# Exact description of quantum stochastic models as quantum resistors

Tony Jin1, João S. Ferreira1, Michele Filippone1,2 and Thierry Giamarchi1
1Department of Quantum Matter Physics, Ecole de Physique University of Geneva,
Quai Ernest-Ansermet 24, CH-1211 Geneva 4, Switzerland  2Université Grenoble
Alpes, CEA, IRIG-MEM-L_Sim, F-38000, Grenoble, France

###### Abstract

We study the transport properties of generic out-of-equilibrium quantum
systems connected to fermionic reservoirs. We develop a new perturbation
scheme in the inverse system size, named $1/N$ expansion, to study a large
class of out of equilibrium diffusive/ohmic systems. The bare theory is
described by a Gaussian action corresponding to a set of independent two level
systems at equilibrium. This allows a simple and compact derivation of the
diffusive current as a first order pertubative term. In addition, we obtain
exact solutions for a large class of quantum stochastic Hamiltonians (QSHs)
with time and space dependent noise, using a self consistent Born diagrammatic
method in the Keldysh representation. We show that these QSHs exhibit
diffusive regimes which are encoded in the Keldysh component of the single
particle Green’s function. The exact solution for these QSHs models confirms
the validity of our system size expansion ansatz, and its efficiency in
capturing the transport properties. We consider in particular three fermionic
models: i) a model with local dephasing ii) the quantum simple symmetric
exclusion process model iii) a model with long-range stochastic hopping. For
i) and ii) we compute the full temperature and dephasing dependence of the
conductance of the system, both for two- and four-points measurements. Our
solution gives access to the regime of finite temperature of the reservoirs
which could not be obtained by previous approaches. For iii), we unveil a
novel ballistic-to-diffusive transition governed by the range and the nature
(quantum or classical) of the hopping. As a by-product, our approach equally
describes the mean behavior of quantum systems under continuous measurement.

##  I Introduction

Diffusion is the transport phenomenon most commonly encountered in nature. It
implies that globally conserved quantities such as energy, charge, spin or
mass spread uniformly all over the system according to Fick/Ohm’s law

$J=-D\nabla n\,,$ (1)

where the diffusion constant $D$ relates the current density $J$ to a
superimposed density gradient $\nabla n$.

Despite its ubiquity, understanding the emergence of classical diffusive
phenomena from underlying quantum mechanical principles is highly non trivial.
Early works based on field theory and perturbative methods [1, 2] pointed out
the possibility that interactions do not necessarily lead to diffusion at
finite temperature, a question addressed then more rigorously by using the
concepts of integrability [3]. These questions have then fueled many exciting
discoveries in low-dimensional interacting systems [4]. A notable example is
the ballistic-to-diffusive transition in quantum integrable XXZ spin chains
[5, 6, 7, 8, 9, 10], which also exhibit a superdiffusive point in the Kardar-
Parisi-Zhang universality class [11, 12, 13, 14, 15]. These discoveries have
motivated the generalized hydrodynamical descriptions of integrable systems
[16, 17], providing an elegant path to the question of diffusion at finite
temperature [18], and paving the way to the description of diffusive phenomena
based on perturbative approaches [19, 20, 21, 22, 23, 24].

The out-of-equilibrium driving protocol illustrated in Fig. 1, where a system
is coupled to external dissipative baths, has been crucial to unveil and
characterize such exotic transport phenomena [6, 25, 7, 26].

Figure 1:  A stationary current $J$ flows in a one-dimensional lattice when
connected to left (L) and right (R) fermionic reservoirs, described by Fermi
distributions $f(\varepsilon)$ with different temperatures $T$ or chemical
potentials $\mu$. The wiggly lines denote dissipative degrees of freedom
acting on the system with rate $\gamma$. For a fixed difference of chemical
potential $\delta\mu=\mu_{L}-\mu_{R}$, dissipative terms are normally
responsible for the Ohmic suppression of the current, $J\propto 1/N$.

.

It allows to study disordered systems [27, 28, 29], uncover novel integrable
structures [6, 30], and show diffusive transport [31, 32, 33, 34, 35, 36].
These open quantum systems [37, 38, 39], are described within the Lindblad
formalism [40, 41], which is actively employed to investigate the exotic
dynamics induced by non-trivial interactions with external degrees of freedom,
such as lattice vibrations, quantum measurements [42, 43, 44, 45, 46, 47],
dephasing [48, 49, 50, 51, 52, 53], losses [54, 55, 56, 57, 58, 59], coupling
to a lightfield [60, 61, 62] and environmental engineering [63].

This research activity is also motivating ongoing experiments, where recent
progress in space- and time-resolved techniques is applied to directly observe
emergent diffusive and exotic dynamics in various quantum systems, including
cold atoms [64, 60, 65, 66, 67], spin chains [68, 69, 70, 71, 72, 73] and
solid-state [74, 75, 76]. In this context, theoretical predictions are usually
made case-by-case, with strong constraints on geometries and driving protocols
[77]. Thus, devising versatile tools to solve generic quantum models that show
diffusion becomes crucial to understand emerging classical Ohmic transport.

In this paper, we develop a novel approach to characterize the bulk transport
properties of quantum resistors which we show to be exact and systematic for a
wide class of quantum stochastic Hamiltonians (QSHs). Our starting point is
the Meir-Wingreen’s formula [78, 79] (MW), which expresses the current $J$ of
a system driven at its boundaries, see Fig. 1, in terms of single-particle
Green’s functions. We show that, for Ohmic systems, the MW formula supports an
expansion of the current in terms of the inverse of the system size $N$. We
illustrate how to perform practically this $1/N$ expansion, which reveals
efficient to derive the diffusive current and the diffusion constant: we
assume that, in the $N\rightarrow\infty$ limit, diffusive lattices admit a
simple description in terms of independently equilibrated sites and
demonstrate that a well-chosen perturbation theory over this trivial state
leads to the desired $1/N$ expansion.

We provide a comprehensive demonstration of the validity of our approach in
the context of QSHs. Relying on diagrammatic methods and out-of-equilibrium
field theory [80], we show that single-particle Green’s functions of QSHs can
be exactly and systematically derived relying on the self-consistent Born
approximation (SCBA) – a generalization of previous results derived for a
dephasing impurity in a thermal bath [50]. Equipped with this exact solution,
and relying on MW formula, we explicitly derive the dissipative current
flowing in the system and show that the Keldysh component of the single
particle Green’s function encodes the Ohmic suppression of the current. Then,
we explicitly derive the asymptotically equilibrated state by “coarse-
graining” of single particle Green’s functions and validate our procedure to
perform the $1/N$ expansion.

We illustrate the effectiveness and versatility of our approach for three
different QSHs of current interest: i) the dephasing model [31, 81, 32, 30,
82]; ii) the quantum symmetric simple exclusion process (QSSEP) [83, 35, 84,
85, 86, 87] and iii) models with stochastic long range hopping [88, 46]. The
case studies (i) and (ii) illustrate the effectiveness of our approach,
providing simple derivations of the current $J$ and of the diffusion constant
$D$, in alternative to approaches relying on matrix-product state [31, 81,
32], integrability [30] or other case-by-case solutions [35, 33].
Additionally, we address previously unexplored regimes, by exactly solving the
out-of-equilibrium problem with fermion reservoirs at arbitrary temperatures
and chemical potentials. Our approach also allows to access two-times
correlators in the stationary state which were not described by previous
studies. For case (iii), we show instead the ability of our approach to
predict novel and non-trivial transport phenomena, namely a displacement of
the ballistic-to-diffusive transition induced by coherent nearest-neighbor
tunneling in one-dimensional chains. A by-product of our analysis is that all
the results presented here apply also for system under continuous measurement,
which are currently attracting a lot of interest in the context of measurement
induced phase transition [42, 88, 44, 46].

Our paper is structured as follows. Section II describes how the MW formula is
a good starting point to build a systematic expansion of the current in terms
of the inverse system size $N$. Section III presents QSH and shows the
exactitude of SCBA for the computation of single-particle self-energies.
Section IV shows how our formalism allows to fully compute the transport
properties of the dephasing model, the QSSEP and the long-range model. Section
V is dedicated to our conclusions and the discussion of the future research
perspectives opened by our work.

##  II Resistive scaling in finite-size boundary driven systems and
perturbative approach

In this section, we introduce generic tools aimed at studying diffusive
transport in boundary-driven setups like those of Fig. 1. For these setups,
the current is given by the MW formula [78]. In the simplified (yet rather
general) situation, where the reservoirs have a constant density of states and
the tunnel exchange of particles does not depend on energy, the MW formula
reads (we assume $e=\hbar=k_{B}=1$):

$J=i\int\frac{d\omega}{2\pi}\text{Tr}\left\\{\frac{1}{2}\big{(}\Gamma_{L}-\Gamma_{R}\big{)}G^{\cal
K}+\right.\\\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.\left[\left(f_{L}-\frac{1}{2}\right)\Gamma_{L}-\left(f_{R}-\frac{1}{2}\right)\Gamma_{R}\right]\big{(}G^{\cal
R}-G^{\cal A}\big{)}\right\\}\,,$ (2)

where $f_{L(R)}(\omega)=[e^{(\omega-\mu_{L(R)})/T_{L(R)}}+1]^{-1}$ are the
Fermi distributions associated to the left and right reservoir with chemical
potentials $\mu_{L(R)}$ and temperatures $T_{L(R)}$. $G^{\cal R/A/K}$ are the
retarded ($\cal R$), advanced ($\cal A$) and Keldysh ($\cal K$) components of
the single-particle Green’s functions of the system. They are defined in time
representation as $G^{\cal
R}_{j,k}(t-t^{\prime})=-i\theta(t-t^{\prime})\langle\\{c_{j}(t),c^{\dagger}_{k}(t^{\prime})\\}\rangle$,
$G^{\cal A}_{j,k}(t-t^{\prime})=[G^{\cal R}_{j,k}(t^{\prime}-t)]^{*}$ and
$G^{\cal
K}_{j,k}(t-t^{\prime})=-i\langle[c_{j}(t),c^{\dagger}_{k}(t^{\prime})]\rangle$,
where the (curly)square brackets indicate (anti)commutation 111The dependence
of the Green’s functions on time differences $t-t^{\prime}$, instead of
separate times $t,t^{\prime}$ is a consequence of the fact that we consider
stationary situations.. $c_{j}$ is the annihilation operator of a spinless
fermion at site $j$. The $\Gamma_{L(R)}$ matrices describe system-reservoirs
couplings.

Our aim is to establish a systematic procedure to compute diffusive current
for large systems. The starting point will be the state of the system in the
thermodynamic limit ($N\rightarrow\infty$). By identifying in the MW formula
(2) the terms leading to Fick’s law (1), we motivate the simple structure of
the problem for an infinite system size. In resistive systems, a fixed
difference of density $\Delta n:=n_{1}-n_{N}$ at the edges of the system
enforces the $1/N$ suppression of the current ($J\propto\nabla n\propto\Delta
n/N$). It is thus natural to perform a perturbative $1/N$ expansion of the
current on the $N\rightarrow\infty$ state. We conjecture a possible
perturbation scheme and show its validity in the context of QSHs.

Without loss of generality, we focus on discrete $1D$ lattice systems of size
$N$ 222The extension to different geometries and additional degrees of freedom
is straightforward.. In this case, the $\Gamma_{L(R)}$ matrices in Eq. (2)
acquire a simple form in position space:
$[\Gamma_{L(R)}]_{j,k}=\Gamma\delta_{j,1(N)}\delta_{j,k}$. We also express the
local densities in terms of Green’s functions, namely $2n_{j}=2\langle
c^{\dagger}_{j}c_{j}\rangle=1-i\int d\omega\,G_{j,j}^{\cal K}(\omega)/(2\pi)$,
which also implies $2i\Delta n=G^{\cal K}_{1,1}(t=0)-G^{\cal
K}_{N,N}(t=0)=\Delta G^{\cal K}$. The MW formula then acquires the more
compact form:

$J=\Gamma\int
d\omega\Big{[}f_{L}(\omega)\mathcal{A}_{L}(\omega)-f_{R}(\omega)\mathcal{A}_{R}(\omega)\Big{]}-\Gamma\Delta
n\,,$ (3)

where we have introduced the local spectral densities
$\mathcal{A}_{L(R)}(\omega)=-\frac{1}{\pi}\mbox{Im}[G^{\cal
R}_{1,1(N,N)}(\omega)]$ and made use of the fact that $\int
d\omega\mathcal{A}_{L(R)}(\omega)=1$.

