# Green functions and correlation functions of a solvable $S=1$ quantum Ising
spin model with dimerization

Zhi-Hua Yang1, Li-Ping Yang2, Hai-Na Wu3, Jianhui Dai1, and Tao Xiang4,2
1Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027,
China
2Institute of Theoretical Physics, Chinese Academy of Science, P.O. Box 2735,
Beijing 100080, China
3College of Science, Northeastern University, Shengyang 110006, China
4Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing
100080, China

###### Abstract

This is a supplementary material of our recent paperyangPRB , where a class of
exactly solvable $S=1$ quantum Ising spin models were studied based on the
hole decomposition scheme. Here we provide some details for the Green
functions, the spin-spin correlation functions, as well as the spin
susceptibility in the presence of dimerization.

Quantum Ising chains, Statistical lattice model; dimerization; quantum phase
transitions

##  I Introduction

In Ref.yangPRB  we have studied a class of the $S=1$ spin chains with the
nearest neighbor Ising coupling and both transverse and longitude single-ion
anisotropy by a combinational use of a hole decomposition scheme and a
recursive method. These models include the first example of the dimerized
$S=1$ quantum spin chain where all the eigen states can be solved exactly. In
this supplementary material we present some detailed derivations for the
physical quantities of the $S=1$ dimerized chain. All the notations are the
same as in Ref.yangPRB . In Sec. II, we discuss the Green functions of the
uniform or dimerized chains, respectively. In Sec. III, we study the
longitudinal spin-spin correlation function at zero- or finite-temperatures.
In Sec. V and VI we list some detailed formulae for the segmented M-matrices
and the partition functions.

##  II Green functions

###  II.1 Green functions of the uniform spin segments

The original $S=1$ quantum Ising model is mapped onto a large family of the
segmented $S=1/2$ transverse Ising models classified by the total number of
holesOitmaa ; YangPRL . These segmented $S=1/2$ models are then solved by
introducing the Bogoliubov fermionic quasi-particle operators
$\eta_{k}^{\dagger}$ and $\eta_{k}$ as defined in Eq. (14) in Ref. yangPRB .
Inversely, we have

$\begin{split}c_{j}^{\dagger}=\sum_{k}\frac{\Phi_{kj}+\Psi_{kj}}{2}\eta_{k}^{\dagger}+\frac{\Phi_{kj}^{*}-\Psi_{kj}^{*}}{2}\eta_{k},\\\
c_{j}=\sum_{k}\frac{\Phi_{kj}^{*}+\Psi_{kj}^{*}}{2}\eta_{k}+\frac{\Phi_{kj}-\Psi_{kj}}{2}\eta_{k}^{\dagger}.\end{split}$

The Green function, or the two-point correlation function, is defined by

$G_{jq}\equiv\langle F_{j}^{(-)}F_{q}^{(+)}\rangle,$ (1)

where $F_{j}^{(\pm)}\equiv c_{j}^{\dagger}\pm c_{j}$ .

For the uniform system, the wavefunctions $\Phi_{kj}$ and $\Psi_{kj}$ can be
taken as real, we have

$\begin{split}F_{j}^{(-)}=\sum_{k}\Psi_{kj}(\eta_{k}^{\dagger}-\eta_{k})~{},\\\
F_{j}^{(+)}=\sum_{k}\Phi_{kj}(\eta_{k}^{\dagger}-\eta_{k})~{}.\end{split}$ (2)

The Green function can be then expressed as

$\displaystyle
G_{jq}(\beta)=-\sum_{k}\Psi_{kj}\Phi_{kq}\tanh[\beta\Lambda(k)/2]~{}.$

Note that $\tanh[\beta\Lambda(k)/2]\rightarrow 1$ at the ground state
($\beta\rightarrow\infty$), so we have

$\displaystyle G_{jq}(\beta\rightarrow\infty)=-\sum_{k}\Psi_{kj}\Phi_{kq}~{}.$

We denote the wavefunctions for the chain with periodic boundary condition
(cyclic) and open boundary condition (free ends) by ($\Phi^{c}$, $\Psi^{c}$)
and ($\Phi^{f}$, $\Psi^{f}$), respectively. Then we have

$\displaystyle\begin{split}\Phi^{c}_{kj}&=\begin{cases}\sqrt{2/l}\sin
jk~{},~{}k>0,\\\ \sqrt{2/l}\cos jk~{},~{}k\leq 0,\end{cases}\\\
\Psi^{c}_{kj}&=-\frac{D}{\Lambda(k)}\left[(1+\lambda\cos
k)\Phi_{kj}^{c}+\lambda\sin k\Phi_{-kj}^{c}\right],\end{split}$ (3)

where $l$ is the length of the segment. The Green function is

$\displaystyle G^{c}_{r}=L_{r}+\lambda L_{r+1},$ (4)

where $r\equiv|j-q|$ and $L_{r}$ was defined in Refs. Lieb61 ; Pfeuty

$L_{r}=\frac{1}{\pi}\int_{0}^{\pi}dk\frac{1}{\sqrt{1+\lambda^{2}+2\lambda\cos
k}}\cos kr.$

Similarly,

$\displaystyle\begin{split}\Phi_{kj}^{f}&=A_{k}\sin(j-q+1)k,~{}\\\
\Psi_{kj}^{f}&=A_{k}\delta_{k}\sin jk,~{}\end{split}$ (5)

where

$\displaystyle A_{k}=\frac{1}{2}\left[2l+1-\frac{\sin{(2l+1)k}}{\sin
k}\right]^{-1/2}.$ (6)

Consequently, we have

$\displaystyle G^{f}_{jq}=-\sum_{k}A_{k}^{2}\delta_{k}\sin jk\sin(j-q+1)k.$
(7)

At the finite temperatures, we need to add the factor
$\tanh[\beta\Lambda(k)/2]$ to Eqs.(4) and (7).

###  II.2 Green functions of the dimerized segments

In the presence of dimerization, the wavefunctions $\Phi_{kj}$ and $\Psi_{kj}$
are complex in general. So we now have,

$\begin{split}F_{j}^{(-)}=\sum_{k}\Psi_{kj}\eta_{k}^{\dagger}-\Psi_{kj}^{*}\eta_{k},\\\
F_{j}^{(+)}=\sum_{k}\Phi_{kj}\eta_{k}^{\dagger}+\Phi_{kj}^{*}\eta_{k}.\end{split}$
(8)

Then, the Green function is expressed by

$\begin{split}G_{jq}=&\sum_{k}(\Psi_{kj}\Phi_{kq}^{*}+\Psi_{kj}^{*}\Phi_{kq})\langle\eta_{k}^{\dagger}\eta_{k}\rangle-\sum_{k}\Psi_{kj}\Phi_{kq}^{*}.\end{split}$
(9)

Where,
$\langle\eta_{k}^{\dagger}\eta_{k}\rangle=[\exp{(\Lambda_{k}/(k_{B}T))}+1]^{-1}$,
satisfying Fermi-Dirac statistics. At the zero temperature, the Green function
can be written as

$\displaystyle G_{jq}=D_{j}Y[j,q]+2J_{j}Y[j+1,q],$ (10)

where

$\displaystyle Y[j,q]$ $\displaystyle=$
$\displaystyle-\sum_{k}\frac{e^{i(j-q)k}}{\Lambda(k)}[1+(-1)^{j+q}\gamma^{*}\gamma$
(11) $\displaystyle~{}~{}~{}~{}~{}~{}~{}+(-1)^{j}\gamma+(-1)^{q}\gamma^{*}].$

The dimerization parameter $\gamma$ is defined by

$\begin{split}\gamma=\frac{1-\tau}{1+\tau}\end{split}$ (12)

with $\tau$ being determined by Eqs. (19) in Ref. yangPRB .

