$\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}$ given in Eqs.
(30) and (99) for RTPs and AOUPs respectively. To begin with, we rewrite
$\bar{F}^{A}_{s}(t)$ given by Eq. (25) as

$\displaystyle\bar{F}^{A}_{s}(t)=\frac{1}{N}\sum_{\alpha=0}^{N-1}e^{-\frac{2\pi
is}{N}\alpha}~{}F_{\alpha}^{A}(t).$ (163)

The unequal time correlation is given by ,

$\displaystyle\langle\bar{F}^{A}_{s}(\tau_{1})\bar{F}^{A^{*}}_{s^{\prime}}(\tau_{2})\rangle_{c}=\sum_{\alpha=1}^{N-1}e^{-\frac{2\pi
i\alpha}{N}(s-s^{\prime})}~{}\langle
F^{A}_{\alpha}(\tau_{1})F^{\\#}_{\beta}(\tau_{2})\rangle_{c}.$ (164)

where $\bar{F}^{\\#~{}*}_{s^{\prime}}(\tau_{2})$ is the complex conjugate of
$\bar{F}^{\\#}_{s^{\prime}}(\tau_{2})$.

###  A.1 For RTP

Recall for RTP, $F_{\alpha}^{RTP}(t)=v_{0}\sigma_{\alpha}(t)$ as shown in Eq.
(3). Substituting this in Eq. (164),

$\displaystyle\langle\bar{F}^{RTP}_{s}(\tau_{1})\bar{F}^{RTP*}_{s^{\prime}}(\tau_{2})\rangle_{c}=\frac{v_{0}^{2}}{N^{2}}\sum_{\alpha,\beta=1}^{N-1}e^{-\frac{2\pi
i}{N}(s\alpha-s^{\prime}\beta)}\langle\sigma_{\alpha}(\tau_{1})\sigma_{\beta}(\tau_{2})\rangle_{c},$
(165)

where $\bar{F}^{RTP*}_{s^{\prime}}(\tau_{2})$ is the complex conjugate of
$\bar{F}^{RTP}_{s^{\prime}}(\tau_{2})$. The correlation
$\langle\sigma_{\alpha}(\tau_{1})\sigma_{\beta}(\tau_{2})\rangle_{c}$ is given
by Eq. (4). Inserting this in Eq. (165) gives,

$\displaystyle\langle\bar{F}^{RTP}_{s}(\tau_{1})\bar{F}^{RTP*}_{s^{\prime}}(\tau_{2})\rangle_{c}=\frac{v_{0}^{2}}{N^{2}}~{}e^{-2\zeta|\tau_{1}-\tau_{2}|}~{}\sum_{\alpha=1}^{N-1}e^{-\frac{2\pi
i\alpha}{N}(s-s^{\prime})}.$ (166)

Proceeding further we note that $\sum_{\alpha=1}^{N-1}e^{-\frac{2\pi
i\alpha}{N}(s-s^{\prime})}=N\delta_{s,s^{\prime}}$. Using this in Eq. (165)
gives,

$\displaystyle\langle\bar{F}^{RTP}_{s}(\tau_{1})\bar{F}^{RTP*}_{s^{\prime}}(\tau_{2})\rangle_{c}=\delta_{s,s^{\prime}}\frac{v_{0}^{2}}{N}~{}e^{-2\zeta|\tau_{1}-\tau_{2}|}.$
(167)

Finally substituting this in
$\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}$ in Eq. (29) and
performing the integrations over $\tau_{1}$ and $\tau_{2}$, we obtain the
result quoted in Eq. (30).

###  A.2 For AOUP

For this case, we have $\bar{F}_{\alpha}^{AOUP}(t)=\lambda_{\alpha}(t)$ where
the evolution of $y_{\alpha}(t)$ is given by Eq. (2). From this equation, it
is easy to show that,

$\displaystyle\langle
F_{\alpha}^{AOUP}(\tau_{1})F_{\beta}^{AOUP}(\tau_{2})\rangle_{c}$
$\displaystyle=\langle\lambda_{\alpha}(\tau_{1})\lambda_{\beta}(\tau_{2})\rangle_{c},$
$\displaystyle=\delta_{\alpha,\beta}\frac{D}{\gamma}\left(e^{-\gamma|\tau_{1}-\tau_{2}|}-e^{-\gamma(\tau_{1}+\tau_{2})}\right).$
(168)

Next we substitute this in
$\langle\bar{F}^{AOUP}_{s}(\tau_{1})\bar{F}^{AOUP*}_{s^{\prime}}(\tau_{2})\rangle_{c}$
in Eq. (164) to obtain,

$\displaystyle\langle\bar{F}^{AOUP}_{s}(\tau_{1})\bar{F}^{AOUP*}_{s^{\prime}}(\tau_{2})\rangle_{c}=\frac{D}{\gamma
N^{2}}\left[e^{-\gamma|\tau_{1}-\tau_{2}|}-e^{-\gamma(\tau_{1}+\tau_{2})}\right]~{}\sum_{\alpha=1}^{N-1}e^{-\frac{2\pi
i\alpha}{N}(s-s^{\prime})}.$ (169)

Again we use $\sum_{\alpha=1}^{N-1}e^{-\frac{2\pi
i\alpha}{N}(s-s^{\prime})}=N\delta_{s,s^{\prime}}$ in this equation to get,

$\displaystyle\langle\bar{F}^{AOUP}_{s}(\tau_{1})\bar{F}^{AOUP*}_{s^{\prime}}(\tau_{2})\rangle_{c}=\delta_{s,s^{\prime}}\frac{D}{\gamma
N}\left[e^{-\gamma|\tau_{1}-\tau_{2}|}-e^{-\gamma(\tau_{1}+\tau_{2})}\right].$
(170)

Substituting this in
$\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}$ in Eq. (29) and
performing the integrations over $\tau_{1}$ and $\tau_{2}$, we obtain the
result quoted in Eq. (99).

##  Appendix B Derivation of $\langle x^{2}_{\alpha}(t)\rangle_{c}^{RTP}$ when
$t\gg\tau_{K}$

In this appendix, we will derive the form of the mean square distance of the
tagged particle which is given by Eq. (36) for the RTP. We begin with the
expression of $\langle x^{2}_{\alpha}(t)\rangle_{c}^{RTP}$ in Eq. (33) and
define the integral

$\displaystyle\mathcal{J}_{0}\left(a\right)$
$\displaystyle=\int_{-\pi}^{\pi}dq\left[\mathcal{G}(a,b_{q},t)\right],$ (171)
$\displaystyle=\int_{-\pi}^{\pi}dq\left[\frac{b_{q}(1+e^{-2b_{q}t})-a(1-e^{-2b_{q}t})}{b_{q}(b_{q}^{2}-a^{2})}\right],$
(172)

where in going from the first line to the second line, we have substituted the
expression of $\mathcal{G}(a,b_{q},t)$ from Eq. (31). Note that
$b_{q}=4K\sin^{2}\left(q/2\right)$. Substituting the form of $b_{q}$ in Eq.
(172), one finds terms like $\sim\int
dq~{}\text{exp}\left[-Kt\sin^{2}(q/2)\right]$ which in the limit $Kt\to\infty$
will be dominated by the small values of $q$. Therefore, we can approximate
$b_{q}\simeq Kq^{2}$.

$\displaystyle\mathcal{J}_{0}(a)$
$\displaystyle\simeq\int_{-\pi}^{\pi}dq\left[\frac{Kq^{2}(1+e^{-2Kq^{2}t})-a(1-e^{-2Kq^{2}t})}{Kq^{2}(K^{2}q^{4}-a^{2})}\right]$
(173)

Changing the variable $w=\sqrt{2Kt}~{}q$ and taking $Kt\to\infty$, we obtain

$\displaystyle\mathcal{J}_{\nu}(a)$
$\displaystyle\simeq\sqrt{\frac{t^{3}}{2K}}\int_{-\infty}^{\infty}dw~{}\mathcal{G}\left(at,\frac{w^{2}}{2},1\right).$
(174)

We rewrite the expression of $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ in
Eq. (33) for RTPs

$\displaystyle\langle
x_{\alpha}^{2}(t)\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}~{}\mathcal{J}_{0}(2\zeta),$
(175)

where $\mathcal{J}_{0}(2\zeta)$ is given by Eq. (172). For $t\gg\frac{1}{K}$,
we use the approximate expression of $\mathcal{J}_{0}(2\zeta)$ in Eq. (174) to
obtain the scaling form quoted in Eq. (36).

##  Appendix C Derivation of $\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$
when $t\gg\tau_{K}$

This appendix deals with the derivation of two-point correlation $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ when $t\gg\tau_{K}$ which is written in
Eq. (45) for RTP. We begin with the expressions of $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ as given in Eq. (44). Looking at these
expressions, we consider the following integral,

$\displaystyle\mathcal{I}_{0}(\beta,a)=\int_{-\pi}^{\pi}dq~{}\cos\left(q\beta\right)\mathcal{G}(a,b_{q},t),$
(176)

Note that $b_{q}=4K\sin^{2}\left(q/2\right)$. We now proceed to evaluate this
integral for $t\gg\frac{1}{K}$. Inserting the form of $\mathcal{G}(m,b_{q},t)$
from Eq. (31) in Eq. (176), we get terms of the form
$\sim\int~{}dq~{}\text{exp}\left(-4Kt\sin^{2}(q/2)\right)$. For $Kt\to\infty$,
such integrals will be dominated by the small values of $q$ which implies we
can approximate $b_{q}\simeq Kq^{2}$.

$\displaystyle\mathcal{I}_{0}(\beta,a)$
$\displaystyle\simeq\int_{-\pi}^{\pi}dq~{}\cos\left(q\beta\right)\left[\frac{Kq^{2}(1+e^{-2Kq^{2}t})-a(1-e^{-2Kq^{2}t})}{Kq^{2}(K^{2}q^{4}-a^{2})}\right].$
(177)

Changing the variable $w=\sqrt{2Kt}~{}q$ and taking $Kt\to\infty$, we get

$\displaystyle\mathcal{I}_{0}(\beta,a)$
$\displaystyle\simeq\sqrt{\frac{t^{3}}{2K}}\int_{-\infty}^{\infty}dw~{}\cos\left(\frac{w~{}\beta}{\sqrt{2Kt}}\right)\mathcal{G}\left(at,\frac{w^{2}}{2},1\right).$
(178)

From the expression of $\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ in Eq.
(44), we observe that the correlation is given by,

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}~{}\mathcal{I}_{0}\left(\beta,2\zeta\right),$
(179)

where $\mathcal{I}_{0}\left(\beta,2\zeta\right)$ is given in Eq. (176). For
$t\gg\frac{1}{K}$, using the approximate expression of
$\mathcal{I}_{0}\left(\beta,2\zeta\right)$ obtained in Eq. (178), we recover
the result quoted in Eq. (45).