The local spectral densities $\mathcal{A}_{L(R)}(\omega)$ exponentially
converge in the thermodynamic limit $N\rightarrow\infty$. This feature is
generally expected and is illustrated in Fig. 8 for different classes of QSHs.
This observation allows to establish that the $1/N$ scaling, proper to
diffusive currents, must entirely arise from $\Delta n$ in (3). The
possibility to ignore the size-dependence of the first term of (3) imposes
strong constraints on the $1/N$ expansion of the difference of density $\Delta
n$ in diffusive systems. If we write this expansion as

$2i\Delta n=\Delta G^{\cal K}=\Delta G^{(\infty)}+\frac{1}{N}\Delta
G^{\prime}+\ldots$ (4)

one notices immediately that the leading term $\Delta G^{(\infty)}$ has to
compensate the first one in (3), implying

$\begin{split}\frac{\Delta G^{(\infty)}}{2i}=\int
d\omega\Big{[}f_{L}(\omega)\mathcal{A}_{1,1}(\omega)-f_{R}(\omega)\mathcal{A}_{N,N}(\omega)\Big{]}\,.\end{split}$
(5)

A sufficient but not necessary condition fulfilling this relation is obtained
by imposing at each boundary:

$\int\frac{d\omega}{2\pi}G_{L(R)}^{\mathcal{K}(\infty)}(\omega)=-i\int
d\omega\tanh\left(\frac{\omega-\mu_{L(R)}}{2T_{L(R)}}\right)\mathcal{A}_{L(R)}(\omega)\,,$
(6)

which will turn out to be satisfied for QSHs. These relations have a simple
and interesting interpretation. In the infinite size limit, the flowing
current is zero and thus the stationary value of the densities at the boundary
can be computed by supposing that they fulfill a _fluctuation-dissipation_
relation or equivalently, that these sites are at equilibrium with the
neighboring reservoirs.

Reinjecting (4) in the MW formula gives the current

$J=i\frac{\Gamma}{2N}\Delta G^{\prime}$ (7)

and as expected, we get the $1/N$ diffusive scaling. This relation tells us
that the information about the diffusion constant is hidden in the $1/N$
correction to the density profile which can be in general a non trivial
quantity to compute. However, we will see in the following that there is a
shorter path to access it via the use of an infinite system size perturbation
theory.

The main idea of the $1/N$ perturbation is to find an effective simple theory
that captures the relevant properties of the system in the $N\to\infty$ limit.
From there, transport quantities are computed _perturbatively_ on top of this
limit theory. To determine this effective theory, we conjecture that there is
a typical length $a$ beyond which two points of the systems can be considered
to be statistically independent. Thus, by coarse-graining the theory over
cells of size $a$, each cell becomes uncoupled and in local equilibrium, see
Fig. 2.

Figure 2:  Cartoon picture of the coarse-graining procedure. On the left,
spacial correlations in the infinite size limit are depicted. These decay
exponentially as a function of the distance and are non-zero only within a
finite length $a$. By coarse-graining the theory over this typical length, we
obtain an effective theory (right) consisting of an ensemble of uncoupled
sites with a finite self-energy at equilibrium.

The reasons motivating such factorization are twofold. First, the current is
suppressed as $1/N$ in the large system size limit, so the infinite size
theory should predict a null stationary current. Second, factorization of
stationary correlations has actually been demonstrated for a certain number of
diffusive toy models, most notably in the context of large deviations and
macroscopic fluctuation theory [91, 92, 83, 35]. For instance, it is known
that the $n^{th}$ connected correlation functions of physical observables,
such as density, generically behaves as $N^{-(n-1)}$. Thus, it is natural to
assume that for $N\to\infty$, correlations must be exponentially decaying over
a length $a$. We will show explicitly that in all of the examples studied,
this factorization in the coarse-grained theory will turn out to be true and
provide an analytic estimation for $a$ in App.F.

We now put these assumptions on formal grounds. Let $\tilde{j}$ and
$\tilde{k}$ be the spatial indices of the coarse-grained theory

$G_{\tilde{j},\tilde{k}}^{\mathcal{R/A/K}}:=\frac{1}{a}\sum_{m,n=0}^{a-1}G_{\tilde{j}a+m,\tilde{k}a+n}^{\mathcal{R/A/K}}\,.$
(8)

The relation between the different components $\mathcal{R},\,\mathcal{A}$ and
$\mathcal{K}$ of the single particle Green’s functions are assumed to describe
uncoupled sites at equilibrium with a local self-energy $\Sigma_{\tilde{j}}$
[80]. These conditions require then local fluctuation-dissipation relations of
the form

$G_{\tilde{j},\tilde{k}}^{\mathcal{K}(\infty)}(\omega)=\delta_{\tilde{j},\tilde{k}}\tanh\left(\frac{\omega-\mu_{\tilde{j}}}{2T_{\tilde{j}}}\right)\Big{[}G_{\tilde{j},\tilde{j}}^{\cal
R}(\omega)-G_{\tilde{j},\tilde{j}}^{\cal A}(\omega)\Big{]},$ (9)

with retarded and advanced Green’s functions which are diagonal in the coarse-
grained space representation

$G^{\cal
R(A)}_{\tilde{j},\tilde{k}}(\omega)=\frac{\delta_{\tilde{j},\tilde{k}}}{\omega-\omega^{0}_{\tilde{j}}\pm\Sigma_{\tilde{j}}(\omega)}\,.$
(10)

These relations fix entirely the stationary property of the system in the
infinite size limit. The specification of the free parameters
$\mu_{\tilde{j}},T_{\tilde{j}},\omega^{0}_{\tilde{j}}$ and
$\Sigma_{\tilde{j}}$ have to be done accordingly to the model under
consideration. We will see that they take a simple form for QSHs, namely the
self-energy $\Sigma_{\tilde{j}}$ is frequency independent and the
$\mu_{\tilde{j}},T_{\tilde{j}}\gg\omega$ limit can be taken taken in Eq. (9),
as expected in the Markovian limit of the dissipative process [79].

To get the current, one needs to go one step further and understand which
terms have to be expanded. The thermodynamic equilibrated theory does not
exhibit transport, thus should be left invariant by the part of the
Hamiltonian that commutes with the conserved quantity, for us the local
particle density. It is then natural to conjecture that the perturbative term
for the current is given by the dynamical part of the theory, that is, the
part of the Hamiltonian $\hat{H}_{{\rm dyn}}$ which does not commute with the
local density. Thus, we conjecture that, at order $1/N$, the current is given
by :

$J=\langle\hat{J}\hat{H}_{\rm{dyn}}\rangle_{\infty}\,,$ (11)

where the $\langle\rangle_{\infty}$ means the expectation value must be taken
with respect to the infinite system size theory. This formula has the
remarkable advantage that its computational complexity is very low since the
coarse-grained theory is Gaussian. We remark that the $1/N$ expansion
presented here is _not_ a standard expansion in the hopping amplitude $\tau$,
since the latter has an exponentially large degenerate manifold of states at
$\tau=0$.

In Sec. IV, we show explicitly how these ideas unfold for QSHs, by comparing
computations done from the $1/N$ theory with the one obtained from the exact
solution that we present in the following Section Sec. III. Understanding to
which extent and under which conditions Eqs. (9,10) and (11) can be applied is
one of the very challenging direction of study, in particular in the context
of interacting quantum systems without bulk dissipative terms.

##  III Validity of the self-consistent Born approximation for Quantum
stochastic Hamiltonians

In this section, we present a class of quantum stochastic models and
associated Liouvillians (12), that describe either stochastic local dephasing
or stochastic jumps of fermionic particles on a graph. The random processes
are defined by a quantum Markov equation also known as a Lindblad equation. We
will show explicitly two ways, exemplified by Eqs. (15) and (62), to associate
an underlying quantum stochastic model to such Lindblad equation, a process
known as _unraveling_ or _dilatation_ [93, 94, 95]. Of particular interest for
us is the description in terms of quantum stochastic Hamiltonians (QSHs) (15).
It provides a way to resum exactly the perturbative series associated to the
stochastic noise, which coincides with the self-consistent Born approximation
(SCBA) for single particle Green’s functions. This method was originally
devised for the particular case of a single-site dephaser in Ref. [50] and we
extend it here to more general situations. We will show in Section IV that,
relying on SCBA, we can derive the diffusive transport properties of these
models and show the validity of the assumptions underpinning the perturbative
$1/N$ expansion presented in Sec. II.

Consider a graph made of discrete points, each corresponding to a site. To
such graph we associate a Markovian process where spinless fermions on a given
site can jump to any other site only if the target site is empty, see Fig. 3.
We define $\gamma_{ij}\geq 0$ as the probability rate associated to the
process of a fermion jumping from $i$ to $j$ and $\gamma_{ji}=\gamma_{ij}$ the
reverse process. The generator of such process is given by the Liouvillian,
which acts on the density matrix $\rho$ of the system:

$\begin{split}{\cal
L}(\rho)=\sum_{i,j}&\gamma_{i,j}\left(2c_{j}^{\dagger}c_{i}\rho
c_{i}^{\dagger}c_{j}-\big{\\{}c_{i}^{\dagger}c_{j}c_{j}^{\dagger}c_{i},\rho\big{\\}}\right)\,.\end{split}$
(12)

The total evolution of the density matrix $\rho$ is in general given by

$\frac{d}{dt}\rho={\cal L}_{0}(\rho)+{\cal L}(\rho)\,,$ (13)

where ${\cal L}_{0}$ generates what we call the _free evolution_ , in the
sense that ${\cal L}_{0}$ is quadratic in the fermion operators $c_{i}$ and
the related spectrum and propagators can be efficiently computed with Wick’s
theorem [96, 97]. Such Liouvillians can generally describe single-particle
Hamiltonians or dissipative processes (coherent hopping, losses,…). We will
consider ${\cal L}(\rho)$ as a perturbation on top of this theory.

There exists a general procedure to see ${\cal L}(\rho)$ as the emergent
_averaged_ dynamics of an underlying microscopic stochastic, yet Hamiltonian,
process. Lifting ${\cal L}(\rho)$ to this stochastic process is known as
_unraveling_ and there is not a unique way of doing so, see Fig. 3. The
stochastic Hamiltonian can be treated as a perturbation in field-theory which
requires the summation of an infinite series. Our strategy is to pick the
relevant stochastic theory for which there exists a simple way to reorganize
the summation and then take the average in order to get the mean evolution.

We now proceed to present the unraveled theory. Let $dH_{t}$ be the stochastic
Hamiltonian increment, generating the evolution, which is defined by

$\left|\psi_{t+dt}\right\rangle=e^{-idH_{t}}\left|\psi_{t}\right\rangle\,.$
(14)

We work in the Itō prescription and consider stochastic Hamiltonians of the
form

$dH_{t}=\sum_{i,j}\sqrt{2\gamma_{i,j}}c_{j}^{\dagger}c_{i}dW_{t}^{i,j}.$ (15)

$W_{t}^{i,j}$ describes a complex noise and we adopt the convention that
$W_{t}^{ij*}=W_{t}^{j,i}$. The corresponding Itō rules are summed up by

$dW_{t}^{i,j}dW_{t}^{k,l}=\delta_{i,l}\delta_{k,j}dt.$ (16)

Using the Itō rules to average over the noise degrees of freedom one recovers
the Liouvillian (12).

Figure 3:  Schematic representation of our random process. The orange box
represents the Lindblad equation (12) which describes random quantum jumps
between sites connected by an arrow. An arrow leaving and arriving at the same
site represents a local dephasing. To a given Lindblad equation, we can
associate multiple stochastic process (blue and green boxes), a process called
_unraveling_ (orange dashed lines). The Lindblad equation is recovered by
averaging over the noisy degrees of freedom (full blue lines). We show that
the unraveling in terms of quantum stochastic Hamiltonian (QSH) is
particularly useful for the diagrammatic expansion of the theory.

Finally, an other point we would like to emphasize concerns the connection to
systems evolving under continuous measurements. Indeed, another way to unravel
(12) is to see it as the average evolution with respect to the measurement
outcomes of a system for which the variables
$c_{j}^{\dagger}c_{i}+c_{i}^{\dagger}c_{j}$ and
$i(c_{i}^{\dagger}c_{j}-c_{j}^{\dagger}c_{i})$ are continuously monitored and
independently measured with rate $\gamma_{i,j}$ [93]. Although the physics is
radically different at the level of a single realisation of the noise, on
average it gives the same result than the prescription (15). Hence, all the
results that will be presented for the mean behavior of our class of
stochastic Hamiltonians also describe the mean behavior of systems subject to
continuous measurements. The unraveling procedure corresponding to continuous
measurements is described in detail in Appendix A.

