# Dynamical evolution in a one-dimensional incommensurate lattice with
$\mathcal{PT}$ symmetry

Zhihao Xu<EMAIL_ADDRESS>Institute of Theoretical Physics and State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University,
Taiyuan 030006, China  Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan 030006, P.R.China  Shu Chen  Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing, 100049, China  Yangtze River Delta Physics Research Center,
Liyang, Jiangsu 213300, China

###### Abstract

We investigate the dynamical evolution of a parity-time ($\mathcal{PT}$)
symmetric extension of the Aubry-André (AA) model, which exhibits the
coincidence of a localization-delocalization transition point with a
$\mathcal{PT}$ symmetry breaking point. One can apply the evolution of the
profile of the wave packet and the long-time survival probability to
distinguish the localization regimes in the $\mathcal{PT}$ symmetric AA model.
The results of the mean displacement show that when the system is in the
$\mathcal{PT}$ symmetry unbroken regime, the wave-packet spreading is
ballistic, which is different from that in the $\mathcal{PT}$ symmetry broken
regime. Furthermore, we discuss the distinctive features of the Loschmidt echo
with the post-quench parameter being localized in different $\mathcal{PT}$
symmetric regimes.

##  I Introduction

Exploration localization induced by disorder is a long-standing research topic
in condensed-matter physics. The Anderson localization induced by random
disorder first proposed by Anderson Anderson  has found its way across a wide
range of different fields, such as cold atomic gasesBilly ; Roati ; Kondov ;
Jendrzejewski ; Semeghini ; Pasek ; Hainaut ; Richard , quantum opticsSperling
; Wiersma ; Aegerter ; Schwartz ; Lahini , acoustic wavesStrybulevych  and
electronic systemsKatsumoto . In comparison with the random disorder cases,
the quasicrystal systems constitute an intermediate phase between periodic
lattices and disorder media, which display long-range order but no
periodicity. A paradigmatic example of a one-dimensional quasicrystal system
is the Aubry-André (AA) model, which has attracted increasing interest in
recent years Aubry ; Thouless ; Sokoloff ; Hofstadter ; CRDean ; Grempel ;
Kohmoto ; SDasSarma ; Lahini1 ; Biddle ; Biddle1 ; Pouranvari ; Aulbach ;
Modugno ; Larcher ; Ingold ; Kraus ; Lang ; Silva ; Zhihao ; Zhihao1 ;
Shiliang ; Basko ; Schreiber ; Iyer ; Bordia . A typical feature of the AA
model is that, the system undergoes a metal-insulator transition when the
amplitude of the quasicrystal potential exceeds a finite critical value, which
is determined by the self-duality property Thouless . The AA model has been
experimentally realized by the cold atomic technique in bichromatic optical
lattices Roati .

On the other hand, thanks to the impressive progress in controlling quantum
matter in recent decades, the realizations of the real-time dynamics of
quantum systems has been achieved on various experimental platforms, such as
ultra-cold atoms in optical lattices, trapped ions, and photonic lattices, and
the inaccessible dynamical phenomena have been researched. In particular, the
dynamical evolution of a wave packet in a disordered system has drawn
considerable interest. One tries to understand the relation between the energy
spectrum and the dynamical propagation of the wave packet Kohmoto1 ; Ostlund ;
Kohmoto2 ; Kohmoto3 ; Abe ; Katsanos ; Geisel ; Ketzmerick ; Huckestein ;
Ketzmerick1 ; ZhenjunZhang ; Dadras ; Sinha ; Santos ; Silberstein . The
dynamical observation of wave-packet evolution and many-body localization in
one-dimensional incommensurate optical lattices has also been reported in
recent works Luschen ; Kohlert ; Zhihao2 . The dynamical phase transition
based on the Loschmidt echo is another topic attracting wide attention Heyl ;
Karrasch ; Canovi ; Jalabert ; Cucchietti ; Gorin ; HTQuan ; Jafari ; Budich ;
Vajna ; Vajna1 ; Sharma ; Bhattacharya ; YangChao ; Kennes ; Szpak ; Jurcevic
; Pagano ; Vogel ; XYGuo ; KWang ; TTian ; KXu ; HYin ; TongLiu ; YanxiaLiu ;
Peotta ; XTong . A dynamical phase transition occurs, when the quench process
goes across the critical point. It corresponds to the vanishing of the
Loschmidt echo, which has been successfully applied in the AA model YangChao
and its extensions Zhihao2 ; YanxiaLiu ; XTong .

Recently, great interest has been devoted to the interplay of non-Hermiticity
and disorder, which brings a new perspective of the localization properties
YanxiaLiu ; Hatano1 ; Hatano2 ; Kolesnikov ; ZhongPingGong ; Tzortzakakis ;
YHuang1 ; YHuang2 ; QBZeng1 ; QBZeng2 ; TongLiu1 ; YanxiaLiu1 ; YanxiaLiu2 ;
YanxiaLiu3 ; Longhi1 ; Longhi2 ; Longhi3 ; CHLiang ; Cortes ; Harter ; Rivolta
; HuiJiang ; DWZhang ; Claes ; LJZhai ; Okuma ; Tzortzakakis1 ; CWang1 ;
Hamazaki ; CHLiu ; Tzortzakakis2 ; Yusipov1 ; Balasubrahmaniyam ; Goldsheid ;
Molinari ; Markum ; Chalker . Non-Hermitian models are found in open systems
exchanging energy or particles with the environment. For a non-Hermitian
system, the non-Hermiticity is generally obtained by introducing nonreciprocal
hopping terms or gain and loss potentials. According to the random matrix
theory, the spectral statistics of non-Hermitian disorder systems exhibits
different properties from the Hermitian ones Goldsheid ; Molinari ; Markum ;
Chalker . The Hatano-Nelson model describing the interplay of the
nonreciprocal hopping and random disorder exhibits a finite localization-
delocalization transition Hatano1 ; Hatano2 ; Kolesnikov ; ZhongPingGong .
Non-Hermitian extensions of the AA models realized by introducing non-
reciprocal hopping or $\mathcal{PT}$ symmetric potential have been
investigated in Refs. YanxiaLiu ; QBZeng1 ; QBZeng2 ; TongLiu1 ; YanxiaLiu1 ;
YanxiaLiu2 ; YanxiaLiu3 ; Longhi1 ; Longhi2 ; Longhi3 ; HuiJiang . For a
$\mathcal{PT}$ symmetric extension of the AA model, one can find the
coincidence of a localization transition point with a $\mathcal{PT}$ symmetry
breaking point by both numerical and analytical calculations Longhi1 ; Longhi2
; HuiJiang . The analytical results show that the localization length in the
insulator phase is independent of energy which is similar to the Hermitian AA
case. On the other hand, the energy spectrum is gapless in the metallic phase
which is unlike the Hermitian AA model Longhi2 . Due to the anomalous energy
spectrum of the $\mathcal{PT}$ symmetric AA model, some interesting questions
arise here: What are the features of the dynamical evolution of the wave
packet in the one-dimensional incommensurate lattice with $\mathcal{PT}$
symmetry? Can the Loschmit echo method be applied to detect the localization
transition for a $\mathcal{PT}$ symmetric AA model?

In this work, to address these questions, we study the dynamical evolution of
the $\mathcal{PT}$ symmetric AA model in different localization regimes by
applying the propagation of the profile of the wave packet, the long-time
survival probability, the evolution of the mean displacement, and the
Loschmidt echo dynamics. As a comparison, we also discuss the case of the
standard AA model. We find that the propagation of the profile of the wave
packet and the long-time survival probability exhibit distinctive features in
different localization regimes for both the Hermitian and non-Hermitian cases
. The behaviors of the evolution of the mean displacement are dependent on the
breaking of the $\mathcal{PT}$ symmetry in the non-Hermitian AA model, while
for the Hermitian case, it displays different diffusion exponents in different
localization regimes. The Loschmidt echo dynamics with the post-quench
parameter localized in the $\mathcal{PT}$ symmetry unbroken regime can be used
to detect the localization transition, which is similar to the Hermitian case.
However, for the post-quench parameter localized in the $\mathcal{PT}$
symmetry broken regime, our results indicate that the detection of the
dynamical phase transition is unavailable using Loschmidt echo dynamics.

##  II Model and Hamiltonian

Figure 1: (Color online) Imaginary part of the energy spectrum
$\mathrm{Im}(E)$ for the $\mathcal{PT}$ symmetric AA model as a function of
the modulation strength $\lambda$. Here, $J=1$ and $L=377$. Figure 2: (Color
online) Scaling of MIPR for the $\mathcal{PT}$ symmetric AA model in (a)
$\lambda<J$, (b) $\lambda>J$, and (c) $\lambda=J$, respectively. The dashed
line indicates a power-law fitting. Here, $J=1$.

We consider a non-Hermitian extension of the AA model with a complex
incommensurate lattice subjected to $\mathcal{PT}$ symmetry, which can be
described by the Hamiltonian

$\hat{H}=-\sum_{j}J(\hat{c}_{j}^{\dagger}\hat{c}_{j+1}+H.c.)+\sum_{j}\lambda_{j}\hat{n}_{j},$
(1)

where $\hat{c}_{j}$ is the annihilation operator of fermions at the $j$-th
site, and $J$ denotes the strength of the hopping term. The on-site
$\mathcal{PT}$ symmetric potential can be written as

$\lambda_{j}=\lambda\exp{(-i2\pi\alpha j)}$ (2)

with the amplitude $\lambda$ and $\alpha$ being irrational, which leads to the
non-Hermiticity of the Hamiltonian (1). The potential is the complexification
of the standard AA model with on-site potential
$\lambda_{j}=2\lambda^{\prime}\cos{(2\pi\alpha j+\varphi)}$ taking
$\varphi=ih$, and the limits $\lambda^{\prime}\to 0$, $h\to\infty$, keeping
$\lambda^{\prime}\exp{(h)}=\lambda$ finite. It can be realized in the silicon
waveguide platform, the cold atomic gases, and ion chains Bylinskii1 ;
Bylinskii2 ; Bonetti ; Benassi ; Mandelli ; Kiethe . Numerical and analytical
results exhibit a metal-insulator phase transition at $\lambda=J$ for the non-
Hermitian system, which also corresponds to $\mathcal{PT}$ symmetric breaking
Longhi1 ; Longhi2 . Figure 1 shows the imaginary part of the energy spectrum
$\mathrm{Im}(E)$ of the $\mathcal{PT}$ symmetric AA model as a function of
$\lambda$. As seen in Fig. 1, when $\lambda<J$, all the energy spectra are
real values, while for $\lambda>J$ the energies become complex.

