 2017, _7_ , 2745.
  * Monkman and Sirker (2020) K. Monkman and J. Sirker, _Phys. Rev. Research_ 2020, _2_ , 043191\.
  * Huda _et al._ (2020) M. N. Huda, S. Kezilebieke, T. Ojanen, R. Drost and P. Liljeroth, _npj Quantum Materials_ 2020, _5_ , 17.
  * Balachandran _et al._ (2013) A. P. Balachandran, T. R. Govindarajan, A. R. de Queiroz and A. F. Reyes-Lega, _Phys. Rev. Lett._ 2013, _110_ , 080503.
  * Friis _et al._ (2013) N. Friis, A. R. Lee and D. E. Bruschi, _Phys. Rev. A_ 2013, _87_ , 022338\.
  * Peschel (2003) I. Peschel, _J. Phys. A Math. Gen._ 2003, _36_ , L205.
  * Cheong and Henley (2004) S.-A. Cheong and C. L. Henley, _Phys. Rev. B_ 2004, _69_ , 075111\.
  * Amico _et al._ (2008) L. Amico, R. Fazio, A. Osterloh and V. Vedral, _Rev. Mod. Phys._ 2008, _80_ , 517.
  * Lo Franco and Compagno (2016) R. Lo Franco and G. Compagno, _Sci. Rep._ 2016, _6_ , 20603\.
  * Shi (2003) Y. Shi, _Phys. Rev. A_ 2003, _67_ , 024301.
  * Zanardi (2002) P. Zanardi, _Phys. Rev. A_ 2002, _65_ , 042101.
  * Wiseman and Vaccaro (2003) H. M. Wiseman and J. A. Vaccaro, _Phys. Rev. Lett._ 2003, _91_ , 097902\.
  * Friis _et al._ (2013) N. Friis, A. R. Lee and D. E. Bruschi, _Phys. Rev. A_ 2013, _87_ , 022338\.
  * Zander and Plastino (2010) C. Zander and A. R. Plastino, _Phys. Rev. A_ 2010, _81_ , 062128\.
  * Gigena and Rossignoli (2015) N. Gigena and R. Rossignoli, _Phys. Rev. A_ 2015, _92_ , 042326\.
  * Majtey _et al._ (2016) A. P. Majtey, P. A. Bouvrie, A. Valdés-Hernández and A. R. Plastino, _Phys. Rev. A_ 2016, _93_ , 032335.
  * Bouvrie _et al._ (2012) P. A. Bouvrie, A. P. Majtey, A. R. Plastino, P. Sanchez-Moreno and J. S. Dehesa, _Eur. Phys. J. D_ 2012, _66_ , 15.
  * Simonović and Nazmitdinov (2015) N. S. Simonović and R. G. Nazmitdinov, _Phys. Rev. A_ 2015, _92_ , 052332.
  * Iemini _et al._ (2015) F. Iemini, T. O. Maciel and R. O. Vianna, _Phys. Rev. B_ 2015, _92_ , 075423.
  * Melnikov and Fedichkin (2016) A. A. Melnikov and L. E. Fedichkin, _Sci. Rep._ 2016, _6_ , 34226.
  * Melnikov and Fedichkin (2018) A. A. Melnikov and L. E. Fedichkin, _AIP Conf. Proc._ 2018, _1936_ , 020025.
  * Di Tullio _et al._ (2018) M. Di Tullio, N. Gigena and R. Rossignoli, _Phys. Rev. A_ 2018, _97_ , 062109.
  * Di Tullio _et al._ (2019) M. Di Tullio, R. Rossignoli, M. Cerezo and N. Gigena, _Phys. Rev. A_ 2019, _100_ , 062104.
  * Bouvrie _et al._ (2019) P. A. Bouvrie, E. Cuestas, I. Roditi and A. P. Majtey, _Phys. Rev. A_ 2019, _99_ , 063601.
  * Bussandri and Majtey (2020) D. G. Bussandri and A. Majtey, A. P. amd Valdés-Hernández, _J. Phys. A: Math. Theo._ 2020, _53_ , 405303.
  * Christov (2021) I. P. Christov, _Entropy_ 2021, _23_ , 868.
  * Ferreira _et al._ (2022) D. L. B. Ferreira, T. O. Maciel, R. O. Vianna and F. Iemini, _Phys. Rev. B_ 2022, _105_ , 115145.
  * Capizzi _et al._ (2022) L. Capizzi, S. Murciano and P. Calabrese, (Preprint) arXiv: 2205.04722, v1, submitted: may 2022.
  * Fraenkel and Goldstein (2022) S. Fraenkel and M. Goldstein, (Preprint) arXiv:12205.12991, v1, submitted: may 2022 .
  * Wootters (1998) W. K. Wootters, _Phys. Rev. Lett._ 1998, _80_ , 2245.
  * Asbóth _et al._ (2016) J. K. Asbóth, L. Oroszlány and A. Pályi, _A Short Course on Topological Insulators_ , Springer International Publishing 2016.
  * Almeida _et al._ (2007) M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro and L. Davidovich, _Science_ 2007, _316_ , 579.
  * Ares _et al._ (2022) F. Ares, J. G. Esteve and F. Falceto, _J. Stat. Mech. Theory Exp._ 2022, _2022_ , 013102.