###  III.1 Self-energy

We show now that the perturbation theory in the stochastic Hamiltonian (15)
can be fully resummed, leading to exact results for single particle Green’s
functions. To perform this task, we rely on the Keldysh path-integral
formalism [80], which describes the dynamics of the system through its action
$S$. The presence of dissipative effects can be naturally included in $S$
using Lindblad formalism [98]. The action gives the Keldysh partition function
${\cal Z}={\rm tr}(\rho_{t})$

$\mathcal{Z}=\int{\cal
D}[\psi^{\pm},\bar{\psi}^{\pm}]e^{iS[\psi^{\pm},\bar{\psi}^{\pm}]}.$ (17)

where $\psi=(\psi^{+},\psi^{-})$ are Grassmann variables defined respectively
on the positive and negative Keldysh time contours $\mathcal{C}_{\pm}$. We
follow the Larkin-Ovchinnikov’s convention 333In our conventions, Larkin-
Ovchinnikov’s rotation reads
$\psi^{1/2}=(\psi^{+}\pm\psi^{-})/\sqrt{2}\,,\bar{\psi}^{1/2}=(\bar{\psi}^{+}\mp\bar{\psi}^{-})/\sqrt{2}$
[123]., in which the Keldysh action $S_{0}$ corresponding to the free-
evolution $\mathcal{L}_{0}$ is expressed in terms of the inverse Green’s
function $\boldsymbol{G}^{-1}$ namely

$\mathcal{S}_{0}=\sum_{i,j}\int\frac{d\omega}{2\pi}\left(\begin{array}[]{cc}\bar{\psi}^{1},&\bar{\psi}^{2}\end{array}\right)_{i}\Big{[}\boldsymbol{G}^{-1}\Big{]}_{i,j}\left(\begin{array}[]{c}\psi^{1}\\\
\psi^{2}\end{array}\right)_{j}\,.$ (18)

All variables in the integral (18) are implicitly assumed to depend on a
single frequency $\omega$, which coincides with the assumption of stationary
behavior, valid for our class of problems. The inverse Green’s function
$\boldsymbol{G}^{-1}$ is itself expressed in terms of the retarded, advanced
and Keldysh green functions $G^{\cal R/A/K}$, defined in Section II:

$\Big{[}\boldsymbol{G}^{-1}\Big{]}=\left(\begin{array}[]{cc}G^{\cal R}&G^{\cal
K}\\\ 0&G^{\cal A}\end{array}\right)^{-1}$ (19)

and whose diagrammatic representations in the time domain are given in Fig. 4.

Figure 4:  Diagrammatic representation of the retarded ($\cal R$), advanced
($\cal A$) and Keldysh ($\cal K$) Green’s function. Time flows from right to
left.

The causality structure of the Keldysh Green functions is enforced by the
suppression of correlators $\langle\psi^{2}\bar{\psi}^{1}\rangle=0$. This
means that a retarded propagator can never become advanced, which pictorially
translates into the fact that a solid line cannot switch to a dashed one.

The action corresponding to the Liouvillian term (12) reads [98]

$S_{\cal L}:=-\int
dt\sum_{i,j}\gamma_{i,j}\left(\bar{\psi}^{1}_{j,t}\psi^{1}_{i,t}\bar{\psi}^{2}_{i,t}\psi^{2}_{j,t}+\bar{\psi}^{1}_{i,t}\psi^{1}_{j,t}\bar{\psi}^{2}_{j,t}\psi^{2}_{i,t}\right)\,.$
(20)

which is a quartic action in the Grassmann fields. At the level of single
particle Green’s functions, the action $S_{\cal L}$ is incorporated through
the self energy $\boldsymbol{\Sigma}$, defined as the sum of all one-particle
irreducible diagrams. As in equilibrium field theory, the Dyson equation
relates the full propagator to the bare propagator and the self energies
$\boldsymbol{\Sigma}$:

$\boldsymbol{G}=\Big{[}\boldsymbol{G}_{0}^{-1}-\boldsymbol{\Sigma}\Big{]}^{-1}\,.$
(21)

To compute the diffusive current from MW formula, $\boldsymbol{\Sigma}$ must
be know to any order; an a priori difficult task given the quartic nature of
the action (20). Instead, rewriting the action at the stochastic level allows
us to exactly derive the self-energy $\boldsymbol{\Sigma}$ and solve this
problem. In the field-theory language, the unraveling procedure exemplified by
Eq. (15) leads to the equivalent action

$S_{{\rm
sto}}=-\sum_{i,j}\int\sqrt{2\gamma_{i,j}}\left(\bar{\psi}_{j,t}^{1}\psi_{i,t}^{1}+\bar{\psi}_{j,t}^{2}\psi_{i,t}^{2}\right)dW_{t}^{i,j}\,,$
(22)

where $S_{\rm sto}$ is related to $S_{\cal L}$ by the average $\mathbb{E}[]$
over the noise degrees of freedom:

$\mathbb{E}[e^{iS_{\rm sto}}]=e^{iS_{\cal L}}\,.$ (23)

In formal terms, this transformation is reminiscent of a Hubbard-Stratonovich
transformation where the action becomes quadratic in terms of the Grassmann
variables. Note that the complexity encoded in Eq. (20) is preserved by the
consequent introduction of the space and time dependent noise $dW_{t}^{i,j}$.
However, the noise correlations imposed by Itō’s rules (16) allow a dramatic
simplification of the diagrammatic expansion in $\gamma_{i,j}$ of the Green
functions within the stochastic formulation. Such simplified structure does
not manifestly appear when working with the Lindbladian (averaged) formulation
of the problem (20) (see Fig. 14 in Appendix B).

The resummation works as follows. In Fig. 5, we show the diagrammatic
expansion of (21) up to second order in the stochastic noise $\gamma_{i,j}$.

Figure 5:  Perturbative series in the Keldysh formalism for our class of
stochastic models. Average quantities are obtained by contracting pairs of
wiggly lines together. Here a wiggly line represents either $dW_{t}^{i,j}$ or
its complex conjugated pair for simplicity. The formulation of the theory in
terms of QSH allows for a simple writing of the perturbative expansion.

The wiggly lines represent $dW_{t}^{i,j}$. Since we are interested in the mean
behavior, we have to take the average over the noise degrees of freedom. This
amounts to contract wiggly lines pair by pair. From the Itō rules (16), we see
that upon contraction, a wiggly line forces the two vertices it connects to
have the same time and position, as illustrated in Fig. 5.

The important consequence is that all the diagrams which present a _crossing_
of the wiggly lines vanish because of the causal structure of the Keldysh’s
Green function, namely that $G^{\cal R}(t,t^{\prime})$ is non zero only for
$t>t^{\prime}$ and conversely for $G^{\cal A}$. For a detailed proof of this
statement, see Appendix B. In particular, the constraints of avoided wiggly
lines establishes the validity of the self-consistent Born approximation
(SCBA) for the self-energy of single particle Green’s function and generalize
the approach presented in Ref. [50]. SCBA allows a simple and compact
derivation of all components as exemplified by the diagrammatic representation
in Fig. 6.

Figure 6:  a) Non-crossing rule for the contraction of wiggly lines. b) Self-
energies for the different Keldysh components.

Namely, we have that in position space

$\boldsymbol{\Sigma}_{i,j}(t,t^{\prime})=\delta_{i,j}\delta(t,t^{\prime})\sum_{k}\gamma_{i,k}\boldsymbol{G}_{k,k}(t,t)\,.$
(24)

For the retarded and advanced components, this relation takes a particularly
simple form since $G_{j,k}^{\cal{R}(\cal{A})}(t,t)=\mp\frac{i}{2}\delta_{j,k}$
in position space. Note that this simple expression is only valid when the two
time indices are taken to be equal and comes entirely from the causal
structure of the Green’s functions in the Keldysh formalism. One way to see
this is to evaluate the step function $\theta(t-t^{\prime})$ for the retarded
and advanced Green’s functions from the discrete version of the path integral
presented in 9.2 of [80]. To get the Keldysh component $G^{\cal K}$, one has
to solve the self-consistent Dyson equation :

$\boldsymbol{G}^{\cal K}=-\boldsymbol{G}^{\cal
R}\left(\left[\boldsymbol{G}_{0}^{-1}\right]^{\cal
K}-\boldsymbol{\Sigma}^{\cal K}\right)\boldsymbol{G}^{\cal A}\,,$ (25)

which is a problem whose complexity only scales polynomially with the number
of degrees of freedom in the system (such as the system size $N$ of the setup
in Fig. 1). This solves the problem entirely at the level of single-particle
correlation functions. Remark that this applies to any model as long as the
bare theory respects a Wick’s theorem and its propagators are known. It allows
a systematic study of quantum systems in the presence of external noisy
degrees of freedom.

This ability to calculate the Keldysh Green’s function is crucial to give an
exact description of out-of-equilibrium transport in dissipative systems, as
we are going to show in the next section.

##  IV Applications

We now proceed to employ the self-consistent approach to showcase our $1/N$
expansion, presented in Sec. II, against a large class of QSHs that display
diffusive transport.

The action describing the out-of-equilibrium setting represented in Fig. 1 has
the form

$S=S_{{\rm Bd}}+S_{0}+S_{{\rm sto}}.$ (26)

The first term in the action, $S_{{\rm Bd}}$, describes the exchange coupling
with gapless non-interacting fermionic reservoirs of chemical potential
$\mu_{L,R}$ and temperature $T_{L,R}$. The corresponding action, under the
assumptions discussed in Section II, was derived for instance in Ref. [79]:

$\displaystyle S_{{\rm Bd}}$
$\displaystyle=i\Gamma\sum_{a=L,R}\int\frac{d\omega}{2\pi}\bar{\psi}_{a}\begin{bmatrix}1&2\tanh\left(\frac{\omega-\mu_{a}}{2T_{a}}\right)\\\
0&-1\end{bmatrix}\psi_{a}\,,$ (27)

where $\psi_{a}$ is a shorthand notation for $(\psi_{a}^{1},\psi_{a}^{2})$,
$L$ designates site 1 and $R$ designates site $N$. The action $S_{0}$ is the
quadratic action related to the intrinsic dynamics of the system, which can
describe various situations from coherent dynamics to single-particle
dissipative gains and losses [79]. In this paper, we will focus on one-
dimensional nearest-neighbour coherent bulk hopping, which is described by the
standard Hamiltonian,

$H_{\tau}:=\tau\sum_{j=1}^{N-1}\left(c_{j}^{\dagger}c_{j+1}+c_{j+1}^{\dagger}c_{j}\right),$
(28)

with $\tau$ the hopping amplitude. The corresponding action reads

$S_{0}=-i\tau\sum_{j}\int\frac{d\omega}{2\pi}\Big{(}\bar{\psi}_{j}^{1}\psi_{j+1}^{1}+\bar{\psi}_{j}^{2}\psi_{j+1}^{2}+\mbox{c.c}\Big{)}\,.$
(29)

The free propagators are directly derived from the previous expressions of the
action and read

$\displaystyle\begin{split}\left[G_{0}^{-1}\right]_{j,k}^{\cal{R}(\cal{A})}(\omega)=&\delta_{j,k}\Big{[}\omega\pm
i\Gamma(\delta_{j,1}+\delta_{j,N})\Big{]}\\\
&\qquad\qquad+\tau(\delta_{j,k+1}+\delta_{j,k-1})\,,\end{split}$ (30)
$\displaystyle\left[G_{0}^{-1}\right]_{j,k}^{\cal K}(\omega)=$ $\displaystyle
2i\Gamma\delta_{j,k}\sum_{a=L,R}\delta_{j,a}\tanh\left(\frac{\omega-\mu_{a}}{2T_{a}}\right)\,.$
(31)

Notice that the reservoirs act, through the hybridization constant $\Gamma$,
as natural regulators of the imaginary components of the non-interacting
problem [80].

Finally $S_{{\rm sto}}$ is the action corresponding to the QSH (22). As
explained in the previous section, the demonstrated validity of SCBA for the
Dyson equation (25) allows to derive exact expressions for the self-energies
(24), and thus for the propagators of the full theory. Such solution allows to
fully determine the transport properties of the system through MW formula (3).
As shown in Section III, Equation (24) implies a particularly simple form for
the advanced and retarded components of the self-energy:

$\Sigma^{\cal R(A)}_{i,j}=\mp
i\delta_{i,j}\delta(t,t^{\prime})\sum_{l}\frac{\gamma_{i,l}}{2}\,.$ (32)

Importantly, in the geometry of Fig. 1, we can derive a compact and explicit
expression of (25) for the diagonal terms $G^{\cal K}(t,t)$

$\vec{G}^{\cal K}=(\mathbb{I}-M)^{-1}\cdot\vec{V}$ (33)

where we introduced the $N$-dimensional vectors

$\displaystyle\vec{G}_{j}^{\cal K}$ $\displaystyle=G_{j,j}^{\cal K}(t,t)\,,$
(34) $\displaystyle\vec{V}_{j}$
$\displaystyle=\frac{2\Gamma}{i}\sum_{a\in\\{L,R\\}}\int\frac{d\omega}{2\pi}G_{j,a}^{\cal
R}G_{a,j}^{\cal A}\tanh\left(\frac{\omega-\mu_{a}}{2T_{a}}\right)\,,$ (35)

and $M$ is an $N\times N$ matrix with elements

$M_{j,k}=\sum_{l}\gamma_{k,l}\int\frac{d\omega}{2\pi}G_{j,l}^{R}G_{l,j}^{A}\,.$
(36)

Notice that only $G^{\mathcal{K}}$ carries information about the biased
reservoirs, as can be seen from (35). The first term in (3) depends
exclusively on spectral functions, which are readily derived from Eqs. (30)
and (32), while Eq. (33) sets, through Eq. (4), the expression of the density
differences at the edges $\Delta n$.

Note that our analysis shows that the matrix $M$ (36) is the key object
encoding information about diffusion and it appears exclusively in the Keldysh
component of the single-particle Green’s function (33). A convenient way to
understand this is to consider systems with single-particle gains and losses
that do not display Ohmic $1/N$ suppression of the current. It was shown in
Ref. [79] that, while (32) remains valid in those systems, the matrix $M$ in
(33) becomes 0 for these systems and the current saturates to a size-
independent value. Thus, having a finite-lifetime in the retarded and advanced
Green’s function is not sufficient to get diffusive transport. The imaginary
contribution to the retarded/advanced self-energy, such as the one in (32),
has the interpretation of a lifetime for the free single-particle excitations
of the system, yet it is the Keldysh component of the self-energy that
describes the consequences of dissipative scattering on the transport
properties of the system. When $M\neq 0$, equation (36) gives us a linear
profile for the density profile, which eventually leads to a $1/N$ diffusive
contribution for the current as discussed in the II.