## References

## References

  * [1] Lieb E H and Mattis D C 1966 Mathematical Physics in One Dimension. Exactly Solvable Models of Interacting Particles (Academic Press, New York)
  * [2] Ha Z N C 1996 Quantum Many-Body System in One Dimension (World Scientific, Singapore)
  * [3] Lebowitz J L 1987 Simple Models of Equilibrium and Nonequilibrium Phenomena (Elsevier, Amsterdam)
  * [4] Wioland H, Lushi E and Goldstein R 2016 New J. Phys. 18 075002
  * [5] Stenhammer J, Nardini C, Nash R, Marenduzzo D and Mozorov A 2017 Phys. Rev. Lett. 119 028005
  * [6] Reichhardt C and Reichhardt C 2014 Soft Matter 10 7502
  * [7] Gans W, Blumen A abd Amann A 1990 Large-Scale Molecular Systems Quantum and Stochastic Aspects - Beyond the Simple Molecular Picture (Plenum Press, New York and London)
  * [8] Ben-Avraham and Doering C R 1988 Phys. Rev. A 37 5007(R)
  * [9] Privman V 1997 Nonequilibrium Statistical Mechanics in One Dimension (Cambridge University Press)
  * [10] Harris T E 1965 J. Appl. Probab. 2 323
  * [11] Jepsen D W 1965 J. Math. Phys. 6 405
  * [12] Jerome K P 1974 Phys. Rev. A 9 557
  * [13] Alexander S and Pincus P 1978 Phys. Rev. B 18 2011
  * [14] Rödenbeck C, Kärger J and Hahn K 1998 Phys. Rev. E 57 4382
  * [15] Barkai E and Silbey R 2009 Phys. Rev. Lett. 102 050602
  * [16] Hegde C, Sabhapandit S and Dhar A 2014 Phys. Rev. Lett. 113 120601
  * [17] Ferrari P A and Fontes L R G 1998 Electron. J. Probab. 3 134
  * [18] Rajesh R and Majumdar S N 2001 Phys. Rev. E 64 036103
  * [19] Cividini J, Kundu A, Majumdar S N and Mukamel D 2016 J. Phys. A: Math. Theor. 49 085002
  * [20] Cividini J, Kundu A, Majumdar S N and Mukamel D 2016 J. Stat. Mech. 053212
  * [21] Kundu A and Cividini J 2016 Europhys. Lett. 115 54003
  * [22] Ponet A, Bénichou O, Démery V and Oshanin G 2018 Phys. Rev. E 97 062119
  * [23] Ponet A, Bénichou O, Démery V and Oshanin G 2019 Phys. Rev. Research 1 033089
  * [24] Gupta V, Nivarthi S S, McCormick A V and Davis H T 1995 Chem. Phys. Lett. 247 596
  * [25] Hahn K, Kärger J and Kukla V 1996 Phys. Rev. Lett. 76 2762
  * [26] Wei Q H, Bechinger C and Leiderer P 2000 Science 287 625
  * [27] Ramaswamy S 2010 Annu. Rev. Condens. Matter Phys. 1 323-345
  * [28] Romanczuk P, Bär M, Ebeling W, Lindner B and Schimansky-Geier L 2012 Eur. Phys. J.: Spec. Top. 202 1-162
  * [29] Marchetti M C, Joanny J F, Ramaswamy S, Liverpool T B, Prost J, Rao M and Simha A R 2013 Rev. Mod. Phys. 85 1143
  * [30] Ramaswamy S 2017 Active matter, J. Stat. Mech. 054002
  * [31] Schweitzer F 2003 Brownian Agents and Active Particles Collective Dynamics in the Natural and Social Sciences (Berlin: Springer)
  * [32] Bechinger C, Di Leonardo R, Löwen H, Reichhardt C, Volpe G and Volpe G 2016 Rev. Mod. Phys. 88 045006
  * [33] Cates M E and Tailleur J 2015 Annu. Rev. Condens. Matter Phys. 6 219-244
  * [34] Gonnella G, Marenduzzo D, Suma A and Tiribocchi A 2015 C. R. Phys. 16 316-331
  * [35] Partridge B and Lee C F 2019 Phys. Rev. Lett. 123 068002
  * [36] Caprini L, Marconi U and Puglisi A 2020 Phys. Rev. Lett. 124 078001
  * [37] Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Lecomte V, Orlandi A, Parisi G, Procaccini A, Viale M and Zdravkovic V 2008 Proc. Natl. Acad. Sci. USA 105 1232-37
  * [38] Katz Y, Tunstrøm K, Ioannou C C, Huepe C and Couzin I D 2011 Proc. Natl. Acad. Sci. USA 108 18720-25
  * [39] Redner G S, Hagan M F and Baskaran A 2013 Phys. Rev.Lett. 110 055701
  * [40] Bricard A, Caussin J B, Desreumaux N, Dauchot O, and Bartolo D 2013 Nature 503 95
  * [41] Solon A P, Fily Y, Baskaran A, Cates M, Kafri Y, Kardar M and Tailleur J 2015 Nature physics 11 673-678
  * [42] Basu U, Majumdar S N, Rosso A and Schehr G 2018 Phys. Rev. E 98 062121
  * [43] Malakar K, Das A, Kundu A, Kumar K V and Dhar A 2020 Phys. Rev. E 101 022610
  * [44] Chaudhuri D and Dhar A arXiv:2005.14234
  * [45] Pototsky A and Stark H 2012 EPL 98 50004
  * [46] Basu U, Majumdar S N, Rosso A and Schehr G 2019 Phy. Rev. E 100 062116
  * [47] Malakar K, Jemseena V, Kundu A, Kumar K V, Sabhapandit S, Majumdar S N, Redner S and Dhar A 2018 J. Stat. Mech. 043215\.
  * [48] Dhar A, Kundu A, Majumdar S N, Sabhapandit S and Schehr G 2019 Phys. Rev. E 99 032132
  * [49] Basu U, Majumdar S N, Rosso A, Sabhapandit S and Schehr G 2020 J. Phys. A: Math. Theor. 53 09LT01
  * [50] Demaerel T and Maes C 2018 Phys. Rev. E 97 032604
  * [51] Basu U, Majumdar S N, Rosso A and Schehr G 2018 Phys. Rev. E 98 062121
  * [52] Angelani L, Di Leonardo R and Paoluzzi M 2014 Eur. Phys. J. E 37 59
  * [53] Scacchi A and Sharma A 2018 Molecular Physics 116 460-464
  * [54] Mori F, Doussal P L, Majumdar S N and Schehr G 2020 Phys. Rev. Lett. 124 090603
  * [55] Singh P and Kundu A 2019 J. Stat. Mech. 083205
  * [56] Hartmann A K, Majumdar S N, Schawe H and Schehr G 2020 J. Stat. Mech. 053401
  * [57] Singh P and Kundu A 2020 arXiv:2011.04716
  * [58] Woillez E, Zhao Y, Kafri Y, Lecomte V and Tailleur J 2019 Phys. Rev. Lett. 122 258001
  * [59] Majumdar S N and Evans M 2018 J. Phys. A: Math. Theor. 51 47\.
  * [60] Kumar V, Sadekar O and Basu U 2020 Phys. Rev. E 102 052129
  * [61] Gradenigo G and Majumdar S N 2019 J. Stat. Mech. 053206
  * [62] Banerjee T, Majumdar S N, Rosso A and Schehr G 2020 Phys. Rev. E 101 052101
  * [63] Santra I, Basu U and Sabhapandit S 2020 Phys. Rev. E 101 062120
  * [64] Doussal P L, Majumdar S N and Schehr G 2020 EPL 130 40002
  * [65] Singh P, Sabhapandit S and Kundu A 2020 J. Stat. Mech. 083207
  * [66] Slowman A B, Evans M and Blythe R 2016 Phys. Rev. Lett. 116 218101
  * [67] Slowman A B, Evans M, and Blythe R 2017 J. Phys. A: Math. Theor. 50 37
  * [68] Mallmin E, Blythe R and Evans M 2019 J. Stat. Mech. 013204
  * [69] Das A, Kundu A and Dhar A 2020 J. Phys. A: Math. Theor. 53 345003
  * [70] Doussal P L, Majumdar S. N. and Schehr G. 2019 Phys. Rev. E 100 012113
  * [71] Cates M E 2019 arXiv:1904.01330
  * [72] Teomy E and Metzler R 2019 J. Phys. A: Math. Theor. 52 385001
  * [73] Teomy E and Metzler R 2019 J. Stat. Mech. 103211
  * [74] Galanti M, Fanelli D and Piazza F 2013 Eur. Phys. J. B 86 456
  * [75] Dolai P, Das A, Kundu A, Dasgupta C, Dhar A and Vijay Kumar K 2020 Soft Matter 16 7077-7087
  * [76] Put S, Berx J and Vanderzande C 2019 J. Stat. Mech. 123205
  * [77] Lizana L, Ambjörnsso T, Taloni A, Barkai E and Lomholt M A 2010 Phys. Rev. E 81 051118
  * [78] Krapivsky P L, Mallick K and Sadhu T 2015 J. Stat. Mech. P09007


# Engineering topological phases of any winding and Chern numbers in extended
Su-Schrieffer-Heeger models

Rakesh Kumar Malakar<EMAIL_ADDRESS>Asim Kumar Ghosh
<EMAIL_ADDRESS>Department of Physics, Jadavpur University,
188 Raja Subodh Chandra Mallik Road, Kolkata 700032, India

###### Abstract

Simple route of engineering topological phases for any desired value of
winding and Chern numbers is found in the Su-Schrieffer-Heeger (SSH) model by
adding a further neighbor hopping term of varying distances. It is known that
the standard SSH model yields a single topological phase with winding number,
$\nu=1$. In this study it is shown that how one can generate topological
phases with any values of winding numbers, for examples, $\nu=\pm 1,\pm 2,\pm
3,\cdots,$ with the presence of a single further neighbor term which preserves
inversion, particle-hole and chiral symmetries. Quench dynamics of the
topological and trivial phases are studied in the presence of a specific
nonlinear term. Another version of SSH model with additional modulating
nearest neighbor and next-nearest-neighbor hopping parameters was introduced
before those exhibit a single topological phase characterized by Chern number,
$\mathcal{C}=\pm 1$. Standard form of inversion, particle-hole and chiral
symmetries are broken in this model. Here this model has been studied in the
presence of several types of parametrization among which for a special case
system is found to yield a series of phases with Chern numbers,
$\mathcal{C}=\pm 1,\pm 2,\pm 3,\cdots.$ In another parametrization, multiple
crossing within the edge states energy lines are found in both trivial and
topological phases. Topological phase diagrams are drawn for every case.
Emergence of spurious topological phases is also reported.

##  I INTRODUCTION

Su-Schrieffer-Heeger (SSH) model is the most popular representative of one-
dimensional (1D) topological insulator which paved the way for studying the
topological phases in the simplest mannerSSH1 ; SSH2 ; SSH3 . Both trivial and
topological insulating phases has been realized in this model by tuning the
ratio of inter and intracell hopping amplitudes in this staggered model
composed by two-site unit cells. Nontrivial phase is observed when this ratio
exceeds unity and it is characterized by nonzero topological invariant known
as winding number ($\nu$) which is connected to the integral of Berry
curvature over the Brillouin zone (BZ) and known as Pancharatnam-Berry (PB)
phase or Zak phase. This nontrivial phase is at the same time associated with
the emergence of symmetry protected zero energy states those are found
localized on both edges of open chain. Transition between those two phases
with nonzero band gap is accompanied by vanishing band gap found at the phase
transition point.

Importance of topological matter lies with the fact that additional
topological robustness in the nontrivial phase protects these systems from any
kind of imperfections present in the materials. This robustness enhances
quantum correlations Chen  and causes higher efficiency in electronic
transport. As a result, topological materials are expected to be more suitable
in the development of quantum processing devices DasSarma .

SSH model was introduced before in a totally different context, when it was
employed to understand the role of solitonic excitations in conducting
polymers, like polyacetylene, etc. Direct measurement of PB phase has been
done recently by mimicking the 1D periodic potential of polyacetylene using
system of ultracold atoms in optical latticesAtala . Signature of
topologically protected pair of bound states is also detected by photonic
quantum walkKitagawa . In addition, properties of tight-binding SSH model has
been experimentally validated in photonic lattice composed of helical
waveguides Rechtsman  and in phononic crystal composed of cylindrical
waveguidesLi3 .

The topological phase in two-band SSH model is defined uniquely by $\nu=1$ for
each band. Besides this one, search of new topological phases preferably with
higher values of $\nu$ continues afterwards by adding further neighbor hopping
terms. Collectively they are called extended SSH (eSSH) models. Emergence of a
new phase with $\nu=-1$ has been demonstrated before by adding a single
further neighbor hopping termLi . By invoking multiple further neighbor
hopping terms new phases of $\nu=2,3,4$ has been generated laterPlatero ;
Ghosh . In this work it is shown that the eSSH model is capable to host
indefinite number of topological phases with a series of different winding
numbers as one wishes. And remarkably, in this series of eSSH models only a
single extra further neighbor hopping term is sufficient for their
realization.

Interestingly, demonstration of topological phases in two-dimensional (2D)
system has been started long before with the discovery of integer quantum Hall
effectTKNN ; Hasan . Subsequently, this phenomenon has been extended to wider
systems by Haldane in his seminal work on a tight-binding model with complex
further neighbor hopping terms formulated on honeycomb latticeHaldane . This
finding gives birth to quantum anomalous Hall (QAH) effect where the magnetic
field is replaced by phase dependent hoppings. This state of matter was
experimentally realized in periodically modulated optical honeycomb
latticeJotzu . For the 2D systems, Chern number ${\mathcal{C}}$, is treated as
the topological invariant. In the two-band Haldane model, topological phase is
defined by ${\mathcal{C}}=\pm 1$, values of opposite signs for the two
different energy bands. Realization of topological phase for higher values of
${\mathcal{C}}$s continues thereafter by either invoking further neighbour
hopping terms Moumita1 ; Moumita2  or imposing periodic driveArghya ; Agarwal
, etc. Experimental realization of QAH phases tunable up to ${\mathcal{C}}=\pm
5$ has been reported recentlyZhao .

In another development, finding of QAH effect breaks its dimensional barrier,
as the realization of this phase makes possible in 1D eSSH model, where
nearest neighbor (NN) and next-nearest-neighbor (NNN) hopping amplitudes are
modulated by two independent cyclic variablesLi2 . Remarkably, in this case,
one of the cyclic variable can be treated like an additional synthetic
dimension. So as a whole, this 1D model behaves like an effective 2D model in
the reciprocal space and at the same time hosts nontrivial topological phases.

Again, in this work, the eSSH models are studied in 2D reciprocal space by
introducing different kind of parametrization in terms of those two cyclic
parameters. And again, it is shown that those models are capable to host
indefinite number of topological phases with a series of different Chern
numbers. Properties of those new phases with higher values of ${\mathcal{C}}$s
have been characterized in details. Article has been organized in the
following manner.

Structure of those eSSH models are described in the section II. Topological
phases of eSSH models are characterized in Sec. III. Four different eSSH
models are introduced here whose topological properties are studied in details
in terms of winding numbers, edge states, and quench dynamics. Models for
phases of higher values of $\nu$ will be generalized at the end of this
section. Topological phases in terms of Chern numbers are studied in Sec. IV.
Several different types of parametrization are introduced and their
topological properties are characterized. Spurious topological phases are
identified. Topological phase diagrams have been drawn in very case and
symmetries of the Hamiltonian are explained. A discussion based on those
results is available in Sec V.

##  II SSH models with further neighbor terms

The standard SSH model SSH1 ; SSH2  is defined on a 1D bipartite lattice where
one primitive cell contains two different sites, A and B. The corresponding
Hamiltonian is described as

$H_{vw}=\sum_{j=1}^{N}\left(v\,c^{\dagger}_{{\rm A},j}c_{{\rm
B},j}+w\,c^{\dagger}_{{\rm A},j+1}c_{{\rm B},j}\right)\\!+{\rm h.c.},$ (1)

where $c_{{\rm A},j}$ and $c_{{\rm B},j}$ stand for the annihilation operators
of electron on sublattices A and B, respectively, in the $j$th primitive cell.
$N$ is the total number of primitive cells where $v$ and $w$ are the intracell
and intercell hopping amplitudes, respectively. Those terms permit hopping
only between the adjacent sites. Energy spectrum of $H_{vw}$ is gapless when
$w=v$, while there is a band gap when $w\neq v$. Among the two gapful regions
around the gapless point one is topologically trivial ($\nu=0$) when $w<v$,
and remarkably as long as $w>v$, this simple model hosts a single nontrivial
topological phase with $\nu=1$.

In 2019, Li and Miroshnichenko Li  showed that a new topological phase with
$\nu=-1$ appears upon introducing additional terms which allow hopping between
sites of A sublattice and nonadjacent sites of B sublattice but only among the
NN primitive cells. A single pair of topological edge states is found to
appear associated with this new phase.

Figure 1: Extended SSH model introduced by Li and Miroshnichenko Li .

