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# Dephasing-assisted transport in a tight-binding chain with a linear
potential

Samuel L. Jacob 1,∗, Laetitia P. Bettman 1, Artur M. Lacerda 1, Krissia
Zawadzki 2, Stephen R. Clark 3, John Goold 1,4, Juan José Mendoza-Arenas 3,5,6

###### Abstract

## 1

An environment interacting with a quantum system can enhance transport through
the suppression of quantum effects responsible for localization. In this
paper, we study the interplay between bulk dephasing and a linear potential in
a boundary-driven tight-binding chain. A linear potential induces Wannier-
Stark localization in the absence of noise, while dephasing induces diffusive
transport in the absence of a tilt. We derive an approximate expression for
the steady-state current as a function of both dephasing and tilt which
closely matches the exact solution for a wide range of parameters. From it, we
find that the maximum current occurs for a dephasing rate equal to the period
of Bloch oscillations in the Wannier-Stark localized system. We also find that
the current displays a maximum as a function of the system size, provided that
the total potential tilt across the chain remains constant. Our results can be
verified in current experimental platforms and represents a step forward in
analytical studies of environment-assisted transport.

##  2 Keywords:

Dephasing-assisted transport, Wannier-Stark localization

##  3 Introduction

The quantum features of an open system interacting with a macroscopic
environment are inevitably destroyed in a process known as decoherence Joos et
al. (2003); Breuer and Petruccione (2007); Ángel Rivas and Huelga (2012).
However, the notion that the environment is always detrimental for quantum
processes – such as information processing and transport – has been challenged
for more than a decade, prompted by investigations of quantum effects in
biological systems Plenio and Huelga (2008); Mohseni et al. (2008); Rebentrost
et al. (2009); Chin et al. (2010). It is now understood that the environment
can assist energy transport in non-interacting quantum systems Plenio and
Huelga (2008); Mohseni et al. (2008); Rebentrost et al. (2009); Chin et al.
(2010); Sinayskiy et al. (2012); Zerah-Harush and Dubi (2020, 2021); Alterman
et al. (2024); Ferreira et al. (2024), an effect which has been experimentally
verified in quantum networks of photons Viciani et al. (2015); Tang et al.
(2024), trapped ions Gorman et al. (2018); Maier et al. (2019) and
superconducting circuits Potočnik et al. (2018). Although several mechanisms
for environment-assisted quantum transport have been proposed and debated
Rebentrost et al. (2009); Chin et al. (2010); Zerah-Harush and Dubi (2020), a
clear mechanism is at play in localized quantum systems – when destructive
interference responsible for localization and transport suppression is
destroyed by the environment, quantum transport is enhanced. This is indeed
the expected impact of environmental coupling on Anderson and Wannier-Stark
localization. In Anderson localized systems, quantum transport is suppressed
as a consequence of lattice disorder Anderson (1958). On the other hand,
Wannier-Stark localization occurs in the presence of a linear lattice
potential such as an electric field Bloch (1929); Zener (1934); Wannier
(1962); in this case, coherent (Bloch) oscillations take place within the
region of localization Bloch (1929); Hartmann et al. (2004); van Nieuwenburg
et al. (2019); Guo et al. (2021). Despite the fact that Wannier-Stark
localized systems subject to noise have been investigated Burkhardt et al.
(2013); Bhakuni et al. (2019); Langlett and Xu (2023); Teretenkov and
Lychkovskiy (2024), the literature on this subject is still scarce.

Environment-assisted transport can also affect the non-equilibrium transport
properties of many-body quantum systems. A common approach to evidence this
effect is to drive the quantum system to a non-equilibrium steady state by
high temperature reservoirs located at the boundaries Bertini et al. (2021);
Landi et al. (2022). The dephasing effect of the environment renders the
transport diffusive, which has been shown to enhance the steady-state current
in boundary-driven systems with disorder Žnidarič (2010); Žnidarič and Horvat
(2013); Žnidarič et al. (2016) or quasi-periodicity Lacerda et al. (2021) and
to modify the transport properties of systems with long-range hopping Sarkar
et al. (2024). Moreover, Markovian boundary-driven non-interacting systems are
amenable to analytical treatment even in the presence of dephasing, with exact
and approximate solutions of steady-state quantities available for tight-
binding chains Žnidarič (2010, 2010, 2011); Žnidarič and Horvat (2013);
Turkeshi and Schiró (2021).