To characterize the localized properties of the $\mathcal{PT}$ symmetric AA
model, we can investigate the fractal dimension of the wave functions $\beta$
defined by $\mathrm{MIPR}\propto L^{-\beta}$, where the mean inverse
participation ratio (MIPR)

$\mathrm{MIPR}=\frac{1}{L}\sum_{n=1}^{L}\sum_{j=1}^{L}\frac{|\psi_{n,j}|^{4}}{|\psi_{n,j}|^{2}}$
(3)

with $\psi_{n,j}$ being the amplitude of the eigenstate $|\psi_{n}\rangle$ of
the eigenvalue $E_{n}$ at the $j$th site and $L$ is the size of the lattice.
It is known that $\beta=0$ for the localized regime, $\beta=1$ for the
extended regime, and $0<\beta<1$ corresponding to a multifractal phase. Figure
2 shows the scaling of the MIPR for the $\mathcal{PT}$ symmetric AA model in
different localized regimes. As shown in Fig. 2(a) for $\lambda/J=0.6$ and
$0.8$ in the extended regime, $\beta=1$ and the MIPRs tend to $0$ with the
increase of $L$. In the localized regime, taking $\lambda/J=1.2$ and $1.5$ as
examples shown in Fig. 2(b), the MIPRs are finite and independent of $L$ with
$\beta=0$. For $\lambda/J=1$ [Fig. 2(c)], when $L\to\infty$, the MIPR
approaches $0$ with $\beta\approx 0.62$. This implies that the non-Hermitian
AA model is localized in the multifractal phase for $\lambda/J=1$. As a
comparison, we briefly recall the main conclusions for the Hermitian AA model,
corresponding to $\lambda_{j}=2\lambda^{\prime}\cos{(2\pi\alpha j)}$, with
$\alpha$ being irrational. The system exhibits a transition from the
delocalized phase for $\lambda^{\prime}<J$ with $\beta=0$ to localized region
for $\lambda^{\prime}>J$ with $\beta=1$, and for $\lambda^{\prime}=J$, the
system is localized in the multifractal phase with $\beta=0.5$ Zhihao2 ;
Geisel . We can see that the $\mathcal{PT}$ symmetric AA model exhibits
similar localization properties to those of the Hermitian one. However, their
dynamical evolutions of both cases show some distinctive behaviors.

In this paper, we study the dynamical behaviors of a non-Hermitian AA model
with $\mathcal{PT}$ symmetric potentials described by the Hamiltonian (1) in
real space on a ring, and we take $J=1$ as the energy unit. The irrational
number $\alpha=(\sqrt{5}-1)/2$ is chosen of which the approximants are
$F_{\mu-1}/F_{\mu}$, with $F_{\mu}$ being the $\mu$th Fibonacci number defined
by $F_{\mu}=F_{\mu-1}+F_{\mu-2}$, with $F_{0}=F_{1}=1$ yielding the size of
the lattice $L=F_{\mu}$.

##  III Wave packet dynamics

Figure 3: (Color online) (a)-(c) Evolutions of the profile of the wave
function $\rho_{j}(t)$ with the non-Hermitian modulation amplitude (a)
$\lambda=0.6$, (b) $\lambda=1.0$, and (c) $\lambda=1.5$, respectively. (d)
Long-time survival probability $P(r)$ for both the non-Hermitian and the
Hermitian cases. For the non-Hermitian cases, we choose $\lambda=0.6$ and
$1.0$ at $t=800$ and $\lambda=1.5$ at $t=400$. For the Hermitian cases, we
choose the Hermitian modulation amplitude $\lambda^{\prime}=0.6$, $1.0$, and
$1.5$ at $t=8000$. Here, $J=1$ and $L=377$.

We first investigate the expansion dynamics of the wave packet
$|\Psi(0)\rangle=|j_{0}\rangle$ initially localized at the center of the
lattice $j_{0}$ governed by the non-Hermitian AA model described by the
Hamiltonian (1). The evolution wave function at time $t$ can be written as

$|\Psi(t)\rangle=\frac{1}{\sqrt{\mathcal{N}}}e^{-i\hat{H}t}|\Psi(0)\rangle,$
(4)

with $\mathcal{N}$ being the normalization coefficient of the time-evolution
wave function $|\Psi(t)\rangle$. Here, we use a normalized wave function in
our $\mathcal{PT}$ symmetric model to avoid an exponential rise of the wave
function with time in the $\mathcal{PT}$ symmetry broken regime. For a given
initial state $|\Psi(0)\rangle$, to numerically calculate the wave function
$|\Psi(t)\rangle$ at time $t$, we divide time $t$ into $\bar{M}$ intervals
with $dt=t/\bar{M}$. In the limit $\bar{M}\to\infty$, $dt\to 0$, and drop the
$O(dt^{2})$ terms. The wave function at time $(\bar{m}+1)dt$ is given by

$|\Psi((\bar{m}+1)dt)\rangle=\frac{(1-iHdt)|\Psi(\bar{m}dt)\rangle}{\langle\Psi(\bar{m}dt)|(1+iHdt)(1-iHdt)|\Psi(\bar{m}dt)\rangle},$

with

$|\Psi(dt)\rangle=\frac{(1-iHdt)|\Psi(0)\rangle}{\langle\Psi(0)|(1+iHdt)(1-iHdt)|\Psi(0)\rangle}.$

and $\bar{m}$ is the iteration times. After the $\bar{M}$ times iteration
processes, we can obtain $|\Psi(t)\rangle$.

To obtain an intuitive picture, we study the profile of the wave function at
time $t$ given by $\rho_{j}(t)=\langle\Psi_{j}(t)|\Psi_{j}(t)\rangle$ shown in
Figs. 3(a)-(c) for different $\lambda$ with $L=F_{13}=377$. For the extended
case [Fig. 3(a) with $\lambda=0.6$], the initial state localized at the center
of the lattice rapidly expands to the whole system, and after some time
intervals, the profile of the wave function exhibits an extended character. In
the multifractal phase, which is shown in Fig. 3(b), the center part of the
profile of the wave function decays with time and keeps finite in our
observation time. The expanding part seems to show a multifractal property
with time. When we turn the strength of quasi-periodic modulation $\lambda$ to
the localized regime, e.g., $\lambda=1.5$ in Fig. 3(c), the wave function
exhibits non-Hermitian jumps between distant sites, which only occurs in the
localized regime of the non-Hermitian system and does not have a Hermitian
analog Wiersma ; Tzortzakakis2 ; Yusipov1 ; Balasubrahmaniyam .

To further observe the dynamical behaviors of the non-Hermitian system in
different phases, we define the long-time survival probability

$P(r)=\lim_{t\to\infty}\sum_{|j-j_{0}|\leq
r/2}\langle\Psi_{j}(t)|\Psi_{j}(t)\rangle,$ (5)

which represents the normalized probability of detecting the wave packet in
sites within the region $(-r/2,r/2)$ in the long-time limit Santos . The
recent works for random matrix models show that $P(r)$ provides crucial
information about localization properties Kravtsov ; Herrera ; Tomasi . Figure
3(d) shows $P(r)$ as a function of $r/L$ for both non-Hermitian and Hermitian
cases with $L=F_{13}=377$ in the long-time limit. When the strength of the
modulation is localized in the extended regime, $P(r)$ of both cases increase
linearly with $r$, _i.e._ , $P(r)\approx r/L$, since the probability of
finding the normalized wave packet at each site is the same for both cases.
For $\lambda=\lambda^{\prime}=1$, $P(r)$ of both cases show the multifractal
feature with $P(r)\propto(r/L)^{\tilde{\beta}}$ and the power-law exponent
$\tilde{\beta}\in(0,1)$. For the Hermitian case in the localized regime with
$\lambda^{\prime}=1.5$ shown in Fig. 3(d), we can see that $P(r)$ is finite at
$r=0$, and it presents an exponential rise, and rapidly reaches $(r/L)^{0}=1$.
However, for the non-Hermitian case with $\lambda=1.5$, the value of $P(r)$
approaches $0$ in the $r/L\ll 1$ limit, and it exhibits an exponential
increase after some $r$, and rapidly reaches $(r/L)^{0}=1$. According to our
results, the long-time survival probability $P(r)$ is proportional to
$(r/L)^{\tilde{\beta}}$, and it can be applied to the non-Hermitian case to
distinguish the long-time dynamical behaviors of the wave packet in different
localization regimes. We also find that $P(r)$ of the non-Hermitian case in
the localized regime presents a vacuum space in the small-$r$ limit that is
different from the Hermitian one.

