SI

# Impact of dephasing on non-equilibrium steady-state transport in fermionic
chains with long-range hopping

Subhajit Sarkar<EMAIL_ADDRESS>Institute of Theoretical Physics,
Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland.  Bijay Kumar
Agarwalla<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Science Education and Research Pune, Dr. Homi Bhabha Road, Ward No. 8, NCL
Colony, Pashan, Pune, Maharashtra 411008, India.  Devendra Singh Bhakuni
<EMAIL_ADDRESS>The Abdus Salam International Centre for Theoretical
Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy  Department of
Physics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel.

###### Abstract

Quantum transport in a non-equilibrium setting plays a fundamental role in
understanding the properties of systems ranging from quantum devices to
biological systems. Dephasing- a key aspect of out-of-equilibrium systems,
arises from the interactions with the noisy environment and can profoundly
modify transport features. Here, we investigate the impact of dephasing on the
non-equilibrium steady-state transport properties of non-interacting fermions
on a one-dimensional lattice with long-range hopping
($\sim\frac{1}{r^{\alpha}}$). We show the emergence of distinct transport
regimes as the long-range hopping parameter $\alpha$ is tuned. In the short-
range limit ($\alpha\gg 1$), transport is diffusive, while for the long-range
limit ($\alpha\sim\mathcal{O}(1)$), we observe a super-diffusive transport
regime. Using the numerical simulation of the Lindblad master equation, and
corroborated with the analysis of the current operator norm, we identify a
critical long-range hopping parameter, $\alpha_{c}\approx 1.5$, below which
super-diffusive transport becomes evident that quickly becomes independent of
the dephasing strength. Interstingly, within the super-diffusive regime, we
find a crossover from logarithmic to power-law system-size dependence in the
non-equilibrium steady-state resistance when $\alpha$ varies from $\alpha\leq
1$ to $\alpha\lesssim 1.5$. Our results, thus, elucidate the intricate balance
between dephasing and unitary dynamics, revealing novel steady-state transport
features.

##  I Introduction

Quantum transport of charge and energy in non-equilibrium settings stands as a
foundational cornerstone in modern physics. This includes phenomena such as
light-harvesting in photosynthesis and other chemical processes in both
chemical and biological systems [1, 2]. It further extends to the exploration
of emerging non-equilibrium states of matter in nano-fabricated quantum
devices [3, 4, 5, 6]. Specifically, charge transport measurements using
quantum dot arrays have led to a multitude of breakthroughs in quantum
physics. These encompass phenomena such as Coulomb blockade, Kondo physics,
superconductivity, quantum-point-contact universal conductance, and even
aspects of quantum computing [7, 8, 4, 9, 10].

The study of quantum transport often focuses on analyzing the behavior of the
mean squared displacement of a wave packet or observing the flow of energy and
charge through a system using a boundary drive [11, 12, 13, 14, 15, 16, 17].
For example, for systems displaying ballistic transport, the mean squared
displacement of a wave packet grows quadratic in time, while in diffusive
systems, its growth is linear in time. Alternatively, for the boundary-driven
systems, if the current in the non-equilibrium steady state (NESS) decreases
by the inverse of system size, i.e., $J_{\infty}(L)\sim L^{-1}$, or remains
constant as the system size increases, it indicates diffusive or ballistic
transport, respectively. The subscript ‘$\infty$’ in $J_{\infty}(L)$ here
refers to the current evaluated in the steady state. Any deviation from these
scaling of mean squared displacement or steady state current signifies
anomalous transport [18, 19, 20, 21].

Building on this, dephasing, introduced via coupling to an environment,
profoundly impacts the nature of quantum transport in lattice systems [15, 16,
17, 22, 23]. In particular, a Zeno-type dephasing characterized by classical
noise [24] can give rise to a phenomenon known as environment-assisted quantum
transport [1, 25, 26, 27]. Additionally, studies employing dynamical measures
have demonstrated that dephasing can cause anomalous transport in localized
systems, manifesting as sub-diffusive, diffusive, or logarithmic behavior
based on the degree of noise sparsity [28, 22, 23]. In relation to a boundary-
driven system in the nearest-neighbor chain of free fermions, recent
investigations point to a crossover from ballistic to diffusive transport by
varying dephasing strength [29, 30].

While most research has focused on systems with short-range hopping, how
transport properties of systems exhibiting hopping beyond nearest-neighbor
changes in the presence of dephasing remains an open question. These long-
range systems, characterized by interactions/hopping that decay as
$r^{-\alpha}$ (where $r$ is the distance between two sites), are widely
observed in nature and can also be engineered in cold-atom and trapped ionic
systems [31, 32, 33, 34, 35]. The presence of long-range hopping leads to
qualitative changes in various physical properties, including the equilibrium
phase, ground state, and dynamic properties [36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46]. For $\alpha>d$, with $d$ being the dimension of the system, the
physics often remains qualitatively the same compared to the short-range
systems. However, the opposite limit of $\alpha<d$ can lead to exciting and
novel features such as logarithmic growth of entanglement [47, 48], heating
suppression [49], light-cone evolution [50, 51], and self-trapping [52].

In this work, we investigate the NESS transport properties of the long-range
systems coupled to dissipative particle injection-extraction baths with a rate
$\Gamma$ corresponding to an infinite bias condition and a dephasing
environment with a rate $\gamma$. Note that previous work focused on these
systems without dephasing, and a sub-diffusive scaling of conductance was
observed under proper tuning of chemical potential bias [53]. A finite bias
window also leads to a substantial non-Markovianity [54, 55], a situation that
needs further investigation.

Here, we analyze the Markovian Lindblad quantum master equation in the steady
state. Remarkably, we observe that above a critical value of the long-range
parameter, $\alpha_{c}\approx 1.5$, the NESS charge current exhibits a
diffusive scaling with the system size, i.e., $J_{\infty}\sim L^{-1}$.
However, beneath the critical parameter $\alpha_{c}$, we observe a captivating
super-diffusive transport regime. This regime displays a power-law
($J_{\infty}\sim L^{-\nu}$, with $\nu<1$) dependence on system size for
$1<\alpha<1.5$. For $\alpha\leq 1$, this super-diffusive transport regime
displays a logarithmic dependence of the current on the system size
($J_{\infty}\sim[\log(L)]^{-1}$), overall indicating an anomalous transport.
The emergence of distinctive transport regimes can be attributed to the
intriguing interplay between long-range hopping and dephasing, a detail we
will delve into in subsequent results sections. Building on these findings, we
also highlight the striking resemblance between the NESS current heat map in
the $\alpha-\gamma$ plane with the entanglement phase diagram of the
measurement-induced phase transition (MIPT) recently illustrated in long-range
hopping systems [56, 57, 58].