These considerations are those underpinning our general discussion about
diffusive transport in Sec. II. We now turn to the case-by-case study of the
specific QSHs depicted in Fig. 7. As said in the Introduction, we will focus
on three one-dimensional models: the dephasing model, the quantum symmetric
simple exclusion process (QSSEP) and models with stochastic long-range
hopping. For the dephasing model, every single point on the lattice is coupled
with itself by the noise. For the QSSEP, the noise couples each point with its
neighbours. For the long-range, a given point is paired to all the rest of the
lattice with a power-law decay as a function of the distance. These processes
are illustrated for all three models in Fig. 7 and we will give more details
about their physical motivations in the related sections.

Figure 7:  Particular 1D discrete cases that will be of interest. Only the
noise contribution is presented in this figure. In the dephasing model, all
the sites are paired with themselves. For the QSSEP, the pairs are between
nearest neighbours. In the long-range model, a given point is linked to all
the rest of the lattice with a coupling decaying as power-law.

Without loss of generality, in the oncoming analysis of the current $J$, we
focus on a linear response regime in the chemical potential bias. We set an
idential temperature for both reservoirs $T_{L}=T_{R}=T$ and
$\mu_{L}\to\mu+\delta\mu,\hskip 10.00002pt\mu_{R}\to\mu-\delta\mu$. We expand
Eq. (3) in $\delta\mu$. One thus obtains, to linear order in $\delta\mu$:

$J=\Gamma\frac{\delta\mu}{2T}\int
d\omega\frac{1}{\cosh^{2}\left(\frac{\omega-\mu}{2T}\right)}\left[\mathcal{A}(\omega)-\frac{\Gamma}{2\pi}\Delta^{\mathcal{K}}(\omega)\right]\,.$
(37)

where $\mathcal{A}(\omega)$ is the edge spectral function, which coincides
with $\mathcal{A}_{L/R}(\omega)$, because of the mirror symmetry of the class
of QSHs that we will consider. The second term can be expressed in the form

$\Delta^{\mathcal{K}}(\omega)=\left[\frac{1}{\mathbb{I}-M}\cdot\vec{W}(\omega)\right]_{1}-\left[\frac{1}{\mathbb{I}-M}\cdot\vec{W}(\omega)\right]_{N}\,,$
(38)

in which $\vec{W}$ is an $N$ dimensional vector whose components are given by
$\vec{W}_{j}(\omega)=G_{j,1}^{\cal R}(\omega)G_{1,j}^{\cal
A}(\omega)-G_{j,N}^{\cal R}(\omega)G_{N,j}^{\cal A}(\omega)$.

###  IV.1 Dephasing model

The dephasing model describes fermions hopping on a 1D lattice while subject
to a random onsite dephasing coming from dissipative interactions with
external degrees of freedom. In the language of Sec. III, this model
corresponds to the case where all the points are paired with themselves, which
results in substituting the rates

$\gamma_{i,j}\rightarrow\,\gamma_{\rm Dph}\delta_{i,j}\,,$ (39)

in Eqs. (12) and (15) (see also Fig. 7). There are various limits in which
this model can be derived. For instance, it can be thought as describing the
effective dynamics of fermions interacting weakly with external bosonic
degrees of freedom within the Born-Markov approximation [38]. In Refs. [31,
81, 32] it was shown, relying on matrix product operator techniques, that the
dephasing model exhibits diffusive transport. Two-times correlators in the XXZ
under dephasing was also studied in [52] and were shown to exhibit a complex
relaxation scheme. For bosonic interacting systems, it was shown that the
addition of an external dephasing could lead to anomalous transport [100,
101]. Additionally, as discussed in Section III, the mean dynamics of this
model coincides with the one where the occupation numbers of fermions on each
site are independently and continuously monitored [102, 45]. For this reason,
the dephasing model has recently attracted a lot of interest as a prototypical
model exhibiting a measurement rate-induced transition in the entanglement
dynamics [43, 44]. Finally, we note that in Ref. [30] a mapping between the
dephasing model and the Fermi-Hubbard model was established. Although we will
not discuss this mapping here, we stress that it implies that our method also
provides the computation of exact quantities valid for equivalent systems
governed by Hubbard Hamiltonians.

The stochastic Hamiltonian for the dephasing model is readily obtained from
the substitution (39), namely

$dH_{t}=\sqrt{2\gamma_{{\rm Dph}}}\sum_{j}\hat{n}_{j}dB_{t}^{j}\,,$ (40)

where $B_{t}$ denotes a real Brownian motion with Itō rule
$dB_{t}^{j}dB_{t}^{k}=\delta_{j,k}dt$. The retarded and advanced self-energies
are obtained from Eq. (32) and read

$\Sigma_{j,k}^{\cal{R}(\cal{A})}(t,t^{\prime})=\mp\frac{i}{2}\gamma_{{\rm
Dph}}\delta_{j,k}\delta(t-t^{\prime})\,.$ (41)

while $G^{\mathcal{R},\mathcal{A}}$ are obtained by inversion of Eq. (30) with
inclusion of the self-energy (41). These functions are symmetric and given by,
for $i\leq j$ [103, 79]:

$G_{i,j}^{\cal{R}/\cal{A}}(\omega)=\frac{(-1)^{i+j}\tau^{j-i}B_{i-1}^{\cal{R}/\cal{A}}B_{N-j}^{\cal{R}/\cal{A}}}{\left[\omega\pm
i\left(\Gamma+\frac{\gamma_{{\rm
Dph}}}{2}\right)\right]B_{N-1}^{\cal{R}/\cal{A}}-\tau^{2}B_{N-2}^{\cal{R}/\cal{A}}}\,,$
(42)

where $B_{i}^{\cal{R}/\cal{A}}=[(r_{+}\pm i\Gamma)r^{i}_{+}-(r_{-}\pm
i\Gamma)r_{-}^{i}]/(r_{+}-r_{-})$ and $r_{\pm}=\left(\omega\pm
i\frac{\gamma}{2}+\sqrt{(\omega\pm i\frac{\gamma_{{\rm
Dph}}}{2})^{2}-4\tau^{2}}\right)/2$.

The related spectral functions at the system edges
$\mathcal{A}(\omega)=\mathcal{A}_{11}(\omega)=\mathcal{A}_{NN}(\omega)$ is
represented in Fig. 8 for different system sizes $N$.

Figure 8:  Edge spectral function $\mathcal{A}(\omega)$ for the dephasing
model (40) in the configuration of Fig. 1 for difference systems sizes $N$.
Darker blue solid lines correspond to larger systems sizes
$N=11,21,51,101,201,501,1001$. We consider only odd values of $N$, as they
ensure the presence of a resonance at $\omega=0$. The inset shows the
exponential convergence of the spectral function at a fixed (odd) system size
$\mathcal{A}_{N}=\mathcal{A}(\omega=0)$ towards its asymptotic value
$\mathcal{A}_{\infty}(\omega)$, obtained from Eq. (43) and corresponding to
the dashed black line in the main plot (for $N\gtrsim 100$ and the parameters
reported in the plot, numerical curves overlap with
$\mathcal{A}_{\infty}(\omega)$).

It displays $N$ peaks corresponding to the eigenspectrum of the system without
dissipation. The width of the peaks is controlled non-trivially by the
hybridization constant $\Gamma$ and the bulk dissipation rate $\gamma_{\rm
Dph}$. Plots for closely related quantities in the $\gamma_{\rm
Dph}\rightarrow 0$ limit can be found in Ref. [79]. In this nondissipative
limit, the height of the peaks does not decay with the system size $N$. On the
contrary, for $\gamma_{\rm Dph}>0$, the peaks vanish in the
$N\rightarrow\infty$ limit, and the spectral function converges exponentially
towards a smooth function $\mathcal{A}_{\infty}(\omega)$ as shown in the inset
of Fig. 8. One can analytically derive $\mathcal{A}_{\infty}(\omega)$, as the
retarded Green function (42) at the edges $G^{\cal R}_{1,1}=G^{\cal R}_{N,N}$
converges to

$\lim_{N\rightarrow\infty}G^{\cal
R}_{1,1}(\omega)=\frac{1}{\omega+i\left(\Gamma+\frac{\gamma_{\rm
Dph}}{2}\right)-\frac{\tau^{2}}{r_{\rm sgn(\omega)}}}\,.$ (43)

The exponential convergence of the edge spectral function is reproduced by all
the other QSHs discussed below and verifies one of the preliminary assumptions
exposed in Section II, identifying the density difference $\Delta n$ as the
term entirely responsible for the $1/N$ suppression of the dissipative current
in (3).

Our approach provides an efficient way to compute the second term in (37),
through an explicit derivation of the matrix $M$:

$M_{j,k}=\gamma_{\rm Dph}\int\frac{d\omega}{2\pi}G_{j,k}^{\cal R}G_{k,j}^{\cal
A}\,.$ (44)

As we detail in Appendix D, the expressions (38), (42) and (44) allow the
efficient derivation of the current (37) up to system sizes $N\simeq
10^{3-4}$. As a consequence, we can systematically study the expected
crossover from a ballistic-to-diffusive regime expected at length scales
$N^{*}\simeq\gamma_{{\rm Dph}}^{-1}$ [31]. See also Appendix E for additional
details.

Two main technical advances of our approach compared to previous studies [31,
32, 104, 26, 97, 81, 105] consist in its ability to naturally address
reservoirs with finite temperatures $T<\infty$, accessing transport regimes
left unexplored by previous studies and to access two-times correlators in the
stationary state. An important consequence of our analysis is that the
rescaled conductance of the system, that we define as
$\mathcal{G}=NJ/\delta\mu$, has a non-trivial dependence on the temperature
$T$ and the dephasing rate $\gamma_{\rm Dph}$, namely

$\mathcal{G}=\lim_{N\rightarrow\infty}\frac{JN}{\delta\mu}=\frac{\eta\tau^{\alpha+\delta}}{T^{\alpha}\gamma_{{\rm
Dph}}^{\delta}}\,.$ (45)

In Fig. 9, we plot the coefficients $(\alpha,\delta,\eta)$ across the
parameter space $(T,\gamma_{{\rm Dph}})$.

Figure 9:  Fitted parameters $(\alpha,\delta,\nu)$ of the rescaled conductance
of the dephasing model as defined in Eq. (45). These values define different
regions in the temperature - dephasing plane with different behaviors for the
conductance, see Eq. (46). The dashed lines are a guide for the eyes to
delimit the regions. The bottom right plot summarizes the characteristic
values of each region.

From the plot, we identify three main diffusive transport regimes,
$R_{\tau,T,\gamma}$, in which these coefficients are different. Note that the
regions are not connected by sharp phase transitions but instead by
crossovers, which appear sharp in logarithmic scale. Deep in the three
regions, the rescaled conductance takes the approximate values

$\mathcal{G}=\begin{cases}\frac{\tau^{2}}{T\gamma_{{\rm
Dph}}}&T\gg\gamma_{{\rm Dph}},\tau\\\ \frac{2.6\tau^{2}}{\gamma_{{\rm
Dph}}^{2}}&\gamma_{{\rm Dph}}\gg T,\tau\\\ \frac{1.3\tau}{\gamma_{{\rm
Dph}}}&\tau\gg\gamma_{{\rm Dph}},T\end{cases}\,.$ (46)

In previous studies carried in the $T\rightarrow\infty$ limit for the
reservoirs, where they can be described as Lindblad injectors [79], the
conductance $\mathcal{G}$ is assumed to be proportional to the bulk diffusion
constant $D$ [4, 21]. The density profiles in the system (see App. E) clearly
show that such interpretation cannot be extended to lower temperatures. The
emergence of coherent effects between the system and its baths leads to
finite-sized boundary effects, which do not allow the determination of the
bulk diffusion constant through Eq. (46). To obtain the bulk diffusion
constant we can use our approach to derive the density profiles _inside the
system_ and far away from its boundaries. We numerically verify Fick’s law (1)
in the bulk and find the diffusion constant to be

$D=\frac{2\tau^{2}}{\gamma_{\rm Dph}}\,,$ (47)

which is double the conductance in the $T\gg\gamma_{{\rm Dph}}$ limit, as
expected. At variance with the rescaled conductances (46), this quantity is
not affected by any boundary effect and it is in agreement with previous
analytical ansatzes, valid in the infinite temperature limit [31]. The
independence of the diffusion constant (47) from the temperature at the
boundaries is a consequence of the stochastic dephasing (40), which locally
brings the system back to an infinite temperature equilibrium state regardless
of boundary conditions. We thus see on this example that our approach allows
to compute both the two- and four-points measurements of the resistance. Even
for diffusive systems, the distinction between the two processes can be
important.