Generally, $\tau$ has two solutions, corresponding to the upper/lower signs of
$\pm$ respectively in Eqs. (19) in Ref. yangPRB . In order to numerically
calculate the Green function, we need to express $Y[j,q]$-function in terms of
real variables. We introduce $p_{1,2}$, $q_{1,2}$ to express complex $\gamma$
as follows.

$\gamma_{1}=p_{1}+iq_{1},~{}~{}~{}\gamma_{2}=p_{2}+iq_{2},$ (13)

$p_{1,2}$ and $q_{1,2}$ are the real and imaginary parts of $\gamma_{1,2}$,
respectively,

$\displaystyle p_{1,2}$ $\displaystyle=$
$\displaystyle\frac{b_{1}^{2}+b_{2}^{2}+4b_{1}b_{2}\cos
2k-\left(\zeta_{1}\mp\zeta_{2}\right)^{2}}{\left[(b_{1}+b_{2})\cos
k-\zeta_{1}\pm\zeta_{2}\right]^{2}+(b_{2}-b_{1})^{2}\sin^{2}k},$
$\displaystyle q_{1,2}$ $\displaystyle=$
$\displaystyle\frac{-2(b_{2}-b_{1})\sin k\left[(b_{1}+b_{2})\cos
k+\zeta_{1}\mp\zeta_{2}\right]}{\left[(b_{1}+b_{2})\cos
k-\zeta_{1}\pm\zeta_{2}\right]^{2}+(b_{2}-b_{1})^{2}\sin^{2}k},$

where the subscript ${1}$ corresponds to the upper case, the subscript $2$
corresponds to the lower case. $\zeta_{1,2}$ are given by

$\displaystyle\zeta_{1}$ $\displaystyle=$ $\displaystyle{(a_{2}-a_{1})}/{2},$
$\displaystyle\zeta_{2}$ $\displaystyle=$
$\displaystyle\Gamma^{2}\sqrt{1-P+Q\cos 2k}~{}.$

where $a_{1}$, $a_{2}$, $P$, $Q$ and $\Gamma$ are defined in Ref.yangPRB .

For convenience, we divide $k$-region $[-\pi,\pi)$ into two subregions: ($I$)
for $[-\pi/2,\pi/2)$ and ($II$) for $[-\pi,-\pi/2)\cup[\pi/2,\pi)$,
respectively. Thus $G_{jq}$ can be expressed by

$G_{jq}=G_{jq}^{(I)}+G_{jq}^{(II)}.$ (14)

In Region ($I$), because of the symmetry between $k$ and $-k$, the Green
function can be reduced in $(0,\pi/2)$,

$\begin{split}G_{jq}^{(I)}=&-\sum_{(0,\pi/2)}\frac{2}{\Lambda_{-1}(k)}\\{D_{j}[1+(-1)^{j+q}(p_{1}^{2}+q_{1}^{2})\\\
&+(-1)^{j}p_{1}+(-1)^{q}p_{1}]\cos(j-q)k\\\
&+2J_{j}[1+(-1)^{j+q+1}(p_{1}^{2}+q_{1}^{2})\\\
&+(-1)^{j+1}p_{1}+(-1)^{q}p_{1}]\cos(j-q+1)k\\}~{}.\end{split}$ (15)

A similar Green function can be obtained for Region ($II$). The function
$Y[j,q]$ can be rewritten as

$\begin{split}Y[j,q]=&-\sum_{(0,\pi/2)}\frac{2}{\Lambda_{-1}(k)}[1+(-1)^{j+q}(p_{1}^{2}+q_{1}^{2})\\\
&+(-1)^{j}p_{1}+(-1)^{q}p_{1}]\cos(j-q)k\\\
&-\sum_{(\pi/2,\pi)}\frac{2}{\Lambda_{-2}(k)}[1+(-1)^{j+q}(p_{2}^{2}+q_{2}^{2})\\\
&+(-1)^{j}p_{2}+(-1)^{q}p_{2}]\cos(j-q)k.\end{split}$ (16)

So it is convenient to express the total Green function Eq. (10) in terms of
$Y[j,q]$. In the dimerization case, there are four such Green functions
associated with the four different parity combinations of the segments.

##  III Correlation functions

###  III.1 Zero temperature

In this subsection, we discuss the spin-spin correlations at zero temperature.
In Ref. yangPRB  we show that the ground state has no hole if
$D_{z}>-\Delta_{h}(0)$, otherwise, it has holes once
$D_{z}\leq-\Delta_{h}(0)$. In the latter case, the holes break the original
chain into segments. We note that only the intra-segment spin-spin
correlations are non-zero.

For $D_{z}>-\Delta_{h}(0)$, the spin-spin correlation function of $S^{z}$ is
defined by $C^{z}_{mn}=\langle\Psi_{0}|S_{m}^{z}S_{n}^{z}|\Psi_{0}\rangle$,
where $|\Psi_{0}\rangle$ is the normalized ground state of the Hamiltonian. By
use of the Jordan-Wigner transformation, one has

$C_{mn}^{z}=\langle\Psi_{0}|F_{m}^{(-)}F_{m+1}^{(+)}F_{m+1}^{(-)}\cdots
F_{n-1}^{(-)}F_{n}^{(+)}|\Psi_{0}\rangle.$ (17)

It is straightforward to show that
$\langle\Psi_{0}|F_{j}^{(\pm)}F_{q}^{(\pm)}|\Psi_{0}\rangle=\pm\delta_{jq}$.
By further utilizing the Wick Theorem, we find that

$C_{mn}^{z}=\left|\begin{array}[]{cccc}G_{m,m+1}&G_{m,m+2}&\cdots&G_{m,n}\\\
G_{m+1,m+1}&G_{m+1,m+2}&\cdots&G_{m+1,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
G_{n-1,m+1}&G_{n-1,m+2}&\cdots&G_{n-1,n}\end{array}\right|,$ (18)

for $n>m$, where,
$G_{jq}=\langle\Psi_{0}|F_{j}^{(-)}F_{q}^{(+)}|\Psi_{0}\rangle=-\langle\Psi_{0}|F_{j}^{(+)}F_{q}^{(-)}|\Psi_{0}\rangle$.

The general expression of $G_{jq}$ is derived in Sec. II.1 for the uniform
chain and in Sec. II.2 for the dimerized chain respectively. In general, one
has

$\displaystyle G_{jq}=D_{j}Y[j,q]+2J_{j}Y[j+1,q],$ (19)

where $Y[j,q]$ is given by Eq. (16). For a uniform system,
$Y[j,q]=Y[q,j]=\frac{1}{D}L_{j-q}$.