The organization of the paper is as follows: We introduce the model
Hamiltonian and provide a formulation of the characterization of the non-
equilibrium transport in Sec.II. We provide the central findings of our work
in Sec. III. Finally, in the last section, we summarize and discuss the
possible connection with MIPT.

##  II Model Hamiltonian and Formulation

We consider a linear chain of $L$ sites hosting non-interacting spinless
Fermions with long-range hoping, as shown in Fig. 1. The model Hamiltonian for
the chain is given by

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L-1}\left[\sum_{r}\frac{J}{r^{\alpha}}(\hat{c}_{j}^{\dagger}\hat{c}_{j+r}+\text{h.c.})\right]=\sum_{i,j=1}^{L-1}\sum_{r}\hat{c}_{i}^{\dagger}\mathbb{H}_{i,j}^{(r)}\hat{c}_{j}$
(1) $\displaystyle\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\
\mathbb{H}_{i,j}^{(r)}=\frac{J}{r^{\alpha}}(\delta_{i,j+r}+\delta_{i+r,j}),$

where $\frac{J}{r^{\alpha}}$ is the long-range hopping strength with the
exponent of the spatial dependence $\alpha$, $r=|i-j|$ being the distance
between lattice sites $i$ and $j$ . The operator norm of the long-range
hopping term is given by $L^{1-\alpha}$, and for $\alpha<1$, it becomes
dominant in the thermodynamic limit [59]. However, one can re-scale such a
term by an $\alpha-$dependent factor $\mathcal{N}_{\alpha}$ factor which is
$\mathcal{N}_{\infty}=2$ and $\mathcal{N}_{0}=L$ [60, 61, 62, 63], experiments
are indeed performed with a finite number of sites where this re-scaling does
not appear naturally [3, 31, 32, 64]. Therefore, we do not consider this re-
scaling in our work, following Ref. [49].

We couple the system to a particle injecting source and a particle extraction
sink to the left and right end, respectively, with a constant rate $\Gamma$.
Additionally, we add Zeno-type dephasing at each site with strength $\gamma$
providing an energy relaxation channel. The equation of motion of the density
matrix of the system is governed by the standard Lindblad quantum master
equation.

$\displaystyle\frac{\partial\rho_{t}}{\partial t}$ $\displaystyle=$
$\displaystyle-i[\hat{\mathcal{H}},\rho_{t}]+\mathcal{D}_{d}[\rho_{t}]+\mathcal{D}_{L}[\rho_{t}]+\mathcal{D}_{R}[\rho_{t}]$
$\displaystyle\mathcal{D}_{d}[\rho_{t}]$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}\sum_{j=1}^{L}\left(\hat{n}_{j}\rho_{t}\hat{n}_{j}-\frac{1}{2}\\{\hat{n}_{j},\rho_{t}\\}\right)$
$\displaystyle\mathcal{D}_{L}[\rho_{t}]$ $\displaystyle=$
$\displaystyle\frac{\Gamma}{2}\left(\hat{c}_{1}^{\dagger}\rho_{t}\hat{c}_{1}-\frac{1}{2}\\{\hat{c}_{1}\hat{c}_{1}^{\dagger},\rho_{t}\\}\right)$
$\displaystyle\mathcal{D}_{R}[\rho_{t}]$ $\displaystyle=$
$\displaystyle\frac{\Gamma}{2}\left(\hat{c}_{L}\rho_{t}\hat{c}_{L}^{\dagger}-\frac{1}{2}\\{\hat{c}_{L}^{\dagger}\hat{c}_{L},\rho_{t}\\}\right),$
(2)

where $\mathcal{D}_{d}[\rho_{t}]$, $\mathcal{D}_{L}[\rho_{t}]$ and
$\mathcal{D}_{R}[\rho_{t}]$ are the Lindblad dissipators correspond to the on-
site dephasing, left and right boundary drive, respectively, and $\hat{n}_{i}$
is the Fermion number operator at site $i$.

Figure 1: (Color online:) Schematic of our set-up: a non-interacting fermionic
lattice chain with long-range hopping (indicated by yellow solid lines) is
placed between two baths (electrodes kept at the infinite bias limit), ‘L’–
left and ‘R’–right injecting and extracting particles at the first and last
sites, respectively. At each site of the chain, an on-site dephasing is
applied that mimics the Zeno-type measurement of number density at that site,
indicated by arrows.

For the long-range Hamiltonian, the local current is defined from the particle
number conservation [27]:
$\displaystyle\frac{d\hat{n}_{j}}{dt}=\hat{J}_{j,in}-\hat{J}_{j,out}$. For the
site connected to the left-lead, the current into the first site is,
$\hat{J}_{1,in}=\mathcal{D}_{L}[\hat{n}_{1}]=\Gamma(1-\hat{n}_{1})$ and the
current out of the first site is,
$\hat{J}_{1,out}=i[\hat{\mathcal{H}},\hat{n}_{1}]$. Similarly, for the last
site connected to the right lead, the current into the last site is,
$\hat{J}_{L,in}=i[\hat{\mathcal{H}},\hat{n}_{L}]$ and the current out of the
last site is $\hat{J}_{L,out}=\mathcal{D}_{R}[\hat{n}_{L}]=\Gamma\hat{n}_{L}$.
For all other sites, i.e., $2\leq j\leq L-1$,
$\hat{J}_{j,in}-\hat{J}_{j,out}=i[\hat{\mathcal{H}},\hat{n}_{j}]$ which gives,

$\displaystyle\hat{J}_{j,in}=-i\sum_{r}\frac{J}{r^{\alpha}}\left(\hat{c}_{j}^{\dagger}\hat{c}_{j-r}-\hat{c}_{j-r}^{\dagger}\hat{c}_{j}\right),$
$\displaystyle\hat{J}_{j,out}=-i\sum_{r}\frac{J}{r^{\alpha}}\left(\hat{c}_{j+r}^{\dagger}\hat{c}_{j}-\hat{c}_{j}^{\dagger}\hat{c}_{j+r}\right).$
(3)

Note that the dephasing dissipator commutes with the local number operator
and, therefore, does not appear in the corresponding equation of motion.

Figure 2: Non-equilibrium steady-state density profile, viz.,
$\langle\hat{n}_{m}(\infty)\rangle$ with respect to the lattice site index,
and for a range of long-range hopping parameter $\alpha$. Upper (a) and lower
(b) panels correspond to two different dephasing strengths, $\gamma=0.05$ and
$\gamma=2.0$, respectively.