To conclude our analysis of the transport in the dephasing model, we note that
the different transport regimes in (46) explicitly depend on the _stationary
bias_ $n_{1}-n_{N}$, which suffers from boundary effects in some regions of
the $(T,\gamma_{{\rm Dph}})$ parameter space. We confirm with our exact
numerical solution that this is indeed the case. This interesting bias
dependence is beyond the scope of the present paper and left for future
studies.

#### 1/N expansion

Let us now show how the diffusion constant (47), that we obtained from our
exact solution, can also be easily derived from the novel $1/N$ perturbative
theory we introduced in Sec. II.

The first step is to fix the action of the infinite size theory $S_{\infty}$
with the aid of the coarse graining procedure. We start by disposing the
elements of $G^{\mathcal{R}/\mathcal{A}/\mathcal{K}}_{i,j}$ as a matrix and
subdivide it in square cells of width $a$. We take the average over all the
terms in the cell to obtain the effective Green function
$G^{\mathcal{R}/\mathcal{A}/\mathcal{K}}_{\tilde{i},\tilde{j}}$, describing
the correlations between the $\tilde{i}$ and $\tilde{j}$th cell. This
procedure is illustrated in Fig. 10-(right) for the retarded Green’s function
and increasing cell size ($a=1$ corresponds to no coarse graining).

Figure 10:  Coarse-graining procedure in the dephasing model,
$\gamma_{\text{Dph}}=1$ for increasing size of the cell, $a$. Left: Real and
imaginary part of the diagonal terms of $G^{\cal R}(\omega)$ for increasing
cell size, $a=1,2,3,4,5,7,12,20,40,50$, respectively from light to dark.
Inset: $G^{\cal K}$ component measured at one-third of the chain and $T=0.1$.
Black lines depict the $1/N$ predictions obtained by inverting the matrix in
Eq. (48). The symmetry around $\omega=0$ is broken as $a$ increases. Right:
Color plot of the absolute value of $G^{R}(\omega=0)$ for the first $20\times
20$ coarse grained cells of a system with $N=2000$ sites, darker colors
represent higher values

As the cell size increases,
$G^{\mathcal{R}/\mathcal{A}/\mathcal{K}}_{\tilde{i},\tilde{j}}$ becomes a
diagonal matrix with the off-diagonal terms vanishing as $1/a$ and
exponentially suppressed with the distance $|\tilde{i}-\tilde{j}|$.

This explicit calculation confirms the diagonal structure of
$G^{\mathcal{R}/\mathcal{A}/\mathcal{K}}$ and the reduction of the action to a
sum of local commuting terms $S_{\infty}=\sum_{\tilde{j}}S_{\tilde{j}}$, where
$S_{\tilde{j}}$ is the action associated to the $\tilde{j}$th cell. To
simplify the notations, we drop the tilde indices from now on and implicitly
assume that the calculations are done in the effective coarse-grained theory.
The diagonal terms of $G^{\mathcal{R}}(\omega=0)$ are depicted in Fig.
10-(left) as function of frequency with $G^{\mathcal{K}}$ shown in the inset.
As $a$ increases, the symmetry center of the functions changes to
$\omega=-2\tau$ converging to the black curves depicting Eqs. (9),(10). As
mentioned before, the only free parameters that need to be fixed in the local
theory are $\mu_{j},T_{j}$ and $\Sigma_{j}(\omega)$. For the dephasing model,
we find that the self energy is simply given by $i\gamma_{\rm Dph}/2$. For a
single site such an imaginary term was shown [79] to coincide with the
effective action of a reservoir within the limit $\mu_{j},T_{j}\to\infty$
while keeping the ratio $\mu_{j}/T_{j}$ fixed. Let $n_{j}$ be the local
density at site $j$, $n_{j}=\frac{1}{2}(1-iG^{K}(t,t))$. Using
$[G^{-1}]^{\mathcal{K}}=-G^{\mathcal{R}-1}G^{\mathcal{K}}G^{\mathcal{A}-1}$
and
$G^{\mathcal{K}}(\omega)=-\tanh{\frac{\mu_{j}}{T_{j}}}(G^{\mathcal{R}}(\omega)-G^{\mathcal{A}}(\omega))$.
Interestingly, at leading order in $1/N$, this relation turns out to be
verified even at the microscopic level, i.e for $a=1$. This tells us that the
local equilibration condition of the infinite size theory is always true in
our case. We furthermore suppose that in the coarse-grained theory, the
expression of the retarded and advanced components will be given by a single-
site two-level system, i.e we suppose the following expression for
$S_{\tilde{j}}$:

$\displaystyle S_{\tilde{j}}$
$\displaystyle=\int\frac{d\omega}{2\pi}(\bar{\psi}_{j}^{1},\bar{\psi}_{j}^{2})\begin{pmatrix}\omega+i\frac{\gamma_{\rm
Dph}}{2}&-i(2n_{j}-1)\gamma_{\rm Dph}\\\ 0&\omega-i\frac{\gamma_{\rm
Dph}}{2}\end{pmatrix}\begin{pmatrix}\psi_{j}^{1}\\\ \psi_{j}^{2}\end{pmatrix}$
(48)

Where we absorbed the $-2\tau$ shift of frequencies in the integral.
Expression (48) is valid in the bulk, independently from any value of $\mu,T$
at the boundaries. We check explicitly that the coarse-grained theory indeed
converges towards $S_{\tilde{j}}$ as $a$ is increased as shown in Fig. 10.

In the path integral formalism, the $1/N$ corrections to the current (11) is
given by

$J=i\langle\hat{J}_{j}[\bar{\psi}^{+},\psi^{+}]S_{{\rm dyn}}\rangle_{\infty}$
(49)

where $\hat{J}$ is the current operator, $\hat{J}[\bar{\psi}^{+},\psi^{+}]$ is
the evaluation of this operator in the fermionic coherent basis on the $+$
Keldysh contour, $\langle\bullet\rangle_{\infty}:=\int{\cal
D}[\psi^{\pm},\bar{\psi}^{\pm}]e^{iS_{\infty}}\bullet$ and $S_{{\rm dyn}}$ is
the Keldysh action (29) associated to the contour integral of $\hat{H}_{{\rm
dyn}}$ defined in (11). Here we have explicitly that $\hat{H}_{{\rm
dyn}}=\tau\sum_{j}c_{j}^{\dagger}c_{j+1}+\rm{h.c}$. The current operator is in
this case :

$\hat{J}_{j}=i\tau(c_{j+1}^{\dagger}c_{j}-c_{j}^{\dagger}c_{j+1}).$ (50)

A straightforward calculation reported in Appendix C then leads to an explicit
derivation of Fick’s law:

$\displaystyle J$ $\displaystyle=-\frac{2\tau^{2}}{\gamma_{{\rm Dph}}}\nabla
n_{j}.$ (51)

where $\nabla$ is the discrete gradient $\nabla n_{j}=n_{j+1}-n_{j}$. Equation
(51), derived from the $1/N$ expansion, coincides with the exact result (47)
in the whole parameter space. Such agreement validates the $1/N$ expansion as
a systematic and efficient procedure to compute diffusion constants. From the
computational point-of-view, note that the $1/N$ expansion did not resort to
any numerical schemes and provided an exact expression of the diffusive
constant, which could not be extracted explicitly from the Dyson equation
(25).

###  IV.2 QSSEP

In this section, we illustrate how our method can also be applied to the study
of the _quantum symmetric simple exclusion process_ (QSSEP) [35].

The QSSEP is a model of fermionic particles that hop on the lattice with
random amplitudes which can be thought as the quantum generalization of
classical exclusion processes [92]. Classical exclusion processes have
attracted a widespread interest over the last decades as they constitute
statistical models with simple rules but a rich behavior that is thought to be
representative of generic properties of non-equilibrium transport. It has been
particularly impactful in the formulation of the macroscopic fluctuation
theory (MFT) [91], which aims at understanding in a generic, thermodynamic
sense, macroscopic systems driven far from equilibrium. It is hoped that the
QSSEP will play a similar role in a quantum version of MFT, which is for now
largely unknown.

We are interested in a model of QSSEP plus the coherent jump Hamiltonian (28)
that was first studied in Ref. [33]. The case of pure QSSEP can be retrieved
in the limit $\tau\to 0$. As for the dephasing model discussed in Sec. IV.1,
we will see that the $1/N$ expansion formalism again offers a simple route to
derive the diffusive current.

As pictured in Fig. 7, the QSSEP couples nearest neighbour sites. It is
derived from Eqs. (12) and (15) by taking the prescription

$\gamma_{i,j}=\gamma_{\rm QS}\,\frac{\delta_{i,j+1}+\delta_{i,j-1}}{2}\,.$
(52)

The associated QSH is

$dH_{t}=\sqrt{\gamma_{\text{QS}}}\sum_{j}\left[c_{j}^{\dagger}c_{j+1}dW_{t}^{j+1,j}+c_{j+1}^{\dagger}c_{j}dW_{t}^{j,j+1}\right]\,.$
(53)

From Eq. (24), we get the advanced and retarded components of the self
energies:

$\Sigma_{j,k}^{\cal{R}(\cal{A})}(t,t^{\prime})=\mp\frac{i}{2}\gamma_{\rm
QS}\delta_{j,k}\delta(t,t^{\prime})\left[1-\frac{\delta_{1,j}+\delta_{j,N}}{2}\right]\,.$
(54)

The retarded and advanced Green functions are given by inserting the bare
propagators (30) and the self energy (54) into the Dyson equation (25) . These
propagators can be directly derived from the ones of the dephasing model by
making the substitutions $\gamma_{\rm Dph}\rightarrow\gamma_{\rm QS}$ and
$\Gamma\rightarrow\Gamma-\gamma_{\rm QS}/2$. As a consequence, all the
considerations made for the spectral function and Fig. 8, in the dephasing
model, equally apply to the QSSEP.

This is not the case for the Keldysh component, where the $M$ matrix has the
different expression 444in the previous expression, if an index is out of
boundary, it must simply be set to $0$, we don’t write that explicitly to
avoid cumbersome notation.

$\displaystyle M_{j,k}=\frac{\gamma_{{\rm
QS}}}{2}\int\frac{d\omega}{2\pi}\big{(}$ $\displaystyle G_{j,k-1}^{\cal
R}G_{k-1,j}^{\cal A}+G_{j,k+1}^{\cal R}G_{k+1,j}^{\cal A}\big{)}.$ (55)

Combining the above equation with (33) allows to obtain $G^{K}$ and allows to
compute the current from (3), or its linearized version (37). For all values
of the parameter space $(T,\gamma_{{\rm QS}})$ the current follows the
relation (see Fig. 11)

$J_{j}=-\left(\frac{\gamma_{{\rm QS}}}{2}+\frac{2\tau^{2}}{\gamma_{{\rm
QS}}}\right)\nabla n_{j}.$ (56)

which tells us that the diffusion constant is $\frac{\gamma_{{\rm
QS}}}{2}+\frac{2\tau^{2}}{\gamma_{{\rm QS}}}$ in agreement with the result
presented in [33].

Figure 11:  Diffusion constant of the QSSEP model as a function of the noise
strength, $\gamma$, for different hopping amplitudes $\tau$ and temperatures
$T$. The results are independent of the latter. The dots are obtained from the
MW formula (3) while dashed lines depict the results of the $1/N$ expansion
$\eqref{eq:currentcoherentQSSEP}$.

For $\tau=0$, this generalizes the result from [35] which was restricted to
boundaries with infinite temperature and chemical potential.

#### 1/N expansion

The expression (56) for the current can also be obtained easily in the $1/N$
perturbative approach illustrated in Sec. II. The action in the infinite size
limit is again of the form (48). From (54) we see that the expression of the
self-energy is similar to the one of the dephasing model by simply replacing
$\gamma_{\rm Dph}$ by $\gamma_{\rm QS}$ up to differences that tend to $0$ in
the infinite size limit. The current operator from site $j$ to $j+1$ in the
bulk is given here by

$\hat{J}_{j}=\frac{\gamma_{{\rm
QS}}}{2}(\hat{n}_{j}-\hat{n}_{j+1})+i\tau(c_{j+1}^{\dagger}c_{j}-c_{j}^{\dagger}c_{j+1}).$
(57)

The first part is easily evaluated to be $-\gamma_{{\rm QS}}\nabla n_{j}/2$ to
first order in $1/N$ in the diffusive limit. For the second part, we simply
need to redo the previous derivation by replacing $\gamma_{{\rm Dph}}$ by
$\gamma_{{\rm QS}}$. The term
$i\tau(c_{j+1}^{\dagger}c_{j}-c_{j}^{\dagger}c_{j+1})$ then becomes
$-\frac{2\tau^{2}}{\gamma_{\rm QS}}(n_{j+1}-n_{j})$ which yields
$\eqref{eq:currentcoherentQSSEP}$.

###  IV.3 Long-range Hopping

Finally we turn to the model with long-range hopping from the noise (see Fig.
7). In this model each particle can jump to any unoccupied site with a
probability rate that decays with the distance as a power law of exponent
$\alpha$. Power-laws appear naturally for instance in quantum simulation with
Rydberg atoms [107, 108, 109] where they emerge because of the dipole-dipole
interactions. Depending on the order of the interactions between atoms,
different power laws can be reached. In the limit $\alpha\to\infty$, we get an
“all-to-all” model, i.e there are random quantum jumps between all sites.
These types of models have recently attracted interest as toy models to
understand the interplay between quantum chaos and quantum information notably
in the context of random unitary circuits [110, 88].