###  III.2 Finite temperatures

At finite temperatures, the contribution from $p\neq 0$-sector should be taken
into account. A recursion formula similar to Eq. (36) in Ref. yangPRB  can be
derived for the correlation function as following

$\displaystyle\sum_{m,n}^{L}C_{mn}^{z}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{Z(L)}\sum_{p=0}^{L}\sum_{l=0}^{L-p}\sum_{m,n}^{l}\alpha^{p}(p+1)\rho_{mn}^{z}$
(20) $\displaystyle z(l)Z^{(p-1)}(L-p-l).$

Where, $\rho_{mn}^{z}$ is the correlation function of individual segments. It
has a similar form with that in Eq. (18), but now $G_{jq}$ should be replaced
by $G_{jq}(\beta)$.

Figure 1: Temperature dependence of the spin-spin correlation function in a
uniform spin chain with $\lambda=1.5$.

In Fig. 1, we plotted the temperature dependence of the spin-spin correlation
function per site, $\sum_{m,n}^{L}C_{mn}^{z}(\beta)/L$. We find that when
$D_{z}\leq-\Delta_{h}(0)$, the correlation function approaches to zero in the
limit $T\rightarrow 0$. This indicates that the ground state is in the hole
condensation phase. On the other hand, when $D_{z}>-\Delta_{h}(0)$, the
correlation function approaches to a finite value (about 0.85 for the two
cases shown in the figure) in the zero temperature limit.

##  IV Spin susceptibility

The spin susceptibility of the $S=1$ QIM can be also calculated using the
recursion formula introduced in the previous section. To do this, one needs to
first evaluate the partition functions of each $S=1/2$ Ising segments in the
applied magnetic field $\xi$, denoted by $z(l_{n},\xi)$. The partition
function of the original $S=1$ QIM is then given by
$Z(L,\xi)=\sum_{p=0}^{L}\sum_{\\{l_{n}\\}}\prod_{n=1}^{p+1}z(l_{n},\xi)\alpha^{p}$.
In terms of the segment magnetization
$m(l_{n},T)=-\frac{1}{\beta}\frac{\partial\ln z(l_{n},\xi)}{\partial\xi}$ and
the segment susceptibility $\chi(l_{n},T)=\frac{\partial
m(l_{n},T)}{\partial\xi}$, the total susceptibility $\chi(T)$ at zero-magnetic
field can be expressed as

$\displaystyle\chi(T)$ $\displaystyle=$
$\displaystyle\frac{1}{Z(L)}\sum_{p=0}^{L}\sum_{l=0}^{L-p}\alpha^{p}(p+1)$
(21) $\displaystyle\chi(l,T)z(l)Z^{(p-1)}(L-p-l).$

Thus the hole decomposition scheme provide an alternative approach to
calculate the susceptibility of the $S=1$ QIM. This approach is efficient
provided that the susceptibilities of the corresponding $S=1/2$ TIM’s with
varying chain length $L$ are available. We note that the susceptibility of the
$S=1/2$ TIM has already been studied by a number of groupsPfeuty70 ;
Ovchinnikov ; skew . So in principle these results could be used in the
numerical study of the susceptibility of the $S=1$ QIM.

##  V Diagonalization of the M-matrix

For a periodic spin chain, the diagonalization of the M-matrix has been
discussed in Sec. IV A in Ref. yangPRB . Here we consider the diagonalization
of this $l\times l$ M-matrix for an open spin chain with the length $l$. The
aim here is to solve the following eigen equation

$\displaystyle M\Phi_{k}=\Lambda^{2}(k)\Phi_{k}$ (22)

in various cases, where $\Phi_{k}(j)$’s take the form of Eqs. (23) in Ref.
yangPRB .

We assume that the two ends of the open chain are located at the sites $r_{1}$
and $r_{2}$, respectively. $r_{1}$ and $r_{2}$ can be either odd or even, so
there are four kinds of $M$-matrices. In the following, we will present the
results for each cases.

###  V.1 $(r_{1},r_{2})=(odd,\,even)$

In this case, the matrix $M$ is defined by

$M=\begin{pmatrix}a_{0}&b_{1}&0&\cdots&0&0\\\ b_{1}&a_{2}&b_{2}&\cdots&0&0\\\
0&b_{2}&a_{1}&\cdots&0&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&a_{1}&b_{1}\\\ 0&0&0&\cdots&b_{1}&a_{2}\end{pmatrix},$ (23)

where $a_{1,2},~{}b_{1,2}$ are defined in the main text and $a_{0}=D_{1}^{2}$.

The energy spectra can be solved following the approach introduced in Section
IV. The result is given by

$\displaystyle\Lambda^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{e^{2ik}-t_{e}e^{-2ik}}[b_{1}\tau(e^{ik}-t_{o}e^{-ik})$
$\displaystyle+a_{2}(e^{2ik}-t_{e}e^{-2ik})+b_{2}\tau(e^{3ik}-t_{o}e^{-3ik})],$

The reflection parameters are

$\displaystyle t_{o}$ $\displaystyle=$ $\displaystyle e^{2i(l+1)k},~{}~{}$
(24) $\displaystyle t_{e}$ $\displaystyle=$
$\displaystyle\frac{t_{o}(b_{1}e^{ik}+b_{2}e^{-ik})}{(b_{1}e^{-ik}+b_{2}e^{ik})}.$

Then, the secular equation is given by

$\displaystyle\left[(a_{2}-a_{1})\pm W\right][b_{1}\sin(l+2)k+b_{2}\sin lk]$
(25) $\displaystyle=$
$\displaystyle\frac{2(a_{0}-a_{1})(b_{1}^{2}+b_{2}^{2}+2b_{1}b_{2}\cos 2k)\sin
lk}{b_{2}},$

where $W$ is defined as in Eq. (20) in Ref. yangPRB .

Other cases can be solved by the same way and the results are listed below.

###  V.2 $(r_{1},r_{2})=(odd,\,odd)$

The reflection parameters $t_{o,e}$ are

$\displaystyle t_{e}$ $\displaystyle=$ $\displaystyle e^{2i(l+1)k},$ (26)
$\displaystyle t_{o}$ $\displaystyle=$
$\displaystyle\frac{t_{e}(b_{1}e^{-ik}+b_{2}e^{ik})}{(b_{1}e^{ik}+b_{2}e^{-ik})}.$

The secular equation is

$\displaystyle\left[(a_{1}-a_{2})\pm
W\right]\left[b_{1}\sin(l-1)k+b_{2}\sin(l+1)k\right]$ (27) $\displaystyle=$
$\displaystyle\frac{2b_{2}(b_{1}^{2}+b_{2}^{2}+2b_{1}b_{2}\cos
2k)\sin(l+1)k}{a_{0}-a_{1}}.$

###  V.3 $(r_{1},r_{2})=(even,\,even)$

The reflection parameters $t_{o,e}$ are

$\displaystyle t_{e}$ $\displaystyle=$ $\displaystyle e^{2i(l+1)k},$ (28)
$\displaystyle t_{o}$ $\displaystyle=$
$\displaystyle\frac{t_{e}(b_{1}e^{ik}+b_{2}e^{-ik})}{(b_{1}e^{-ik}+b_{2}e^{ik})}.$

The secular equation is

$\displaystyle\left[(a_{2}-a_{1})\pm
W\right]\left[b_{1}\sin(l+1)k+b_{2}\sin(l-1)k\right]$ (29) $\displaystyle=$
$\displaystyle\frac{2b_{1}(b_{1}^{2}+b_{2}^{2}+2b_{1}b_{2}\cos
2k)\sin(l+1)k}{a_{3}-a_{2}}$

where, $a_{3}=D_{2}$.