This particular choice of sites between which the hopping is being permitted
is found to preserve the chiral symmetry of the resulting system. If $z$ be
the amplitude of this additional hopping, total Hamiltonian can be expressed
as

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{vw}+H_{z},$
$\displaystyle H_{z}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}z\,c^{\dagger}_{{\rm A},j}c_{{\rm B},j+1}+{\rm
h.c.},$ (2)

and distribution of winding numbers in the parameter space of the system is
given by

$\nu=\left\\{\begin{array}[]{ll}0,&|w+z|<v,\\\\[3.00003pt] 1,&|w+z|>v\;{\rm
and}\;w>z,\\\\[3.00003pt] -1,&|w+z|>v\;{\rm and}\;w<z.\end{array}\right.$ (3)

In another study, Pérez-González et al showed that an additional topological
phase with $\nu=2$ emerges in the presence of more than one further neighbor
hopping terms Platero . Two distinct pairs of topological edge states are
found to appear. In the presence of multiple further neighbor hopping terms,
topological phases with higher winding numbers, say, up to $\nu=4$ has been
reported so far Ghosh .

In this study we are going to show that a single additional hopping term is
sufficient to produce the topological phases with any value of winding number
as one wish. Topological phases with higher values of winding numbers can be
generated by systematically increasing the separation between the sites among
which hopping is taken into account. Multiple pairs of edge states consistent
with the value of $\nu$ are found to appear.

##  III Topological phases in terms of winding numbers

In order to generate the topological phases with any values of winding numbers
in the most simple way, two different types of eSSH models are introduced,
however, both of them include a single further neighbor hopping term. Two
different types of Hamiltonians are termed as ‘A-B’ and ‘B-A’ depending on the
ordering of the sublattice sites and they are noted as
$H^{\rm{A\mbox{-}B}}_{z,n}$ and $H^{\rm{B\mbox{-}A}}_{z,n}$, respectively,
where $(n-1)$ is the number of intermediate primitive cells covering under the
hopping distance and $z$ is the amplitude of the further neighbour hopping. In
this nomenclature, Hamiltonian $H_{z}$ in Eq. 2 can be specified as
$H^{\rm{A\mbox{-}B}}_{z,1}$. However, hopping only between different
sublattices is allowed in this case. This type of hopping term preserves the
particle-hole and inversion symmetries. Conservation of those symmetries means
the preservation of chiral symmetry in addition. Now the topological
properties of four eSSH models will be studied in great details. Among them
four models, two are of type ‘B-A’ and the remaining two are of type ‘A-B’.

###  III.1 Topological phases for $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,2}$

Total Hamiltonian in this case is expressed as

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{vw}+H^{\rm{B\mbox{-}A}}_{z,2},$ $\displaystyle H^{\rm{B\mbox{-}A}}_{z,2}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{N}z\,c^{\dagger}_{{\rm B},j}c_{{\rm
A},j+2}+{\rm h.c.},$ (4)

where the hopping term extends over one intermediate primitive cell, which is
shown in Fig. 2.

Figure 2: Extended SSH model describing the hopping in
$H^{\rm{B\mbox{-}A}}_{z,2}$.

Under the Fourier transformations,

$\displaystyle c_{{\rm A},j}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\rm k\in{\rm BZ}}a_{\rm k}\,e^{i\rm
kj},$ $\displaystyle c_{{\rm B},j}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\rm k\in{\rm BZ}}b_{\rm k}\,e^{i\rm
kj},$

where the summation extends over BZ zone, and assuming periodic boundary
condition (PBC), the Hamiltonian in the k-space becomes

$H=\sum_{\rm k\in{\rm BZ}}\Psi^{\dagger}_{\rm k}H({\rm k})\psi_{\rm k},$

for which $\Psi^{\dagger}_{\rm k}=[a^{\dagger}_{\rm k}\;b^{\dagger}_{\rm k}],$
and $H(\rm k)=\boldsymbol{g}(\rm k)\cdot\boldsymbol{\sigma}$. Here,
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$, are the Pauli
matrices, and assuming the unit lattice parameter, ($a=1$),

$\boldsymbol{g}(\rm k)\equiv\left\\{\begin{array}[]{l}g_{x}=v+w\cos{\\!(\rm
k)}+z\cos{\\!(2\rm k)},\\\\[3.00003pt] g_{y}=w\sin{\\!(\rm k)}+z\sin{\\!(2\rm
k)},\\\\[3.00003pt] g_{z}=0.\end{array}\right.$

It can be shown that $H({\rm k})$ satisfies the following transformation
relations under the three different operators:

$\left\\{\begin{array}[]{l}\mathcal{P}H({\rm k})\mathcal{P}^{-1}=-H(-{\rm
k}),\\\\[3.99994pt] \sigma_{x}H({\rm k})\sigma_{x}=H(-{\rm k}),\\\\[3.99994pt]
\sigma_{z}H({\rm k})\sigma_{z}=-H({\rm k}),\end{array}\right.$

where $\mathcal{P}=\mathcal{K}\sigma_{z}$ and $\mathcal{K}$ is the complex
conjugation operator. Those relations correspond to the conservation of
particle-hole, inversion and chiral symmetries. The last relation is satisfied
as a consequence of the first two.

$\boldsymbol{g}(\rm k)$ can be spanned as a vector in the $g_{x}\mbox{-}g_{y}$
complex plane, due to the conservation of chiral symmetry. As a result, the
dispersion relation can be expressed as $E_{\pm}({\rm k})=\pm|g({\rm k})|$,
or, $E_{\pm}({\rm k})=\pm\sqrt{v^{2}+w^{2}+z^{2}+2[vw\cos{\\!(\rm
k)}+vz\cos{\\!(2\rm k)}+wz\cos{\\!(\rm k)}]}$. Dispersions are symmetric
around the energy, $E=0$, since the Hamiltonian preserves particle-hole
symmetry. Variation of dispersion relation, $E_{+}({\rm k})$, with $w/|v+z|$
for $v=1$, $z=1/2$ and $v=3/4$, $z=1$ are shown in Fig. 3 (a) and (b),
respectively. The lower band, $E_{-}({\rm k})$ is not drawn. The figures in
(a) and (b) are serving as prototype figures for $v/z>1$ and $v/z<1$,
respectively. Dispersions comprise of one broad peak when $w/|v+z|\leq 1$ for
both the cases $v/z>1$ and $v/z<1$. Band gap vanishes at the BZ boundaries,
${\rm k}=\pm\pi$, when $w=|v+z|$, for both also. As a result, $\nu$ is
undefined at the point when $w=|v+z|$.

Figure 3: Dispersion relation for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,2}$ when
$v/z>1$, (a) and $v/z<1$, (b).

###  III.2 Winding number, $\nu$

Tip of the vector $\boldsymbol{g}(\rm k)$ traces out closed loops in the
$g_{x}\mbox{-}g_{y}$ plane if k runs from $-\pi$ to $\pi$ on the BZ. Winding
number is defined to enumerate the number of closed looping around the origin
of the plane. Mathematically it is expressed as

$\nu=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[\boldsymbol{\hat{g}}(\rm
k)\times\frac{d}{d\rm k}\,\boldsymbol{\hat{g}}(\rm k)\right]_{z}\\!d{\rm k},$

where $\boldsymbol{\hat{g}}(\rm k)=\boldsymbol{g}(\rm k)/|\boldsymbol{g}(\rm
k)|.$ Two distinct topological phases, with $\nu=1,2$ are found for this case
in the parameter space as

$\nu=\left\\{\begin{array}[]{ll}1,&w>|z+v|,\\\\[3.00003pt] 0,&w<|z+v|,\;{\rm
and}\;v>z,\\\\[3.00003pt] 2,&w<|z+v|,\;{\rm and}\;v<z,\end{array}\right.$ (5)

and those are associated with a number of topological phase transitions.

For examples, when $v>z$, a transition takes place at $w=|v+z|$, separating
trivial phase, $\nu=0$ for $w<|v+z|$ and topological phase, $\nu=1$ for
$w>|v+z|$. Whereas, transition occurs at the same point between two
topological phases when $v<z$. In this case, the phase for $w>|v+z|$ is marked
by $\nu=1$, while that for $w<|v+z|$ is identified by $\nu=2$. In all cases
transition takes place between the phases with energy gap, and obviously, gap
closes at the transition point, $w=|v+z|$. Winding number is found undefined
when $w<|v+z|$ for $v=z$ as the band gap closes there.

The parametric plot of winding diagrams in the $g_{x}\mbox{-}g_{y}$ complex
plane are shown in Fig. 4. Four figures are drawn for (a) $v=0.5$, $w=1.2$,
$z=0.3$, (b) $v=0.5$, $w=0.4$, $z=0.3$, (c) $v=0.3$, $w=0.4$, $z=0.5$, and (c)
$v=0.3$, $w=0.4$, $z=0.3$. Arrow head indicates the direction of move of the
the vector $\boldsymbol{g}({\rm k})$ for an infinitesimal increment of ${\rm
k}$. Tip of $\boldsymbol{g}({\rm k})$ moves in counterclockwise direction over
all the closed contours. The contours in (a) and (c) enclose the the origin,
while that in (b) does not. On the other hand, contour in (d) passes over the
origin. The curve passes around the origin once in (a) and twice in (c). Those
figures serve as the prototype windings for the four different regions,
$w>|v+z|$, $v>z$ for $\nu=1$, $w<|v+z|$, $v>z$ for $\nu=0$, $w<|v+z|$, $v<z$,
for $\nu=2$, and $w<|v+z|$, $v=z$, where $\nu$ is undefined. There is gap for
the first three cases while the spectrum is gapless for the last.

Figure 4: Parametric winding diagrams in the $g_{x}\mbox{-}g_{y}$ plane for
the Hamiltonian, $H^{\rm{B\mbox{-}A}}_{z,2}$. Four figures are drawn for (a)
$v=0.5$, $w=1.2$, $z=0.3$, (b) $v=0.5$, $w=0.4$, $z=0.3$, (c) $v=0.3$,
$w=0.4$, $z=0.5$, and (c) $v=0.3$, $w=0.4$, $z=0.3$.

Figure 5: Bulk and edge state energies for
$H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,2}$ when $v/z>1$, (a) for $v=0.7$, $z=0.3$,
and $v/z<1$, (b) for $v=0.3$, $z=0.7$. Variation of energy with
$w/|v\\!+\\!z|$ are shown for the lattice of 200 sites. Figure 6: Probability
density of edge states for $H^{\rm{B\mbox{-}A}}_{z,2}$: (a) for $v=0.25$,
$w=2.5$, $z=0.25$, one pair of edge states, (b) for $v=0.25$, $w=0.25$,
$z=2.5$, two pairs of edge states. Figures are drawn for the lattice of 142
sites.

Variation of bulk-edge state energies with respect to $w/|v+z|$ is shown in
Fig. 5 as long as $w/|v+z|\leq\pm 2$. A single pair of zero energy edge states
survives when $w>|v\\!+\\!z|$ as shown in (a). No edge state is there in this
system when $w<|v\\!+\\!z|$ and and $v>z$. In contrast, zero energy edge
states are always there when $w>|v\\!+\\!z|$ and $v<z$ which is shown in Fig.
5 (b). Actually, a single pair of zero energy edge states survives when
$w>|v\\!+\\!z|$, and two pairs of edge states are there when $w<|v\\!+\\!z|$
and $v<z$. The figures are drawn for lattice of sites 200. All these results
are consistent with the bulk-boundary correspondence rule.

In order to confirm the presence of zero energy edge states, probability
density of those states are drawn in Fig. 6 for the lattice of 142 sites. Two
figures are drawn for two distinct topological phases. In the upper panel (a),
probability density of two distinct edge states with $E=0$ are shown when
$v=0.25$, $w=2.5$, $z=0.25$, as these values conform to the conditions,
$w>|v\\!+\\!z|$. Probability density of one edge state exhibits sharp peak at
site $m=1$ and another one at site $m=142$. This corresponds to the
topological phase with $\nu=1$. On the other hand, for $w<|v\\!+\\!z|$ and
$v<z$, probability density of four distinct edge states with $E=0$ are shown
in the lower panel (b) when $v=0.25$, $w=0.25$, $z=2.5$, as these values are
in accordance to the last conditions. Probability density of four orthogonal
edge states exhibit sharp peak at sites $m=1$, $m=3$, $m=140$, and $m=142$. It
indicates that zero energy states near the left edge are localized on the A
sublattice, while those close to the right edge are localized on the B
sublattice. This result is in accordance to the topological phase with
$\nu=2$.

An extensive phase diagram of the total Hamiltonian including
$H^{\rm{B\mbox{-}A}}_{z,2}$ is shown in Fig 7. Here contour plot for $\nu$ is
drawn in the $v$-$w/|v+z|$ space. Presence of two distinct topological phases,
$\nu=1$ and 2 along with the trivial phase, $\nu=0$ are shown in green, blue
and red, respectively. The horizontal line is drawn at $v=1$ or $v/z=1$ since
this diagram is drawn for $v+z=2$. The line segment within the points
$w/(v+z)=\pm 1$ separates the trivial phase from the topological phase with
$\nu=2$. Hence phase transition occurs around this segment. Another
topological phase with $\nu=1$ appears beyond the two vertical lines drawn at
$w/(v+z)=\pm 1$. They separate topological phases with $\nu=1$ and 2 when
$v/z<1$ and topological ($\nu=1$) and trivial phase when $v/z>1$. So the
system undergoes phase transition around those straight lines. Band gap
vanishes over those lines as well as on the line segment.

Figure 7: Topological phase diagram for the Hamiltonian,
$H^{\rm{B\mbox{-}A}}_{z,2}$. Trivial phase is shown by blue ($\nu=0$) while
two distinct topological phases are shown by green ($\nu=1$) and red
($\nu=2$). This diagram is drawn for $v+z=2.$ The horizontal line indicates
the value $v=1$ or $v/z=1.$

###  III.3 Quenched dynamics in the presence of nonlinear terms

Now the effect of nonlinearity on the topological phase will be studied
following the method developed by Ezawa Ezawa1 . Schrödinger equation for a
Hamiltonian matrix $M$ spanned on a lattice composed of $L$ sites can be
written as ($\hbar=1$),

$i\,\frac{\partial\psi_{l}}{\partial t}+\sum_{m=1}^{L}M_{lm}\,\psi_{l}=0.$ (6)

It actually comprises of $L$ coupled linear equations and governs the time
evolution of the system where $M_{lm}$ is recognized as the element of hopping
matrix in case of tight-binding model. This system hosts the topological as
well as trivial phases for different parameter regime.