Figure 4: (Color online) Time evolution of the mean displacement $\sigma(t)$
for the non-Hermitian cases marked by the solid lines and the Hermitian cases
marked by the dashed lines. The black dotted lines represent the power-law
fitting curves. (a1) and (a2) are for $\lambda=\lambda^{\prime}=0.6$. (b1) and
(b2) are for $\lambda=\lambda^{\prime}=1.0$. (c1) and (c2) are for
$\lambda=\lambda^{\prime}=1.5$. $\lambda$ and $\lambda^{\prime}$ are the non-
Hermitian and Hermitian modulation amplitudes, respectively. The top row
corresponds to the evolution in the short-time stage, and the bottom row
exhibits the long-time evolution processes. Here, $J=1$ and $L=377$.

The wave-packet spreading dynamics can be described by the evolution of the
mean displacement ZhenjunZhang ; CMDai  given by

$\sigma(t)=\sqrt{\sum_{j=1}^{L}\langle\Psi_{j}(t)|(j-j_{0})^{2}|\Psi_{j}(t)\rangle}.$
(6)

In general, for a long expansion time in a Hermitian case, the mean
displacement obeys the power law $\sigma(t)\sim t^{\tilde{\gamma}}$. Apart
from ballistic spread for $\tilde{\gamma}=1$ and localization or diffusive
transport for $\tilde{\gamma}=0$ or $1/2$, subdiffusion for
$0<\tilde{\gamma}<1/2$ and superdiffusion for $1/2<\tilde{\gamma}<1$ can occur
in some quasi-periodic lattices. When $\tilde{\gamma}>1$, the initial
localized wave packet exhibits a hyperdiffusion transport for a certain time
scale. Figure 4 shows the time evolution of $\sigma(t)$ with an initial
localized wave packet $|j_{0}\rangle$ at the center of the lattice $j_{0}$.
The evolution of $\sigma(t)$ in different localization regimes for both the
non-Hermitian and Hermitian cases is marked by solid lines and dashed lines,
respectively. During the first stage of evolution shown in Figs. 4(a1)-4(c1)
with $\lambda=\lambda^{\prime}=0.6$, $1.0$, and $1.5$, respectively, the wave
packet spreading is ballistic with the same power-law indices
$\tilde{\gamma}=1$ independent of the hermiticity of the system and the
strength of the modulation. The long-time evolution of $\sigma(t)$ is shown in
Figs. 4(a2)-4(c2). During the second stage, as shown in Fig. 4(a2) with
$\lambda=\lambda^{\prime}=0.6$, both cases present ballistic diffusion with
$\tilde{\gamma}=1$. When the on-site potential amplitude is localized in a
multifractal regime, the diffusion exponent $\tilde{\gamma}=0.5$ for the
Hermitian case, while it still exhibits a ballistic diffusion
($\tilde{\gamma}=1$) for the non-Hermitian case with $\lambda=1$. For
$\lambda=\lambda^{\prime}>1$ cases [$\lambda=\lambda^{\prime}=1.5$ shown in
Fig. 4(c2)], after the first stage spreading, $\sigma(t)$ exhibits an
oscillating characteristic and the diffusion exponent $\tilde{\gamma}=0$.
However, the non-Hermitian case is quite different. After the ballistic
spreading stage, $\sigma(t)$ enters into a temporary stage whose behavior is
similar to that in the second stage of the Hermitian case. It then presents a
hyperdiffusion with $\tilde{\gamma}\approx 2$ in a short-time interval, which
corresponds to a non-Hermitian jump process Wiersma ; Tzortzakakis2 ; Yusipov1
; Balasubrahmaniyam . The diffusion exponent $\tilde{\gamma}>1$ in this stage
is dependent on the value of $\lambda$ [see also Fig. 5(d)]. Finally, the wave
packet seems to be frozen with $\tilde{\gamma}=0$. According to our results,
we can see that the evolution of the mean displacement $\sigma(t)$ shows a
ballistic spreading for the non-Hermitian AA model (1) in the $\mathcal{PT}$
symmetry unbroken regime independent of its localization properties, which is
different from the Hermitian case. However, when the non-Hermitian AA model
enters into the localized regime corresponding to the $\mathcal{PT}$ symmetry
broken regime, $\sigma(t)$ successively undergoes ballistic diffusion to
localization to hyperdiffusion and back to localization. The results of
$\sigma(t)$ suggest that the dynamics in the $\mathcal{PT}$ symmetry unbroken
regime and the broken regime display distinctive behaviors for the non-
Hermitian AA model with $\mathcal{PT}$ symmetry. We also consider the
evolution of the mean displacement $\sigma(t)$ for different strengths of the
hopping term $J$ when $\lambda$ is set as the unit energy shown in Appendix A.
It displays similar results to those discussed above.

Figure 5: (Color online) Time evolution of the mean displacement $\sigma(t)$
with different $L$ and the non-Hermitian modulation strength (a)
$\lambda=0.8$, (b) $\lambda=1.0$, (c) $\lambda=1.5$, and (d) $\lambda=3$,
respectively. Here, $J=1$.

Figure 5 shows the time evolution of the mean displacement $\sigma(t)$ with
different lattice lengths $L$ and $\lambda$ to show the size-induced physics
in our system. As shown in Fig. 5(a) with $\lambda=0.8$ in the extended
regime, both cases with $L=F_{13}=377$ and $L=F_{16}=1597$ exhibit the same
dynamic feature, _i.e._ , the diffusion exponent $\tilde{\gamma}=1$. For the
case with $\lambda=1$, $\sigma(t)\sim t$ independent of the lattice length $L$
shown in Fig. 5(b). When the parameter $\lambda$ enters into the
$\mathcal{PT}$ symmetry broken localization regime, $\sigma(t)$ for $L=377$
and $1597$ with the initial wave packet localized at the center of the lattice
shows similar dynamical features to those shown in Figs. 5(c) and 5(d). As
seen in Figs. 5(c) with $\lambda=1.5$ ($\tilde{\gamma}\approx 2$) and 5(d)
with $\lambda=3$ ($\tilde{\gamma}\approx 5$), a remarkable point is that the
$\lambda$-dependent diffusion exponent $\tilde{\gamma}>1$ of the
hyperdiffusion process increases with the increase of the non-Hermitian
modulation strength. In Appendix B, we also discuss the cases of the chain’s
length deviating from a Fibonacci number. When the wave packet spreads near
the boundaries, the evolution of the initially localized wave packet with the
parameter $\lambda$ in the extended and multifractal regimes brings about a
sudden rise in a short time, and then the wave packet is frozen at the
boundaries, which is different from the one with the chain’s length being a
Fibonacci number. When $\lambda$ is localized in the $\mathcal{PT}$ symmetry
broken regime, the evolution process is $L$-dependent. Investigations on the
size-dependent features in quasi-periodic chains have been reported in Refs.
Znidaric1 ; Varma ; Znidaric2 ; Purkayastha ; Znidaric3 ; Purkayastha1 . Our
results imply that for the case with a chain’s length being a Fibonacci
number, the dynamical evolution for different $L$ shows a similar feature.

##  IV Loschmidt echo dynamics

The Loschmidt echo plays a significant role in characterizing the dynamical
signature of the quantum phase transition. It was shown that the Loschmidt
echo evolution could characterize the localization-delocalization transition
in a Hermitian AA model Zhihao2 ; YangChao . When the initial and post-quench
systems are located in the same localization regime, the Loschmidt echo will
oscillate without decaying to zero in a long time. However, if they are
located in different localization regimes, the Loschmidt echo will decay and
reach near $0$ at some time intervals. However, study of the Loschmidt echo
dynamics for a $\mathcal{PT}$ symmetric AA model has not been demonstrated.

In this section, we focus on the quench dynamics of the non-Hermitian AA model
described by the Hamiltonian (1). The system is initially prepared in an
eigenstate $|\Psi(t_{0})\rangle$ of the Hamiltonian $\hat{H}(\lambda^{i})$ at
time $t_{0}$ with $\langle\Psi(t_{0})|\Psi(t_{0})\rangle=1$, and then suddenly
quenched to the final Hamiltonian $\hat{H}(\lambda^{f})$. We define the return
amplitude

$G(t;\lambda^{i},\lambda^{f})=\langle\Psi(t_{0})|\Psi(t)\rangle$ (7)

where

$|\Psi(t)\rangle=\frac{1}{\sqrt{\mathcal{N}}}e^{-i(t-t_{0})\hat{H}(\lambda^{f})}|\Psi(t_{0})\rangle,$
(8)

with $\mathcal{N}$ being the normalization coefficient of the time-evolution
wave function $|\Psi(t)\rangle$, and setting $t_{0}=0$. The behavior of the
return probability (Loschmidt echo) can be described by

$\mathcal{L}(t;\lambda^{i},\lambda^{f})=|G(t;\lambda^{i},\lambda^{f})|^{2},$
(9)

where the superscripts $i$ and $f$ correspond to before and after the quench
process, respectively. The distinctive dynamics in different $\mathcal{PT}$
symmetric regimes suggest us the we should study the quench processes in
$\lambda^{f}\leq J$ and $\lambda^{f}>J$, respectively.

###  IV.1 Quench processes with $\lambda^{f}\leq J$

Figure 6: (Color online) Evolution of the Loschmidt echo. The initial state is
chosen to be the eigenstate of the energy with the lowest real part at
different pre-quenched modulation amplitude $\lambda^{i}$, and the final
Hamiltonian with the parameter (a) $\lambda^{f}=0$, (b) $\lambda^{f}=0.6$, and
(c) $\lambda^{f}=1$, respectively. The inset of (a) shows the evolution of
dynamical free energy $f(t)$ for different $\lambda^{i}$ and $\lambda^{f}=0$.
Here, $J=1$ and $L=610$.  Figure 7: (Color online) Evolution of the Loschmidt
echo with the final modulation amplitude $\lambda^{f}=0.6$. The initial states
are chosen to be the given eigenstates with the energies $E_{\tilde{n}}$ for
(a) $\lambda^{i}=0.8$, (b) $\lambda^{i}=1.0$, and (c) $\lambda^{i}=1.5$,
respectively. The real part of the energies is ordered in ascending order.
Here, $J=1$ and $L=377$.