###  V.4 $(r_{1},r_{2})=(even,\,odd)$

The reflection parameters $t_{o,e}$ are

$\displaystyle t_{o}=e^{2i(l+1)k},$ (30) $\displaystyle
t_{e}=\frac{t_{o}(b_{1}e^{-ik}+b_{2}e^{ik})}{(b_{1}e^{ik}+b_{2}e^{-ik})}.$

The secular equation is

$\displaystyle\left[(a_{1}-a_{2})\pm W\right]$ (31) $\displaystyle=$
$\displaystyle\frac{2b_{1}[b_{1}\sin(lk)+b_{2}\sin(l+2)k]}{a_{3}-a_{2}}.$

##  VI The partition functions of segments

The partition function of individual segment of length $l$ and parity
$(r_{1},r_{2})$ (defined in Sec. V) is given by

$z_{(r_{1},r_{2})}(l)=\prod_{\begin{subarray}{c}k_{1}\in(0,\pi/2),\\\
k_{2}\in(\pi/2,\pi)\end{subarray}}\cosh\left[\frac{\beta\Lambda_{1}(k_{1})}{2}\right]\cosh\left[\frac{\beta\Lambda_{2}(k_{2})}{2}\right],$
(32)

where, $k_{1,2}$ satisfy the corresponding secular equations.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China, the national program for basic research of China (the 973 program), the
PCSIRT (IRT-0754), and SRFDP (No.J20050335118) of Education Ministry of China.

## References

  * (1) Z.H. Yang, L.P. Yang, H.N. Wu, J. Dai, and T. Xiang, Phys. Rev. B 79, 214427(2009).
  * (2) J. Oitmaa and A.M.A. von Brasch, Phys. Rev. B 67, 172402 (2003).
  * (3) Z.H. Yang, L.P. Yang, J. Dai, and T. Xiang, Phys. Rev. Lett. 100, 067203 (2008).
  * (4) E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. N.Y. 16, 407 (1961).
  * (5) P. Pfeuty, Ann. Phys. N.Y. 57, 79 (1970).
  * (6) R.J. Elliott, P. Pfeuty and C. Wood, Phys. Rev. Lett. 25, 443 (1970).
  * (7) A. A. Ovchinnikov, D. V. Dmitriev, V.Ya. Krivnov, and V. O. Cheranovskii, Phys. Rev. B 68, 214406 (2003) .
  * (8) H. C. Fogedby, J. Phys. C. 11, 2801 (1978).


# Delayed Recognition; the Co-citation Perspective

The Author  Wenxi Zhao Dmitriy Korobskiy George Chacko
Netelabs, NET ESolutions Corporation (an NTT DATA Company)
McLean, VA 22102

###### Abstract

A Sleeping Beauty is a publication that is apparently unrecognized for some
period of time before experiencing sudden recognition by citation. Various
reasons, including resistance to new ideas, have been attributed to such
delayed recognition. We examine this phenomenon in the special case of co-
citations, which represent new ideas generated through the combination of
existing ones. Using relatively stringent selection criteria derived from the
work of others, we analyze a very large dataset of over 940 million unique co-
cited article pairs, and identified 1,196 cases of delayed co-citations. We
further classify these 1,196 cases with respect to amplitude, rate of
citation, and disciplinary origin and discuss alternative approaches towards
identifying such instances.

##  1 Introduction

The term ‘Sleeping Beauty’ has been used to describe an article that is not
well cited in the early years after its publication but experiences a sharp
increase in the rate at which it is subsequently cited (van Raan, 2004). An
implication is that the new concept presented in such an article is ‘ahead of
its time’ and resistance to it delays recognition. Other causes for resistance
and delayed recognition have been postulated that include (i) information
overload from the large amount of information available, (ii) modest
communication skills of authors, (iii) insufficient promotion of ideas, (iv)
conflict with existing theory and experimental data, (v) the author’s position
in the social hierarchy of science, (vi) multiple discovery, (vii) the
management structures of scientific institutions, (viii), and the conservative
nature of establishments (Barber, 1961; Merton, 1963; Cole, 1970; Garfield,
1970, 1980). The Sleeping Beauty phenomenon, and variants of it, have been
studied and debated with some degree of agreement that a fraction of the
scientific literature exhibits citation kinetics that suggest delayed but
eventual recognition of new ideas (Glänzel et al., 2003; Glänzel & Garfield,
2004; van Raan, 2004; Redner, 2005; Braun et al., 2010; Li, 2014; Ke et al.,
2015; Li & Ye, 2016; Song et al., 2018; Sugimoto & Mostafa, 2018; Ye &
Bornmann, 2018; van Raan & Winnink, 2019).

Various approaches have been used to identify Sleeping Beauties and variants
of it. Depth of sleep, length of sleep, and awake intensity as variables (van
Raan, 2004), the Gini coefficient to examine later years of citation history
(Li et al., 2014), a parameter-free beauty coefficient (Ke et al., 2015),
positional measures (Costas et al., 2010), and the citation angle by Ye &
Bornmann (2018). While earlier studies examined small datasets, subsequent
ones considered large samples of the literature, for example, 22 million
publications in Ke et al. (2015).

The research cited above has focused on single publications, however, new
ideas also result from combining two previously independent ones. The
recognition of such novelty through combination can be examined by co-citation
analysis (Marshakova-Shaikevich, 1973; Small, 1973; Uzzi et al., 2013; Boyack
& Klavans, 2014; Wang et al., 2017; Bradley et al., 2020). Tracing co-
citations, therefore, provides another lens with which to study delayed
recognition. In precedent is a somewhat related use of co-citation analysis by
Zong et al. (2018); Teixeira et al. (2017) who sought to identify the so-
called ‘princes’ that awaken Sleeping Beauties.

The measurement of delayed recognition by co-citation has been briefly
explored by Devarakonda et al. (2020) in a study of 33.6 million reference
pairs. The authors used simplified criteria derived from prior Sleeping Beauty
studies on single publications (Ke et al., 2015; van Raan, 2004; van Raan &
Winnink, 2019), reported 24 co-cited pairs all in the 99th percentile of co-
citation frequencies, and proposed the term _delayed co-citations_ for such
cases. This initial exploration, albeit at scale, only considered reference
pairs where each member of a pair was in the 99th percentile of highly cited
articles in Scopus. In this article, we extend the work delayed co-citation to
a much larger dataset, approximately 940 million pairs of articles. We refine
the criteria in Devarakonda et al. (2020) and identify co-cited article pairs
that exhibit delayed recognition using modifications of the techniques of van
Raan (2004); van Raan & Winnink (2019) and Ke et al. (2015). We also ask
whether delayed co-citations are derived from Sleeping Beauty publications.