In this paper, we study dephasing-assisted transport in a boundary-driven,
tight-binding chain with a linear potential. We make use of a steady-state
ansatz elaborated in Refs. Žnidarič (2010, 2010, 2011); Žnidarič and Horvat
(2013) to numerically access the relevant steady-state quantities for up to
one thousand chain sites. By performing a minimal approximation, we derive an
analytical expression for the steady-state current which matches the exact
dynamics for a vast range of parameters. From it, we find that the current is
maximized at a dephasing rate equal to the period of Bloch oscillations in a
Wannier-Stark localized system. We also find that the current displays a
maximum as a function of the system size, provided that the total potential
tilt across the chain is kept constant. Our results present a significant
contribution in the analytics of environment-assisted transport and can be
experimentally verified in several platforms.

The paper is organized as follows. In Sec. 4 we introduce the model and the
steady-state ansatz used in our study. In Sec. 5 we present and discuss our
results for dephasing-assisted transport. In Sec. 6 we present the conclusion
and outlook of our work. Sec. 7 is the supplementary material, containing
technical details referenced throughout the main text.

##  4 Setup

###  4.1 The model

We study a non-interacting, one-dimensional lattice of $L$ sites with
Hamiltonian given by the $XX$ model

$\displaystyle
H=J\sum_{j=1}^{L-1}~{}(\sigma_{j}^{x}\sigma^{x}_{j+1}+\sigma_{j}^{y}\sigma^{y}_{j+1})+\sum_{j=1}^{L}\epsilon_{j}\sigma_{j}^{z}\;.$
(1)

Here, $\\{\sigma^{x}_{j},\sigma^{y}_{j},\sigma^{z}_{j}\\}_{j=1}^{L}$ are Pauli
matrices, $J$ is the hopping term and $\epsilon_{j}$ is a local field which
will be specified shortly. In order to induce a non-equilibrium state in the
system, we couple the boundary sites to high-temperature reservoirs with
different chemical potentials; in addition, each site is also exposed to its
own local reservoir which induces dephasing. All the reservoirs are assumed to
be ideal and couple weakly to the system. The dynamics of the system’s density
operator $\rho(t)$ at time $t$ is given by the Lindblad master equation Breuer
and Petruccione (2007); Ángel Rivas and Huelga (2012); Landi et al. (2022)

$\displaystyle\frac{d\rho(t)}{dt}=-\frac{i}{\hbar}[H,\rho(t)]+\sum_{\alpha=l,r,d}\mathcal{L}^{\mathrm{\alpha}}[\rho(t)]\;.$
(2)

Here the superoperator $\mathcal{L}^{\alpha}$ describes dissipation induced by
the reservoirs $\alpha=l,r,d$ corresponding to left boundary, right boundary
and dephasing, respectively. It is a sum of local Lindblad jump operators of
the form

$\displaystyle\mathcal{L}^{\alpha}~{}[\cdot]=\sum_{j,\beta}L^{\alpha\beta}_{j}\cdot
L_{j}^{\alpha\beta\dagger}-\frac{1}{2}\big{\\{}L_{j}^{\alpha\beta\dagger}L^{\alpha\beta}_{j},\cdot\big{\\}}\;,$
(3)

where $j$ labels the site acted on by the reservoir $\alpha$ and $\beta$
labels different jump operators on that same site. The effect of each boundary
reservoir ($\alpha=l,r$) is represented by two jump operators