Figure 8: Quench dynamics for $H^{\rm{B\mbox{-}A}}_{z,2}$ when $\zeta=0.5$,
(a) for $v=0.25$, $w=2.5$, $z=0.25$, (b) for $v=2.5$, $w=0.25$, $z=0.25$, (c)
for $v=0.25$, $w=0.25$, $z=2.5$. Figures are drawn for lattice with 20 sites.

The eigenvalue equation for the hopping matrix, $M$ is written as

$M\bar{\phi}_{q}=E_{q}\bar{\phi}_{q},\;1\leq q\leq L,$ (7)

where $q$ serves as the quantum index. Hence, time evolution of the model is
governs by the solution of Eq. 6 as

$\bar{\phi}_{q}(t)=e^{-itE_{q}}\,\bar{\phi}_{q}(t),$ (8)

since the Schrödinger equation eventually turns into a set of decoupled
equations

$i\,\frac{\partial\bar{\phi}_{q}}{\partial
t}+\sum_{m=1}^{L}M_{lm}\,\bar{\phi}_{q}=0.$ (9)

The variation of energies $E_{q}$ with $w/|v+z|$ for two different topological
phases have been shown in the Fig. 5, when the hopping matrix is constituted
for the Hamiltonian defined in Eq. 4 for the lattice of sites $L=200$.
Topological phases are always protected by the zero energy edge states by
virtue of particle-hole symmetry of the system. As a result, no time evolution
of those localized states is permissible according to the Eq. 8. In light of
this fact, time evolution of the edge states in the presence of additional
nonlinear term will be studied.

The Schrödinger equation in the presence of nonlinear term for the one-
dimensional tight-binding model of hopping matrix $M_{lm}$ is defined by

$i\frac{\partial\psi_{l}}{\partial
t}+\sum_{m}^{L}M_{lm}\psi_{l}+\zeta|\psi_{l}|^{2}\psi_{l}=0,$ (10)

where the effect of nonlinearity is controlled by the parameter $\zeta$.
Explicit form of the set of coupled nonlinear first order differential
equation for finite chain of $L$ sites and for the Hamiltonian defined in Eq.
4 with open boundary condition (OBC) is given by

$\displaystyle i\,\frac{\partial\psi_{2j-1}}{\partial t}$ $\displaystyle=$
$\displaystyle v(\psi_{2j}-\psi_{2j-1})+w(\psi_{2j-2}-\psi_{2j-1})$
$\displaystyle+z(\psi_{2j-4}-\psi_{2j-1})-\zeta|\psi_{2j-1}|^{2}\psi_{2j-1},$
$\displaystyle\vdots\quad\;$ $\displaystyle=$
$\displaystyle\qquad\qquad\vdots$ (11) $\displaystyle
i\,\frac{\partial\psi_{2j}}{\partial t}$ $\displaystyle=$ $\displaystyle
w(\psi_{2j+1}-\psi_{2j})+v(\psi_{2j-1}-\psi_{2j})$
$\displaystyle+z(\psi_{2j+3}-\psi_{2j})-\zeta|\psi_{2j}|^{2}\psi_{2j},$

where $j$ denotes the cell index which ultimately generates $L$ number of
coupled equations each one for every sites. Differential equations for odd and
even sites are different since the translational symmetry of one lattice unit
is broken.

The fate of the topological state when $\zeta\neq 0$ will be studied in terms
of the time evolution of the nonlinear system by imposing an initial
condition,

$\psi_{l}(t)=\delta_{l,m}\quad{\rm when}\;\;t=0.$

It means a delta-function like pulse at the $m$-th site is given initially.
Henceforth dynamics of the resulting nonlinear system will be examined by
maintaining the conservation rule imposed by the equation,

$\sum_{l=1}^{L}|\psi_{l}(t)|^{2}={\rm constant}.$ (12)

Value of the constant may be fixed on depending the choice of the initial
conditions. The initial conditions by turn depend on the value of winding
number for a particular topological phase. It is shown that topological phase
defined in the linear system is robust against the introduction of the
nonlinear term as long as $\zeta<1$ since the quenching of the edge states are
observed.

As the general solution of the Eq. can be expanded as

$\psi_{l}(t)=\sum_{q}c_{q}(t)\bar{\phi}_{q}(t),$

the initial state can be expressed as

$\psi_{l}(0)=\delta_{l,m}=\sum_{q}c_{q}\bar{\phi}_{q}(0).$ (13)

The topological phase of the linear system is always protected by the presence
of zero energy edge (localized) states. So keeping in mind the position of
edge states, initial condition is imposed either by $l=1$ or $l=L$ when
$\nu=1$. Here $l=1$ ($l=L$) denotes the leftmost (rightmost) site of the
lattice. Now for $l=1$, initial condition turns out as
$\psi_{l}(0)=\delta_{l,1}$. Right hand side of the Eq. 13 may be simplified by
labeling the zero energy state by $\bar{\phi}_{1}$ with $E_{1}=0$ in Eq. 7.
So, at $t=0$,

$\psi_{1}(0)=c_{1}\bar{\phi}_{1}(0).$ (14)

As $E_{1}=0$, $\bar{\phi}_{q}(t)=\bar{\phi}_{q}(0)$, which leads to the fact
that

$\psi_{1}(t)=c_{1}\bar{\phi}_{1}(0).$

It means no time evolution of the edge states is there or in other words a
nonzero probability amplitude at the edge site remains at any time. It
corresponds to the quenching of the edge states. No such quenching is possible
for the bulk states by virtue of their non-zero energy, ($E_{p}\neq 0$).

In contrast, zero-energy localized states are absent in the trivial phase, and
all the states are found to extend within the bulk. As a result, quenching
dynamics of edge states may serve as an alternative numerical tool to
distinguish the topological and trivial phases by investigating the effect of
nonlinear component on the initial condition.

Quenching of edge states for the nonlinear system is shown in Fig. 8, by
solving the set of Eq. 11, for $L=20$ when $\zeta=0.5$. Contour plot for the
time evolution of the absolute value of complex amplitude, $|\psi_{l}(t)|$, is
drawn for every site, $l=1,2,3,\cdots,20$, which is shown along the horizontal
axis. Three contour plots are shown (a) for $v=0.25$, $w=2.5$, $z=0.25$, (b)
for $v=2.5$, $w=0.25$, $z=0.25$, (c) for $v=0.25$, $w=0.25$, $z=2.5$, where
(b) indicates trivial phase while (a) and (c) for the topological phases of
$\nu=1$ and $\nu=2$. Initial condition is set by $\psi_{l}(0)=\delta_{l,m}$,
where $m=1,3,18,20$. Which means the initial pulse is given only at those
sites. As a result, conservation rule follows the relation,
$\sum_{l=1}^{L}|\psi_{l}(t)|^{2}=4$.

Time evolution is explored for the span $0\leq t\leq 20$, which is plotted
along the vertical axis. The diagram clearly indicates that probability
amplitudes for $l=1,20$, i. e., $|\psi_{1}(t)|$ and $|\psi_{20}(t)|$ survive
with time in (a). So the edge states bound to the topological phase with
$\nu=1$ exhibit their quenching. No such quenching is found for the trivial
phase as shown in (b). Quenching of four edge states, $|\psi_{l}(t)|$,
$l=1,3,18,20$ are found in (c) which correspond to the topological phase with
$\nu=2$. So for the lattice with $L$ sites, quenching are found for the
amplitude with sites $l=1,3,L\\!-\\!2,L$. It is true that the diagram
exhibiting the quenching of edge states will be different if the initial
conditions are made different from this set. However, this particular choice
of initial conditions is considered from the previous knowledge of locations
for the peaks of probability density of edge states as shown in Fig. 6. Hence
the quench dynamics provide another route for distinguishing topological and
trivial phases for a system.

###  III.4 Topological phases for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,2}$

Total Hamiltonian in this case is

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{vw}+H^{\rm{B\mbox{-}A}}_{z,2},$ $\displaystyle H^{\rm{A\mbox{-}B}}_{z,2}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{N}z\,c^{\dagger}_{{\rm A},j}c_{{\rm
B},j+2}+{\rm h.c.},$ (15)

where the hopping term one again extends over one intermediate primitive cell,
which is shown in Fig. 9. As a result,

$\boldsymbol{g}(\rm k)\equiv\left\\{\begin{array}[]{l}g_{x}=v+w\cos{\\!(\rm
k)}+z\cos{\\!(2\rm k)},\\\\[3.00003pt] g_{y}=w\sin{\\!(\rm k)}-z\sin{\\!(2\rm
k)},\\\\[3.00003pt] g_{z}=0.\end{array}\right.$

Figure 9: Extended SSH model describing the hopping in
$H^{\rm{A\mbox{-}B}}_{z,2}$.

Dispersion relation in this case is $E_{\pm}({\rm
k})=\pm\sqrt{v^{2}+w^{2}+z^{2}+2[vw\cos{\\!(\rm k)}+vz\cos{\\!(2\rm
k)}+wz\cos{\\!(3\rm k)}]}$. Variation of dispersion relation, $E_{+}({\rm
k})$, with $w/|v+z|$ for $v=1$, $z=1/2$ and $v=1/2$, $z=1$ are shown in Fig.
10 (a) and (b), respectively. Those are serving as prototype figures for
$v/z>1$ and $v/z<1$, respectively. Dispersions comprise of three peaks for any
values of the parameters, $v$, $w$ and $z$. Like the previous case, band gap
vanishes at ${\rm k}=\pm\pi$, and ${\rm k}=$, when $w=|v+z|$, for both the
cases $v/z>1$ and $v/z<1$. As a result, $\nu$ is undefined again at the point
when $w=|v+z|$. But in the region $w<|v+z|$ for $v<z$, the system undergoes an
additional phase transition at the point defined by the set of equations,
$E_{\pm}({\rm k})=0$, and $\frac{dE_{\pm}({\rm k})}{d{\rm k}}=0$, which will
be discussed later.

Figure 10: Dispersion relation for $H^{\rm{A\mbox{-}B}}_{z,2}$ when $v/z>1$,
(a) and $v/z<1$, (b).

In this case also two different topological phases appear in the parameter
space as given below those are separated by phase transition lines.

$\nu=\left\\{\begin{array}[]{ll}1,&w>|z+v|,\\\\[3.00003pt] 0,&w<|z+v|,\;{\rm
and}\;v>z,\\\\[3.00003pt] -2,0&w<|z+v|,\;{\rm and}\;v<z.\end{array}\right.$
(16)

Topological phase with $\nu=1$ exists as along as the relation $w>|z+v|$ holds
irrespective of individual values of $v$ and $z$. Another nontrivial phase
with $\nu=-2$ appears in a limited region for $w<|z+v|$ and $v<z$, separated
by trivial phase. The equation of phase transition line can be obtained by
satisfying the conditions, $E_{\pm}({\rm k})=0$, and $\frac{dE_{\pm}({\rm
k})}{d{\rm k}}=0$. Anyway, this model hosts the new topological phase with
$\nu=-2$.

Figure 11: Parametric winding diagrams in the $g_{x}\mbox{-}g_{y}$ plane for
the Hamiltonian, $H^{\rm{A\mbox{-}B}}_{z,2}$. Four figures are drawn for (a)
$v=0.5$, $w=1.2$, $z=0.3$, (b) $v=0.5$, $w=0.5$, $z=0.3$, (c) $v=0.3$,
$w=0.2$, $z=0.5$, and (c) $v=0.45$, $w=0.6$, $z=0.5$.

The parametric plot of winding for the tip of the vector, $\boldsymbol{g}$ in
the $g_{x}\mbox{-}g_{y}$ complex plane are shown in Fig. 11. Four figures are
drawn for (a) $v=0.5$, $w=1.2$, $z=0.3$, (b) $v=0.5$, $w=0.5$, $z=0.3$, (c)
$v=0.3$, $w=0.2$, $z=0.5$, and (d) $v=0.45$, $w=0.6$, $z=0.5$. Those figures
serve as the prototype contours for the four different regions, $w>|v+z|$, for
$\nu=1$, $w<|v+z|$, $v>z$ for $\nu=0$, $w<|v+z|$, $v<z$, for $\nu=-2$ and
$\nu=0$. $\boldsymbol{g}({\rm k})$ traces the closed contour in clockwise
direction for (a) and (b) while it is counter clockwise for (c) and (d). Curve
encloses the origin once in (a) and twice in (c) but in opposite direction
which corresponds to winding numbers of opposite sign. Nonzero band gap is
there for all the cases.

Variation of bulk-edge state energies with respect to $w/|v+z|$ is shown in
Fig. 12 for the regime $-2\leq(w/|v+z|)\leq 2$. A single pair of zero energy
edge states is there when $w>|v+z|$ as shown in (a). No edge state is there in
this system when $w<|v+z|$ and and $v>z$. However, two pairs of zero energy
edge states appear in a region around the point $w/|v+z|=0$ when $w>|v+z|$ and
$v<z$ which is shown in Fig. 12 (b). This particular region is surrounded by a
trivial phase as long as $-1\leq(w/|v+z|)\leq 1$. The figures are drawn for
lattice of sites 200, and the results confirm the existence of edge states in
the topological phases.

Figure 12: Bulk and edge state energies for
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,2}$ when $v/z>1$, (a) for $v=0.7$, $z=0.3$,
and $v/z<1$, (b) for $v=0.3$, $z=0.7$.

To make sure the presence of zero energy edge states, probability density of
those states are drawn in Fig. 13 for the lattice of 150 sites. Two figures
are drawn for two distinct topological phases. In the upper panel (a),
probability density of two distinct edge states with $E=0$ are shown when
$v=0.25$, $w=2.5$, $z=0.25$. Those values are selected for satisfying the
conditions, $w>|v+z|$. Probability density of one edge state exhibits sharp
peak at site $m=1$ and another one at site $m=150$. This corresponds to the
topological phase with $\nu=1$. On the other hand, for $w<|v+z|$ and $v<z$,
probability density of four distinct zero energy edge states are shown in the
lower panel (b) when $v=0.25$, $w=0.25$, $z=2.5$. Probability density of four
orthogonal edge states exhibit sharp peak at sites $m=2$, $m=4$, $m=147$, and
$m=149$. In this case zero energy states close to the left edge are localized
on the B sublattice, while those close to the right edge are localized on the
A sublattice. The difference on localization with the previous case attributes
to the change of sign in winding number, as the new topological phase of
$\nu=-2$ appears with opposite sign.