##  2 MATERIALS AND METHODS

We have previously described a dataset of 33.6 million cited pairs each
belonging to the top 1% of cited articles in the Scopus bibliography
(Devarakonda et al., 2020, Figure 2). In the present study, we include all co-
cited pairs from references cited by articles published in Scopus in the 11
year period, 1985-1995, not only those drawn from the top 1% of cited
articles. We developed methods to manage the expected volume of data using a
combination of SQL, Cypher, and Python. Our code for parsing and updating
Scopus XML data, a PostgreSQL schema for Scopus data, SQL, Cypher, and Python
scripts used in this study are freely available from a Github repository
(Korobskiy et al., 2019).

To assemble and analyze a working dataset, we first exported 95,524,693
publication records from Scopus (all citation types) as a citation graph
consisting of an edgelist and a nodelist, imported these data into a graph
database (Neo4j) treating publications as nodes and citations as edges. After
creating indexes to improve performance, we selected all publications of
citation type ‘article’ published in the years 1985-1995 (inclusive of both)
that had at least five cited references each. In counting references, we only
considered references with complete Scopus records. Incomplete references and
those with cryptic placeholder identifiers were removed from the dataset. We
also filtered rare cases in the data where a publication cites itself, or if
the publication date of a cited reference was missing or greater than the
publication date of its citing article. Selection of publications with at
least 5 references was performed after curating references.

After initial comparison of SQL vs Cypher, we chose, on the basis of
simplicity and performance, to use Cypher queries in Neo4j to generate all
pairwise $n\choose 2$ combinations of an article’s cited references. We de-
duplicated these pairs across all articles to assemble a dataset of $\sim$940
million pairs (940,357,633 pairs), roughly 28 times larger than the dataset in
Devarakonda et al. (2020). We then calculated the frequency of co-cited pairs
by dividing the data and processing batches in parallel using Neo4j and the
GNU Parallel utility. After tuning experiments on a test set of 1 million
pairs using a Neo4j 4.0 in a Centos 7.5 virtual machine with 128 Gb of RAM and
16 vCPUs in the Microsoft Azure environment, we set the batch size to 1,000
pairs and the degree of parallelization to 15 cores. Under these conditions,
it took roughly 11 min to compute co-citation frequencies for a batch of 1,000
pairs. We divided these 940 million pairs into 9 subsets of around 100 million
pairs each and processed them at the rate of approximately 19 hours per
subset.

In illustration, the simple Cypher query for calculating co-citation
frequencies of pairs in Neo4j is shown below. The input to the query is a csv
file containing two columns of article identifiers with each row representing
a co-cited pair.

⬇

UNWIND $input_data AS row

MATCH (a:Publication{

node_id: row.cited_1})<–(p)–>(b:Publication{node_id:row.cited_2})

RETURN row.cited_1 AS cited_1, row.cited_2 AS cited_2,

count(p) AS scopus_frequency;

Frequencies thus calculated, were loaded back into PostgreSQL. For kinetic
analysis, we selected all pairs with a co-citation frequency $>=$ 100 and
calculated the kinetics of citation accumulation from the first possible year
of co-citation for each pair through the year 2018, again in Neo4j. Finally,
for continuity, we set zero as the frequency for all years between the first
possible year of co-citation and the last co-cited year (2018), with missing
frequency counts. Minor differences between the data in Devarakonda et al.
(2020) are due to more current data in Scopus in our study, and computing
kinetic data through 2018 in this study. We compared small samples between the
two datasets and confirmed that these minor differences in co-citation
frequencies could be bridged by including citations from publications in 2019
and later.

After generating a dataset of 940 million pairs, we applied three relatively
conservative conditions to identify co-cited pairs of interest: (i) a minimum
peak (annual) co-citation frequency for a pair of at least 20 (ii) a minimum
total co-citation frequency of at least 100 (iii) a requirement both members
of a co-cited pair should be published no earlier than 1970. We then
identified delayed co-citation cases by setting two more conditions: (i) a
minimum sleeping duration of 10 years as measured from the first possible year
of co-citation (the more recent publication year of the two articles), (ii)
during this sleeping period of 10 years or more, the average co-citation
frequency should be at most 1 with no more than 2 co-citations in any one
year.

We also calculated the slope between the co-citation frequency of the
awakening year and the peak frequency and modified the Beauty Coefficient (Ke
et al., 2015; Devarakonda et al., 2020), which was designed to measure
kinetics in single publications, to be relevant to co-citations by treating
the first possible year of co-citation equivalently to the year of publication
for a single article (Devarakonda et al., 2020).

To identify, single Sleeping Beauty publications, we narrowed the criteria of
van Raan & Winnink (2019) to consider only a single sleeping period of 10
years or greater; depth of sleep (average citation rate during sleep) of at
most 1; an awakening period of 5 years; and an average co-citation frequency
during the awakening period (which is defined as awakening citation intensity
by van Raan) of at least 5. We also calculated the Beauty Coefficient (Ke et
al., 2015) for all single publications for comparison.

##  3 RESULTS AND DISCUSSION

In this study of delayed co-citations, we first examined cited references from
3,433,578 publications in the Scopus database. The criteria for selection of
these publications were that they were classified as ‘article’, that they were
published in the period 1985-1995, and they contained at least 5 cited
references each. We generated all possible co-cited pairs for the references
in these articles and de-duplicated them across articles. since the same
reference pair can occur in more than one article. Then we measured the co-
citation frequency of each pair across the entire Scopus database by counting
all co-citation events from the first possible year of co-citation onwards,
Fig 1,Table 1).

Figure 1: Frequencies of $\sim$940 million co-cited pairs drawn from Scopus 1985-1995. Pairwise combinations, $n\choose 2$, of references from articles indexed in Scopus (1985-1995), were generated as described in Materials and Methods. Total co-citation frequencies for these pairs, ranged from 1 to 52,471 with a median frequency of 1. The empirical cumulative distribution function (ECDF) was calculated from 940,357,633 co-citation frequencies and plotted against co-citation frequencies on a log2 scale. Table 1: Distribution of 940 million Co-citation Frequencies. The count of co-cited pairs in each frequency class as well as the percentage relative to the total number of 940,357,633 is shown. Counts include the lower bound in each class and exclude the upper bound. Add legend details _f_ Interval | Count | Percentage
---|---|---
$<=$ 2 | 790,189,114 | 84.03
2 -4 | 82,022,893 | 8.72
4 -8 | 41,772,728 | 4.44
8 -16 | 17,749,436 | 1.89
16-32 | 6,429,234 | 0.68
32-64 | 1,704,908 | 0.18
64-128 | 385,923 | 0.041
128-256 | 81,164 | 0.0086
256-512 | 17,150 | 0.0018
512-1024 | 3,777 | 0.00040
1024-2048 | 948 | 0.00010
$>$ 2048 | 358 | 0.000038

The data in Fig 1 show a highly skewed distribution of co-citation frequencies
across a large dataset. Roughly 84% of the pairs have a total co-citation
frequency of 2 or less, and the 99th percentile is 16 although each pair had
at least 10 years to accumulate co-citations. Even for a pair of articles from
the most recent year in our data, 1995, this frequency of 16 corresponds to
less than one co-citation per year on average. Thus, only a small fraction of
pairs in these data have co-citation frequencies greater than 2 per year. One
might consider that the reasons advanced for delayed recognition described in
the Introduction could also contribute to such modest recognition or even
acknowledgment of non-merit.