We first consider the case of the parameter after the quench process localized
in the $\lambda^{f}\leq J$ region, which is a $\mathcal{PT}$ symmetry unbroken
regime with all energy spectra being real. For the limiting case of the quench
process from $\lambda^{i}\to\infty$ to $\lambda^{f}=0$, the initial state is
localized in a single site,
$|\psi_{m}^{\lambda^{i}\to\infty}\rangle=\sum_{j}\delta_{jm}\hat{c}_{j}^{\dagger}|0\rangle=|m\rangle$,
and the corresponding eigenenergy $E_{m}=\lambda^{i}\exp{(-i2\pi\alpha m)}$.
By performing a quench process to $\lambda^{f}=0$, the eigenstates for the
case of $\lambda^{f}=0$ can be written as
$|\psi_{k}^{\lambda^{f}=0}\rangle=1/\sqrt{L}\sum_{j}e^{ikj}\hat{c}_{j}^{\dagger}|0\rangle=|k\rangle$,
with the wave vector $k=2\pi l/L$ ($l=1,2,\cdots,L$), and the corresponding
eigenvalue is $E_{k}=2J\cos{k}$. The return amplitude is

$\displaystyle G(t;\infty,0)$ $\displaystyle=\langle
m|e^{-iH(\lambda^{f})t}|m\rangle=\sum_{k}\langle
m|e^{-iH(\lambda^{f})t}|k\rangle\langle k|m\rangle$
$\displaystyle=\frac{1}{L}\sum_{k}e^{-i2Jt\cos{(k)}}.$ (10)

In the thermodynamic limit ($L\to\infty$), $G(t)=\mathrm{J}_{0}(2Jt)$ with
$\mathrm{J}_{0}(x)$ being the zero-order Bessel function of the first kind. It
is clear that the zeros of the Loschmidt echo $t^{*}$ occur at the halfway
point of the zeros of $\mathrm{J}_{0}(x)$. The emergence of the zeros of the
Loschmidt echo implies that the initial and final states are localized in
different localized regimes, according to previous results in the Hermitian
cases. Figure 6(a) shows the evolution of the Loschmidt echo with
$\lambda^{f}=0$ and different $\lambda^{i}>J$. In this case, the Loschmidt
echoes $\mathcal{L}(t)$ for $\lambda^{i}=100$, $10$, $5$, and $3$ oscillate at
the same frequency as the analytical result. This implies that the frequencies
of the Loschmidt echo are not sensitive to the initial parameter $\lambda^{i}$
as long as $\lambda^{i}$ is large enough. To see the zeros of $\mathcal{L}(t)$
more clearly, we introduce the dynamical free energy
$f(t)=-\ln{\mathcal{L}(t)}$, which is shown in the inset of Fig. 6(a). It will
be divergent at the dynamical phase transition time $t^{*}$. In the large
$\lambda^{i}$ limit, $f(t)$ exhibits obvious peaks at $t=t^{*}$ that almost
completely overlap the analytical result. With the decrease of $\lambda^{i}$,
the change of the peak positions of $f(t)$ is tiny, while the peak amplitudes
of $f(t)$ decrease.

The analytical result for the limit case of the quench process from
$\lambda^{i}=\infty$ to $\lambda^{f}=0$ is close to the $0$ of the Loschmidt
echo. Now, we consider the general cases in which $\lambda^{i}$ and
$\lambda^{f}$ deviate from the limit case for $\lambda^{f}\leq J$. Figures
6(b) and 6(c) show the evolution of the Loschmidt echo with different
$\lambda^{i}$, the parameter of the final Hamiltonian deviating from zero, and
the initial state being chosen to be the eigenstate of the energy with the
lowest real part at different $\lambda^{i}$. In Fig. 6(b), we choose
$\lambda^{f}=0.6$, where all the eigenstates are extended. For
$\lambda^{i}=0.2$ and $0.8$, $\mathcal{L}(t)$ oscillates and has a positive
lower bound that never approaches $0$ during the evolution. However, if
$\lambda^{i}>J$, $\mathcal{L}(t)$ approaches $0$ after some time intervals
[see Fig. 6(b) for $\lambda^{f}=1.5$ and $2$]. Figure 6(c) shows the dynamics
of the Loschmidt echo with $\lambda^{f}=1$ where the system is localized in
the multifractal phase. As seen in Fig. 6(c), the long-time evolution of the
Loschmidt echo approaches $0$ for the initial states localized in either the
extended ($\lambda^{i}=0.6$ and $0.8$) or the localized ($\lambda^{i}=1.2$ and
$1.5$) regime.

To show that our result is independent of the initial eigenstate’s choice, we
consider the different eigenstates as the initial states to calculate the
evolution of the Loschmidt echo. As a concrete example, we choose
$\lambda^{f}=0.6$, and the initial states are the $\tilde{n}$-th eigenstates
of the corresponding eigenenergies $E_{\tilde{n}}$ with $\tilde{n}=100$,
$150$, $200$, $250$, and $300$ as shown in Fig. 7. The real part of the
$E_{\tilde{n}}$ is ordered in ascending order. For the case of
$\lambda^{i}=0.8$ in the extended regime shown in Fig. 7(a), the Loschmidt
echoes for different initial states oscillate and never approach $0$. However,
for $\lambda^{i}=1.0$ and $1.5$ localized in different regimes from the one
with $\lambda^{f}=0.6$ [see Figs. 7(b) and 7(c), respectively], the dynamics
of the Loschmidt echo for different initial eigenstates decay in an
oscillating way and can always touch near $0$ in some time intervals.

In conclusion, for the case of the parameter of the post-quench Hamiltonian
localized in the $\lambda\leq J$ region where all the eigenenergies are real,
the behaviors of the Loschmidt echo are similar to the Hermitian cases in that
the dynamical signature of the localized transition can be characterized by
the emergence of zero points in the evolution of the Loschmidt echo.

###  IV.2 Quench processes with $\lambda^{f}>J$

Figure 8: (Color online) (a) and (b) Evolution of the Loschmidt echo for the
large final modulation amplitude $\lambda^{f}$ with $L=610$, and the initial
state being chosen to be the ground state at $\lambda^{i}=0$. (a)
$\mathcal{L}(t;0,\lambda^{f})$ versus $t$. (b) $\mathcal{L}(t;0,\lambda^{f})$
versus the rescaled time $\lambda^{f}t$. (c) Evolution of the Loschmidt echo
for finite $\lambda^{f}$ with $L=377$, and the initial state being chosen to
be the wave packet localized at the center of the lattice. (d) Log-log plot of
$1-\mathcal{L}(t;\lambda^{i},\lambda^{f})$ as a function of time $t$ for
$\lambda^{f}=1.5$ with $L=377$, and the initial state being chosen to be the
wave packet localized at the center of the lattice. The black dashed line
represents a power-law fitting. Here, $J=1$.

In this subsection, we consider the case of the final Hamiltonian with the
parameter $\lambda^{f}>J$, where all the eigenstates are localized and the
$\mathcal{PT}$ symmetry is broken. First, an analytical calculation is
considered for the quench process with $\lambda^{i}=0$ and
$\lambda^{f}=\infty$. For $\lambda^{i}=0$, the system is initially prepared in
a plane-wave state $|k\rangle$ with the eigenvalue $E_{k}=2J\cos{k}$. By
performing a sudden quench to $\hat{H}(\lambda^{f})$ in the limit of
$\lambda^{f}\to\infty$, the eigenstates of $\hat{H}(\lambda^{f})$ are
localized in a single site $|m\rangle$, and the corresponding eigenenergy is
$E_{m}=\lambda^{f}e^{-i2\pi\alpha m}$. The evolution wave function can be
written as

$\displaystyle|\Psi(t;\lambda^{i}=0,\lambda^{f}\to\infty)\rangle$
$\displaystyle=\frac{1}{\sqrt{\mathcal{N}}}e^{-i\hat{H}(\lambda^{f})t}|k\rangle$
$\displaystyle=\frac{1}{\sqrt{\mathcal{N}L}}\sum_{m}e^{ikm}e^{-i\lambda^{f}te^{-i2\pi\alpha
m}}|m\rangle.$ (11)

In the large $L\to\infty$ limit, we can obtain

$G(t;0,\infty)=\frac{1}{\sqrt{\mathcal{N}}}\sum_{\nu=-\infty}^{\infty}(-1)^{\nu}\mathrm{J}_{\nu}(\lambda^{f}t)\mathrm{I}_{\nu}(\lambda^{f}t),$
(12)

where $\mathcal{N}=\mathrm{I}_{0}(2\lambda^{f}t)$, with $\mathrm{J}_{\nu}(x)$
being the Bessel function of the first kind of the index $\nu$, and
$\mathrm{I}_{\nu}(x)$ being the modified Bessel function of the first kind of
$\nu$ order. Figures 8(a) and 8(b) show the evolution of the Loschmidt echo
with the initial state chosen to be the ground state of
$\hat{H}(\lambda^{i}=0)$ and large $\lambda^{f}$ as a function of $t$, and the
rescaled time $\lambda^{f}t$, respectively. The Loschmidt echoes for
$\lambda^{f}=10$, $30$, and $100$ rapidly decay to $0$ with time, and the
larger $\lambda^{f}$ is, the faster $\mathcal{L}(t)$ decays. They nearly
overlap with the analytical result
$\mathcal{L}(t;0,\infty)=1/\mathcal{N}|\sum_{\nu}(-1)^{\nu}\mathrm{J}_{\nu}(\lambda^{f}t)\mathrm{I}_{\nu}(\lambda^{f}t)|^{2}$
by rescaling the time $\lambda^{f}t$ shown in Fig. 8(b).