Figure 13: Probability density of edge states for
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,2}$: (a) for $v=0.25$, $w=2.5$, $z=0.25$, one
pair of edge state, (b) for $v=0.25$, $w=0.25$, $z=2.5$, two pairs of edge
states. Figures are drawn for the lattice of 150 sites.  Figure 14:
Topological phase diagram for the Hamiltonian,
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,2}$. Two distinct topological phases are
shown by blue ($\nu=1$) and red ($\nu=-2$). The remaining portion is trivial
($\nu=0$). This diagram is drawn for $v+z=2.$ The horizontal line indicates
the value $v=1$ or $v/z=1.$

A comprehensive phase diagram for this model is shown in Fig 14 where contour
plot for $\nu$ is drawn in the $v$-$w/|v+z|$ space. Variation of the
parameters is made by maintaining the constraint $v+z=2$. Existence of two
different topological phases, $\nu=1$ and $-2$ along with the trivial phase,
$\nu=0$ are shown in three different colours. The horizontal line is drawn at
$v/z=1$, above which topological phase with $\nu=-2$ does not survive. This
phase exists over the line segment, $-1\leq w/(v+z)\leq+1$, when $v=0$.
However the length of this segment reduces symmetrically around $w/(v+z)=0$
and vanishes at the point $v=z$. The boundary lines of those phases can be
obtained by simultaneously solving the Eqs. $E_{\pm}({\rm k})=0$, and
$\frac{dE_{\pm}({\rm k})}{d{\rm k}}=0$. As a result, the transition lines are
given by the two solutions of quadratic equation,
$v^{2}+w^{2}+z^{2}+2\\{vwp+vz(2p^{2}-1)+wzp(4p^{2}-3)\\}=0$, where
$p=\cos^{-1}{\left(\frac{-vz\pm\sqrt{v^{2}z^{2}-3w^{2}z(v-3z)}}{6wz}\right)}$,
along with the constraint, $v+z=2$. Those curved lines are symmetric around
the straight line $w/(v+z)=0$ and meet at the point, $w/(v+z)=0$, $v=1$.
Another topological phase with $\nu=1$ appears beyond the two vertical lines
drawn at $w/(v+z)=\pm 1$. They separate topological phases with $\nu=1$ from
the trivial phase. So the system undergoes phase transition around those
straight lines.

As the quenching of edge states provides their exact location more clearly,
dynamics of the edge states in the presence of nonlinear terms for the
topological phases of this model will be discussed. The set of coupled
nonlinear first order differential equation for finite chain of $L$ sites and
for the Hamiltonian defined in Eq. 15 with OBC is explicitly given by

$\displaystyle i\,\frac{\partial\psi_{2j-1}}{\partial t}$ $\displaystyle=$
$\displaystyle v(\psi_{2j}-\psi_{2j-1})+w(\psi_{2j-2}-\psi_{2j-1})$
$\displaystyle+z(\psi_{2j+4}-\psi_{2j-1})-\zeta|\psi_{2j-1}|^{2}\psi_{2j-1},$
$\displaystyle\vdots\quad\;$ $\displaystyle=$
$\displaystyle\qquad\qquad\vdots$ (17) $\displaystyle
i\,\frac{\partial\psi_{2j}}{\partial t}$ $\displaystyle=$ $\displaystyle
w(\psi_{2j+1}-\psi_{2j})+v(\psi_{2j-1}-\psi_{2j})$
$\displaystyle+z(\psi_{2j-5}-\psi_{2j})-\zeta|\psi_{2j}|^{2}\psi_{2j}.$

Figure 15: Quench dynamics for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,2}$ when
$\zeta=0.5$, (a) for $v=0.25$, $w=2.5$, $z=0.25$, (b) for $v=2.5$, $w=0.25$,
$z=0.25$, (c) for $v=0.25$, $w=0.25$, $z=2.5$. Figures are drawn for lattice
with 20 sites.

Quenching of edge states for the nonlinear system is shown in Fig. 15, by
solving the set of Eq. 17, for $L=20$ when $\zeta=0.5$. Contour plot for the
time evolution of $|\psi_{l}(t)|$, is drawn for every site which is shown
along the horizontal axis. Three contour plots are shown (a) for $v=0.25$,
$w=2.5$, $z=0.25$, (b) for $v=2.5$, $w=0.25$, $z=0.25$, (c) for $v=0.25$,
$w=0.25$, $z=2.5$, where (b) indicates trivial phase as before while (a) and
(c) for the topological phases of $\nu=1$ and $\nu=-2$, respectively. In this
case, initial condition is set by $\psi_{l}(0)=\delta_{l,m}$, where
$m=2,4,17,19$. As a result, conservation rule follows the same equation as
before, $\sum_{l=1}^{L}|\psi_{l}(t)|^{2}=4$.

Evolution of the system is explored for the time span $0\leq t\leq 20$, where
the time is plotted along the vertical axis. The diagram in (a) clearly
indicates that probability amplitudes for $l=1,20$, i. e., $|\psi_{1}(t)|$ and
$|\psi_{20}(t)|$ survive with time. So the edge states bound to the
topological phase with $\nu=1$ exhibit their quenching. No such quenching is
found for the trivial phase as shown in (b). Those results are similar to the
previous case, although the respective figures are qualitatively different.
Quenching of four edge states, $|\psi_{l}(t)|$, when $l=2,4,17,19$ are found
in (c) which correspond to the topological phase with $\nu=-2$. In contrast to
this result, quenching of four edge states, for $l=1,3,18,20$ are found when
$\nu=2$, as discussed in the previous model. It means quenching over A and B
sublattices interchange their edges with the change of sign in $\nu$. Thus
quenching are found for the amplitude on sites, $l=2,4,L\\!-\\!3,L\\!-\\!1$,
for any arbitrary length of lattice.

###  III.5 Topological phases for $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,3}$

Now the total Hamiltonian is

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{vw}+H^{\rm{B\mbox{-}A}}_{z,3},$ $\displaystyle H^{\rm{B\mbox{-}A}}_{z,3}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{N}z\,c^{\dagger}_{{\rm B},j}c_{{\rm
A},j+3}+{\rm h.c.},$ (18)

where the hopping term extends over two intermediate primitive cell, which is
shown in Fig. 16.

Figure 16: Extended SSH model describing the hopping in
$H^{\rm{B\mbox{-}A}}_{z,3}$.

In this model every cell is connected to the third NN cell by the hopping
parameter $z$. The $\boldsymbol{g}(\rm k)$ vector assumes the form,

$\boldsymbol{g}(\rm k)\equiv\left\\{\begin{array}[]{l}g_{x}=v+w\cos{\\!(\rm
k)}+z\cos{\\!(3\rm k)},\\\\[3.00003pt] g_{y}=w\sin{\\!(\rm k)}+z\sin{\\!(3\rm
k)},\\\\[3.00003pt] g_{z}=0.\end{array}\right.$

Dispersion relation in this case is $E_{\pm}({\rm
k})=\pm\sqrt{v^{2}+w^{2}+z^{2}+2[vw\cos{\\!(\rm k)}+vz\cos{\\!(3\rm
k)}+wz\cos{\\!(2\rm k)}]}$. Variation of dispersion relation, $E_{+}({\rm
k})$, with $v/|w+z|$ for $w=1$, $z=1/2$ and $w=1/2$, $z=1$ are shown in Fig.
17 (a) and (b), respectively. Those are serving as prototype figures for $w>z$
and $w<z$, respectively. Dispersions comprise of three broad peaks when
$v/|w+z|\leq 1$ for both the cases $w/z>1$ and $w/z<1$. Band gap vanishes at
the BZ boundaries, ${\rm k}=\pm\pi$, when $v=|w+z|$, for both also. As a
result, $\nu$ is undefined at the point when $v=|w+z|$. The dispersions
plotted in Figs. 10 and 17 look alike although they are different in a sense
that they are plotted with respect to different parameters, say, $w/|v+z|$ in
Fig. 10 and $v/|w+z|$ in Fig. 17. This similarity attributes to the fact that
dispersions for the Hamiltonians in Eqs. 15 and 18 are interchangeable upon
interchange of $v$ and $w$.

Figure 17: Dispersion relation for $H^{\rm{B\mbox{-}A}}_{z,3}$ when $v/z>1$,
(a) and $v/z<1$, (b).

In this case also two different types of topological phases with $\nu=1$ and 3
appear in the parameter space as given below and they are separated by phase
transition lines.

$\nu=\left\\{\begin{array}[]{ll}0,&v>|w+z|,\\\\[3.00003pt] 1,&v<|w+z|,\;{\rm
and}\;w>z,\\\\[3.00003pt] 3,1,&v<|w+z|,\;{\rm and}\;w<z.\end{array}\right.$
(19)

The system is trivial as long as $v>|w+z|$, irrespective of the values of $w$
and $z$. Topological phase with $\nu=1$ exists when the relations $v<|w+z|$
and $w>z$ do hold. Another nontrivial phase with $\nu=3$ appears in a limited
region for $v<|w+z|$ and $w<z$, separated by the topological phase with
$\nu=1$. It means the phase with $\nu=1$ emerges for $v<|w+z|$ for both $w>z$
and $w<z$. The equation of phase transition lines can be obtained by
satisfying the conditions, $E_{\pm}({\rm k})=0$, and $\frac{dE_{\pm}({\rm
k})}{d{\rm k}}=0$. So, this model hosts the new topological phase with
$\nu=3$.

Figure 18: Parametric winding diagrams in the $g_{x}\mbox{-}g_{y}$ plane for
the Hamiltonian, $H^{\rm{B\mbox{-}A}}_{z,3}$. Four figures are drawn for (a)
$v=1.1$, $w=0.6$, $z=0.4$, (b) $v=0.7$, $w=0.6$, $z=0.4$, (c) $v=0.2$,
$w=0.4$, $z=0.6$, and (c) $v=0.7$, $w=0.4$, $z=0.6$.

The parametric plot of winding for the tip of the vector, $\boldsymbol{g}$ in
the $g_{x}\mbox{-}g_{y}$ complex plane are shown in Fig. 18. Four figures are
drawn for (a) $v=1.1$, $w=0.6$, $z=0.4$, (b) $v=0.7$, $w=0.6$, $z=0.4$, (c)
$v=0.2$, $w=0.4$, $z=0.6$, and (d) $v=0.7$, $w=0.4$, $z=0.6$. All the curves
traverse in the counter clockwise direction as a result all the winding
numbers are positive. Those figures serve as the prototype contours for the
four different regions, $v>|w+z|$, for $\nu=1$, $v<|w+z|$, $w>z$ for $\nu=1$,
$v<|w+z|$, $w<z$, for $\nu=3$ and $\nu=1$. $\boldsymbol{g}({\rm k})$. Nonzero
band gap is there for all the cases.

Variation of bulk-edge state energies with respect to $v/|w+z|$ is shown in
Fig. 19 for the regime $-2\leq(v/|w+z|)\leq 2$. A single pair of zero energy
edge states is there when $v>|w+z|$ as shown in (a). No edge state is there in
this system when $v<|w+z|$ and and $w>z$. However, three pairs of zero energy
edge states appear in a region around the point $v/|w+z|=0$ when $w>|v+z|$ and
$v<z$ which is shown in Fig. 19 (b). This particular region is surrounded by a
trivial phase as long as $-1\leq(v/|w+z|)\leq 1$. The figures are drawn for
lattice of sites 200, and the results confirm the existence of edge states in
the topological phases.

Figure 19: Edge states for $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,3}$ when $w/z>1$,
(a) for $w=0.7$, $z=0.3$, and $w/z<1$, (b) for $w=0.3$, $z=0.7$. Figure 20:
Probability density of edge states for $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,3}$:
(a) for $v=0.25$, $w=2.5$, $z=0.25$, one pair of edge states, (b) for
$v=0.25$, $w=0.25$, $z=2.5$, three pairs of edge states. Figures are drawn for
200 sites.

In order to confirm the existence of zero energy edge states, probability
density of those states are drawn in Fig. 20 for the lattice of 200 sites. Two
figures are drawn for two distinct topological phases. In the upper panel (a),
probability density of two distinct edge states with $E=0$ are shown when
$v=0.25$, $w=2.5$, $z=0.25$. Those values are selected for satisfying the
conditions, $v<|w+z|$. Probability density of one edge state exhibits sharp
peak at site $m=1$ and another one at site $m=200$. This corresponds to the
topological phase with $\nu=1$. On the other hand, for $v<|w+z|$ and $w<z$,
probability density of four distinct zero energy edge states are shown in the
lower panel (b) when $v=0.25$, $w=0.25$, $z=2.5$. Probability density of six
orthonormal edge states exhibit sharp peak at sites $m=1,2,3,196,198$ and 200.
This result is in accordance to the topological phase of $\nu=3$. Here
localization of zero energy states are found on A sublattice near left edge
and B sublattice near right edge.