Our approach is based on analyzing the single particle correlation matrix
$\mathbb{C}$ with matrix elements
$C_{n,m}(t)=\text{Tr}[\rho(t)\hat{c}_{n}\hat{c}_{m}^{\dagger}]$, which for the
non-interacting systems can be computed more efficiently [65, 66, 67, 29].
Moreover, it allows calculating the non-equilibrium steady state density
profile and the current even for on-site dephasing cases. The equation of
motion for the correlation matrix is given by,

$\frac{\partial\mathbb{C}}{\partial
t}=-i[\mathbb{H},\mathbb{C}]-\\{\mathbb{D},\mathbb{C}\\}+\mathbb{P},$ (4)

where
$\displaystyle(\mathbb{D})_{m,k}=\frac{\delta_{m,k}}{2}\left(\gamma+\Gamma[\delta_{m,1}+\delta_{m,N}]\right)$,
and $\displaystyle(\mathbb{P})_{m,k}=\delta_{m,k}\left(\gamma
C_{m,n}(t)+\Gamma\delta_{m,N}\delta_{k,N}\right)$. Furthermore, (4) can be re-
written as $\displaystyle\frac{\partial\mathbb{C}}{\partial
t}=-i\mathbb{H}_{\text{eff}}\mathbb{C}+i\mathbb{C}\mathbb{H}_{\text{eff}}^{\dagger}+\mathbb{P}$,
where $\mathbb{H}_{\text{eff}}$ corresponds to the effective single-particle
non-Hermitian Hamiltonian,

$\displaystyle\mathcal{H}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{H}}-i\frac{\gamma}{2}\sum_{j=1}^{L}\hat{n}_{j}-i\frac{\Gamma}{2}\left([1-\hat{n}_{1}+\hat{n}_{L}]\right)$
(5) $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{L}\hat{c}_{i}^{\dagger}\mathbb{H}^{i,j}_{\text{eff}}\hat{c}_{j}.$

The spectral properties of this non-Hermitian Hamiltonian determine the
stationary values of the particle density and current [65, 66, 67, 29]. The
stationary state solution corresponds to $\frac{\partial\mathbb{C}}{\partial
t}=0$ is given by, $\displaystyle\mathbb{C}(\infty)=\int_{0}^{\infty}d\tau
e^{-i\mathbb{H}_{\text{eff}}\tau}\mathbb{P}(\infty)e^{i\mathbb{H}_{\text{eff}}^{\dagger}\tau}$.
Then the eigendecomposition of
$\mathbb{H}_{\text{eff}}=\sum_{p}\lambda_{p}|\phi_{R}(p)\rangle\langle\phi_{L}(p)|$
in terms of bi-orthogonal eigenvectors
$\langle\phi_{L}(p)|\phi_{R}(q)\rangle=\delta_{p,q}$, leads to the steady
state values of the matrix elements of $\mathbb{C}$ given by,

$\displaystyle C_{i,j}(\infty)$ $\displaystyle=$
$\displaystyle\Gamma\Theta_{i,j,L}+\gamma\sum_{k=1}^{L}\Theta_{i,j,k}C_{k,k}(\infty)$
$\displaystyle\Theta_{i,j,k}$ $\displaystyle=$
$\displaystyle-\sum_{p,q=1}^{L}\frac{\phi_{i}^{R}(p)[\phi_{l}^{L}(p)]^{*}[\phi_{j}^{R}(q)]^{*}\phi_{l}^{L}(q)}{i(\lambda_{p}-\lambda_{q}^{*})}.$
(6)

Using (II) one can obtain the NESS charge density and current using

$\displaystyle\langle\hat{n}_{m}(\infty)\rangle$ $\displaystyle=$
$\displaystyle 1-C_{m,m}(\infty),$ (7) $\displaystyle
J_{\infty}=\langle\hat{J}_{m}(\infty)\rangle$ $\displaystyle=$
$\displaystyle\sum_{r}\frac{2J}{r^{\alpha}}\Im[C_{m,m-r}(\infty)].$ (8)

In what follows, we will use current and resistance
($R_{\infty}=J_{\infty}^{-1}$) interchangeably in analyzing NESS transport and
show our numerical findings.

Figure 3: Non-equilibrium steady-state current $J_{\infty}$ corresponding to
several values of $\alpha$ as a function of dephasing strength $\gamma$ for
system size $L=1024$, in log-log scale. Black dashed lines correspond to
$J_{\infty}\sim\frac{1}{\gamma}$ scaling.  Figure 4: The heat map of current
$J_{\infty}=\langle\hat{J}_{L,out}\rangle$. x- and y-axis plot long-range
hopping exponent $\alpha$ and dephasing $\gamma$, and the color-bar represents
the magnitude of $J_{\infty}$ for three different length of the lattice.

##  III Results

We focus on the NESS density profile throughout the lattice chain and the
transport current from the right lead, viz.,
$J_{\infty}=\Gamma\langle\hat{n}_{L}(\infty)\rangle$. We study the system size
scaling of the NESS current to characterize the various transport regimes. We
fix the hopping amplitude $J=1$. The dephasing strength $\gamma$ and particle
injection-extraction rate $\Gamma$ are taken in the unit of $J$. In all the
following analysis, we consider the system-lead coupling to be $\Gamma=1$. It
is a rather benign parameter compared to $\gamma$ and $\alpha$ within our
infinite bias set-up as it only determines the overall magnitude of the
current.

NESS density profile and current: We first study the steady state density
profile of the system and plot $\langle\hat{n}_{m}(\infty)\rangle$ as a
function of the site index ‘$m$’ for a range of long-range parameter $\alpha$
(shown in the color bar) in Fig. 2. We consider the dephasing strengths
$\gamma=0.05$ [upper panel (a)] and $\gamma=2.0$ [lower panel (b)],
respectively. For $\alpha\neq 0$, and small dephasing strength $\gamma=0.05$,
the density profile shows a trend towards a linear profile for larger
$\alpha$-values starting from a non-linear profile at smaller $\alpha$-values.
Although, $\langle\hat{n}_{m}(\infty)\rangle$ shows the same trend towards a
linear profile for larger dephasing, e.g., $\gamma=2.0$ in Fig. 2,
interestingly, $\langle\hat{n}_{m}(\infty)\rangle$ is more flattened in the
bulk of the chain compared to the level of flatness seen for $\gamma=0.05$,
thereby suggesting an enhanced transport due to the dephasing. This is
opposite to the expectation, i.e., in the absence of dephasing, a chain with
nearest-neighbor hopping exhibits a flat density profile in bulk with $\langle
n_{m}(\infty)\rangle\approx 0.5$, a hallmark of ballistic transport and, for
stronger dephasing strength, the density profile scales as $\langle
n_{m}(\infty)\rangle\propto 1/L$ suggesting a suppression in transport
(diffusive transport). Such contrasting features hint towards the emergence of
an interesting anomalous transport regime in the presence of long-range
hopping.