Another limit is also considered, i.e., the parameter $\lambda^{i}\to\infty$
of the Hamiltonian (1) quenched to finite $\lambda^{f}>J$. As an example, we
set the initial state to be the wave packet localized at the center of the
lattice $|j_{0}\rangle$, which is shown in Figs. 8(c) and 8(d). It is known
that in the $\lambda^{f}\to\infty$ limit, $\mathcal{L}(t;\infty,\infty)=1$,
due to the initial state being the eigenstate of the post-quench Hamiltonian.
When $\lambda^{f}$ deviates from the infinite value, the initial state
$|j_{0}\rangle$ is no longer the eigenstate of $\hat{H}(\lambda^{f})$, and the
excited single-channel has a superposition of various eigenstates. No matter
how small the superposition with the initial condition is, the mode with the
largest imaginary part among them will dominate after a finite evolution time,
and a non-Hermitian jump process occurs that will induce an evident change of
the return amplitude. Hence, for a finite $\lambda^{f}$, the Loschmidt echo no
longer remains uniform, and it will present complex features. As seen in Fig.
8(c), the Loschmidt echo first decays to a finite value $\mathcal{L}_{c}$ in a
short time $\tilde{t}_{1}$ corresponding to a finite overlap between the
evolution wave function $|\Psi(t_{1})\rangle$ and $|j_{0}\rangle$.
$\mathcal{L}(t;\lambda^{i},\lambda^{f})$ then displays an oscillation around
the limited value $\mathcal{L}_{c}$ in a certain time interval
$\Delta\tilde{t}$. $\mathcal{L}_{c}$ and $\Delta\tilde{t}$ depend on
$\lambda^{f}$. In the large-$\lambda^{f}$ limit, the value of
$\mathcal{L}_{c}$ tends to $1$ and $\Delta\tilde{t}\to 0$. With the decrease
of $\lambda^{f}$, $\mathcal{L}_{c}$ decreases, while $\Delta\tilde{t}$
increases in the large-$\lambda^{f}$ case and decreases for a finite
$\lambda^{f}$. Finally, after the temporary localization process, the
Loschmidt echo evolution displays a $\lambda^{f}$-dependent damping to $0$,
which shows that the overlap of the final state and $|j_{0}\rangle$ tends to
$0$, and it corresponds to the emergence of a non-Hermitian jump process.
Figure 8(d) shows the log-log plot of
$1-\mathcal{L}(t;\lambda^{i},\lambda^{f})$ as the function of time $t$ for
$\lambda^{f}=1.5$. We can see that the Loschmidt echo presents similar
behaviors to the mean displacement $\sigma(t)$ in this limit, and the same
power-law indices are found in the corresponding evolution stages.

Figure 9: (Color online) Evolution of the Loschmidt echo with
$\lambda^{f}=1.5$. The initial states are chosen to be the given eigenstates
with the energies $E_{\tilde{n}}$ for (a) $\lambda^{i}=0.6$, (b)
$\lambda^{i}=1.0$, and (c) $\lambda^{i}=1.2$, respectively. The energies
$E_{\tilde{n}}$ have been ordered according to increasing values of the real
part. Here, $J=1$ and $L=377$.

The results of two limits suggest that the existence of zero points during the
Loschmidt echo evolution seems to no longer be a dynamical signature of the
localization-delocalization transition for the case of $\lambda^{f}>J$. To
clarify such a conjecture, we calculate the evolution of the Loschmidt echo
with $\lambda^{f}=1.5$ for different $\lambda^{i}$ and $\tilde{n}$, with the
real part of the energies in ascending order, which is shown in Fig. 9. When
$\lambda^{i}$ is localized in the extended ($\lambda^{i}=0.6$) or the
multifractal regime ($\lambda^{i}=1.0$), the evolution of the Loschmidt echo
exhibits a series of zero points for different $\lambda^{i}$ and $\tilde{n}$.
However, for the initial state being an eigenstate of $\hat{H}(\lambda^{i})$
with $\lambda^{i}>J$, e.g., $\lambda^{i}=1.2$ shown in Fig. 9(c), the
Loschmidt echo is strongly dependent on the initial state, and the conclusion
of the Hermitian case is no longer available for the non-Hermitian one in the
$\mathcal{PT}$ symmetry unbroken regime.

##  V Conclusion

In this paper, we study the dynamics evolution of the non-Hermitian AA model
with $\mathcal{PT}$ symmetry. The $\mathcal{PT}$ symmetric AA model exhibits a
$\mathcal{PT}$ symmetry broken point at $\lambda=J$. When $\lambda<J$, all the
eigenenergies are real and the corresponding eigenstates are extended, while
for $\lambda>J$, the complex energies emerges and the corresponding states are
localized. The states at the transition point are multifractal. We can apply
the evolution of the profile of the wave function and the long-time survival
probability to distinguish the localization properties of the system. The
evolution of the mean displacement displays distinctive behaviors in different
$\mathcal{PT}$ symmetric regimes, which is also available for the evolution of
the Loschmidt echo. According to our calculation, when the post-quench
parameter is localized in the $\mathcal{PT}$ symmetry unbroken regime, the
behaviors of the Loschmidt echo are similar to the Hermitian cases in that the
dynamical signature of the localization-delocalization transition can be
characterized by the emergence of a series of zero points in the evolution of
the Loschmidt echo. However, when the post-quench parameter is localized in
the $\mathcal{PT}$ symmetry broken regime, the dynamical detection of the
transition point by the Loschmidt echo method is unavailable. Similar
conclusions for the Loschmidt echo in some other $\mathcal{PT}$ symmetric
systems without disorder have been reported Znojil ; Krejcirik ; KWang . Our
results can be easily examined in the silicon waveguide platform, cold atomic
gases, and ion chains Bylinskii1 ; Bylinskii2 ; Bonetti ; Benassi ; Mandelli ;
Kiethe .

###### Acknowledgements.

Z. Xu is supported by the NSFC (Grants No. 11604188 and No. 12047571), Beijing
National Laboratory for Condensed Matter Physics, and STIP of Higher Education
Institutions in Shanxi under Grant No. 2019L0097. S. Chen is supported by the
National Key Research and Development Program of China (2016YFA0300600 and
2016YFA0302104), NSFC under Grants No.11974413, and the Strategic Priority
Research Program of Chinese Academy of Sciences under Grant No. XDB33000000.
This work is also supported by NSF for Shanxi Province Grant No.1331KSC.

## Appendix A: Effects of $J$ with fixed $\lambda$

Figure 10: (Color online) Time evolution of the mean displacement $\sigma(t)$
with the hopping strength (a) $J=1.5$, (b) $J=1.0$, and (c) $J=0.5$,
respectively. Here, $\lambda=1$ and $L=377$.

In this appendix, we consider $\lambda$ as the unit energy, and the evolution
of the mean displacement $\sigma(t)$ with different strengths of the hopping
term is shown in Fig. 10. In this case, when $J>1$, the system is localized in
the extended regime corresponding to a ballistic diffusion with
$\tilde{\gamma}=1$ seen in Fig. 10(a). For the multifractal case [Fig. 10(b)],
which is also studied in the main text, it displays the diffusion exponent
$\tilde{\gamma}=1$. When the system is in the localization regime [Fig. 10(c)
with $J=0.5$], the similar structure of $\sigma(t)$ is detected as shown in
main text with $J=1$ and $\lambda=1.5$, though the diffusion exponent
$\tilde{\gamma}\approx 3$ in the hyperdiffusion stage. For the case in the
$\mathcal{PT}$ symmetry broken regime, $\tilde{\gamma}$ is dependent on
$\lambda/J$ for a hyperdiffusion.

## Appendix B: Chain’s length deviating from a Fibonacci number

Figure 11: (Color online) Time evolution of the mean displacement $\sigma(t)$
with different $L$ and the non-Hermitian modulation strength (a)
$\lambda=0.6$, (b) $\lambda=1.0$, and (c) $\lambda=1.5$, respectively. Here,
$J=1$. Figure 12: (Color online) Evolutions of the profile of the wave
function $\rho_{j}(t)$ with the non-Hermitian modulation amplitude (a)
$\lambda=0.6$ and (b) $\lambda=1.0$, respectively. Here, $J=1$ and $L=501$.

In this appendix, we consider the dynamics of the non-Hermitian AA model
described by the Hamiltonian (1) and (2) in the main text with the chain’s
length deviating from a Fibonacci number. This is another useful direction to
study the transport in quasi-periodic potentials Znidaric1 ; Varma ; Znidaric2
; Purkayastha ; Znidaric3 ; Purkayastha1 . Figure 11 shows the time evolution
of the mean displacement $\sigma(t)$ with the chain’s length $L=501$ and
$1201$. The initial wave packet is localized at the center of the lattice. As
shown in Figs. 11 (a) and 11(b) with $\lambda=0.6$ and $1.0$, respectively, we
can find that before the wave packet spreading near the boundaries,
$\sigma(t)\propto t$. When the evolution of the wave packet is near the
boundaries, the mean displacement brings about a sudden rise to a finite value
in a short time. To intuitively see the phenomenon displaying in the extended
and multifractal regimes, we show the evolution of the profile of the wave
packet $\rho_{j}(t)$ in Fig. 12. The evolution of the packet in both regimes
exhibits similar features to the chain’s length being a Fibonacci number
before the wave packet reaching the boundaries. Then the wave packet is frozen
at the boundaries corresponding to a stationary value of $\sigma(t)$. When the
localized wave packet spreads in the $\mathcal{PT}$ symmetry broken regime,
$\sigma(t)$ exhibits an $L$-dependent feature, which is shown in Fig. 11(c).