Beyond a high level understanding of the distribution of co-citation
frequencies, however, we are interested in frequently co-cited publications,
which are derived from highly cited publications (Small, 1973), and are of
interest to the community. Thus, we subset the data using a conservative
threshold of 100 for total co-citation frequency along with a peak annual co-
citation frequency of at least 20. These criteria are analogous to those
proposed by van Raan (van Raan, 2004) and Redner (Redner, 2005). After
applying these two further restrictions, the number of co-cited pairs is
reduced to 51,613 (approximately 0.055% of the total number of pairs).

To find cases of delayed co-citation, we applied the following conditions to
these 51,613 pairs: (i) a co-cited paper should have slept for at least $10$
years and received no more than $2$ co-citations in each year during this
sleeping period, which is defined as as the number of years from the first
possible co-cited year to the first year that the pair receives more than $2$
co-citations. To be considered as a Sleeping Beauty, the awakening period that
follows the sleeping period is characterized by (ii) a peak annual co-citation
frequency of at least 20. These criteria when collectively applied, identified
1,196 cases of delayed co-citation, whose characteristics are summarized in
Table 2.

Table 2: Summary Statistics of 1,196 Delayed Co-citation Pairs. Criteria for selection were a minimum sleeping period of 10 years and a minimum peak of 20 citations in any year. | Total Frequency | Sleep Duration | Slope | Beauty Coefficient*
---|---|---|---|---
Min | 20.00 | 10.00 | 0.21 | 34.21
Q1 | 22.00 | 11.00 | 1.23 | 89.40
Median | 26.00 | 14.00 | 1.7000 | 128.53
Mean | 34.06 | 15.11 | 2.40 | 167.63
3rd Qu | 36.00 | 17.00 | 2.67 | 190.93
Max | 296.00 | 38.00 | 38.00 | 1678.62

Interestingly, these 1,196 pairs are derived from only 1,267 of a possible
2,392 individual publications indicating that some members of frequently co-
cited pairs are found in multiple pairs. Indeed, we have previously noted that
a pair of articles concerning methods in biochemistry, contribute to over
40,000 different co-cited pairs of frequency $>=$ 10 (Devarakonda et al.,
2020).

A logical question is whether any of these 1,267 individual publications would
be classified as Sleeping Beauties. Applying van Raan’s criteria (Materials
and Methods), we identify 128 of these 1,267 publications as Sleeping
Beauties. Interestingly, 27 of the 1,196 delayed co-citation pairs were cases
where both members were Sleeping Beauties. Of these, the 1978 article by
Rassias titled ‘On the stability of the linear mapping in Banach spaces’ was a
member of four different pairs. Thus, delayed recognition can occur without a
requirement that at least one member of a co-cited pair with delayed
recognition should have Sleeping Beauty characteristics. These observations
also suggest that while high-referencing fields such as biology (Small &
Greenlee, 1980) might be advantaged by our selection criteria, the thresholds
we set do not entirely exclude other fields. Accordingly, continuing this work
with field normalization of co-citation frequencies, to the extent possible,
is warranted.

In contrast to co-citation frequencies for delayed co-citations (Fig. 2),
which range from 20-260; citation counts for the 1,267 publications that
contribute to these 1,196 delayed co-citations range from 121 to 190,832 with
72 of these publications having citation counts of greater than 10,000.

However, other co-citation frequencies do exceed the seemingly modest
frequencies noted for delayed co-citations. For example, Becke (1993) and Lee
et al. (1988), a pair of articles from the field of physical chemistry, have
been co-cited over 51,000 times but do not exhibit delayed citation kinetics.
It should also be noted that these articles have individually been cited over
70,000 times each. Similarly, 1,357 pairs from the data shown in Fig 1 have
co-citation frequencies greater than 1,000.

We observe (Fig 1), that the 90th, 95th, and 99th percentiles of co-citation
frequencies in our dataset are 4, 6, and 16 respectively. In comparison. the
90th, 95th, and 99th percentile of citation frequencies of $\sim$10.7 million
publications of type ‘article’ in Scopus, published in the years 1970-1995,
are 58, 96, and 254 respectively (roughly ten fold greater). What emerges is
that delayed co-citations tend to have frequency profiles that are lower than
those of other co-cited pairs, and single publications. This is not unexpected
since co-cited frequencies cannot exceed the citation frequencies of the
publications in these pairs but it does suggest that seemingly low co-citation
frequencies should not be overlooked.

To examine rates of awakening, we also calculated the slope between the co-
citation frequency in the first awakening year and the frequency of the peak
year and noted a fairly broad range of slopes with a mean of 2.4 (Table 2).
The kinetics of co-citation are visualized in Fig 2, for three examples with
the maximum slope, the mean slope, and the minimum slope observed.

Figure 2: Kinetics of Co-citation Frequencies for Delayed Co-citations. Three
sample plots are shown from 1,196 delayed co-citations selected for maximum
slope (left panel). mean slope(middle panel), and minimum slope(right panel)
of a line connecting the co-citation frequency of the awakening year to the
co-citation frequency of the peak year . Total co-citation frequencies for
these three plots were 131, 174, and 254, with peaks of 22, 22, and 23, and
slopes of NA, 2.38, and 0.21 respectively. The red triangle marks the
awakening year and the dotted blue line, the slope. The slope in the left
panel is NA since the peak year is the awakening year. The article pairs shown
above are (i) Spacetime as a membrane in higher dimensions (Gibbons 1987) & An
exotic class of Kaluza-Klein models (Visser 1985), (ii) Formulation of the
reaction coordinate (Fukui 1070) & Ab initio effective core potentials for
molecular calculations. Potentials for main group elements Na to Bi (Wadt &
Hay 1985), (iii) A proposed grading system for arteriovenous malformations
(Spetzler 1986) & Arteriovenous malformations of the brain: Natural history in
unoperated patients (Crawford et al. 1986).

Of 1,196 delayed co-citations, the slope could not be computed for 10 pairs
because the peak year was the year of awakening. This small number of cases,
suggest sudden recognition of the concepts represented by these pairs (Table
3. These 10 pairs span the areas of LED technology, cosmology, immunology,
psychology, and computational science. One publication from 1985 titled, “An
exotic class of Kaluza-Klein models” appears in 3 of 10 pairs and the author
himself refers, in 1999, to ‘renewed interest due to the explosion of activity
in the non compact extra dimensions variant of the Kaluza Klein model’
(Visser, 1999).