Figure 21: Topological phase diagram for the Hamiltonian,
$H^{\rm{B\mbox{-}A}}_{z,3}$. Trivial phase is shown by red ($\nu=0$) while two
distinct topological phases are shown by yellow ($\nu=1$) and blue ($\nu=3$).
This diagram is drawn for $w+z=2.$ The horizontal line indicates the value
$w=1$ or $w/z=1.$

A rigorous phase diagram for this model is shown in Fig 21 where contour plot
for $\nu$ is drawn in the $w$-$v/|w+z|$ space. Variation of the parameters is
made by maintaining the constraint $w+z=2$. Existence of two different
topological phases, $\nu=1$ and $3$ along with the trivial phase, $\nu=0$ are
shown in yellow, blue and red. The horizontal line is drawn at $w/z=1$, above
which topological phase with $\nu=3$ does not survive. This phase exists over
the line segment, $-1\leq v/(w+z)\leq+1$, when $w=0$. However the length of
this segment reduces symmetrically around $v/(w+z)=0$ and vanishes at the
point $w=z$. The boundary lines of separation of those phases can be obtained
as before by solving the Eqs. $E_{\pm}({\rm k})=0$, and $\frac{dE_{\pm}({\rm
k})}{d{\rm k}}=0$. Combinations of those two equations leads to a quadratic
equations, $v^{2}+w^{2}+z^{2}+2\\{vwp+wz(2p^{2}-1)+vzp(4p^{2}-3)\\}=0$, where
$p=\cos^{-1}{\left(\frac{-wz\pm\sqrt{w^{2}z^{2}-3v^{2}z(w-3z)}}{6vz}\right)}$.
Two solutions of this equation along with the constraint, $w+z=2$, yields the
equation of phase transition lines. Those curved lines are symmetric around
the straight line $v/(w+z)=0$.

Trivial phase ($\nu=0$) appears beyond the two vertical lines drawn at
$v/(w+z)=\pm 1$. They separate topological phases with $\nu=1$ and 3 from the
trivial phase. So the system undergoes phase transition around those straight
lines. The structure of this phase diagram looks similar to that shown in Fig.
14. However, a closer scrutiny will reveal that the positions of topological
phases are different. At the same time parameters plotted along the two axes
are also different. Topological phase with $\nu=-2$ is replaced by that of
$\nu=3$ and the trivial phase ($\nu=0$) and another topological phase with
$\nu=1$ interchanges their positions.

According to the formalism for quenching of edge states as discussed before,
dynamics of the edge states in the presence of the same nonlinear terms for
the topological phases of this model has been studied. The set of coupled
nonlinear equation for chain of $L/2$ unit cells and for the Hamiltonian
defined in Eq. 18 with OBC is given by

$\displaystyle i\,\frac{\partial\psi_{2j-1}}{\partial t}$ $\displaystyle=$
$\displaystyle v(\psi_{2j}-\psi_{2j-1})+w(\psi_{2j-2}-\psi_{2j-1})$
$\displaystyle+z(\psi_{2j-6}-\psi_{2j-1})-\zeta|\psi_{2j-1}|^{2}\psi_{2j-1},$
$\displaystyle\vdots\quad\;$ $\displaystyle=$
$\displaystyle\qquad\qquad\vdots$ (20) $\displaystyle
i\,\frac{\partial\psi_{2j}}{\partial t}$ $\displaystyle=$ $\displaystyle
w(\psi_{2j+1}-\psi_{2j})+v(\psi_{2j-1}-\psi_{2j})$
$\displaystyle+z(\psi_{2j+5}-\psi_{2j})-\zeta|\psi_{2j}|^{2}\psi_{2j}.$

Figure 22: Quench dynamics for $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,3}$ when
$\zeta=0.5$, (a) for $v=2.5$, $w=0.25$, $z=0.25$, (b) for $v=0.25$, $w=2.5$,
$z=0.25$, (c) for $v=0.25$, $w=0.25$, $z=2.5$. Figures are drawn for lattice
of 20 sites.

Evolution of edge states for the nonlinear system is shown in Fig. 22, by
solving the set of Eq. 20, for $L=20$ when $\zeta=0.5$. Contour plot for the
time evolution of $|\psi_{l}(t)|$, is drawn for every site which is shown
along the horizontal axis. Three contour plots are shown (a) for $v=2.5$,
$w=0.25$, $z=0.25$, (b) for $v=0.25$, $w=2.5$, $z=0.25$, (c) for $v=0.25$,
$w=0.25$, $z=2.5$, where (a) indicates trivial phase as before while (b) and
(c) for the topological phases of $\nu=1$ and $\nu=3$, respectively. In this
case, initial condition is set by $\psi_{l}(0)=\delta_{l,m}$, where
$m=1,3,5,16,18,20$. As a result, conservation rule is modified by the
equation, $\sum_{l=1}^{L}|\psi_{l}(t)|^{2}=6$ for every case.

Evolution of the system is explored for the time range $0\leq t\leq 20$, as
shown along the vertical axis. The diagram in (b) clearly indicates that
probability amplitudes for $l=1,20$, i. e., $|\psi_{1}(t)|$ and
$|\psi_{20}(t)|$ survive with time. So the edge states bound to the
topological phase with $\nu=1$ exhibit their quenching. Obviously, no such
quenching is found for any site in the trivial phase as shown in (a).
Quenching of amplitudes of wave function for six sites, $|\psi_{l}(t)|$, when
$l=1,3,5,16,18,20$ are found in (c) which correspond to the topological phase
with $\nu=3$. So, quenching will be found for the amplitude with sites
$l=1,3,5,L\\!-\\!4,L\\!-\\!2,L$, in case of chain of length $L$. The number of
quenched sites increases with the increase of $\nu$.

###  III.6 Topological phases for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$

Total Hamiltonian now is

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3},$ $\displaystyle H^{\rm{A\mbox{-}B}}_{z,3}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{N}z\,c^{\dagger}_{{\rm A},j}c_{{\rm
B},j+3}+{\rm h.c.},$ (21)

where the hopping term extends over two intermediate primitive cells, as shown
in Fig. 23.

Figure 23: Extended SSH model describing the hopping in
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$.

The components of $\boldsymbol{g}(\rm k)$ in this case are

$\boldsymbol{g}(\rm k)\equiv\left\\{\begin{array}[]{l}g_{x}=v+w\cos{\\!(\rm
k)}+z\cos{\\!(3\rm k)},\\\\[3.00003pt] g_{y}=w\sin{\\!(\rm k)}-z\sin{\\!(3\rm
k)},\\\\[3.00003pt] g_{z}=0.\end{array}\right.$

The corresponding dispersion relation is $E_{\pm}({\rm
k})=\pm\sqrt{v^{2}+w^{2}+z^{2}+2[vw\cos{\\!(\rm k)}+vz\cos{\\!(3\rm
k)}+wz\cos{\\!(4\rm k)}]}$. Variation of dispersion relation, $E_{+}({\rm
k})$, with $v/|w+z|$ for $w=1$, $z=1/2$ and $w=1/2$, $z=1$ are shown in Fig.
24 (a) and (b), respectively for the region $-2\leq v/|w+z|\leq+2$. Figures
for $w>z$ and $w<z$ will be of similar shape as shown in (a) and (b).
Dispersions exhibits three broad peaks in the regions away from the point
$v/|w+z|=1$ for both the cases $w/z>1$ and $w/z<1$. Again the band gap
vanishes at the BZ boundaries, ${\rm k}=\pm\pi$, when $v=|w+z|$, for both
also. As a result, $\nu$ is undefined at the point when $v=|w+z|$. The
dispersions plotted in Figs. 10 and 17 look alike although they are different
in a sense that they are plotted with respect to different parameters, say,
$w/|v+z|$ in Fig. 10 and $v/|w+z|$ in Fig. 17. This similarity attributes to
the fact that dispersions for the Hamiltonians in Eqs. 4 and 18 are
interchangeable upon interchange of $v$ and $w$.

Figure 24: Dispersion relation for $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,3}$ when
$v/z>1$, (a) and $v/z<1$, (b).

This time three different nontrivial phases with $\nu=+1,-1$ and 3 appear in
the parameter space as given below those are separated by distinct phase
transition lines.

$\nu=\left\\{\begin{array}[]{ll}0,&v>|w+z|,\\\\[3.00003pt] 1,-1&v<|w+z|,\;{\rm
and}\;w>z,\\\\[3.00003pt] -3,-1,&v<|w+z|,\;{\rm and}\;w<z.\end{array}\right.$
(22)

Trivial phase exists when $v>|w+z|$. A pair of distinct topological phase with
$\nu=\pm 1$ emerges when $v<|w+z|$ and $w>z$. Another pair of nontrivial phase
with $\nu=-3$ and $-1$ appears $v<|w+z|$ when $w<z$. It means the phase with
$\nu=-1$ emerges two different regions when $v<|w+z|$ but for both $w>z$ and
$w<z$. The location of this transition point can be determined by satisfying
the conditions, $E_{\pm}({\rm k})=0$, and $\frac{dE_{\pm}({\rm k})}{d{\rm
k}}=0$. So, this model is capable to host the new topological phase with
$\nu=-3$.

The parametric plot of winding for the tip of the vector, $\boldsymbol{g}$ in
the $g_{x}\mbox{-}g_{y}$ plane are shown in Fig. 25. Four figures are drawn
for (a) $v=1.0$, $w=0.5$, $z=0.4$, (b) $v=0.3$, $w=0.6$, $z=0.5$, (c) $v=0.2$,
$w=0.3$, $z=0.6$, and (d) $v=0.5$, $w=0.3$, $z=0.4$. Curves shown in (a) and
(d) traverse along the counter clockwise direction, while those in (b) and (c)
traverse along the clockwise direction. As a result the winding number for (d)
is positive, but that for (b) and (c) is negative. Those figures can be
regarded as prototype contours for the four different regions. For examples,
the phase is trivial ($\nu=0$), for any values of $v$, $w$ and $z$ as long as
$v>|w+z|$. On the other hand, four different regions are identified when
$v<|w+z|$, where distinct topological phases appear. A pair of phases appear
for $w>z$ and another pair for $w<z$. Nonzero band gap exists for all the
cases.

Variation of bulk-edge state energies with respect to $v/|w+z|$ is shown in
Fig. 26 for the regime $-2\leq(v/|w+z|)\leq 2$. Zero energy edge states
emerges as long as $-1\leq(v/|w+z|)\leq 1$, which is consistent to the
previous observation. So, no edge state is there in this system when $v<|w+z|$
or when $v>|w+z|$. However, three pairs of zero energy edge states appear in a
region around the point $v/|w+z|=0$ when $v<|w+z|$ and $w<z$ which is shown in
Fig. 26 (b). This particular region is surrounded by another topological phase
with $\nu=-1$ as long as $-1\leq(v/|w+z|)\leq 1$. The figures are drawn for
lattice of sites 200, and the results conform to the bulk-edge correspondence
rule in the topological phases.

Figure 25: Parametric winding diagrams in the $g_{x}\mbox{-}g_{y}$ plane for
the Hamiltonian, $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$. Four figures are drawn
for (a) $v=1.0$, $w=0.5$, $z=0.4$, (b) $v=0.3$, $w=0.6$, $z=0.5$, (c) $v=0.2$,
$w=0.3$, $z=0.6$, and (c) $v=0.5$, $w=0.3$, $z=0.4$.

In order to confirm the existence of zero energy edge states, probability
density of those states are drawn in Fig. 27 for the lattice of 200 sites. Two
figures are drawn for two distinct topological phases. In the upper panel (a),
probability density of two distinct edge states with $E=0$ are shown when
$v=0.25$, $w=2.5$, $z=0.25$. Those values are selected for satisfying the
conditions, $v<|w+z|$. Probability density of one edge state exhibits sharp
peak at site $m=1$ and another one at site $m=200$. This corresponds to the
topological phase with $\nu=1$. On the other hand, for $v<|w+z|$ and $w<z$,
probability density of four distinct zero energy edge states are shown in the
lower panel (b) when $v=0.25$, $w=0.25$, $z=2.5$. Probability density of six
orthonormal edge states exhibit sharp peak at sites $m=2,4,6,195,197$ and 199.
This result is in accordance to the topological phase of $\nu=-3$.
Localization of zero energy states are found on B sublattice near left edge
and A sublattice near right edge.

Figure 26: Edge states for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$ when $w/z>1$,
(a) for $w=0.6$, $z=0.4$, and $w/z<1$, (b) for $w=0.3$, $z=0.7$. Figure 27:
Probability density of edge states for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$:
(a) for $v=0.25$, $w=2.5$, $z=0.25$, one pair of edge states, (b) for
$v=0.25$, $w=0.25$, $z=2.5$, three pairs of edge states. Figures are drawn for
232 sites.

A rigorous phase diagram for this model is shown in Fig 28, where contour plot
for $\nu$ is drawn in the $w$-$v/|w+z|$ space. Variation of the parameters is
made by maintaining the constraint $w+z=2$. Existence of three different
topological phases, $\nu=\pm 1$ and $-3$ along with the trivial phase, $\nu=0$
are shown in four different colors. The horizontal line is drawn at $w/z=1$,
which separates the topological phase with $\nu=1$ from that with $\nu=-3$.
Three topolgical phases meet at the point, $v/(w+z)=0$, $w=1$. All the
topological phases remain within the region bounded by the vertical lines
drawn at $v/(w+z)=\pm 1$. So the trivial phase lies beyond the region $-1\leq
v/(w+z)\leq+1$ for any value of $w$. The curved boundary lines separating the
nontrivial phases can be obtained from the solutions of the Eqs. $E_{\pm}({\rm
k})=0$, and $\frac{dE_{\pm}({\rm k})}{d{\rm k}}=0$. Those equations yield a
cubic equation, whose solutions along with the constraint, $w+z=2$, provides
the equation of transition lines.

Figure 28: Topological phase diagram for the Hamiltonian
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$. Three distinct topological phases are
shown by blue ($\nu=1$), green ($\nu=-1$) and red ($\nu=-3$). The remaining
portion is trivial ($\nu=0$). This diagram is drawn for $w+z=2.$ The
horizontal line indicates the value $w=1$ or $w/z=1.$

According to the formalism for quenching of edge states as discussed before,
dynamics of the edge states in the presence of nonlinear terms for the
topological phases of this model will be discussed. The set of coupled
nonlinear equation for chain of $L$ lattice sites and for the Hamiltonian
defined in Eq. 21 with OBC is given by

$\displaystyle i\,\frac{\partial\psi_{2j-1}}{\partial t}$ $\displaystyle=$
$\displaystyle v(\psi_{2j}-\psi_{2j-1})+w(\psi_{2j-2}-\psi_{2j-1})$
$\displaystyle+z(\psi_{2j+6}-\psi_{2j-1})-\zeta|\psi_{2j-1}|^{2}\psi_{2j-1},$
$\displaystyle\vdots\quad\;$ $\displaystyle=$
$\displaystyle\qquad\qquad\vdots$ (23) $\displaystyle
i\,\frac{\partial\psi_{2j}}{\partial t}$ $\displaystyle=$ $\displaystyle
w(\psi_{2j+1}-\psi_{2j})+v(\psi_{2j-1}-\psi_{2j})$
$\displaystyle+z(\psi_{2j-7}-\psi_{2j})-\zeta|\psi_{2j}|^{2}\psi_{2j}.$

Figure 29: Quench dynamics for $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,3}$ when
$\zeta=0.5$, (a) for $v=2.5$, $w=0.25$, $z=0.25$, (b) for $v=0.25$, $w=2.5$,
$z=0.25$, (c) for $v=0.25$, $w=0.25$, $z=2.5$. Figures are drawn for lattice
of 20 sites.