$\displaystyle L^{l\pm}_{1}=\sqrt{\frac{\Gamma(1\pm
f)}{2}}\sigma_{1}^{\pm}\hskip 7.11317pt,\hskip
7.11317ptL^{r\pm}_{L}=\sqrt{\frac{\Gamma(1\mp f)}{2}}\sigma_{L}^{\pm}\;,$ (4)

where $\Gamma$ is the coupling rate at the boundaries, $f$ is the chemical
potential bias and $\sigma_{j}^{\pm}=(\sigma_{j}^{x}\pm i\sigma_{j}^{y})/2$.
For forward bias ($0\leq f\leq 1$) excitations are mostly created on the first
site and annihilated on the last site; the opposite reasoning applies for
reverse bias ($-1\leq f\leq 0$). In order to describe the effects of
dephasing, we consider $L$ jump operators acting on individual sites as if
each of them were coupled to its own reservoir

$\displaystyle L_{j}^{d}=\sqrt{\frac{\gamma}{2}}\sigma^{z}_{j}\hskip
7.11317pt\;,\hskip 7.11317ptj=1,2,...,L\;,$ (5)

where $\gamma$ is the dephasing rate.

The key observables in the system are the magnetization and its associated
current. We denote by $\langle A\rangle(t)\equiv\mathrm{Tr}[A\rho(t)]$ the
expectation value of an observable $A$ at time $t$. Differentiating with
respect to time and using Eq. (2) yields a continuity equation (see Sec. 7.1
for more details)

$\displaystyle\frac{d\langle\sigma^{z}_{j}\rangle(t)}{dt}$
$\displaystyle=\langle I_{j-1}\rangle(t)-\langle
I_{j}\rangle(t)+\delta_{j,1}\Gamma[f-\langle\sigma^{z}_{j}\rangle(t)]-\delta_{j,L}\Gamma[f+\langle\sigma^{z}_{j}\rangle(t)]\;\;,$
(6)

involving the expectation value of the magnetization current $\langle
I_{j}\rangle(t)$ flowing from $j$ to $j+1$, where

$\displaystyle
I_{j}=\frac{2J}{\hbar}(\sigma_{j}^{x}\sigma_{j+1}^{y}-\sigma_{j}^{y}\sigma_{j+1}^{x})\;$
(7)

is the corresponding current operator. Note that the dephasing reservoir does
not change the average magnetization and therefore does not contribute
explicitly to Eq. (6), although it still affects the average magnetization and
currents implicitly. Moreover, the boundary reservoirs do not contribute to
the expression of the current in the bulk of the chain.

Figure 1: A schematic diagram of the system under study. The lattice is
modeled as an $XX$ spin chain, characterised by a hopping amplitude $J$ and
on-site field linearly decaying in steps of $U$ across the chain. The boundary
sites are coupled to separate reservoirs that inject/eject spin excitations
controlled by a bias $f$, while in the bulk each spin is coupled to its own
local reservoir which induces dephasing.

When the steady state is reached, the left-hand side of Eqs. (2) and (6)
vanishes. We denote by $\langle
A\rangle_{\infty}\equiv\mathrm{Tr}[A\rho(\infty)]$ the expectation value of
$A$ at the steady state $\rho(\infty)$. From Eq. (6) we obtain the following
equations for the steady-state current

$\displaystyle\langle I_{1}\rangle_{\infty}$
$\displaystyle=\Gamma(f-\langle\sigma^{z}_{1}\rangle_{\infty})\;,$ (8)
$\displaystyle\langle I_{j-1}\rangle_{\infty}$ $\displaystyle=\langle
I_{j}\rangle_{\infty}\;,\hskip 42.67912ptj=2,3,\dots,L-1$ (9)
$\displaystyle\langle I_{L-1}\rangle_{\infty}$
$\displaystyle=-\Gamma(f+\langle\sigma^{z}_{L}\rangle_{\infty})\;.$ (10)

This current is uniform across the chain, with its value dictated by the
average magnetization at the boundaries; from now on, we remove the subscript
and denote it simply by $\langle I\rangle_{\infty}$. In the absence of any
bias ($f=0$), the boundary reservoirs induce an infinite temperature
(maximally-mixed) steady state $\rho(\infty)=\mathbb{I}/2^{L}$, in which the
values of magnetization and current all vanish.