To explore the regime of unusual transport, we study the variation of the NESS
current $J_{\infty}$ for different values of dephasing $\gamma$ and long-range
parameter $\alpha$ for a fixed system size $L=1024$. Fig. 3 plots $J_{\infty}$
as a function of dephasing strength $\gamma$. We observe that the current gets
suppressed with increased dephasing for all the values of $\alpha$. This is
clearly seen in Fig. 3 (black dashed lines) for $\alpha=1.5$ and higher, where
current decreases monotonically with $\gamma$, i.e.,
$J_{\infty}\propto\frac{1}{\gamma}$. For the short-range hopping model, in the
absence of dephasing, the transport is always ballistic. Finite dephasing is
detrimental to this ballistic transport and plays a role in in-elastic
scattering [29]. On the other side, for $\alpha\ll 1$, current initially
decreases with increasing dephasing and, interestingly, settles down to a
plateau regime for a considerable range $10^{-2}\lesssim\gamma\lesssim 10^{1}$
of dephasing. With increasing $\alpha$, the range of $\gamma$ over which this
interesting plateau regime appears starts to shrink and eventually disappears
for $\alpha\geq 1.5$. The plateau regime emerges due to the intricate
interplay of long-range hopping and dephasing. In other words, due to long-
range hopping, particles can now evade the in-elastic scattering induced by
the dephasing more easily and, thereby, can possibly deviate from standard
diffusive transport.

In the following, we elaborate further on the appearance of the plateau and
restrict ourselves to the dephasing strength $0.1<\gamma<2.0$, for which the
$J_{\infty}$ plateaued in Fig. 3. We plot a heat map of the $J_{\infty}$ as a
function of long-range parameter $\alpha$ and dephasing $\gamma$ in Fig. 4,
for system sizes $L=512,\leavevmode\nobreak\ 768$, and $1024$, respectively.
Clearly, comparing the plots corresponding to all these chain lengths, we find
that apart from the magnitude of $J_{\infty}$, other features do not
distinctively differ. This indicates no significant finite size effect in
$J_{\infty}$ with the system size in the physical picture to follow.
Interestingly, the heat map of the NESS current on the $\alpha-\gamma$ plane
closely resembles the phase diagram of entanglement entropy, which
characterizes the phenomenon of measurement-induced phase transition [58], and
thus hints at a possible interesting connection between the MIPT and quantum
transport.

Firstly, we observe that NESS current becomes almost negligible for very small
values of $\alpha\rightarrow 0$. In this limit, the system manifests all-to-
all coupling that hinders transport and incites cooperative shielding, akin to
Anderson localization [68]. Intriguingly, this cooperative shielding is robust
against the dephasing and system size $L$, as evident from Fig. 4. Then, for a
finite $\alpha$, we note a surge in NESS current, peaking near $\alpha\approx
0.6$ for values of $\gamma>0.1$ and remains almost independent of $\gamma$.
Recall that this is reminiscent of the plateau observed in Fig. 3 for larger
$\gamma$ values for $\alpha=0.65$. On further increasing the value of
$\alpha$, the plateau regime shrinks and eventually disappears for
$\alpha>1.5$ irrespective of $\gamma$, as seen from Fig. 4.

Figure 5: NESS resistance $R_{\infty}=1/J_{\infty}$ as a function of system
size $L$, (a) on a log-linear scale for $\alpha=0.9$ and $1.0$, (b) on a log-
log scale from $\alpha=1.1$ to $2.0$ (bottom to top) in an increment of $0.1$.

Nature of the NESS transport: To understand the nature of the underlying NESS
transport, we examine the system size scaling of the NESS resistance, denoted
as $R_{\infty}=1/J_{\infty}$, for various values of $\alpha$. Note that the
scaling results presented below are almost insensitive for $\gamma>0.1$. We,
therefore, present results for $\gamma=10$. Depending on the system size
scaling of $R_{\infty}$ different transport regimes are classified, namely,
for ballistic transport $R_{\infty}\sim L^{0}$ [69], for diffusive or normal
transport $R_{\infty}\sim L$ [29]. Anything away from these scaling relations
is often categorized as anomalous transport, for example, $R_{\infty}\sim
L^{\nu}$ with $\nu>1$ it is sub-diffusive [53] whereas for $\nu<1$ it is
super-diffusive.

In Fig. 5, we clearly demonstrate the system size scaling of $R_{\infty}$ for
different values of $\alpha$ to analyze the regimes of transport. In Fig. 5(a)
we first concentrate on $\alpha\leq 1$. We plot $R_{\infty}$ as a function of
system size $L$ for $\alpha=0.9$ and $\alpha=1.0$. Interestingly, we observe
logarithmic system size scaling, $R_{\infty}\sim\log(L)$ as clear from the
straight-line fitting of the numerical data on a log-linear scale. This
clearly demonstrates an anomalous super-diffusive transport regime for
$\alpha\leq 1$. Next, in Fig. 5(b) we focus on the case of $\alpha>1$. We plot
$R_{\infty}$ as a function of $L$ for $1.0<\alpha\leq 2$. Remarkably, in this
regime of $\alpha$ we observe a power-law dependence: $R_{\infty}\sim L^{\nu}$
with an $\alpha$-dependent exponent $\nu$. This is evident from the plot of
$R_{\infty}$ on a log-log scale. Therefore, it is clear from Fig. 5(a) and (b)
that as $\alpha$ increases beyond the value 1, the scaling of $R_{\infty}$
changes from a logarithmic to a power-law behavior in system size.

Figure 6: Plot of the system size scaling exponent $\nu$ corresponding to the
fit $R_{\infty}\sim L^{\nu}$ as a function of $\alpha$ for $\alpha>1.0$ and
different values of dephasing strength $\gamma$. The error bars are associated
with the fitting of $R_{\infty}\sim L^{\nu}$.

To further elaborate on the nature of the transport for $\alpha>1$, we plot
the dependence of the exponent $\nu$ with $\alpha$ in Fig. 6. Our numerical
calculations show that $\nu$ increases with increasing $\alpha$ before
saturating beyond $\alpha\gtrsim 1.6$. It is worth mentioning that the super-
diffusive transport for $0.0<\alpha\lesssim 1.6$ is extremely robust against
the dephasing strength. However, a finite amount of dephasing along with the
long-range hopping is necessary to observe the emergence of these intriguing
transport regimes. Remarkably, for $\alpha\gtrsim 1.6$, a diffusive transport
regime sets in independent of the value of dephasing. However, we show below
that the transition from super-diffusive to diffusive transport in the
thermodynamic limit happens exactly at $\alpha=1.5$. We believe this
difference in the value of $\alpha$ appears due to the finite size effect.