Evolution of edge states for the nonlinear system is shown in Fig. 29, by
solving the set of Eq. 23, for $L=20$ when $\zeta=0.5$. Contour plot for the
time evolution of $|\psi_{l}(t)|$, is drawn for every site which is shown
along the horizontal axis. Three contour plots are shown (a) for $v=2.5$,
$w=0.25$, $z=0.25$, (b) for $v=0.25$, $w=2.5$, $z=0.25$, (c) for $v=0.25$,
$w=0.25$, $z=2.5$, where (a) indicates trivial phase as before while (b) and
(c) for the topological phases of $\nu=1$ and $\nu=3$, respectively. In this
case, initial condition is set by $\psi_{l}(0)=\delta_{l,m}$, where
$m=1,3,5,16,18,20$. As a result, conservation rule is modified by the
equation, $\sum_{l=1}^{L}|\psi_{l}(t)|^{2}=6$, for every case.

Evolution of the system is explored for the time range $0\leq t\leq 20$, as
shown along the vertical axis. The diagram in (b) clearly indicates that
probability amplitudes for $l=1,20$, i. e., $|\psi_{1}(t)|$ and
$|\psi_{20}(t)|$ survive with time. So the edge states bound to the
topological phase with $\nu=1$ exhibit their quenching. Obviously, no such
quenching is found for any site in the trivial phase as shown in (a).
Quenching of amplitudes of wave function for six sites, $|\psi_{l}(t)|$, when
$l=2,4,6,15,17,19$ are found in (c) which correspond to the topological phase
with $\nu=-3$. Therefore, quenching will be found in general for the amplitude
on sites $l=2,4,6,L\\!-\\!5,L\\!-\\!3,L\\!-\\!1$, if a chain of length $L$ is
considered. Again, the quenched sites for A and B sublattices interchange the
edges with respect to the last case. So the phase with higher values of $\nu$
can be studied where the quenching of absolute value of the probability
amplitude for higher number of sites close to the ends of the lattice is
observed.

Summarizing the results of above findings it is concluded that the method
proposed in this work by constructing series of Hamiltonians,
$H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,m}$, and
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,m}$, with $m=1,2,3,\cdots$, topological
phases with $\nu=\pm 1,\pm 2,\pm 3,\cdots,$ can be realized. Only one further
neighbour hopping term of strength $z$ is introduced within the standard SSH
model, whose extend is determined by the integer $m$. In this formulation, $m$
denotes the extend of further neighbor hopping which passes over $m$
intermediate cells. With the increase of $m$, topological phases of higher
values of $\nu$ will be realized. Hamiltonians of the type
$H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,m}$, yield phases of positive winding number
only, while that of type $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,m}$, yield those of
negative winding numbers along with $\nu=0$ and $\nu=+1$. A list of
Hamiltonians along with the winding number of accompanying phases are given in
Tab. 1.

$m$ |  $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,m}$ | $H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,m}$
---|---|---
1 | $\nu=0,1$ | $\nu=-1,0,1$
2 | $\nu=0,1,2$ | $\nu=-2,0,1$
3 | $\nu=0,1,3$ | $\nu=-3-1,0,1$
4 | $\nu=0,1,2,4$ | $\nu=-4,-2,0,1$
5 | $\nu=0,1,3,5$ | $\nu=-5,-3,-1,0,1$
$\vdots$ | $\vdots$ | $\vdots$
$2p$ | $\nu=0,1,2,\cdots\\!,2p$ | $\nu=-2p,-2p\\!+\\!2,\cdots\\!,-2,0,1$
$2p\\!+\\!1$ | $\nu\\!=\\!0,1,3,\cdots\\!,2p\\!+\\!1$ | $\nu\\!=\\!-2p\\!-\\!1,-2p\\!+\\!1,\cdots\\!,-1,0,1$
Table 1: Distribution of winding numbers with the value of $m$ for the
Hamiltonians $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,m}$ and
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,m}$.

From this table it is evident that, topological phase with $\nu=1$ and the
trivial phase are present in every case. Apart from those common phases, an
‘odd-even’ effect for the values of $\nu$ is found with respect to $m$. In
addition to $\nu=0,1$ only even (odd) values of $\nu$ appear when $m$ is even
(odd). This is true for both ‘B-A’ and ‘A-B’ types of Hamiltonians. Also
number of nontrivial phases increases with the value of $m$ for both types of
Hamiltonians. More precisely, the number of nontrivial phase is equal to the
value of ($m-2$) and ($m-1$), respectively for ‘B-A’ and ‘A-B’ types of
Hamiltonians when m is odd. For even $m$, both ‘B-A’ and ‘A-B’ types yield
($m-1$) number of topological phase.

Another interesting finding is that in order to realize the topological phase
of the largest possible values of winding numbers from a specific Hamiltonian,
value of $z$ is to be made larger than the individual values of $v$ and $w$.
More elaborately, it is known that maximum value of $\nu$ for the topological
phase host by $H=H_{vw}+H^{\rm{B\mbox{-}A}}_{z,m}$ is $+m$, while that host by
$H=H_{vw}+H^{\rm{A\mbox{-}B}}_{z,m}$ is $-m$. However this particular phase
cannot be generated from the relevant Hamiltonian by substituting any value of
$v/z$ and $w/z$. By examining the phase diagrams as depicted in the Figs. 7,
14, 21, and 28, it can be concluded that phase of the maximum value of $\nu$
can be achieved easily by choosing the values of the parameters such a way
that they must satisfy the relations: $v/z\rightarrow 0$, and $w/z\rightarrow
0$. These limiting values indicate that phase with the maximum value of $\nu$
emerges when both the inter and intracell hopping strength are much weaker
than that of the further neighbour cells.

Figures (b) in 6, 13, 20 and 27 indicate that localization of symmetry-
protected zero energy states are found on A and B sublattices in a different
way for phases with +ve and -ve values of $\nu$. For $\nu>0$, localization are
found on A sublattice for left edge and on B sublattice for right edge. But
the localization occurs in a opposite way when $\nu<0$.

##  IV Topological phases in terms of Chern numbers

In this case further neighbour hoppings are allowed within the same types of
sublattice, means between A-A and B-B types of sites. In addition, further
neighbour hoppings are limited between the NN cells. The simplest model which
exhibits topological phases with any values of Chern numbers is defined by the
Hamiltonian,

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{vw}+H_{t},$
$\displaystyle H_{t}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\left(t_{a}\,c^{\dagger}_{{\rm A},j}c_{{\rm
A},j+1}+t_{b}\,c^{\dagger}_{{\rm B},j}c_{{\rm B},j+1}\right)\\!+{\rm h.c.},$
(24)

where $t_{a}$ and $t_{b}$ denote respectively the hopping parameters between
A-A and B-B types of sites belonging to the NN cells. The model is described
in the Fig. 30.

Figure 30: Extended SSH model describing the hopping for the Hamiltonian in
Eq. 24.

Assuming PBC, the Hamiltonian in the k-space becomes $H({\rm k})=g_{I}({\rm
k})I+\boldsymbol{g}({\rm k})\cdot\boldsymbol{\sigma}$, where $I$ is the
$2\times 2$ identity matrix, $g_{I}({\rm k})=(t_{a}+t_{b})\cos{\\!(\rm k)}$,
and

$\boldsymbol{g}(\rm k)\equiv\left\\{\begin{array}[]{l}g_{x}=v+w\cos{\\!(\rm
k)},\\\\[3.00003pt] g_{y}=w\sin{\\!(\rm k)},\\\\[3.00003pt]
g_{z}=(t_{a}-t_{b})\cos{\\!(\rm k)}.\end{array}\right.$

Since $\boldsymbol{g}(\rm k)$ is a three-component vector, chiral symmetry is
not preserved for this model. In addition particle-hole and inversion
symmetries are not preserved. The standard norms of those symmetries for 1D
system now satisfy

$\left\\{\begin{array}[]{l}\mathcal{P}H({\rm k})\mathcal{P}^{-1}\neq-H(-{\rm
k}),\\\\[3.99994pt] \sigma_{x}H({\rm k})\sigma_{x}\neq H(-{\rm
k}),\\\\[3.99994pt] \sigma_{z}H({\rm k})\sigma_{z}\neq-H({\rm
k}).\end{array}\right.$

However, those symmetries in the ${\rm k}$-$\theta$ space can be restored by
choosing the hopping terms accordingly. No topological phase in terms of
nonzero winding number is present in this case. However, this model is capable
to behave like an effective 2D model in the virtual momentum space if the
amplitude of hopping parameters are modulated cyclically in terms of two
additional angular parameters $\theta$ and $\phi$. Li et al introduced this
model and reported the emergence of topological phases characterized by the
Chern numbers of Haldane like two-band 2D system Li2 . Parametrization behind
the realization of this phase is

$\left\\{\begin{array}[]{l}v=t[1+\delta\cos{\\!(\theta)}],\\\\[3.00003pt]
w=t[1-\delta\cos{\\!(\theta)}],\\\\[3.00003pt]
t_{a}=h\,\cos{\\!(\theta+\phi)},\\\\[3.00003pt]
t_{b}=h\,\cos{\\!(\theta+\phi)}.\end{array}\right.$ (25)

The Hamiltonian with this parametrization preserve the inversion symmetry in
terms of two variables ${\rm k}$ and $\theta$, since

$\sigma_{x}H({\rm k},\theta)\sigma_{x}=H(-{\rm k},-\theta),$

for any values of $\phi$, as long as $\theta\neq 0$. In addition, Hamiltonian
preserves the mirror symmetry with respect to $\theta$, when $\phi=0,\pm
m\pi$, where $m$ is integer. Let $\mathcal{M}_{\theta}$ be the operator for
mirror symmetry and it acts as, $\mathcal{M}_{\theta}g(\theta)=g(-\theta)$,
where $g(\theta)$ is an arbitrary function of $\theta$. Hamiltonian obeys the
relation,

$\mathcal{M}_{\theta}H({\rm k},\theta)\mathcal{M}_{\theta}^{-1}=H({\rm
k},\theta),$

only for $\phi=0,\pm m\pi$. Topological phase emerges when this symmetry is
broken. However, Hamiltonian preserves the mirror symmetry with respect to
both $\theta$, and $\phi$. It means, if $\mathcal{M}_{\theta,\phi}$ be the
operator of that symmetry, Hamiltonian holds the relation:

$\mathcal{M}_{\theta,\phi}H({\rm
k},\theta)\mathcal{M}_{\theta,\phi}^{-1}=H({\rm k},\theta).$

Interestingly, topological phase of 2D system is realized in this 1D model
when $\theta$ is allowed to vary from $-\pi$ to $\pi$, for a specific value of
another angular variable $\phi$. For example, $\mathcal{C}=\pm 1$ is realized
when $0<\phi<\pi$, while $\mathcal{C}=\mp 1$ when $-\pi<\phi<0$. Mirror
symmetry with respect to $\mathcal{M}_{\theta}$ is broken in this entire
regime, but that with respect to $\mathcal{M}_{\theta,\phi}$ is preserved.
Band gap vanishes for $\phi=0$, as well as band inversion occurs around this
point. Hence these two distinct topological phases are realized in this model.
No other phase is realized if the further neighbour hopping extends beyond the
NN primitive cells for the parametrization defined in Eq. 25. However, band
inversion is found to take place if further neighbor hopping between NN cells
is replaced by NNN cells and this phenomenon occurs recursively if the further
neighbor hopping terms extend beyond NNN cells successively in addition.

In this study, new topological phases other than $\mathcal{C}=\pm 1$ and $\mp
1$ have been obtained in a very simple way in which the angular variable
$\theta$ is replaced by ($f\theta$) where $f$ may assume any values. Another
cyclic parameter $\phi$ also depends on $f$ as shown below. Higher values of
$f$ lead to phases with higher values of $\mathcal{C}$. In other words for the
realization of phases with $\mathcal{C}=\pm n$ and $\mp n$, with
$n=1,2,3,\cdots,$ sequentially, the following parametrization is implemented,

$\left\\{\begin{array}[]{l}v=t[1+\delta\cos{\\!(f\theta)}],\\\\[3.00003pt]
w=t[1-\delta\cos{\\!(f\theta)}],\\\\[3.00003pt]
t_{a}=h\,\cos{\\!(f\theta+\phi)},\\\\[3.00003pt]
t_{b}=h\,\cos{\\!(f\theta-\phi)},\end{array}\right.$ (26)

where $\phi=(f\pm\frac{1}{2})\pi$. The particle-hole symmetry is preserved in
the ${\rm k}$-$\theta$ space, since the relation,

$\mathcal{P}H({\rm k},\theta)\mathcal{P}^{-1}=-H(-{\rm k},-\theta),$

holds only when $f$ is integer.