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#  Gender and the influence of research environment in topic selection of
early-career faculty in STEM

Lluís Danús1,2, Robert H. Davis3,
Roger Guimerà4,1,∗, Marta Sales-Pardo1,∗
1Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona,
Catalonia
2Annenberg School for Communication, University of Pennsylvania, Philadelphia,
United States
3 Department of Chemical and Biological Engineering, University of Colorado,
Boulder, Colorado, United States
4ICREA, Barcelona, Catalonia
∗To whom correspondence should be addressed;
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
> We study the influence that research environments have in shaping careers of
> early-career faculty in terms of their research portfolio. We find that
> departments exert an attractive force over early-career newcomer faculty,
> who after their incorporation increase their within-department
> collaborations, and work on topics closer to those of incumbent faculty.
> However, these collaborations are not gender blind: Newcomers collaborate
> less than expected with female senior incumbents. The analysis of
> departments grouped by fraction of female incumbents reveals that female
> newcomers in departments with above the median fractions of female
> incumbents tend to select research topics farther from their department than
> female newcomers in the remaining departments —a difference we do not
> observe for male newcomers. Our results suggest a relationship between the
> collaboration deficit with female incumbents and the selection of research
> topics of female early-faculty, thus highlighting the importance of studying
> research environments to fully understand gender differences in academia.

## Introduction

The topics of scientific research are constantly evolving (?): New topics gain
attention while others languish in response to external pressures, including
funding frameworks and societal needs (?, ?). For individual researchers, the
selection of a portfolio of research topics is a critical decision that has
direct impact in the evolution of their scientific careers. This is specially
true for early-career scientists and young faculty, whose future professional
stability hinges upon their early choices (?). Despite the importance of such
early choices, we still know very little about which are the factors that
affect the selection of research topics by individuals beyond global trends.

Changing and expanding the research portfolio is a common trait of scientific
careers of researchers in STEM fields (?). However, when selecting possible
research topics, individuals need to consider the trade-off between innovation
and exploitation of exiting topics: While the former potentially has high
rewards but implies high risk, the latter offers milder recognition but also
involves lower risk (?). Another factor that plays a role in shaping
scientific careers is establishing collaborations: Collaboration with top-tier
institutions and with prominent researchers increases impact and can help in
career promotion (?, ?). However, because collaborative team sizes are largely
driven by the amount of resources available, putting an emphasis on
collaboration can lead to gender segregation in different fields of study
within the same area (?, ?). Indeed, gender is another factor that has a
strong impact in the scientific career of individuals. Leaving leaky pipelines
apart (?), female faculty often publish fewer articles and receive less grant
money (?, ?, ?), are promoted at a later career stage than their males
colleagues (?), are more likely to experience issues when discussing
authorship (?), and are given less credit for their contributions (?).

Unfortunately, studies looking at the evolution of scientific careers rarely
consider the host institution or department as one of the factors playing an
important role in the development of early-career faculty. Although formal
collaborations, like coauthoring a publication, are the most recognizable way
through which researchers share their ideas with each other, departments are
collections of people who are exposed to similar scientific influences by, for
example, attending the same seminars and informal meetings. Indeed,
researchers within the same institution are more likely to collaborate (?).
These collaborations are typically face-to-face, resulting in lower
communication costs (?) and having a higher chance to spark creativity (?).
However, the effect that research environments have in shaping research
careers and whether this effect has gender disparities have not been assessed.

Figure 1: Study design. We use three different types of datasets: (i)
“background datatsets,” which comprise all the abstracts of articles published
in generalist journals in each of the fields of study (Materials and Methods);
(ii) a “hiring dataset” that lists researcher who were offered a position at
three institutions; and (iii) “gender datasets” for each field, which comprise
the starting year and gender of all the faculty currently affiliated at
selected highly ranked departments within the five fields of study (Materials
and Methods) (?). First, for each of the researchers in the dataset, we look
at the collaboration network with incumbent department members before and
after the offer/starting year. We then compare these collaborations between
accepting and declining researchers (with offers from the same departments)
and between male and female faculty over time. Second, we build a bipartite
network of documents and words for each background dataset and obtain research
topics for each field. Using the list of publications of each researcher, we
obtain the projection of each researcher onto topic space. This process allows
us to make comparisons between researchers at different stages in their
careers.

Here, we aim precisely to cover this gap (Fig. 1). Our assumption is that
departments and research institutions expose researchers to certain research
questions and approaches, and they become incubators for novel ideas through
collaboration among faculty members. To investigate this, we consider two
cohorts of early-career researchers. The first cohort comprises young
researchers who were offered an assistant professorship in a chemical
engineering department; some of them declined and some accepted. The second
cohort comprises early-career faculty in top departments in Europe and the
United States in the fields of chemical engineering, biology, physics,
mechanical engineering, and civil engineering. The analysis of the first
cohort confirms that researchers who accepted an assistant professorship
develop more collaborations with incumbent faculty in the department than
those candidates who did not accept the offer, and it shows that the
researchers who accepted offers are also more likely to converge towards the
research topics of the department, even when discarding papers co-authored
with other faculty in the department. The analysis of the second cohort
reveals that, except in physics, there is a systematic collaboration deficit
between newcomer faculty (both male and female) and female senior incumbent
faculty. We also find a striking difference between early-career male and
female newcomer faculty across fields with the aforementioned collaboration
deficit—female newcomer faculty in departments with below-median fractions of
female incumbent faculty tend to converge less to department research topics
than do female newcomers in departments with above-median fractions of female
incumbent faculty. However, we see no such differences in the selection of
research topics for male newcomers.

Our findings show that, while departments exert an attractive force on early-
career faculty, this force has different effects on male and female newcomer
faculty. While more work still needs to be done to elucidate the reasons
behind such disparities, our findings highlight the importance of studying
research environments to fully understand gender differences in academia.

## Results

### Effects of joining a department on collaborations and research topics

We start by quantifying the effect of joining a department on the
collaborations and research topics of early-career faculty. To this end, we
analyze the hiring history of three different chemical engineering departments
in the United States between 2007 and 2017 (Fig. 1). Each of the three
datasets contains a list of young researchers who were offered an assistant
professor position in one of the departments and their response (accepted or
declined), as well as the publications of those researchers and of all other
faculty in the department, before and after the hiring offer. The candidates
who accepted the offer give us the opportunity to analyze the effect of
joining the department, in terms of both their number of within-department
collaborators and their research topics. The researchers who declined allow us
to control for potential confounding effects, such as field-wise shifts in
research topics.

Figure 2: Changes in departmental collaborations and research topics after
joining a department. (A) Accumulated within-department distinct collaborators
over time with respect to the year the offer to join the department was made
for researchers who accepted (green) and who declined (red). The shadowed area
corresponds to the 95% confidence interval. (B) Comparison of topic
distributions before (dotted lines) and after (solid lines) the year that the
offer to join the department was made for researchers who accepted and for
those who declined. The gray solid area corresponds to the background of the
department at the time of the offer. All distributions are normalized with
respect to the global distribution of the background dataset. The black dashed
line represents the same topic proportion as that of the background. (C)
Fraction of researchers whose topic distribution after the offer was made
converges to that of the department – researchers who declined the offer, red;
researchers who accepted, green. Stars indicate statistical significance
obtained from randomizing the accepted and declined labels in our dataset
(***: 1%, **: 5%, *: 10%, n.s.: not significant). Note that in the analysis we
exclude papers that are published in collaboration with department faculty,
and, therefore, the effects we report are not a direct consequence of those
collaborations.

As expected, the decision to join a department affects the number of
collaborations of a researcher with members of that department. Figure 2A
shows that, for the three departments considered here, a new faculty member
joining the department increases the number of new collaborators within the
department to an average of 5 collaborators after 10 years. By contrast,
researchers who declined the offers barely collaborate with members of the
department after declining. We surmise that these new collaborations, as well
as other forms of scientific socialization within the department, bring the
early-career newcomer in contact with new methodologies, research questions,
and ideas. To analyze the influence of the department on the research
interests of the newcomers, we analyze the change in research topics by means
of topic modeling of article abstracts (see Methods), before and after the
offer to join the department. In particular, we compare the distribution over
topics of each researcher with that of the department (Fig. 2B). We observe
that, before joining a department, the early-career faculty who accepted and
those who declined were similarly close to the topic distribution of the
department (Fig. S1). However, after joining a department, newcomers tend to
shift away from topics that are not popular in the department and towards
others that are more prominent, even when we exclude direct collaborations
(that is, direct co-authorship) with other department members. This result is
in contrast to those who declined, who are less likely to converge towards the
topics that are prominent in the department.

To quantify this effect, we compare the distribution of research topics of
researchers and departments using information-theoretic metrics. In
particular, we compute the Jensen-Shannon distance (see Methods) between the
topic distributions of the early-career faculty (before and after the offer,
accepted or declined) and that of the department (that is, of incumbent
faculty before the offer). We exclude from this analysis all publications that
involve co-authorship between the early-career faculty and incumbent faculty
in the department; therefore, the convergence in topic distribution we observe
is not an immediate effect of those collaborations. Researchers converge
towards the department if the change in distance with respect to the
department is negative (they become closer after acceptance or declination)
and diverge from the department if the change is positive. We find that the
fraction of accepting early-career faculty who converge towards the department
is 60% (Fig. 2C), while the fraction of declining early-career faculty that
converge towards the department is only 32%, both significant at the 5%
confidence level when compared to the null hypothesis of researchers randomly
moving towards or away from the department and therefore converging or
diverging with equal probability. The difference between the two fractions is
also significant.