To substantiate the above numerical observation that a diffusive transport
emerges only for $\alpha>1.5$, we show that the behavior of the current
operator norm (in the absence of the dephasing), analogously to the
Hamiltonian operator norm discussed in Ref. [58], can provide more insightful
information. We start by writing down the operator for current into the site
$L$ as

$\displaystyle\hat{J}_{L,in}$ $\displaystyle=$
$\displaystyle-i\sum_{r=1}^{L-1}\frac{J}{r^{\alpha}}\left(\hat{c}_{L-r}^{\dagger}\hat{c}_{L}-\hat{c}_{L}^{\dagger}\hat{c}_{L-r}\right).$
(9)

The corresponding operator norm
$||J_{L}||=\text{Tr}\left(\sqrt{\hat{J}_{L}^{\dagger}\hat{J}_{L}}\right)$
exhibit the following system-size dependence (see Appendix A for the details
of the calculation),

$\displaystyle||J_{L}||$ $\displaystyle\sim$
$\displaystyle\frac{L^{3/2-\alpha}}{\sqrt{(2\alpha-1)}(3/2-\alpha)},\leavevmode\nobreak\
\text{for }\alpha<1.5,$ (10) $\displaystyle\sim$ $\displaystyle\text{constant
independent of $L$ for }\alpha>1.5.$

This suggests that the current operator norm, representing the maximum rate of
coherent particle transport, remains constant and independent of $L$ for
$\alpha>1.5$. Therefore, for a finite system size, a sufficiently strong
dephasing strength can suppress coherent transport and lead to diffusive
behavior. In fact, the scaling behavior of $||J_{L}||$ mentioned above implies
that, for $\alpha>1.5$ in the thermodynamic limit, any non-zero dephasing can
induce diffusion similar to the short-range hopping model [70].

###

On the other hand, the current operator norm diverges when $\alpha<1.5$,
implying that, in the thermodynamic limit, any amount of dephasing strength is
insufficient to yield diffusive (or slower than diffusive, e.g., sub-
diffusive) transport. Furthermore, the divergent system size scaling also
suggests that, in the presence of finite (and large) dephasing, transport
cannot be ballistic either in the thermodynamic limit. This leaves us with the
only possibility of super-diffusive transport in the thermodynamic limit, even
in the presence of finite (and large) dephasing. Therefore, the behavior of
the operator norm in the absence of dephasing suggests that $\alpha=1.5$ holds
a special significance, aiding our understanding of the transition from super-
diffusive to diffusive transport in the presence of dephasing.

##  IV Summary and Discussion

In summary, using the Markovian Lindblad quantum master equation, we examined
the NESS transport properties of non-interacting fermions on a one-dimensional
lattice with $\frac{1}{r^{\alpha}}$-long-range hopping, subjected to on-site
dephasing. A non-trivial interplay between the long-range hopping and the
dephasing strength leads to the emergence of distinctive regimes of NESS
transport as one tunes the long-range exponent $\alpha$. Specifically, we
demonstrate that for $\alpha>1.5$, the transport is diffusive/normal, whereas,
for $\alpha<1.5$, the transport is super-diffusive. Furthermore, we observe
two different system-size scaling within the super-diffusive regime; namely,
for $1<\alpha<1.5$, we find a power law system size dependence with an
$\alpha$-dependent exponent $\nu<1$ and for $\alpha\leq 1$, we observe
inverse-logarithmic system size scaling for the current. Remarkably, the
super-diffusive regime is robust against the dephasing strength. We further
provide analytical insights to support our numerical findings by analyzing the
current operator norm.

It is worth pointing out that the heat map of the NESS current on the
$\alpha-\gamma$ plane, as presented in Fig. 4, bears a close resemblance to
the phase diagram of entanglement measures related to the MIPT[58]. The NESS
transport features we delved into stem from the steady-state solution of the
Lindblad quantum master equation, whereas, in the context of MIPT, the
transition of entanglement entropy from an area law phase to a volume law
phase emerges through a specific unraveling of the Lindblad master equation,
namely the quantum state diffusion. For the free Fermion case in one dimension
(without the boundary drive), MIPT appears at the long-range exponent
$\alpha=1.5$ [58]. Intriguingly, in our setup, a transition from super-
diffusive to diffusive regime also appears at the same long-range exponent
value, i.e., $\alpha=1.5$. Given that number-conserving local dephasing in our
setup acts like a Zeno-type local number density measurement, our results hint
at a connection between the MIPT and the transition from anomalous to normal
observed in the NESS current.

However, our identification of transport as a possible signature of the
underlying MIPT is not caveat-free. Indeed, the absence of a direct link
between mutual information and particle current renders such an identification
non-trivial, positing an intriguing avenue for future research. Relevantly,
prior numerical analyses have pointed out interesting connections between NESS
transport current and entanglement measures, like mutual information and
concurrence, particularly in quantum dot set-ups [71, 72, 73]. Establishing a
more rigorous link could pave the way for discerning MIPT in experiments via
current measurements.

##  Appendix A Operator norm and explanation of super-diffusive to diffusive
crossover

Here we would like to prove the scaling of $||J_{L}||$ corresponding to (10).
We start by writing down the operator for current into the site $L$ as

$\displaystyle\hat{J}_{L}$ $\displaystyle=$
$\displaystyle-i\sum_{r=1}^{L-1}\frac{1}{r^{\alpha}}\left(\hat{c}_{L-r}^{\dagger}\hat{c}_{L}-\hat{c}_{L}^{\dagger}\hat{c}_{L-r}\right),$
(11) $\displaystyle=$
$\displaystyle-i\sum_{j=1}^{L-1}\frac{1}{(j-L)^{\alpha}}\left(\hat{c}_{j}^{\dagger}\hat{c}_{L}-\hat{c}_{L}^{\dagger}\hat{c}_{j}\right)$
$\displaystyle=$
$\displaystyle-i\sum_{j=1}^{L-1}\frac{1}{(j-L)^{\alpha}}\hat{O}_{j}.$

We can then calculate the upper bound of the operator norm with respect to the
trace norm
$||J_{L}||=\text{Tr}\left(\sqrt{\hat{J}_{L}^{\dagger}\hat{J}_{L}}\right)$
which is given by,

$\displaystyle||J_{L}||$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{L-1}\frac{1}{|j-L|^{\alpha}}||\hat{O}_{j}||\leq\sum_{j=1}^{L-1}\frac{1}{|j-L|^{\alpha}}$
(12) $\displaystyle=$
$\displaystyle\left[\sum_{j,k=1}^{L-1}\frac{1}{|(j-L)(k-L)|^{\alpha}}\right]^{1/2},$

where we have used the triangle inequality for the operator norm,
$\displaystyle||\sum_{j}\hat{A}_{j}||\leq\sum_{j}||\hat{A}_{j}||$, where
$\hat{A}_{j}$ is an operator defined on a normed Hilbert space, in obtaining
the first inequality in (12), and also used the upper bound of the operator
norm $||\hat{O}_{j}||\leq 1$. Next, we can show that the following
inequalities hold,