Table 3: Co-cited pairs with peak frequency in the the first year of awakening Years | Title
---|---
1974 | Fundamental energy gap of GaN from photoluminescence excitation spectra
1971 | Absorption, reflectance, and luminescence of GaN epitaxial layers
1986 | Dimensional reduction caused by a cosmological constant
1985 | An exotic class of Kaluza-Klein models
1987 | Spacetime as a membrane in higher dimensions
1985 | An exotic class of Kaluza-Klein models
1985 | An exotic class of Kaluza-Klein models
1985 | Do we live inside a domain wall?
1971 | Mental rotation of three-dimensional objects
1976 | Demonstration of a mental analog of an external rotation
1974 | Biologic and clinical significance of cryoglobulins. A report of 86 cases
1980 | Mixed cryoglobulinemia: Clinical aspects and long-term follow-up of 40 patients
1977 | Imitation of Facial and Manual Gestures by Human Neonates
1979 | Matching behavior in the young infant.
1978 | Cognitive determinants of fixation location during picture viewing
1979 | Framing pictures: The role of knowledge in automatized encoding and memory for gist
1983 | Parst: A system of fortran routines for calculating molecular structure parameters (truncated)
1983 | On enantiomorph‐polarity estimation
1980 | Toward a positive theory of consumer choice
1973 | On the psychology of prediction

We also examined lesser co-citation frequencies, between 20 and 100, and
observed 5,928,815 pairs. After removing pairs with (i) less than 10 years of
kinetic data (the difference between publication year and peak year is less
than 10 years) (ii) a negative Beauty Coefficient, which describes articles
whose citations growing linearly with time or with a citation trajectory that
is a concave function of time, (iii) without at least one peak of frequency
20, then the number reduced to 13,057 pairs. Of these 12,920 had only a single
peak of 20 or greater that may be similar to ‘flash in the pan’ citations (Li,
2013; Ye & Bornmann, 2018). Given our focus on frequently co-cited pairs, we
did not study these further.

An appealing alternative approach for delayed co-citations and Sleeping
Beauties is the Beauty Coefficient. We have previously modified (Devarakonda
et al., 2020) the Beauty Coefficient (Ke et al., 2015) designed to measure
kinetics in single publications, to be useful to the case of co-cited pairs.
We computed the Beauty Coefficient for these 1,196 pairs observing a range of
34.21-1678.62. These data are summarized in Table 2. Given co-citation
frequencies being generally lower than citation frequencies, the top 15 Beauty
Coefficient values of the 1,196 delayed co-citations range from
712.47-1678.62, which appear comparable to the top 15 described by Ke, all
above 2,000.

Ke and colleagues comment that parameterized approaches in preceding studies
have suffered from being somewhat arbitrary. The comment is fair, but
arbitrariness may not have impeded discovery, for example Redner’s work on the
physics literature (Redner, 2005) with its selection threshold of 250
citations. Further, while the Beauty Coefficient is parameter free, the choice
of selection threshold is left to the user leaving the door open for arbitrary
selection thresholds. We consider this a strength of the measure since it can
be used in contextual studies. The approach of van Raan is also intuitive and
flexible but does not consider the maximum number of citations received as an
important parameter to be tuned. The cases with a sleeping period of ten
years, and a citation rate of 5 for the next 5 years, would satisfy
requirements for a Sleeping Beauty but are perhaps less noteworthy.

Finally, to ask which fields these 1,196 delayed co-citations are found in, we
mapped them to the All Science Journal Classification (ASJC) maintained by
Scopus, which consists of 27 major subject area categories. The data are
represented in Figure 3 but should be interpreted in the light of these
subject area labels being derived from journals and that an article may have
more than one label. Even so, the data suggest that delayed co-citations, as
we define them in our dataset are largely drawn from the domain of
biochemistry, genetics, and molecular biology followed by physics, computer
science, chemistry, and engineering.

Figure 3: Disciplinary composition of 1,196 Delayed Co-citations. Each node
represents a major subject area in the Scopus ASJC classification. Node size
is scaled to the number articles in a given subject area. Edge thickness
indicates the number of pairs that have one member in one each of the two
nodes connected by the edge. Major subject areas are abbreviated in the
graphic: _MTH_ (Mathematics); _IMM_ (Immunology and Microbiology); _HP_
(Health Professions); _GEN_ (General); _ENS_ (Environmental Science); _ENG_
(Engineering); _EPS_ (Earth & Planetary Sciences); _DCS_ (Decision Sciences);
_MAT_ (Material Sciences); _CEN_ (Chemical Engineering); _PSY_ (Psychology);
_PHY_ (Physics and Astronomy); _NEU_ (Neuroscience); _CS_( Computer Science);
_A &H_ (Arts and Humanities); _SS_ (Social Sciences); _MED_ (Medicine); _EGY_
(Energy); _CHE_ (Chemistry); _ABS_ (Agricultural & Biological Sciences);
_BGMB_ (Biochemistry, Genetics & Molecular Biology); _BMA_ (Business,
Management, and Accounting); _EEF_ (Economics, Econometrics and Finance) _PTP_
(Pharmacology, Toxicology & Pharmaceutics)

##  4 CONCLUSION

In a large-scale exploration of the kinetics of co-citation (more than 940
million unique article pairs), we have identified 1,196 cases of delayed co-
citation using criteria largely derived from the work of van Raan and Ke. We
acknowledge that our selection criteria, while guided by positional statistics
and intuitive preference, suffers from some degree of arbitrariness. With all
bibliometric data, coverage and data quality also influence discovery. Thus,
we have tried to identify co-cited pairs of higher frequency since the trends
in such cases are more likely to be reproducible across other data sources.
Relaxing these conditions, will identify additional cases. Our goal was to
identify a set of delayed co-cited pairs that can be studied, in the longer
term, to understand the reasons for the patterns of citation. This future task
will require a greater understanding of the fields in which such delayed co-
citations occurred and ideally should be coupled to qualitative techniques.

## Conflict of Interest Statement

Data used in this study derive from the ERNIE project, which involves a
collaboration with Elsevier. The content of this publication is solely the
responsibility of the authors and does not necessarily represent the official
views of the National Institutes of Health or Elsevier. Elsevier staff did not
have a role in design, manuscript-writing, or review and interpretation of
results.

## Author Contributions

Wenxi Zhao: Conceptualization; Methodology; Investigation; Writing—Review and
Editing. Dmitriy Korobskiy: Methodology; Writing – Review and Editing; George
Chacko: Conceptualization; Methodology; Investigation; Writing—Original Draft;
Writing—Review and Editing; Funding Acquisition, Resources; Supervision.

## Funding

Research and development reported in this publication was partially supported
by federal funds from the National Institute on Drug Abuse (NIDA), National
Institutes of Health, U.S. Department of Health and Human Services, under
Contract Nos. HHSN271201700053C (N43DA-17-1216) and HHSN271201800040C
(N44DA-18-1216).

## Acknowledgments

We thank our Elsevier colleagues for their support of the ERNIE project.

## Data Availability Statement

Data used in this study are restricted by a license from Elsevier Inc.
Interested persons with a license for these data can use the code on our
Github repository (Korobskiy et al., 2019) to reproduce our findings.