### Departmental influence on male and female early-career researchers

We have established that joining a department has a statistically significant
effect on both the collaborations and the research topics of early-career
faculty. However, Fig. 2C also reveals that 40% of the newcomer faculty who
joined one of the three departments considered in the first cohort did not
converge towards it, suggesting that the influence of the department is not
equal for everyone. Given that studies in the literature have shown stark
differences in gender when it comes to scientific output, attribution and
authorship (?, ?, ?, ?), we further surmise that the research environment can
also have a different effect depending on gender.

To test this hypothesis, we analyze data from the most prominent departments
in North America and Europe (according to the QS World University Rankings
(?)) in five different fields: chemical engineering, physics, biology,
mechanical engineering, and civil engineering (Fig. 1; Data; Supplementary
Tables S1–5). For each one of these departments, we identified the tenure-
track researchers (assistant, associate and full professors) and collected
their publication and career data. With this information, we construct
faculty-level collaboration networks and compute the topic distribution of
each researcher. These data, together with information about gender and the
year in which each researcher joined the department, allow us to compare the
effects of the department on the newcomer faculty. Since we can expect more-
established researchers to be less permeable to new research topics, even when
joining a new department, we focus on researchers who had less than 30 papers
published at the time of joining. Other than this restriction, we consider as
newcomers all faculty who joined one of the departments considered after the
year 2000, and as incumbents those faulty having joined before.

Figure 3: Overall absence of significant gender differences in within-
department collaborations after joining a chemical engineering department. (A)
Accumulated within-department distinct collaborators with respect to the year
of incorporation to the department for male (orange) and female (purple)
researchers. The shadowed area corresponds to the 95% confidence interval. (B)
Fraction of publications that are with departmental collaborators. (C)
Accumulated number of publications. (D) Fraction of authors with no
collaborators within the department. The same analyses for the fields on
biology, physics, mechanical and civil engineering can be found in
Supplementary Figs. S2-5.

After joining a department, male and female researchers do not display
significant differences across fields in the overall number of departmental
collaborators or in the fraction of their publications that involve
collaborations with other faculty in the department (Fig. 3A-B for chemical
engineering and Supplementary Figs. S2-5 A-B for other fields). We also find
that, while female newcomers tend to have a smaller output than male
newcomers, differences are not significant in the majority of fields we study
(except for biology). There are also no systematic differences in terms of the
fraction of researchers who have no within-department collaborations (Fig. 3
C-D; Supplementary Figs. S2–5 C-D).

#### Gender differences in the selection of collaborations

Figure 4: The gendered nature of within-department collaborations in chemical
engineering departments. Cumulative fraction of within-department
collaborators by gender and career stage (purple represents females and orange
represents males). The shadowed area represents the 95% confidence interval.
The gray dashed line represents the expected fraction of collaborations if we
randomize female and male labels in each department. (A) Male and female
newcomers with male incumbents. (B) Male newcomers with (junior and senior)
female incumbents. (C) Female newcomers with (junior and senior) female
incumbents. (D) Male and female newcomers with senior male incumbents. (E)
Male newcomers with senior female incumbents. (F) Female newcomers with senior
female incumbents. (G) Male and female newcomers with junior male incumbents.
(H) Male newcomers with junior male incumbents. (I) Female newcomers with
junior female incumbents. For all other fields, see Supplementary Figs. S6-S9.

Despite the absence of differences described above for the overall number of
collaborators of early-faculty within their new departments, we next show that
there exist differences in the way male and female newcomers’ collaborations
are distributed among junior and senior, male and female incumbents. We focus
on this distinction because senior incumbent faculty are in general more
influential within departments, so that early-career faculty might favor
collaboration with and be more affected by senior professors within the
department. Since women are typically underrepresented in academic senior
positions (?), we surmise that there might be effects mediated by the
interaction of gender and seniority.

Figure 4A-C represents, for chemical engineering (see Supplementary Figs.
S6-9A-C for other fields), the cumulative fraction of within-department
collaborators (that is, the number of collaborators of the early-career
faculty members divided by the number of collaborators available), separated
by the gender of the new faculty member and the gender of the incumbent
collaborator, but ignoring the seniority of the incumbent. Female and male
early-career faculty have indistinguishable fractions of male incumbent
collaborators, and neither group deviates from the null expectation of
selecting collaborators without taking gender into account (Fig. 4A;
Supplementary Figs. S6-9A). For chemical engineering, we observe that, as time
goes by, both male and female newcomer researchers seem to collaborate with a
lower fraction of female incumbent faculty than expected (Fig. 4B-C). However,
this observation is not consistent across fields (Supplementary Figs.
S6-9B-C).

Next, we introduce seniority into the analysis. In particular, we examine the
effect of gender and seniority in collaborations by separating collaborations
with senior incumbents (those who have a career at least 10 years longer than
the early-career faculty member) and junior incumbents (those who have a
career at most 10 years longer than the early-career faculty member). We find
that, despite there being no systematic differences in overall collaboration
rates between newcomers and incumbents disaggregated by gender, there are
systematic gender differences across fields in the way newcomers distribute
collaborations between senior and junior incumbents (Fig. 4D-I and Fig. 5).
Specifically, we find that as time goes by both male and female newcomers have
a clear, systematic tendency to collaborate with a lower fraction of senior
female incumbents than expected Fig. 5. There is only one exception to this
rule—in physics departments, female newcomers collaborate with the expected
fraction of female senior faculty. The reason why this deficit in
collaboration with senior female incumbents is not always discernible in
overall (senior and junior) collaboration rates with female incumbents is
that, in some fields, newcomers collaborate with a slightly larger fraction of
female junior incumbents than expected.

Figure 5: The collaboration deficit with senior female incumbents. Cumulative
fraction of within-department collaborators of senior female incumbent faculty
with newcomer faculty separated by gender (purple: females; orange: males).
The shaded area represents the 95% confidence interval. The dashed line
represents the expected fraction of collaborations if we randomize female and
male labels in each department. (A, F) Chemical engineering. (B, G) Physics.
(C, H) Biology. (D, I) Mechanical engineering. (E, J) Civil engineering.

By contrast, male and female newcomers collaborate with the expected fraction
of male senior incumbents (except in biology in which all newcomers
collaborate with a lower fraction than expected of senior researchers
regardless of gender) (Fig. 4D; Supplementary Figs. S6-9). We do not find any
collaboration deficit with junior incumbents, either (Fig. 4G-H; Supplementary
Figs. S6-9G-H).

#### Gender and convergence to department topics

So far, our results show that there is a systematic gender bias in the way
female and male newcomers establish collaborations with senior faculty within
their departments (with the only exception of female newcomers in physics). We
posit that the existence of such a gender bias could also imply that there is
a gendered difference in the way departments exert influence over newcomer
faculty. Indeed, senior faculty are expected to be the most influential among
incumbent faculty, as they can offer more experienced counsel and guidance to
young researchers. At the same time, incumbents of the same gender as the
newcomers may exert more influence than incumbents of the opposite gender (?,
?, ?). In that case, we would expect that a deficit of collaborations with
senior incumbent females may result in a loss of influence exerted by the
department on newcomer females, a loss that would increase in departments with
a larger fraction of incumbent females, where the deficit is felt more
strongly.

Figure 6: The fraction of incumbent females in a department is related to
differences in the selection of research topics by female newcomers. Mean
Jensen-Shannon distance difference between newcomer and department research
topics for newcomers before and 5 years after joining a department. For each
field, we divide departments into two groups according to the fraction of
female incumbent faculty (above and below median, respectively; see legend).
Each bar represents the mean over all departments in a given field and group
(A and C), or in the aggregate for all fields (B and D). The upper row
illustrates the results for female newcomers, while the bottom row presents
data for male newcomers. Error bars represent the standard error. Stars
indicate statistical significance of the difference of means between the two
groups obtained by randomizing the department membership to the above and
below median groups (***: 1%, **: 5%, *: 10%, n.s.: not significant). Note
that, in the analysis, we exclude papers that are in collaboration with
department faculty, and, therefore, the effects we report are not a direct
consequence of those collaborations (Methods).

To test this scenario in all fields, we separate departments into two groups
accorting to whether they have above-median or below-median fractions of
female incumbent faculty. Then, we look at the difference in distance between
the research topics of newcomers and their department at the time of joining
and five years after having joined the department. By separating newcomers by
gender, we observe clear differences that are in agreement with our above
scenario. In departments with an above-median fraction of incumbent females,
newcomer females tend to converge less to the department topics than those in
departments with a below-median fraction of incumbent female researchers
(Fig.6A). This indicates that in departments with higher fraction of females
the influence of the department on female newcomers is weaker. This
observation holds for all the fields we study except for physics; there,
newcomer females joining departments with above-median fraction of female
incumbent researchers become closer to the department. This result is also
consistent with our scenario—since female newcomers in physics do not have a
collaboration deficit with incumbent females (Fig. 5G), there is no loss in
the influence of female incumbents on female newcomers therefore. Then, the
higher the fraction of females, the more female newcomers would be expected to
converge towards the topics of the department.

Note that, due to sample size, the statistical significance for each field is
moderate, but becomes highly significant if we pool all fields together (Fig.
6B). By contrast, and also consistent with our scenario, we do not observe any
significant differences across fields with the convergence towards the
research topics of the department for male newcomer researchers, neither at an
individual nor at an aggregate level (Fig. 6C,D).

This difference between female and male newcomers is especially striking if we
note that, at the time of joining the department, female newcomers are not
further from the department in topic space than male newcomers are (Fig. S10).
Our analysis shows that, while we can observe little overall differences in
the way within-department collaborations evolve for female and male newcomers,
the nuance of how these collaborations are distributed affect how their
research topics evolve differently after joining.