$\displaystyle||J_{L}||$ $\displaystyle\leq$
$\displaystyle\sqrt{\sum_{j,k=1}^{L-1}\frac{1}{|(j-L)(k-L)|^{\alpha}}}\leq\sqrt{\sum_{j=1}^{L-1}\sum_{\begin{subarray}{c}k=1\\\
j\neq k\end{subarray}}^{L-1}\frac{1}{|j-k|^{2\alpha}}}.$ (13)
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{L-1}\left[\sum_{\begin{subarray}{c}k=1\\\ k\neq
j\end{subarray}}^{L-1}\frac{1}{|j-k|^{2\alpha}}\right]^{1/2},$

We further confirm this by plotting the various summations of Eq.(13) in Fig.
7. It can be seen that the inequality indeed holds for every value of $\alpha$
and differs only by a constant factor that is independent of the system size.

Figure 7: Numerical value of the upper bound of the operator norm verifying
the inequalities in (13), for $L=6000$.

Converting the discrete summations of (13) to integral we find in the large
$L$ limit,

$\displaystyle||J_{L}||\leq\int_{x=1}^{L}dx\left[\int_{y=2}^{L-1}\frac{dy}{|x-y|^{2\alpha}}\right]^{1/2}$
(14) $\displaystyle=$
$\displaystyle\left[\int_{y=2}^{L-1}\frac{dy}{|1-y|^{2\alpha}}\right]^{1/2}+\int_{x=2}^{L}dx\left[\int_{y=1}^{L-1}\frac{dy}{|x-y|^{2\alpha}}\right]^{1/2}$
$\displaystyle\approx$
$\displaystyle\sqrt{\frac{(1-L^{1-2\alpha})}{(2\alpha-1)}}+\frac{L^{3/2-\alpha}}{\sqrt{(2\alpha-1)}(3/2-\alpha)},$

where we have identified in the continuum limit $i\rightarrow x$ and
$j\rightarrow y$, and implicitly assumed $\alpha>0.5$.

For $\alpha\geq 1.5\leavevmode\nobreak\ (=3/2+\epsilon)$ (with $\epsilon\geq
0$ is just a parameter) the first term gives, $\sim\frac{1}{\sqrt{2\alpha-1}}$
and the second term gives,
$-\frac{1}{L^{\epsilon}\sqrt{(2\alpha-1)}\epsilon}$, making the second term
vanishing faster and leaving the dominant scaling to be a constant independent
of the chain-length $L$. For $\alpha<1.5$ in equation (14), the dominant
scaling comes from,
$\displaystyle\sim\frac{L^{3/2-\alpha}}{\sqrt{(2\alpha-1)}(3/2-\alpha)}$.

Therefore, we arrive at the following important leading order behavior of the
current operator norm,

$\displaystyle||J_{L}||$ $\displaystyle\sim$
$\displaystyle\frac{L^{3/2-\alpha}}{\sqrt{(2\alpha-1)}(3/2-\alpha)}\leavevmode\nobreak\
(0\leq\alpha<1.5)$ (15) $\displaystyle\sim$ $\displaystyle
L\leavevmode\nobreak\ \text{independent, for }\alpha>1.5.$

This operator norm scaling signifies that for $\alpha>1.5$ the current
operator norm is upper bounded by a $L-$independent constant whereas, for
$\alpha<1.5$, it is upper bounded by a power-law scaling with the system size,
a representative of a divergent behavior.

## Acknowledgment

SS acknowledges G. Wójtowicz, M. Zwolak, M. M. Rams, M. Dalmonte, and Y. Dubi
for useful comments and suggestions, and funding from the National Science
Center, Poland, under project 2020/38/E/ST3/00150. BKA acknowledges the
MATRICS grant MTR/2020/000472 from SERB, Government of India, and the Shastri
Indo-Canadian Institute for providing financial support for this research work
in the form of a Shastri Institutional Collaborative Research Grant (SICRG).
BKA. would also like to acknowledge funding from the National Mission on
Interdisciplinary Cyber-Physical Systems (NM-ICPS) of the Department of
Science and Technology, Govt. of India through the I-HUB Quantum Technology
Foundation, Pune, India. BKA and SS would like to thank the International
Centre for Theoretical Sciences (ICTS) for organizing the program -
Periodically and quasi-periodically driven complex systems (code:
ICTS/pdcs2023/6) where certain interesting discussions related to this project
took place.