For the long-range QSH we have

$\gamma_{i,j}=(1-\delta_{i,j})\frac{\gamma_{\text{LR}}}{\mathcal{N}_{\alpha}|j-k|^{\alpha}}$
(58)

and the corresponding Hamiltonian is

$dH_{t}=\sum_{j\neq
k}\sqrt{\frac{2\gamma_{\text{LR}}}{\mathcal{N_{\alpha}}|j-k|^{\alpha}}}c_{j}^{\dagger}c_{k}dW_{t}^{k,j}.$
(59)

where $\mathcal{N_{\alpha}}=2\sum_{k=1}^{N/2}k^{-\alpha}$ is a suitable
normalization condition such that $\mathcal{N}_{\infty}=2$ and
$\mathcal{N}_{0}=N$. The limiting cases of this model are the QSSEP and "all-
to-all" model, respectively $\alpha=0$ and $\alpha\rightarrow\infty$.

For the long-range hopping the expression of the retarded(advanced) self-
energy is

$\Sigma_{j,k}^{\cal
R(A)}(t,t^{\prime})=\mp\delta_{j,k}\delta(t-t^{\prime})\frac{i}{2}\sum_{l\neq
j}\frac{\gamma_{\rm LR}}{\mathcal{N}_{\alpha}|j-l|^{\alpha}}.$ (60)

As before, injecting the bare propagators (30), (31) and (60) in (25) yields
$G^{\cal R(A)}$. As illustrated in Fig. 15 in Appendix C.3, this form of the
self-energy is equivalent to the one derived for the dephasing model (41),
with the only difference that the effective dephasing rate $\gamma$ becomes
site-dependent because of the presence of boundaries connected to reservoirs .
We verified that the exponential convergence of the spectral function
illustrated in Fig. 8, equally applies, as expected, for this model as well.

The $M$ matrix is

$M_{j,k}=\sum_{l\neq
k}\int\frac{d\omega}{2\pi}G^{\mathcal{R}}_{j,l}(\omega)G^{\mathcal{A}}_{l,j}(\omega)\frac{\gamma_{\rm
LR}}{\mathcal{N}_{\alpha}|k-l|^{\alpha}}$ (61)

which combined to (33) yields $G^{K}$.

In the absence of coherent hopping, there is a simple argument to conjecture a
phase transition in the transport properties of the system at $\alpha=3$. If
one considers the stochastic process (59) alone, its average has a simple
interpretation as a classical Markov process, where the probability for a
fermion at site $0$ to jump to site $j$ during a timestep $\Delta t$, given
that the target site $j$ is empty, is $p_{j}:=\frac{\gamma_{\rm
LR}}{\mathcal{N}_{\alpha}|j|^{\alpha}}\Delta t$. For a single particle, this
defines a random walk whose variance is given by $v:=\sum_{j}p_{j}j^{2}$ which
is related to the diffusion constant via $D=v/\Delta t$. This diverges at
least logarithmically for $\alpha\leq 3$. However, note that there is no
simple reasoning to understand what happens if one were to study the model
with the coherent hopping term as, a priori, a purely classical analysis does
not hold anymore.

For the numerical computations, we fix $\gamma_{\text{LR}}=1$ and $T=1000$ but
the results are independent of the latter. In Fig. 12, we show the dependence
of the linear response current with the system size for different values of
$\alpha$.

Figure 12:  Scaling of the linear response current as a function of the system
size $N$, for varying power-law coefficients $\alpha$ in the long-range
hopping Hamiltonian (59). The saturation of $J_{\text{LRH}}$ to finite values,
for $\alpha\ll 1$, signals a ballistic regime of transport, which contrasts
with the diffusive regime observed for $\alpha\gg 1$, where $J_{\rm LRH}$
vanishes as $N^{-1}$, as highlighted by the black dashed line.

When $\alpha$ is small, the current saturates in the $N\rightarrow\infty$
limit, while for large values of $\alpha$ it decays as $N^{-1}$, as depicted
in dashed gray line. This a signature of a ballistic-to-diffusive transition
that occurs at a finite value of $\alpha$.

To characterize this transition further, we look at the order parameter
$D^{-1}=-\lim_{N\rightarrow\infty}\nabla n/J$. For diffusive systems, $D^{-1}$
is the inverse of the diffusion constant and should be zero for ballistic
systems. In App. E, we discuss the numerical fitting required to obtain
$D^{-1}$ from a finite-size scaling analysis. $D^{-1}$ undergoes a second
order phase transition at a critical power $\alpha_{c}\approx 2.87$ (see the
dark-blue dots in Fig. 13). When approaching the transition from the diffusive
region, the diffusion constant diverges as $D\sim(\alpha-\alpha_{c})^{1.21}$
(see the gray dashed line in Fig. 13). It is quite remarkable and
counterintuitive that setting $\tau\neq 0$ pushes the diffusive regime to
values of $\alpha<3$ instead of the opposite. A naive reasoning would suggest
that the addition of a coherent hopping term would push the ballistic phase to
values of $\alpha$ larger than the classical estimate ($\alpha=3$), as a
finite $\tau$ would favor the coherent propagation of single particles across
the system. We observe that the opposite is surprisingly true, and we leave
the exploration of this effect to future investigations.

#### 1/N expansion

For $\alpha>\alpha_{c}$, the $1/N$ expansion is valid and we can compute
$D^{-1}$ in the limit of infinite temperature. The action in the infinite
system size is again of the form (48) and the lifetime is fixed by (60).

Unlike the previous models, there is no simple analytic expression for the
diffusion constant since its derivation depends on the system size. We provide
a detailed derivation of the diffusive current in App. C. In Fig. 13, we
depict the results of the $1/N$ expansion for various system sizes (full
lines) and overlap them with the numeric solution of (37) (dots).

Figure 13:  Second order phase transition in the long-range hopping of
$D^{-1}=-\lim_{N\rightarrow\infty}\nabla n/J_{\text{LRH}}$ as a function of
$\alpha$ and $\gamma_{\text{LR}}=1$. Dots represent the numerical solution of
(37) while full lines depict the $1/N$ expansion’s predictions; both results
overlap. The $N\rightarrow\infty$ limit is obtained via the fitting procedure
detailed in Appendix E. The gray dashed line highlights the divergence of the
diffusion constant as $D\sim(\alpha-\alpha_{c})^{1.21}$.

Both methods agree up to machine precision which may be an indication that the
$1/N$ perturbative approach is surprisingly exact even in the ballistic
regime, $\alpha<\alpha_{c}$.

As already highlighted above, the interplay between transport and coherence
gives rise to a rich physics in the long-range hopping model, but
understanding it in depth is beyond the goals of this paper and will be
addressed in a subsequent work.

##  V Conclusion

In this work, we provided a comprehensive analysis of the large system size
properties of diffusive quantum systems driven out-of-equilibrium by boundary
reservoirs. In particular, we showed that diffusive quantum systems can be
described by an effective and simple equilibrated Gaussian theory, which
allows a systematic way to compute their diffusive transport properties via an
expansion in the inverse system size. We illustrated the correctness of our
$1/N$ expansion by comparing to exact results we obtained, using a self-
consistent Born method, for a large class of quantum stochastic Hamiltonians
which show diffusive behavior. In particular, the self-consistent approach
allowed us to explicitly derive the structure of the effective Gaussian
theory, which consists of decoupled sites with a finite lifetime and where the
effective equilibration and diffusivity is entirely encoded in the Keldysh
component of local correlations.

As an illustration of the effectiveness of our approach, we computed the
current in three models that have been of interest in the recent literature:
the dephasing model, the QSSEP and a model with stochastic long-range hopping.
For the dephasing model and the QSSEP, we illustrated the ability of our
approach to extend the study of transport to situations with boundaries at
finite temperatures and arbitrary chemical potentials. This allowed us to show
how dissipative processes restore effective infinite temperature behavior in
the bulk and explicitly derive the effective Gaussian theory via a coarse-
graining procedure. For the long-range hopping model, our analysis unveiled
that coherent hopping processes trigger diffusive behavior in regimes where
transport would be ballistic in the exclusive presence of stochastic long-
range hopping. This counter-intuitive phenomenon is a remarkable example of
the non-trivial interplay between coherent and dissipative dynamics in open
quantum systems, which could be efficiently addressed based on the self-
consistent approach.

The validity of the self-consistent Born approximation for our class of
stochastic Hamiltonians provides in principle the solution to the noisy
version of any model whose bare action is Gaussian. Our proof is not limited
by stationary behavior or by the one-dimensional geometry of the problems
addressed in this paper, but can be extended to time-dependent and higher
dimensional problems as well. This possibility opens interesting perspectives
for the investigation of novel phenomena in a large class of problems.
Extension of our approach could be devised to study quantum asymmetric
exclusion processes [111, 112, 113], spin and heat transport, the dynamics
after a quench, fluctuations on top and relaxation to stationary states and
their extensions to ladder geometries or with non-trivial topological
structure. These settings have been for the moment largely untractable, or
were solved by case by case methods, for which we provided here an unified
framework.

An important issue raised by our work consists in showing whether our
description equally holds and provides technical advantage for studying the
emergence of resistive behavior triggered by intrinsic many-body interactions
with unitary dynamics, where the breaking of integrability leads to diffusive
transport [1, 2, 3, 4, 19, 20, 21, 22, 23, 24]. A priori, the arguments
presented in Section II apply for any quantum systems which follows a local
Fick’s law and, as such, they have the potential for very broad applications.
Additionally, it is commonly accepted that the phenomenology of diffusion is
associated with integrability breaking and subsequent approach to thermal
equilibrium [114, 115, 116, 117, 118]. Understanding if and how our approach
can help make this link clearer is an exciting open question. In this respect,
we also note that, because of the existing mapping between the Fermi-Hubbard
and the dephasing model [30], the self-consistent Born approximation allows to
compute exact quantities in the Fermi-Hubbard model. As far as we know, exact
solutions for this model were only obtained in the framework of the Bethe
Ansatz and it is thus interesting that a seemingly unrelated approach allows
to obtain exact quantities as well. Whether a connection exists between the
two approaches and whether the exact summation allows to compute quantities
out of reach of the Bethe ansatz are interesting open questions.

###### Acknowledgements.

We thank L. Mazza for useful suggestions during the writing of the manuscript.
This work has been supported by the Swiss National Science Foundation under
Division II. J.F. and M.F. acknowledge support from the FNS/SNF Ambizione
Grant PZ00P2_174038. We also thank X. Turkeshi and M. Schiró for making us
aware, in the final phase of the writing of this manuscript, of their work
[105] before publication, where a study of the dephasing model from the point
of view of Green’s function has also been performed.

##  Appendix A Unraveling to continuous measurement

In this appendix, we discuss the unraveling of Eq. (12) to a quantum
stochastic differential equation describing a system under continuous
monitoring. In the Itō prescription the stochastic equation of motion of a
quantum system subject to continuous measurement of an observable
$O+O^{\dagger}$ at rate $\gamma$ is given by [95]

$d\rho={\cal
L}_{0}(\rho)+\frac{\gamma}{2}L_{O}(\rho)+\sqrt{\frac{\gamma}{2}}D_{O}(\rho)dB_{t}$
(62)

where $\mathcal{L}_{0}$ describes the dynamics in absence of measurement,
$L_{O}(\rho)=(O\rho O^{\dagger}-\frac{1}{2}(O^{\dagger}O\rho+\rho
O^{\dagger}O))$ and $D_{O}(\rho)=O\rho+\rho O^{\dagger}-\rho{\rm
tr}(O\rho+\rho O^{\dagger})$. If we assume that at each link we have two
independent measurement processes 1 and 2 with the same rate $2\gamma_{i,j}$
and $O_{1,i,j}:=c_{j}^{\dagger}c_{i}$ and $O_{2,i,j}:=ic_{j}^{\dagger}c_{i}$.
The corresponding measured observables are
$O_{1,i,j}+O_{1,i,j}^{\dagger}=c_{j}^{\dagger}c_{i}+c_{i}^{\dagger}c_{j}$ and
$O_{2,i,j}+O_{2,i,j}^{\dagger}=i(c_{j}^{\dagger}c_{i}-c_{i}^{\dagger}c_{j})$,
namely the so-called bond density and the current. It is straightforward to
see that averaging out (62), we get (12) again.

##  Appendix B Proof of the non-crossing rule

We want to prove that for all stochastic Hamiltonians of the form given by
(15), the only non vanishing diagrams in the averaged perturbative expansion
of the retarded, advanced and Keldysh Green functions are those for which
there is no crossing.