## References

  * Barber (1961) Barber, B. (1961).  Resistance by scientists to scientific discovery.  Science, 134, 596–602.
  * Becke (1993) Becke, A. D. (1993).  Density‐functional thermochemistry. III. The role of exact exchange.  The Journal of Chemical Physics, 98(7), 5648–5652.  Publisher: American Institute of Physics.
  * Boyack & Klavans (2014) Boyack, K., & Klavans, R. (2014).  Atypical combinations are confounded by disciplinary effects.  In International Conference on Science and Technology Indicators, (pp. 49–58). Leiden, Netherlands: CWTS-Leiden University.
  * Bradley et al. (2020) Bradley, J., Devarakonda, S., Davey, A., Korobskiy, D., Liu, S., Lakhdar-Hamina, D., Warnow, T., & Chacko, G. (2020).  Co-citations in context: Disciplinary heterogeneity is relevant.  Quantitative Science Studies, (pp. 1–13).
  * Braun et al. (2010) Braun, T., Glänzel, W., & Schubert, A. (2010).  On Sleeping Beauties, Princes and other tales of citation distributions ….  Research Evaluation, 19(3), 195–202.
  * Cole (1970) Cole, S. (1970).  Professional Standing and the Reception of Scientific Discoveries.  American Journal of Sociology, 76(2), 286–306.
URL https://www.jstor.org/stable/2775594

  * Costas et al. (2010) Costas, R., van Leeuwen, T. N., & van Raan, A. F. (2010).  Is scientific literature subject to a ’sell-by-date’? a general methodology to analyze the ’durability’of scientific documents.  Journal of the American Society for Information Science and Technology, 61(2), 329–339.
  * Devarakonda et al. (2020) Devarakonda, S., Bradley, J., Korobskiy, D., Warnow, T., & Chacko, G. (2020).  Preprint: Frequently co-cited publications: Features and kinetics.  ArXiv.  http://arxiv.org/abs/2005.04793, In Press (Quantitative Science Studies).
URL http://arxiv.org/abs/2005.04793

  * Garfield (1970) Garfield, E. (1970).  Would Mendel’s work have been ignored if the Science Citation Index was available 100 years ago?  Essays of an Information Scientist, 1, 69–70.
  * Garfield (1980) Garfield, E. (1980).  Premature Discovery or Delayed Recognition - Why?  Essays of an Information Scientist, 4, 488–493.
  * Glänzel & Garfield (2004) Glänzel, W., & Garfield, E. (2004).  The myth of delayed recognition.  Scientist, 18(11), 8.
  * Glänzel et al. (2003) Glänzel, W., Schlemmer, B., & Thijs, B. (2003).  Better late than never? on the chance to become highly cited only beyond the standard bibliometric time horizon.  Katholieke Universiteit Leuven, Open Access publications from Katholieke Universiteit Leuven, 58.
  * Ke et al. (2015) Ke, Q., Ferrara, E., Radicchi, F., & Flammini, A. (2015).  Defining and identifying Sleeping Beauties in science.  Proceedings of the National Academy of Sciences, 112(24), 7426–7431.
  * Korobskiy et al. (2019) Korobskiy, D., Davey, A., Liu, S., Devarakonda, S., & Chacko, G. (2019).  Enhanced Research Network Informatics Environment (ERNIE).  Github repository, NET ESolutions Corporation.
URL https://github.com/NETESOLUTIONS/ERNIE

  * Lee et al. (1988) Lee, C., Yang, W., & Parr, R. G. (1988).  Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density.  Physical Review B, 37(2), 785–789.  Publisher: American Physical Society.
  * Li (2013) Li, J. (2013).  Citation curves of “all-elements-sleeping-beauties”: “flash in the pan” first and then “delayed recognition”.  Scientometrics, 100, 595–601.
  * Li (2014) Li, J. (2014).  Citation curves of “all-elements-sleeping-beauties”: “flash in the pan” first and then “delayed recognition”.  Scientometrics, 100, 595–601.
  * Li et al. (2014) Li, J., Shi, D., Zhao, S. X., & Ye, F. Y. (2014).  A study of the “heartbeat spectra” for “sleeping beauties”.  Journal of Informetrics, 8, 493–502.
  * Li & Ye (2016) Li, J., & Ye, F. Y. (2016).  Distinguishing sleeping beauties in science.  Scientometrics, 108(2), 821–828.
  * Marshakova-Shaikevich (1973) Marshakova-Shaikevich, I. (1973).  System of Document Connections Based on References.  Scientific and Technical Information Serial of VINITI, 6(2), 3–8.
URL http://garfield.library.upenn.edu/marshakova/marshakovanauchtechn1973.pdf

  * Merton (1963) Merton, R. (1963).  Resistance to the systematic study of multiple discoveries in science.  European Journal of Sociology, 4(2), 37–282.
  * Redner (2005) Redner, S. (2005).  Citation statistics from 110 years of physical review.  Physics Today, 58, 49–54.
  * Small (1973) Small, H. (1973).  Co-citation in the scientific literature: A new measure of the relationship between two documents.  Journal of the American Society for Information Science, 24(4), 265–269.
  * Small & Greenlee (1980) Small, H., & Greenlee, E. (1980).  Citation context analysis of a co-citation cluster: Recombinant-DNA.  Scientometrics, 2(4), 277–301.
  * Song et al. (2018) Song, Y., Situ, F., Zhu, H., & Lei, J. (2018).  To be the prince to wake up sleeping beauty: the rediscovery of the delayed recognition studies.  Scientometrics, 117(1), 9–24.
  * Sugimoto & Mostafa (2018) Sugimoto, C., & Mostafa, J. (2018).  A Note of Concern and Context: On CarefulUse of Terminologies.  Journal of the Association for Information Science and Technology, 69, 347–348.
  * Teixeira et al. (2017) Teixeira, A. A., Vieira, P. C., & Abreu, A. P. (2017).  Sleeping beauties and their princes in innovation studies.  Scientometrics, 110(2), 541–580.
  * Uzzi et al. (2013) Uzzi, B., Mukherjee, S., Stringer, M., & Jones, B. (2013).  Atypical Combinations and Scientific Impact.  Science, 342(6157), 468–472.
  * van Raan (2004) van Raan, A. F. J. (2004).  Sleeping Beauties in Science.  Scientometrics, 59(3), 467–472.
  * van Raan & Winnink (2019) van Raan, A. F. J., & Winnink, J. J. (2019).  The occurrence of ‘Sleeping Beauty’ publications in medical research: Their scientific impact and technological relevance.  PLOS ONE, 14(10), 1–34.
  * Visser (1999) Visser, M. (1999).  An exotic class of Kaluza-Klein models.  ArXiv.  ArXiv:hep-th/9910093.
  * Wang et al. (2017) Wang, J., Veugelers, R., & Stephan, P. (2017).  Bias against novelty in science: A cautionary tale for users of bibliometric indicators.  Research Policy, 46(8), 1416–1436.
  * Ye & Bornmann (2018) Ye, F. Y., & Bornmann, L. (2018).  “smart girls” versus “sleeping beauties” in the sciences: The identification of instant and delayed recognition by using the citation angle.  Journal of the Association for Information Science and Technology, 69(3), 359–367.
  * Zong et al. (2018) Zong, Z., Liu, X., & Fang, H. (2018).  Sleeping beauties with no prince based on the co-citation criterion.  Scientometrics, 117(3), 1841–1852.