## References

  * Harush and Dubi [2021] E. Z. Harush and Y. Dubi, Do photosynthetic complexes use quantum coherence to increase their efficiency? probably not, Science Advances 7, eabc4631 (2021).
  * Mattioni  _et al._ [2021] A. Mattioni, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, Design principles for long-range energy transfer at room temperature, Phys. Rev. X 11, 041003 (2021).
  * Maier  _et al._ [2019] C. Maier, T. Brydges, P. Jurcevic, N. Trautmann, C. Hempel, B. P. Lanyon, P. Hauke, R. Blatt, and C. F. Roos, Environment-assisted quantum transport in a 10-qubit network, Phys. Rev. Lett. 122, 050501 (2019).
  * Barthelemy and Vandersypen [2013] P. Barthelemy and L. M. K. Vandersypen, Quantum dot systems: a versatile platform for quantum simulations, Annalen der Physik 525, 808 (2013).
  * Sarkar and Dubi [2022a] S. Sarkar and Y. Dubi, Emergence and dynamical stability of a charge time-crystal in a current-carrying quantum dot simulator, Nano Letters 22, 4445 (2022a), pMID: 35580301.
  * Sarkar and Dubi [2022b] S. Sarkar and Y. Dubi, Signatures of discrete time-crystallinity in transport through an open fermionic chain, Communications Physics 5, 1 (2022b).
  * Field  _et al._ [1993] M. Field, C. G. Smith, M. Pepper, D. A. Ritchie, J. E. F. Frost, G. A. C. Jones, and D. G. Hasko, Measurements of coulomb blockade with a noninvasive voltage probe, Phys. Rev. Lett. 70, 1311 (1993).
  * Byrnes  _et al._ [2008] T. Byrnes, N. Y. Kim, K. Kusudo, and Y. Yamamoto, Quantum simulation of fermi-hubbard models in semiconductor quantum-dot arrays, Phys. Rev. B 78, 075320 (2008).
  * Deshpande  _et al._ [2010] V. V. Deshpande, M. Bockrath, L. I. Glazman, and A. Yacoby, Electron liquids and solids in one dimension, Nature 464, 209 (2010).
  * Loss and DiVincenzo [1998] D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57, 120 (1998).
  * Steinigeweg  _et al._ [2009] R. Steinigeweg, H. Wichterich, and J. Gemmer, Density dynamics from current auto-correlations at finite time- and length-scales, Europhysics Letters 88, 10004 (2009).
  * Bar Lev  _et al._ [2015] Y. Bar Lev, G. Cohen, and D. R. Reichman, Absence of Diffusion in an Interacting System of Spinless Fermions on a One-Dimensional Disordered Lattice, Phys. Rev. Lett. 114, 100601 (2015).
  * Steinigeweg  _et al._ [2017] R. Steinigeweg, F. Jin, D. Schmidtke, H. De Raedt, K. Michielsen, and J. Gemmer, Real-time broadening of nonequilibrium density profiles and the role of the specific initial-state realization, Phys. Rev. B 95, 035155 (2017).
  * Luitz and Lev [2017] D. J. Luitz and Y. B. Lev, The ergodic side of the many-body localization transition, Annalen der Physik 529, 1600350 (2017).
  * Žnidarič [2010] M. Žnidarič, Dephasing-induced diffusive transport in the anisotropic heisenberg model, New Journal of Physics 12, 043001 (2010).
  * Žnidarič and Horvat [2013] M. Žnidarič and M. Horvat, Transport in a disordered tight-binding chain with dephasing, The European Physical Journal B 86, 67 (2013).
  * Žnidarič  _et al._ [2017] M. Žnidarič, J. J. Mendoza-Arenas, S. R. Clark, and J. Goold, Dephasing enhanced spin transport in the ergodic phase of a many-body localizable system, Annalen der Physik 529, 1600298 (2017).
  * Bertini  _et al._ [2021] B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R. Steinigeweg, and M. Žnidarič, Finite-temperature transport in one-dimensional quantum lattice models, Rev. Mod. Phys. 93, 025003 (2021).
  * Eisler [2011] V. Eisler, Crossover between ballistic and diffusive transport: the quantum exclusion process, Journal of Statistical Mechanics: Theory and Experiment 2011, P06007 (2011).
  * Esposito and Gaspard [2005a] M. Esposito and P. Gaspard, Emergence of diffusion in finite quantum systems, Phys. Rev. B 71, 214302 (2005a).
  * Esposito and Gaspard [2005b] M. Esposito and P. Gaspard, Exactly solvable model of quantum diffusion, Journal of Statistical Physics 121, 463 (2005b).
  * LM Lezama and Bar Lev [2022] T. LM Lezama and Y. Bar Lev, Logarithmic, noise-induced dynamics in the Anderson insulator, SciPost Phys. 12, 174 (2022).
  * Bhakuni  _et al._ [2023] D. S. Bhakuni, T. L. Lezama, and Y. B. Lev, Noise-induced transport in the Aubry-André-Harper model, arXiv preprint 10.48550/arXiv.2307.06373 (2023).
  * Gu and Franco [2019] B. Gu and I. Franco, When can quantum decoherence be mimicked by classical noise?, The Journal of Chemical Physics 151, 014109 (2019).
  * Zerah-Harush and Dubi [2020] E. Zerah-Harush and Y. Dubi, Effects of disorder and interactions in environment assisted quantum transport, Phys. Rev. Res. 2, 023294 (2020).
  * Rebentrost  _et al._ [2009] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik, Environment-assisted quantum transport, New Journal of Physics 11, 033003 (2009).
  * Sarkar and Dubi [2020] S. Sarkar and Y. Dubi, Environment-assisted and environment-hampered efficiency at maximum power in a molecular photocell, The Journal of Physical Chemistry C 124, 15115 (2020).
  * Gopalakrishnan  _et al._ [2017] S. Gopalakrishnan, K. R. Islam, and M. Knap, Noise-induced subdiffusion in strongly localized quantum systems, Phys. Rev. Lett. 119, 046601 (2017).
  * Turkeshi and Schiró [2021] X. Turkeshi and M. Schiró, Diffusion and thermalization in a boundary-driven dephasing model, Phys. Rev. B 104, 144301 (2021).
  * Saha  _et al._ [2022] M. Saha, B. P. Venkatesh, and B. K. Agarwalla, Quantum transport in quasiperiodic lattice systems in the presence of büttiker probes, Phys. Rev. B 105, 224204 (2022).
  * Richerme  _et al._ [2014] P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions, Nature 511, 198 (2014).
  * Jurcevic  _et al._ [2014] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body system, Nature 511, 202 (2014).
  * Gring  _et al._ [2012] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. Adu Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science 337, 1318 (2012).
  * Morong  _et al._ [2021] W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Observation of stark many-body localization without disorder, Nature 599, 393 (2021).
  * Kyprianidis  _et al._ [2021] A. Kyprianidis, F. Machado, W. Morong, P. Becker, K. S. Collins, D. V. Else, L. Feng, P. W. Hess, C. Nayak, G. Pagano, N. Y. Yao, and C. Monroe, Observation of a prethermal discrete time crystal, Science 372, 1192 (2021).
  * Kosterlitz [1976] J. M. Kosterlitz, Phase transitions in long-range ferromagnetic chains, Phys. Rev. Lett. 37, 1577 (1976).
  * Kuwahara and Saito [2020a] T. Kuwahara and K. Saito, Area law of noncritical ground states in 1d long-range interacting systems, Nature Communications 11, 4478 (2020a).