This statement only relies on the causality structure of the retarded and
advanced Green functions, i.e

$\displaystyle G^{\cal R}(t,t^{\prime})$ $\displaystyle=0\text{\text{ if
}t$<t^{\prime}$,}$ (63) $\displaystyle G^{\cal A}(t,t^{\prime})$
$\displaystyle=0\text{ if t$>t^{\prime}$}.$ (64)

Let $\langle\bullet\rangle_{0}$ denote the average with respect to a quadratic
theory. First, we remark that the causality structure of a given propagator
depends only on its incoming edge and outgoing edge, and thus

$\displaystyle G(t,t^{\prime})$
$\displaystyle:=\langle\psi^{1}(t)f[\psi^{1},\bar{\psi}^{1},\psi^{2},\bar{\psi}^{2}]\bar{\psi}^{1}(t^{\prime})\rangle_{0}=0\text{
for $t<t^{\prime}$},$ (65) $\displaystyle G^{\prime}(t,t^{\prime})$
$\displaystyle:=\langle\psi^{2}(t)f[\psi^{1},\bar{\psi}^{1},\psi^{2},\bar{\psi}^{2}]\bar{\psi}^{2}(t^{\prime})\rangle_{0}=0\text{
for $t>t^{\prime}$.}$ (66)

where $f[\psi^{1},\bar{\psi}^{1},\psi^{2},\bar{\psi}^{2}]$ is an arbitrary
polynomial in the Grassman variables coming from the expansion of the
stochastic action. This is straightforward to show starting from the action
(22): starting from an incoming full (dashed) line, one cannot switch at any
point to a dashed (full) line. Hence, the causality structure is preserved for
each line and thus for the whole propagator. Direct inspection of these
diagrams show that there cannot be any crossing when contracting the noise
terms, as it would lead to a contradiction in the time-orderings. There is
only a single one particle irreducible diagram made of a single loop. This
establishes the non-crossing result for the retarded and advanced components.

For the Keldysh components, a case by case examination of all possible
crossings that are depicted on Fig. 14 where the labels $A,B,C,D$ denote
generic product of free propagators is needed.

Figure 14:  All possible crossings for the Keldysh component of the Green’s
function with the contracted versions on the right. The red lines highlight
the part of the diagram violating the causality structure and are responsible
for making the diagram vanish.

For each one of these diagrams, there is always a subpart that shows an
incompatibility (shown in red in Fig. 14) in the time orderings causing the
whole diagram to vanish. This establishes the non-crossing result for the
Keldysh propagator.

##  Appendix C Computation of the current in the 1/N expansion

In this appendix, we compute the current in the dephasing, QSSEP and long-
range model using the perturbative theory in inverse system size presented in
Sec. II.

###  C.1 Dephasing model

For the dephasing model, the definition of the current in the bulk from site
$j$ to $j+1$ is given by

$\hat{J}_{j}=i\tau(c_{j+1}^{\dagger}c_{j}-c_{j}^{\dagger}c_{j+1}).$ (67)

The expectation value of $\hat{J}_{j}$ in the stationary state is given by

$\begin{split}J_{j}(t):=&{\rm tr}(\hat{J}_{j}\rho_{t})\\\
=&i\tau\langle\bar{\psi}_{j+1}^{+}(t)\psi_{j}^{+}(t)-\bar{\psi}_{j}^{+}(t)\psi_{j+1}^{+}(t)\rangle\\\
=&i\frac{\tau}{2}\langle\big{(}\psi_{j+1}^{1}\bar{\psi}_{j}^{1}+\psi_{j+1}^{1}\bar{\psi}_{j}^{2}+\psi_{j+1}^{2}\bar{\psi}_{j}^{2}\\\
&-(\psi_{j}^{1}\bar{\psi}_{j+1}^{1}+\psi_{j}^{1}\bar{\psi}_{j+1}^{2}+\psi_{j}^{2}\bar{\psi}_{j+1}^{2})\big{)}_{t}\rangle\end{split}$
(68)

where we used the Larkin rotation and removed the terms
$\psi^{2}\bar{\psi}^{1}$ as they are always $0$ for causality reasons.

Using the action associated to the coherent jump $S_{\tau}$

$S_{\tau}=-\tau\int
dt^{\prime}\sum_{j}\big{(}\bar{\psi}_{j}^{1}\psi_{j+1}^{1}+\bar{\psi}_{j}^{2}\psi_{j+1}^{2}+\bar{\psi}_{j+1}^{1}\psi_{j}^{1}+\bar{\psi}_{j+1}^{2}\psi_{j}^{2}\big{)}_{t^{\prime}}$
(69)

we get, from (49), to leading order in $\frac{1}{N}$:

$J_{j}(t)=\frac{\tau^{2}}{2}\int
dt^{\prime}\langle\big{(}\psi_{j+1}^{1}\bar{\psi}_{j}^{1}+\psi_{j+1}^{1}\bar{\psi}_{j}^{2}+\psi_{j+1}^{2}\bar{\psi}_{j}^{2}\\\
-(\psi_{j}^{1}\bar{\psi}_{j+1}^{1}+\psi_{j}^{1}\bar{\psi}_{j+1}^{2}+\psi_{j}^{2}\bar{\psi}_{j+1}^{2})\big{)}_{t}\\\
\big{(}\bar{\psi}_{j}^{1}\psi_{j+1}^{1}+\bar{\psi}_{j}^{2}\psi_{j+1}^{2}+\bar{\psi}_{j+1}^{1}\psi_{j}^{1}+\bar{\psi}_{j+1}^{2}\psi_{j}^{2}\big{)}_{t^{\prime}}\rangle_{\infty}$
(70)

where $\langle\rangle_{\infty}$ means the average with respect to the bare
action in the infinite size limit, where all the sites are uncorrelated.

Using Wick’s theorem and that
$\langle\psi_{j}^{a}\bar{\psi}_{j+1}^{b}\rangle_{\infty}=0$, the previous
equation greatly simplifies :

$J_{j}=-\frac{\tau^{2}}{2}\int\frac{d\omega}{2\pi}\big{(}G_{j+1,j+1}^{\cal
R}(\omega)G_{j,j}^{\cal K}(\omega)+G_{j,j}^{\cal A}(\omega)G_{j+1,j+1}^{\cal
K}(\omega)\\\ -G_{j,j}^{\cal R}(\omega)G_{j+1,j+1}^{\cal
K}(\omega)-G_{j+1,j+1}^{\cal A}(\omega)G_{j,j}^{\cal K}(\omega)\big{)}$ (71)

We can now use the bare action of individual sites (in presence of the
dephasing noise):

$S_{j}=\int\frac{d\omega}{2\pi}(\bar{\psi}_{j}^{1},\bar{\psi}_{j}^{2})\begin{pmatrix}\omega+i\frac{\gamma_{{\rm
Dph}}}{2}&-i(2n_{j}-1)\gamma_{{\rm Dph}}\\\ 0&\omega-i\frac{\gamma_{{\rm
Dph}}}{2}\end{pmatrix}\begin{pmatrix}\psi_{j}^{1}\\\
\psi_{j}^{2}\end{pmatrix}$ (72)

to obtain the explicit expression of the current

$\begin{split}J_{j}&=\tau^{2}\int\frac{d\omega}{2\pi}\bigg{(}\frac{\gamma_{{\rm
Dph}}}{(\omega^{2}+(\frac{\gamma_{{\rm
Dph}}}{2})^{2})^{2}}\bigg{)}\frac{i\gamma_{{\rm
Dph}}}{2}(2i(n_{j+1}-n_{j}))\\\ &=-\frac{2\tau^{2}}{\gamma_{{\rm Dph}}}\nabla
n_{j}\end{split}$ (73)

from which we immediately read the diffusion constant
$D=\frac{2\tau^{2}}{\gamma_{\rm Dph}}$.

###  C.2 QSSEP

For the QSSEP, the self-energy for an individual site is
$\Sigma_{j}(\omega)=\gamma_{{\rm QS}}-\frac{\gamma_{{\rm
QS}}}{2}(\delta_{j,1}+\delta_{j,N})$. The current in the bulk is given by

$\hat{J}_{j}=\frac{\gamma_{{\rm
QS}}}{2}(\hat{n}_{j}-\hat{n}_{j+1})+i\tau(c_{j}^{\dagger}c_{j+1}-c_{j+1}^{\dagger}c_{j})$
(74)

The first part of the current already scales like $1/N$ at order $0$ in the
$S_{\tau}$ expansion. The second term is evaluated in the same fashion as for
the dephasing model. This leads to

$J_{j}=-\left(\frac{\gamma_{{\rm QS}}}{2}+\frac{2\tau^{2}}{\gamma_{{\rm
QS}}}\right)\nabla n_{j}+O\left(\frac{1}{N^{2}}\right)$ (75)

and $D=\frac{\gamma_{{\rm QS}}}{2}+\frac{2\tau^{2}}{\gamma_{{\rm QS}}}$.

###  C.3 Long-range hopping

For the long-range hopping model, the local current is defined from the local
conservation equation of the particle number :

$\frac{d}{dt}\hat{n}_{j}:=\hat{J}_{j}^{{\rm inc}}-\hat{J}_{j}^{{\rm out}}$
(76)

with

$\displaystyle\hat{J}_{j}^{{\rm inc}}$
$\displaystyle=\sum_{k<j}\frac{{\cal\gamma_{{\rm LR}}}}{{\cal
N}_{\alpha}|k-j|^{\alpha}}(\hat{n}_{k}-\hat{n}_{j})+i\tau(c_{j-1}^{\dagger}c_{j}-c_{j}^{\dagger}c_{j-1}),$
(77) $\displaystyle\hat{J}_{j}^{{\rm out}}$
$\displaystyle=\sum_{k>j}\frac{{\cal\gamma_{{\rm LR}}}}{{\cal
N}_{\alpha}|k-j|^{\alpha}}(\hat{n}_{j}-\hat{n}_{k})+i\tau(c_{j}^{\dagger}c_{j+1}-c_{j+1}^{\dagger}c_{j}).$
(78)

Recall the expression of the self-energy at site $j$ (60) :

$\Sigma_{j}=\frac{\gamma_{{\rm LR}}}{2{\cal N}_{\alpha}}\sum_{k\neq
j}\frac{1}{|k-j|^{\alpha}}.$ (79)

which is depicted in Fig. 15.

Figure 15:  Dependence on the site index $j$ of the self-energy in the long-
range model for a chain of $N=100$ sites and different values of the exponent
$\alpha$ of the noise, see (79). The $\alpha\rightarrow\infty$ limit
corresponds to the QSSEP.

To get the current with the $1/N$ expansion, we take, as for the previous
model, the $0^{{\rm th}}$ order term in the first term in the expression of
the current and the first order term in the second part. We obtain:

$\displaystyle J_{j}^{{\rm inc}}=$ $\displaystyle\sum_{k<j}\frac{\gamma_{{\rm
LR}}}{{\cal N}_{\alpha}|k-j|^{\alpha}}(n_{k}-n_{j})$
$\displaystyle+\frac{2\tau^{2}}{\Sigma_{j-1}+\Sigma_{j}}(n_{j-1}-n_{j})\text{
for }j\in[2,N],$ (80) $\displaystyle\hat{J}_{j}^{{\rm out}}=$
$\displaystyle\sum_{k>j}\frac{{\cal\gamma_{{\rm LR}}}}{{\cal
N}_{\alpha}|k-j|^{\alpha}}(n_{j}-n_{k})$
$\displaystyle+i\tau(n_{j}-n_{j+1})\text{ for }j\in[1,N-1].$ (81)

For simplicity, we give in this paper only the expressions for the infinite
temperature and chemical potential boundary conditions which amount to take
Lindblad injecting and extracting terms (see [79]). The current at the
boundaries is then given by:

$\displaystyle J_{1}^{{\rm in}}$
$\displaystyle=\alpha_{L}(1-n_{1})-\beta_{L}n_{1},$ (82) $\displaystyle
J_{N}^{{\rm out}}$ $\displaystyle=-\alpha_{R}(1-n_{N})+\beta_{R}n_{N}.$ (83)

In the stationary state we have that $\forall j\in[1,N]$, $J_{j}^{{\rm
in}}=J_{j}^{{\rm out}}$ which leads to the following system of linear equation
to solve in order to get the density profile :

${\cal M}.\vec{n}=\vec{v}$ (84)

where $\vec{n}$ and $\vec{v}$ are $N$-dimensional vectors with elements
$n_{j}$ and ${\cal M}$ is an $N\times N$ matrix such that

$\begin{split}{\cal M}_{j,k}=&\frac{\gamma_{{\rm LR}}}{{\cal
N}_{\alpha}|k-j|^{\alpha}}(1-\delta_{j,k})\\\
&+\frac{2\tau^{2}}{\Sigma_{j}+\Sigma_{j+1}}(\delta_{k,j+1}-\delta_{j,k}(1-\delta_{j,N}))\\\
&+\frac{2\tau^{2}}{\Sigma_{j}+\Sigma_{j-1}}(\delta_{k,j-1}-\delta_{j,k}(1-\delta_{j,1}))\\\
&-\delta_{j,k}(\sum_{k\neq j}\frac{\gamma_{{\rm LR}}}{{\cal
N}_{\alpha}|k-j|^{\alpha}})\\\
&-\delta_{j,k}\delta_{j,1}(\alpha_{L}+\beta_{L})-\delta_{j,k}\delta_{j,N}(\alpha_{R}+\beta_{R})\end{split}$
(85)

and

$v_{j}=-\delta_{j,1}\alpha_{L}-\delta_{j,N}\alpha_{R}$ (86)

Figure 16:  Scaling of the current as a function of the system size in the
dephasing model. From left to right: $\gamma=10^{-3},10^{-1},10^{1},10^{3}$.
As the dephasing increases, diffusion sets in at smaller system sizes. The
vanishing dependence of $J$ with the temperature indicates the crossover into
the $R_{\gamma}$ region (46).