The system is found to host nontrivial phases with

$\mathcal{C}=\left\\{\begin{array}[]{lll}\pm
n,&(n\\!-\\!\frac{1}{2})\\!<\\!f\\!<\\!(n\\!+\\!\frac{1}{2}),&{\rm
when}\;n=1,3,5,\cdots,\\\\[3.00003pt] \mp
n,&(n\\!-\\!\frac{1}{2})\\!<\\!f\\!<\\!(n\\!+\\!\frac{1}{2}),&{\rm
when}\;n=2,4,6,\cdots,\end{array}\right.$

when $\phi=(f+\frac{1}{2})\pi$. On the other hand, it yields

$\mathcal{C}=\left\\{\begin{array}[]{lll}\mp
n,&(n\\!-\\!\frac{1}{2})\\!<\\!f\\!<\\!(n\\!+\\!\frac{1}{2}),&{\rm
when}\;n=1,3,5,\cdots,\\\\[3.00003pt] \pm
n,&(n\\!-\\!\frac{1}{2})\\!<\\!f\\!<\\!(n\\!+\\!\frac{1}{2}),&{\rm
when}\;n=2,4,6,\cdots,\end{array}\right.$

when $\phi=(f-\frac{1}{2})\pi$. Interestingly, those topological phase emerges
when mirror symmetry with respect to $\mathcal{M}_{\theta}$ is broken.
$\mathcal{C}$ is undefined when
$n=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots,$ as band gap closes at those
points when $\phi=(f\pm\frac{1}{2})\pi$.

Chern number can be defined in this virtual reciprocal space as

$\mathcal{C}=\frac{1}{2\pi}\iint_{\rm BZ}d{\rm
k}d\theta\,(\partial_{\theta}A_{\rm k}-\partial_{\rm k}A_{\theta}),$ (27)

where the Berry phase, $A_{\nu}=i\langle{\rm k}\theta|\partial_{\nu}|{\rm
k}\theta\rangle$, with $\nu={\rm k},\,\theta$ and $|{\rm k}\theta\rangle$ is
the Bloch state. Integration is performed over the BZ in the 2D reciprocal
space. The reciprocal space is called virtual in a sense that no parameter in
the real space can be connected to the angular variable $\theta$, as on
contrary the wave number ${\rm k}$ corresponds to the reciprocal of the
lattice parameter for the real space. In order to find the Chern number the
integral in Eq 27 is numerically evaluated Suzuki .

Appearance of topological phases with increasing Chern numbers with the
increase of $f$ is shown in Fig. 31. Topological phases $\mathcal{C}=\pm 1,\pm
2,\pm 3,\pm 4$ and $\pm 5$ are shown here. New phases with higher values of
$\mathcal{C}$s might appear with the increase of $f$. Chern number is
undefined at the intermediate points when
$n=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots$. This phase diagram is
independent of the value of $t$, $\delta$ and $h$.

Figure 31: Variation of $\mathcal{C}$ with $f$ for the Hamiltonian defined in
Eq. 30 with the parameters in Eq. 26 is shown. Phases with higher values of
$\mathcal{C}$ will appear with the increase of $f$. Solid blue line for
$\phi=(n+\frac{1}{2})\pi$, and dashed red line for $\phi=(n-\frac{1}{2})\pi$.

Figure 32: Variation of energy with $\theta$ for the Hamiltonian defined in
Eq. 30 and the parametrization defined in Eq. 26 are drawn in the upper panel.
Figures are drawn for $t=1$, $\delta=0.5$, $h=0.2$, $\phi=(n+\frac{1}{2})\pi$
and $n=2$ in (a), $n=3$ in (b), $n=4$ in (c). Figures are drawn for lattice of
100 sites. Probability density of the corresponding edge states are shown in
the lower panel. $\theta=\pi/2$ when $n=2$ in (d), $\theta=(\pi-1)/2$ when
$n=3$ in (e), and $\theta=\pi/4$ when $n=4$ in (f).

In order to visualize the edge states, variation of energy with $\theta$ is
shown in the upper panel of Fig. 32. A chain of 100 sites are considered.
Figures are drawn for $t=1$, $\delta=0.5$, $h=0.2$, $\phi=(n+\frac{1}{2})\pi$
and $n=2$ in (a), $n=3$ in (b), $n=4$ in (c). $n$ pairs of edge states are
found for each case when $\mathcal{C}=\pm n$. Those results are consistent
with the ‘bulk-boundary correspondence’ rule which states that: chern number
is equal to the number of pair of edge states in the gap for the two-band
model Hatsugai1 ; Hatsugai2 ; Mook . Probability density of a specific pair
edge states for a definite value of $\theta$ is shown in the respective lower
panels. For example, $\theta=\pi/2$ when $n=2$ in (d), $\theta=(\pi-1)/2$ when
$n=3$ in (e), and $\theta=\pi/4$ when $n=4$ in (f). Variation of energy with
$\theta$ is shown Fig. 33 when $\phi=0$ for $t=1$, $\delta=0.5$, $h=0.2$.
Value of $\mathcal{C}$ is undefined as there is no band gap when $\phi=0$.
Particle-hole symmetry is not preserved in this case.

Figure 33: Variation of energy with $\theta$ for the Hamiltonian defined in
Eq. 30 and the parametrization defined in Eq. 26 are drawn when $\phi=0$.
Figures are drawn for $t=1$, $\delta=0.5$, $h=0.2$, and $n=2$ in (a), $n=3$ in
(b), $n=4$ in (c). Figures are drawn for lattice of 100 sites.

For other type of parametrization, where

$\left\\{\begin{array}[]{l}v=t[1+\delta\cos{\\!(\theta)}],\\\\[3.00003pt]
w=t[1-\delta\cos{\\!(\theta)}],\\\\[3.00003pt]
t_{a}=h\,\cos{\\!(n\theta+\phi)},\\\\[3.00003pt]
t_{b}=h\,\sin{\\!(n\theta+\phi)},\end{array}\right.$ (28)

a single topological phase with $\mathcal{C}=\pm 1$ always appear when
$n=1,3,5,\cdots$. $\mathcal{C}$ is undefined when $n=2,4,6,\cdots$ as band gap
closes for the even $n$. Interestingly edge states appear for the trivial
phases instead. Not only that, multiple crossing of the edge state energy
lines are found for both trivial and topological phases where the number of
crossing points are exactly equal to the value of $n$. Bulk-edge energy
dispersion and probability density of the edge states for this parametrization
is shown in Fig. 34. In the upper panel bulk-edge energy variation with
$\theta$ is shown for $n=2$ in (a), $n=3$ in (b), and $n=4$ in (c). Values of
the fixed parameters are $t=1$, $\delta=0.5$, $h=0.2$, and $\phi=\pi/2$. Lower
panel shows the probability density of edge states in (d), (e) and (f) for the
respective cases. Figures are drawn for lattice of 100 sites.

Figure 34: Variation of energy with $\theta$ for the Hamiltonian defined in
Eq. 30 and the parametrization defined in Eq. 28 are drawn in the upper panel.
Figures are drawn for $t=1$, $\delta=0.5$, $h=0.2$, $\phi=\pi/2$ and $n=2$ in
(a), $n=3$ in (b), $n=4$ in (c). Figures are drawn for lattice of 100 sites.
Probability density of the corresponding edge states when $\theta=\pi$ are
shown in the lower panel for $n=2$ in (d), $n=3$ in (e), and $n=4$ in (f).
Figure 35: Variation of $\mathcal{C}$ in the $f$-$\phi$ phase space for the
Hamiltonian defined in Eq. 30 with the parameters in Eq. 26 is shown but when
$-\pi<\phi<\pi$.

Variation of $\mathcal{C}$ in the $f$-$\phi$ phase space for the Hamiltonian
defined in Eq. 30 with the parameters in Eq. 26 is shown in Fig 35 but when
$-\pi<\phi<\pi$. In this particular case $\phi$ does not depend on the
parameter $f$. Phases with different values of $\mathcal{C}$ will appear with
the increase of $f$ along the vertical axis. However, in this case phases with
different $\mathcal{C}$s are not separated by band gap closing as well as no
symmetry is broken. It means that symmetries with respect to both
$\mathcal{M}_{\theta}$ and $\mathcal{M}_{\theta,\phi}$ are preserved with the
variation of $f$. So those states are no more topological in nature. In
contrast, phases across the line $\phi=0$ are separated by zero band gap.

##  V Discussion

In this work, emergence of two different series of topological phases with
$\nu=\pm 1,\pm 2,\pm 3,\cdots,$ and $\mathcal{C}=\pm 1,\pm 2,\pm 3,\cdots,$
has been successfully demonstrated using the 1D eSSH models. The manuscript is
composed of two main parts, say Sec IV and III, where the major results are
described. In the first case, a single further neighbor hopping term beyond NN
is found enough for the realization of the series of new phase where particle-
hole and inversion symmetries are preserved. It means those phases can be
realized by adding multiple further neighbor terms as well. In the second
case, a single pair of NNN hopping terms is found sufficient for the
realization of the series of new phase where the standard forms of particle-
hole and inversion symmetries for 1D are not preserved.

Four different eSSH models in the first case have been considered in which
extend of further neighbor hopping is limited by $z=2,3$ and their properties
have been studied rigorously. Topological properties have been characterized
in terms of winding number, edge states and dynamics in the presence of
additional nonlinear term. Finally the results are generalized for $z>2$,
where phases with higher values of $\nu$ appear. Comprehensive phase diagram
are drawn, where the equations of phase transition lines are obtained.

It is known that topological interface states emerge when two lattices with
different topological phases are joined. Nowadays study on these interface
states in phononic crystals have been initiated Li3 . So the results obtained
for these tight-binding models will become helpful for constructing the
phononic model in order to realized the topological phase of any desired
winding number, as well as to study the properties of interface states. In
addition, the most simple route for demonstration of topological phase of
higher winding numbers using systems of ultracold atoms in optical lattice can
be obtained by mimicking the structure of these tight-binding models.

In the second case, emergence of topological phase with any value of
$\mathcal{C}$ is discussed when the NN and NNN hopping terms in the two-band
eSSH models are expressed in terms of two angular variables, $\theta$ and
$\phi$. Particle-hole and inversion symmetries are preserved in the ${\rm
k}$-$\theta$ space. Three different types of parametrization are employed when
$\theta$ and $\phi$ depend on another parameter $f$ in three different ways. A
specific parametrization gives rise to a nontrivial phase of any value of
$\mathcal{C}$ which is controlled by the parameter $f$. Phase transition
points are marked on the phase diagram where the band gap vanishes.
Experimental realization of these phases using systems of ultracold atoms in
optical lattice is possible as discussed hereLi2 . System hosts topological
phases with $\mathcal{C}=\pm 1,\mp 1$, for another parameterization, but with
peculiar types of edge states. Here, multiple crossing between the edge states
lines is found within the band gap. This particular phenomenon of multiply-
crossed edge states as shown in Fig 34 is not reported before. However, fake
topological phases with series of different $\mathcal{C}$s appear in the third
kind of parametrization, as the band gap does not vanish at the transition
points.

##  VI ACKNOWLEDGMENTS

RKM acknowledges the DST/INSPIRE Fellowship/2019/IF190085. Authors acknowledge
the fruitful discussion with Oindrila Deb.

## References

  * (1) W. Su, J. Schrieffer and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
  * (2) W. -P. Su, J. Schrieffer and A. Heeger, Phys. Rev. B 22, 2099 (1980). https://link.aps.org/doi/10.1103/PhysRevB.22.2099
  * (3) Heeger A. J., Kivelson S., Schrieffer J. R. and Su W. -P., Rev. Mod. Phys. 60, 781 (1988).
  * (4) Y.-X. Chen and S.-W. Li, Quantum correlations in topo- logical quantum phase transitions, Phys. Rev. A 81, 032120 (2010).
  * (5) J. D. Sau, R. M. Lutchyn, S. Tewari, S. Das Sarma, Generic new platform for topological quan- tum computation using semiconductor heterostructures, Phys. Rev. Lett. 104, 040502 (2010).
  * (6) M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Nature Physics 9, 795 (2013).
  * (7) T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rud- ner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nature communications 3, 1 (2012).
  * (8) Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M and Szameit A 2013 Photonic Floquet topological insulators Nature 496, 196 (2013)
  * (9) Xin Li, Yan Meng, Xiaoxiao Wu, Sheng Yan, Yingzhou Huang, Shuxia Wang, and Weijia Wen, Appl. Phys. Lett. 113, 203501 (2018)
  * (10) C Li and A E Miroshnichenko, Physics, 1, 2-16 (2019) https://doi.org/10.3390/physics1010002
  * (11) B Pérez-González, M Bello, A ómez-Lón and G Platero, arXiv:1802.03973 (2018)
  * (12) A. Ghosh, A. M. Martin and S. Majumder, arXiv:2303.00269 (2023)
  * (13) Thouless D. J., Kohomoto M., Nightingale P. and den Nijs M., Phys. Rev. Lett. 49, 405 (1982).
  * (14) M. Z. Hasan and C. L. Kane, Reviews of modern physics 82, 3045 (2010).
  * (15) Haldane F. D. M., Phys. Rev. Lett. 61, 2015 (1988).
  * (16) Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D and Esslinger T 2014 Experimental realization of the topo- logical Haldane model with ultracold fermions Nature 515, 237 (2014)
  * (17) M. Deb and A. K. Ghosh, J. Phys.: Condens. Matter 31 345601 (2019)
  * (18) M. Deb and A. K. Ghosh, J. Magn. Magn. Mater. 533 167968 (2021)
  * (19) A. Sil and A. K. Ghosh, J. Phys.: Condens. Matter 31 245601 (2019)
  * (20) A. Agarwal and J. N. Bandyopadhyay, J. Phys.: Condens. Matter 34 305401 (2022)
  * (21) Y.-F. Zhao, R. Zhang, R. Mei, L.-J. Zhou, H. Yi, Y.-Q. Zhang, J. Yu, R. Xiao, K. Wang, N. Samarth, M. H. W. Chan, C.-X. Liu, and C.-Z. Chang, Tuning the Chern number in quantum anomalous Hall insulators, Nature 588, 419 (2020).
  * (22) L. Li, Z. Xu, and S. Chen, Topological phases of generalized Su-Schrieffer-Heeger models, Phys. Rev. B 89, 085111 (2014).
  * (23) M Ezawa, Phys. Rev. B 104, 235420 (2021).
  * (24) Hatsugai Y., Phys. Rev. Lett. 71, 3697 (1993).
  * (25) Hatsugai Y., Phys. Rev. B 48, 11851 (1993).
  * (26) Mook A., Henk J. and Mertig I., Phys. Rev. B 90, 024412 (2014).
  * (27) T. Fukui, Y. Hatsugai and H. Suzuki, Journal of the Physical Society of Japan 74, 1674 (2005).