  * Koffel  _et al._ [2012] T. Koffel, M. Lewenstein, and L. Tagliacozzo, Entanglement entropy for the long-range ising chain in a transverse field, Phys. Rev. Lett. 109, 267203 (2012).
  * Vodola  _et al._ [2014] D. Vodola, L. Lepori, E. Ercolessi, A. V. Gorshkov, and G. Pupillo, Kitaev chains with long-range pairing, Phys. Rev. Lett. 113, 156402 (2014).
  * Vodola  _et al._ [2015] D. Vodola, L. Lepori, E. Ercolessi, and G. Pupillo, Long-range ising and kitaev models: phases, correlations and edge modes, New Journal of Physics 18, 015001 (2015).
  * Chen and Lucas [2019] C.-F. Chen and A. Lucas, Finite speed of quantum scrambling with long range interactions, Phys. Rev. Lett. 123, 250605 (2019).
  * Tran  _et al._ [2020] M. C. Tran, C.-F. Chen, A. Ehrenberg, A. Y. Guo, A. Deshpande, Y. Hong, Z.-X. Gong, A. V. Gorshkov, and A. Lucas, Hierarchy of linear light cones with long-range interactions, Phys. Rev. X 10, 031009 (2020).
  * Kuwahara and Saito [2020b] T. Kuwahara and K. Saito, Strictly linear light cones in long-range interacting systems of arbitrary dimensions, Phys. Rev. X 10, 031010 (2020b).
  * Zhou  _et al._ [2020] T. Zhou, S. Xu, X. Chen, A. Guo, and B. Swingle, Operator lévy flight: Light cones in chaotic long-range interacting systems, Phys. Rev. Lett. 124, 180601 (2020).
  * Kuwahara and Saito [2021] T. Kuwahara and K. Saito, Absence of fast scrambling in thermodynamically stable long-range interacting systems, Phys. Rev. Lett. 126, 030604 (2021).
  * Nandkishore and Sondhi [2017] R. M. Nandkishore and S. L. Sondhi, Many-body localization with long-range interactions, Phys. Rev. X 7, 041021 (2017).
  * Schachenmayer  _et al._ [2013] J. Schachenmayer, B. P. Lanyon, C. F. Roos, and A. J. Daley, Entanglement growth in quench dynamics with variable range interactions, Phys. Rev. X 3, 1 (2013).
  * Lerose and Pappalardi [2020] A. Lerose and S. Pappalardi, Origin of the slow growth of entanglement entropy in long-range interacting spin systems, Phys. Rev. Res. 2, 012041 (2020).
  * Bhakuni  _et al._ [2021] D. S. Bhakuni, L. F. Santos, and Y. B. Lev, Suppression of heating by long-range interactions in periodically driven spin chains, Phys. Rev. B 104, L140301 (2021).
  * Santos  _et al._ [2016] L. F. Santos, F. Borgonovi, and G. L. Celardo, Cooperative Shielding in Many-Body Systems with Long-Range Interaction, Phys. Rev. Lett. 116, 250402 (2016).
  * Storch  _et al._ [2015] D. M. Storch, M. Van Den Worm, and M. Kastner, Interplay of soundcone and supersonic propagation in lattice models with power law interactions, New J. Phys. 17, 063021 (2015).
  * Nazareno and de Brito [1999] H. N. Nazareno and P. E. de Brito, Long-range interactions and nonextensivity in one-dimensional systems, Phys. Rev. B 60, 4629 (1999).
  * Purkayastha  _et al._ [2021] A. Purkayastha, M. Saha, and B. K. Agarwalla, Subdiffusive phases in open clean long-range systems, Phys. Rev. Lett. 127, 240601 (2021).
  * Wójtowicz  _et al._ [2020] G. Wójtowicz, J. E. Elenewski, M. M. Rams, and M. Zwolak, Open-system tensor networks and kramers’ crossover for quantum transport, Phys. Rev. A 101, 050301 (2020).
  * Wójtowicz  _et al._ [2023] G. Wójtowicz, A. Purkayastha, M. Zwolak, and M. M. Rams, Accumulative reservoir construction: Bridging continuously relaxed and periodically refreshed extended reservoirs, Phys. Rev. B 107, 035150 (2023).
  * Block  _et al._ [2022] M. Block, Y. Bao, S. Choi, E. Altman, and N. Y. Yao, Measurement-induced transition in long-range interacting quantum circuits, Phys. Rev. Lett. 128, 010604 (2022).
  * Müller  _et al._ [2022] T. Müller, S. Diehl, and M. Buchhold, Measurement-induced dark state phase transitions in long-ranged fermion systems, Phys. Rev. Lett. 128, 010605 (2022).
  * Minato  _et al._ [2022] T. Minato, K. Sugimoto, T. Kuwahara, and K. Saito, Fate of measurement-induced phase transition in long-range interactions, Phys. Rev. Lett. 128, 010603 (2022).
  * Defenu [2021] N. Defenu, Metastability and discrete spectrum of long-range systems, Proceedings of the National Academy of Sciences 118, e2101785118 (2021).
  * Kastner [2011] M. Kastner, Diverging equilibration times in long-range quantum spin models, Phys. Rev. Lett. 106, 130601 (2011).
  * Kastner [2017] M. Kastner, N -scaling of timescales in long-range N -body quantum systems, J. Stat. Mech. Theory Exp. 2017, 014003 (2017).
  * Bachelard and Kastner [2013] R. Bachelard and M. Kastner, Universal threshold for the dynamical behavior of lattice systems with long-range interactions, Phys. Rev. Lett. 110, 170603 (2013).
  * Jin  _et al._ [2022] T. Jin, J. a. S. Ferreira, M. Filippone, and T. Giamarchi, Exact description of quantum stochastic models as quantum resistors, Phys. Rev. Research 4, 013109 (2022).
  * Neyenhuis  _et al._ [2017] B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z. X. Gong, A. V. Gorshkov, and C. Monroe, Observation of prethermalization in long-range interacting spin chains, Sci. Adv. 3, e1700672 (2017).
  * Elenewski  _et al._ [2017] J. E. Elenewski, D. Gruss, and M. Zwolak, Communication: Master equations for electron transport: The limits of the Markovian limit, The Journal of Chemical Physics 147, 151101 (2017).
  * Zwolak [2020] M. Zwolak, Analytic expressions for the steady-state current with finite extended reservoirs, The Journal of Chemical Physics 153, 224107 (2020).
  * Varma  _et al._ [2017] V. K. Varma, C. de Mulatier, and M. Žnidarič, Fractality in nonequilibrium steady states of quasiperiodic systems, Phys. Rev. E 96, 032130 (2017).
  * Celardo  _et al._ [2016] G. L. Celardo, R. Kaiser, and F. Borgonovi, Shielding and localization in the presence of long-range hopping, Phys. Rev. B 94, 144206 (2016).
  * Dhar and Roy [2006] A. Dhar and D. Roy, Heat transport in harmonic lattices, Journal of Statistical Physics 125, 801 (2006).
  * Žnidarič [2010] M. Žnidarič, Exact solution for a diffusive nonequilibrium steady state of an open quantum chain, Journal of Statistical Mechanics: Theory and Experiment 2010, L05002 (2010).
  * Sharma and Rabani [2015] A. Sharma and E. Rabani, Landauer current and mutual information, Phys. Rev. B 91, 085121 (2015).
  * Sable  _et al._ [2018] H. S. Sable, D. S. Bhakuni, and A. Sharma, Landauer current and mutual information in a bosonic quantum dot, Journal of Physics: Conference Series 964, 012007 (2018).
  * Dey  _et al._ [2019] A. Dey, D. S. Bhakuni, B. K. Agarwalla, and A. Sharma, Quantum entanglement and transport in a non-equilibrium interacting double-dot system: The curious role of degeneracy, Journal of Physics: Condensed Matter 32, 075603 (2019).