##  Appendix D Numerical implementation

In this appendix, we present some important elements of the numerical
implementation. The first step to compute any presented result is to stabilize
and efficiently evaluate $G^{\cal{R}(\cal{A})}(\omega)$ at any $\omega$. For
the case of a uniform stochastic noise (e.g. free system, dephasing), a naive
use of (42) would require evaluating the ratio of two polynomials of order
$\mathcal{O}(N)$, a notoriously difficult task for large $N$ using floating
point arithmetics. A possible solution would be to resort to arbitrary-
precision arithmetic but this would entail a heavy speed cost.

We used for the results of the present paper the fact that
$G^{\cal{R}(\cal{A})}(\omega)$ can be written as a ratio of polynomials and
therefore, decomposed into a product of monomials $G^{\cal
R}(\omega)\sim\prod_{j}(\omega-z_{j})/\prod_{i}(\omega-p_{i})$. To efficiently
find the zeros and poles of $G^{\cal R}$ 555The poles and zeros of $G^{\cal
A}$ are the conjugate of $G^{\cal R}$, we note that the inverse of $G^{\cal
R}$ is a simple tridiagonal matrix with a generic form

$T^{-1}(\omega)=\left(\begin{array}[]{cccc}\omega+a_{1}&b_{1}&0&0\\\
b_{1}^{*}&\omega+a_{2}&\ddots&0\\\ 0&\ddots&\ddots&b_{N-1}\\\
0&0&b_{N-1}^{*}&\omega+a_{N}\end{array}\right)$ (87)

whose inverse is given by [120]

$T_{i,j}(\omega)=\begin{cases}(-1)^{i+j}b_{i}...b_{j-1}\theta_{i-1}\phi_{j+1}/\theta_{L}&i<j\\\
\theta_{i-1}\phi_{j+1}/\theta_{L}&i=j\\\
(-1)^{i+j}b_{j}^{*}...b_{i-1}^{*}\theta_{j-1}\phi_{i+1}/\theta_{L}&i>j\end{cases}$
(88)

where
$\theta_{i}=(\omega+a_{i})\theta_{i-1}-\left|b_{i-1}\right|^{2}\theta_{i-2}$
and $\phi_{i}=(\omega+a_{i})\phi_{i+1}-\left|b_{i}\right|^{2}\phi_{i+2}$.
Therefore, computing the poles and zeros of $G^{R}$ requires computing all the
zeros of the sequences $\left\\{\phi_{i},\theta_{i}\right\\}_{i=0}^{L+1}$, a
task that can be done efficiently. If the matrix is invariant under a
reflection along the anti-diagonal, it is enough to compute a single sequence
instead, $\phi_{i}=\theta_{L+1-i}$. This is always the case in the models
studied in the present paper. Since $a_{i}$ does not depend on $\omega$,
$\phi_{i}$ is a polynomial of degree $i$ with the initial conditions defined
as $\phi_{0}=1$ and $\phi_{1}=\omega+a_{1}$. One can efficiently find all the
roots $\\{z_{k}\\}_{k=1}^{i}$ of $\phi_{i}$ using a Weierstrass-like recursive
method [121, 122], see Eqs. 89 and 90 for a second and fourth order scheme

$\displaystyle z_{k}^{(2)}$ $\displaystyle=z_{k}-\frac{W_{k}}{\prod_{k\neq
j}(z_{k}-z_{j})}=z_{k}-C_{k}^{(2)}$ (89) $\displaystyle z_{k}^{(4)}$
$\displaystyle=z_{k}-\frac{W_{k}}{1-\sum_{k\neq
j}\frac{W_{j}}{z_{k}-W_{k}-z_{j}}}=z_{k}-C_{k}^{(4)}$ (90) $\displaystyle
W_{k}$ $\displaystyle=\phi_{i}(z_{k})$

where $W_{k}$ is the Weierstrass weight. We chose these derivative-free
schemes to avoid computing explicit derivatives that would slow down the
computation. Choosing the correct initial condition is critical to the success
of the scheme. To find the roots of $\phi_{i}$, we initialize the scheme with
the roots of $\phi_{i-1}$ plus an extra root. We empirically found that the
extra root should have a random position close to the middle root (after
sorting by the real part) to guarantee the best convergence. This initial
choice can still fail when some roots are located very far way from the
others, which occurs for example for the model QSSEP. This happens when, at
some step in the iteration, two roots coalesce and $C_{k}^{(i)}$ diverges
strongly. In order to stablize this divergence, we introduce a damping factor
$\kappa$ that suppresses large corrections
$z_{i}^{(k)}=z_{i}-C_{i}^{(k)}e^{-\max|C_{i}^{(k)}|/\kappa}$. $\kappa$ is a
purely empirically value, which we typically take as $\kappa=\max(|b|)$. The
role of $\kappa$ is to slow down the algorithm and allow the coalescing roots
to separate. Our root-searching algorithm has thus two parts: a quick search
using a second-order damped scheme, followed by a fourth-order damped scheme
to precisely locate the roots. Once all the roots are recovered, we generate
the new matrix $\tilde{T}$ obtained from the estimates of the roots. We
consider that $\tilde{T}$ is a good estimate only when
$\max\left|T^{-1}(0)\cdot\tilde{T}^{-1}(0)\right|<10^{-10}$. With the
exception of the QSSEP, we find a typical value
$\max\left|T^{-1}(0)\cdot\tilde{T}^{-1}(0)\right|\sim 10^{-13}$ for any system
size.

Once the poles and zeros of $G^{\cal{R}(\cal{A})}(\omega)$ are computed, we
proceed to compute $G^{\cal K}$ using (33). To evaluate the $M$ matrix, we
resort to the residue theorem. If the poles of $G^{\cal{R}(\cal{A})}$ are
simple poles, the sum over residues can be computed in parallel only requiring
the evaluation of the monomials $\left\\{(\omega-z_{k})\right\\}$. We note
that while each monomial $(\omega-z_{k})$ is of order unity, a sequential
multiplication can lead to overflown errors in the limit of large $N$. To
avoid this problem, we multiply the monomials at random. If the algorithm
fails to, within machine precision, separate two roots, the residue is
computed from the contour integral instead.

The last step to compute $G^{\cal K}$ and the current $J$, is to perform the
frequency integral convoluted with $\cosh^{-2}(\frac{\omega-\mu}{2T})$. This
is done by evaluating the integral using a discrete integration scheme instead
of residue theorem. Since the thermal dependence is only encoded in the
$\cosh^{-2}(\frac{\omega-\mu}{2T})$, discretizing the integral allows us deal
with different $(T,\mu)$ values at no significant cost. We carefully verify
that the mesh is fine enough to guarantee convergence of the integral at any
$(T,\mu)$.

##  Appendix E Finite-size scaling

In this section, we detail the finite-size scaling analysis necessary to plot
Figs. 9,11 and 13.

The presence of a dephasing term is not enough to ensure that the system
behaves diffusely at any system size. Signatures of diffusive transport such
as $J\sim 1/N$, only emerge at a characteristic dephasing length, $N^{*}\sim
1/\gamma$. At short system sizes, or short time-scales, the system behaves as
if it was ballistic. In Fig. 16 we highlight this ballistic-to-diffusive
transition for different values of the dephasing and temperature in the baths.
At small dephasing values, one cannot reliably extract the diffusion constant
by fitting a a straight line to Fig. 16.

Figure 17:  Diffusion constant of the dephasing model at different
$(T,\gamma)$ values. In general, the diffusion constant decays with the
inverse system size, which we exploit to extract the $N\rightarrow\infty$
limit from a non-linear fit, dashed lines.

Instead, to extract the relevant information in the $N\to\infty$, we use the
fact that the diffusion constant has itself a $1/N$ scaling [21] when measured
in the middle of the chain. In the QSSEP and dephasing model, we use this
result to perform non-linear fits to $D$ as shown in Fig. 17.

In this figure we plot the diffusion constant of the dephasing model as
measured in the middle of the chain for increasing system sizes and different
$(T,\mu)$ values. The dashed lines depict the non-linear fit of the function
$a+b/(N+c)$ with $a,b,c$ fitting parameters. We find that most observables in
these models exhibit $1/N$ corrections as discussed in [21]. The speed of
convergence however depends on the point in the phase-space $(T,\mu)$, with
region $R_{\tau}$ (see Fig. 9) showing the slowest convergence. This is a
consequence of the effects of the bath discussed in the main text. Deep in the
$\tau$-dominated regime, we observe the breaking of Fick’s law near the edges
as shown in Fig. 18.

Since this effect only occurs in a finite portion of the system close to the
edges, the convergence is only slowed down. We thus evaluate $D$ in the middle
of the chain to mitigate its effects and get a better accuracy.

For the long-range model, one needs a different approach to obtain the
$N\rightarrow\infty$ limit correctly, especially when close to the ballistic-
diffusive transition described in the main text. A tentative form for the
finite size extrapolation is provided by the solution of the diffusion
equation for single particle under a random walk with long-range hopping
discussed in Sec. IV.3 which gives a diffusion constant
$D=H^{(\alpha-2)}_{L}$, where $H^{(r)}_{x}$ is the generalized Harmonic
number. We find that a fit
$D^{-1}=\left(aH^{(\alpha-b+1)}_{N+|c|}\right)^{-1}$, correctly captures the
finite-size dependence of $D^{-1}$ for all $\alpha$ values. The fitting
parameters $a,b,c$ respectively describe the amplitude, critical exponent and
possible finite-size corrections. In Fig. 19, we depict $D^{-1}$ against the
result of the fit, respectively dots and dashed lines. The best fitting
parameters are plotted in the inset. The quality of the fit allows us to
conjecture that, at the transition point, the diffusion constant diverges
logarithmically $D_{LR}(\alpha=\alpha_{c})\sim H^{(1)}_{N}\to\log(N)$.

Figure 18:  Density profiles of a chain of size $N=2000$ in the diffusive
regime for the different regions $R_{\tau,\gamma,T}$ shown in Fig. 9. The
breaking of Fick’s law is limited to a non-extensive number of sites near the
edge. Figure 19:  Inverse of the diffusion constant in the long-range model
for different powers of $\alpha$. Dashed lines are fits to
$D^{-1}=\left(aH^{(\alpha-b+1)}_{N+|c|}\right)^{-1}$. The results of the
fitting are depicted in the inset.

##  Appendix F Coarse-grain length $a$

In this section, we analytically estimate the coarse-grain length $a$ from the
correlation length of the dephasing model. Due to Eq.(9), it is enough to
estimate $a$ from a single Green function, in this case the retarded
component. The starting point is the analytic expression of the elements
$G^{R}_{i,j}$ in the bulk of the chain. For large systems, the boundaries
become irrelevant and the good basis of the problem is the momenta basis. In
$k$-space, the self-energy takes a diagonal form

$\displaystyle\Sigma_{k,k^{\prime}}^{R}$
$\displaystyle=\left(-\frac{i\gamma}{2}\right)\delta_{k,k^{\prime}}.$ (91)

For the QSSEP, there are cross-diagonal terms in momentum that vanish as $1/L$
and can be safely ignored. Since both self-energy and Hamiltonian are diagonal
in the momenta basis, one has

$\displaystyle G^{R}_{k,k^{\prime}}$
$\displaystyle=\delta_{k,k^{\prime}}\frac{1}{\omega-\epsilon_{k}-\Sigma_{k,k^{\prime}}},$
(92)

where $\epsilon_{k}=2\tau\cos(k)$ is the eigenenergy of the bulk Hamiltonian.
To find the retarded function in position space, we take the Fourier transform
with the continuum limit for $k$

$\displaystyle G^{R}_{r,r^{\prime}}$
$\displaystyle=\int\frac{dk}{2\pi}\frac{e^{-ik(r-r^{\prime})}}{\omega-2\tau\cos
k+i\gamma/2}.$ (93)

The integral can be solved using the residue theorem and, after some lengthy
yet simple manipulations, we find a compact formula

$\displaystyle G^{R}_{r,r^{\prime}}$
$\displaystyle=\frac{i^{|r-r^{\prime}|-1}}{2\tau\cos y}e^{iy|r-r^{\prime}|},$
(94)

where $y=\arcsin\frac{\omega+i\gamma/2}{2\tau}$ is a complex variable with
$\text{Im}(y(\omega))>0$. Therefore, in the dephasing model an estimate for
the correlation length is given by

$\displaystyle\xi=\frac{1}{\min\big{(}\text{Im}\big{(}\arcsin\frac{\omega+i\gamma/2}{2\tau}\big{)}\big{)}}=\frac{1}{\text{arcsinh}\frac{\gamma}{4\tau}},$
(95)

In the limit of small dephasing $\gamma$, we have $\xi=4\tau/\gamma$ which
serves as an estimate for the coarse-grain length $a\sim\tau/\gamma$. As
expected, $a$ should be of the order of the dephasing length $N^{*}\sim
1/\gamma$.

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