## Discussion

Our results show that working environments, and in particular hiring
departments, exert a subtle yet quantifiable attractive force in the way young
faculty shape their portfolio of scientific topics. While constant contact
with department colleagues can lead to collaborations that affect the topic-
selection process, the convergence towards department topics is apparent even
when we do not take direct collaborations into account. This finding
unmistakably points to the fact that the local research environment permeates
into the shaping of academic careers beyond strict paper co-authorship. This
finding is in some ways surprising, because departments typically do not
actively seek to affect researchers portfolios (quite on the contrary, top
departments would likely expect their researchers to pursue independent,
fruitful research careers), but they inevitably do have an impact on topic
choices.

Our results also show that female and male young faculty do not respond to
departmental pressures in the same way. While women seem to have the same
overall collaboration patterns as men, the deficit in collaborations with
senior female faculty does not seem to have the same impact on both—in
departments with a larger fraction of incumbent females, female newcomers (in
fields with a deficit in collaborations with senior female incumbents)
converge less to department topics, whereas the convergence of newcomer males
in the same situation is independent of the fraction of incumbent women. These
rather stark differences are both unexpected and worrisome, and they could be
explained by the fact that the influence of senior incumbent on newcomer
faculty has a gender homophily dimension that only has discernible effects on
women. Most broadly, our results point toward yet another gender difference in
academia with unknown causes and unknown consequences down the road.

In recent years, we have witnessed structural changes in many departments (?,
?), with an increase in the fraction of female faculty, and yet large gender
differences still exist in terms of promotion and credit—female researchers
are typically given less credit for their contributions to research (?), and
female faculty typically get promoted at later stages in their careers (?).
These observations could suggest that women might look for different topic
portfolios and different collaboration strategies as mechanisms to receive
credit that could lead to future promotion, a mechanism that male faculty
might feel less pressured to use.

Studies from female academics in the 1990s already discussed that young women
choosing the academic track did not necessarily find more support in
departments with a larger presence of women faculty, and that some women felt
the need to stand out on their own rather than clustering with other female
faculty (?, ?). It is unclear if female faculty currently feel the same way as
in the 1990’s and whether or not there are idiosyncratic differences that
could explain the deficits in collaboration we observe. The obvious questions
moving forward are therefore what are the reasons for these collaboration
deficits and what are the consequences of the differences in research topic
selection that we observe? As many hiring departments make efforts and put
policies in place to help reduce the gender gap in STEM departments, we cannot
forget that hiring young female academics is not the endpoint but the start to
close the gender gap in academia. The department is the research environment
where young faculty should flourish, but right now this environment is not
gender blind.

## Materials and methods

### Data collection

##### Corpus of articles used to obtain the background of research topics in
chemical engineering

We collected the abstracts of all the 36,093 articles published in the _AIChE
Journal (AIChE)_ and the _Chemical Engineering Journal (CHEJ)_ since their
inception until 2021, available on Scopus. We selected these two journals
because they are the most representative, broad-scope chemical engineering
journals of North America and Europe, respectively.

##### Corpus of articles used to obtain the background of research topics in
physics

We collected the abstracts of all the 120,990 articles published in _Physical
Review Letters (PRL)_ since its inception until 2021, available on Scopus.

##### Corpus of articles used to obtain the background of research topics in
biology

We collected the abstracts of all the 19,006 articles published in _Cell_
since its inception until 2021, available on Scopus.

##### Corpus of articles used to obtain the background of research topics in
mechanical engineering

We collected the abstracts of all the 45,210 articles published in the
_International Journal of Mechanical Sciences (IJMS)_ , _International Journal
of Machine Tools and Manufacture (IJMTM)_ , _Journal of Sound and Vibration
(JSV)_ , _International Journal of Plasticity (IJP)_ , and in the _Mechanical
Systems and Signal Processing (MSSP)_ since their inception until 2021,
available on Scopus.

##### Corpus of articles used to obtain the background of research topics in
civil engineering

We collected the abstracts of all the 41,481 articles published in the
_Building and Environment (BE)_ , _Coastal engineering (CE)_ , _Energy and
Buildings (EB)_ , _Engineering Structures (ES)_ , and the _Journal of
Construction Engineering and Management (JCM)_ since their inception until
2021, available on Scopus.

##### Hiring data

Data about faculty hires were collected by R.H.D. and comprise the offers made
to junior candidates by three different chemical engineering faculties in the
United States. These data comprise more than 120 offers to researchers
suitable to cover the position and the response to that offer. For each one of
the candidates, we collected their full list of publications and retrieved the
abstracts of these publications from Scopus. We also obtained faulty rosters
of the three hiring departments at the moment the offer was made and retrieved
full lists of publications and abstracts for these researchers from Scopus.

##### Faculty gender data

We first collected the list of the top 50 institutions across all fields from
_QS World University Rankings 2021_ (?) and selected those 136 departments
based either in North America or Europe (Supplementary Tables S1-S5) for which
an academic staff webpage was available. Then, for a total of 6,221 faculty in
these departments, we downloaded their publication data, including abstracts
and assigned them a binary gender based on their name.

### Construction of the space of research topics

##### Stopword removal

As a first step, we pre-processed the abstracts to remove stopwords, those
words such as prepositions or articles whose removal does not affect the
meaning of a text. Instead of using a bag-of-words methodology, we followed
the information-theoretical approach developed by (?), which allows us to
identify stopwords in a context-dependent manner. Specifically, this
methodology computes how informative a word is in the whole corpus by
comparing the mutual information of the observed distribution of the
occurrences of that word within the corpus to that of a random distribution of
word occurrences. This approach allows us to define a threshold beyond which
words are not considered informative. To better adjust this threshold, we ran
iteratively our topic modeling under different values for informative words
until groups composed by a majority of general words such as _was, been,
then…_ vanished and all the words under this threshold for information content
were removed, leading to a final corpus of 54 words per document on average.

##### hSBM topic modeling

With the background corpus of abstracts, we constructed a bipartite network
with two types of nodes: words and documents. To obtain groups of documents
and of words (research topics), we fit the network to a Stochastic Block Model
(SBM). SBMs are a class of generative models for networks (?, ?, ?). In SBMs,
nodes are assumed to belong to groups, and node-to-node connectivities are
defined by the group memberships of the nodes alone. In particular, if nodes
$i$ and $j$ belong to groups $g_{i}$ and $g_{j}$, respectively, then the
probability that they are connected is given by a fixed quantity
$p_{g_{i}g_{j}}$, which is identical for all other pairs in $g_{i}$ and
$g_{j}$. The degree-corrected stochastic block model (SBM-DC) (?) is a variant
of the SBM that allows for each node to have a different propensity to create
links with other nodes, thus allowing nodes in the same group to have broad
degree (connectivity) distributions despite having the same connectivity
patterns.

Because group memberships are typically unknown, it is necessary to infer them
from the observed connections in a given network. The most plausible partition
of the nodes into groups (the partition that maximizes the Bayesian posterior
over partitions (?)) is also the one with the minimum description length
(MDL), that is, the one that most compresses the observed connections (?). To
obtain the MDL partition, one needs to specify a prior distribution over
partitions. A hierarchical prior, which assumes that there is a nested
hierarchy of groups of nodes, tends to yield shorter descriptions lengths than
more uninformative priors. The SBM with such hierarchical priors is often
referred to as the hierarchical SBM (hSBM) (?).

Approaches to topic modeling using SBMs overcome traditional topic modeling
techniques such as LDA in recovering the text structure, as shown in (?). To
compute the hSBM of the word-document network, we used the implementation in
graph-tool Python module.

##### Background research topics

From the fit of the bipartite network of words and documents (abstracts)
obtained from the background corpus to an hSBM we obtain a list of topics,
that is a list of groups of words that are statistically equivalent in the way
they are used in the documents. Each field required different levels in the
hierarchy to obtain a good coarse-grained representation of the content of
abstracts, resulting in an average of 20 topics per field.

##### Individual and departmental topic distributions

To build the topic distribution of an individual during a certain period of
time, we consider all the pre-processed abstracts of articles authored by that
researcher in that period of time. Because each word is associated to a topic,
we can then construct a frequency vector of use of the $\mathcal{N}$ topics in
the set of abstracts, which we call the topic distribution of an individual
researcher.

To build a department’s background topic distributions, we compute the
distributions of senior faculty (i.e. those faculty that were at the
department prior to 2000) according to the topics inferred by the hSBM. Once
we have the topic distributions of all researchers, we calculate the average
of all of them and use it as the average distribution before the beginning of
the study. The general topic distribution for all the departments and fields
under study is shown in (Fig. S11-15).

### Jensen-Shannon distance

To calculate the distance between two topic distributions _p_ and _q_ , we
compute the Jensen-Shannon distance, which is defined as

${\rm JSD}=\sqrt{\frac{{\rm KL}(p||m)+{\rm KL}(q||m)}{2}}\;,$ (1)

where _m_ is the pointwise mean of _p_ and _q_ and ${\rm KL}$ is the Kullback-
Leibler divergence between $p$ $({\rm or}\,q)$ and $m$. To prevent size
effects in the estimate of the ${\rm KL}$ (an issue common to the estimation
of information-theoretic metrics when the samples are small and some bins
(topics) have very few observations), we compare, a random sample of 10
abstracts from each researcher to a random sample of the same size from a pool
of papers by the authors of the department before the hiring process.

## Acknowledgments

This research was funded by project PID2022-142600NB-I00 from MCIN/
AEI/10.13039/501100011033 FEDER, UE and by the Government of Catalonia
(2021SGR-633).

### Authors contributions

LD and RD obtained data. RD, RG and MSP designed research. LD, RG, and MSP
analyzed data. All authors wrote the manuscript.

### Competing interests

The authors have no competing interests.

#### Data and materials availability

Publication data are available from Scopus. Topic distributions of authors and
departments needed to reproduce all results in the manuscript are available
upon request.

## Supplementary materials

Figs. S1 to S15
Tables S1 and S5

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