###  4.2 Steady-state ansatz

In order to study the transport properties of the model at the steady state,
both numerically and analytically, we make use of a perturbative ansatz
characterized by an expansion in terms of potential bias $f$ as Žnidarič
(2010, 2011); Žnidarič and Horvat (2013)

$\displaystyle\rho$
$\displaystyle=\frac{1}{2^{L}}\big{[}\mathbb{I}+f(H+B)+\mathcal{O}(f^{2})\big{]}\;,$
(11) $\displaystyle H$
$\displaystyle=\sum_{r=1}^{L}\sum_{j=1}^{L+1-r}h^{(r)}_{j}H^{(r)}_{j}\;,$ (12)
$\displaystyle B$
$\displaystyle=\sum_{r=2}^{L}\sum_{j=1}^{L+1-r}b^{(r)}_{j}B^{(r)}_{j}\;,$ (13)

where $h_{j}^{(r)}$ and $b_{j}^{(r)}$ are expansion coefficients, and the
corresponding operators are given by

$\displaystyle H^{(1)}_{j}$ $\displaystyle=-\sigma^{z}_{j}\hskip 7.11317pt,$
$\displaystyle H^{(r)}_{j}$
$\displaystyle=\sigma^{x}_{j}Z^{(r-2)}_{j+1}\sigma^{x}_{j+r-1}+\sigma^{y}_{j}Z^{(r-2)}_{j+1}\sigma^{y}_{j+r-1}\hskip
7.11317pt,\hskip 7.11317ptr\geq 2$ $\displaystyle B^{(r)}_{j}$
$\displaystyle=\sigma^{x}_{j}Z^{(r-2)}_{j+1}\sigma^{y}_{j+r-1}-\sigma^{y}_{j}Z^{(r-2)}_{j+1}\sigma^{x}_{j+r-1}\hskip
7.11317pt,\hskip 7.11317ptr\geq 2\;.$ (14)

These operators are thus strings of Pauli matrices starting at site $j$ and
having length $r$, where
$Z^{(r-2)}_{k+1}=\sigma^{z}_{k+1}\sigma^{z}_{k+2}~{}...~{}\sigma^{z}_{k+r-3}\sigma^{z}_{k+r-2}$
are strings of Pauli $z$ matrices of length $r-2$. The defining characteristic
of this ansatz is that all operators in the expansion are orthogonal according
to the Hilbert-Schmidt inner product $\langle
A,B\rangle\equiv\mathrm{Tr}[A^{\dagger}B]$. This implies that the expectation
values of $H^{(r)}_{j}$ and $B^{(r)}_{j}$ are exactly determined within first
order in $f$. In particular, $H_{j}^{(1)}$ and $B_{j}^{(2)}$ are respectively
proportional to the magnetization and current

$\displaystyle\langle\sigma^{z}_{j}\rangle_{\infty}=-fh^{(1)}_{j}\hskip
7.11317pt,\hskip 7.11317pt\langle
I_{j}\rangle_{\infty}=f~{}\frac{4Jb^{(2)}_{j}}{\hbar}\;.$ (15)

Once we determine the first order expansion coefficients $h^{(r)}_{j}$ and
$b^{(r)}_{j}$, we have access to the exact current and magnetization profile
Žnidarič (2010, 2011); Žnidarič and Horvat (2013); Mendoza-Arenas and Clark
(2024). In order to obtain them, we insert Eq. (11) into Eq. (2) and set the
left-hand side to zero as demanded at stationarity (see Refs. Žnidarič (2010);
Žnidarič and Horvat (2013) and Sec. 7.2 for more details). This gives rise to
a set of equations for the coefficients

$\displaystyle\hbar\Gamma(1+h^{(1)}_{1})-4Jb_{1}^{(2)}=0\;,$ (16)
$\displaystyle\hbar\Gamma(1-h^{(1)}_{L})-4Jb_{L-1}^{(2)}=0\;,$ (17)
$\displaystyle b_{j}^{(2)}-b_{j-1}^{(2)}=0\;,\hskip 7.11317ptj=2,...,L-1\;,$
(18)

for $r=1$. Note that Eqs. (16)–(18) just express the fact that the current is
uniform in the steady state, with its value dictated by the boundary
reservoirs as discussed before. For $r\geq 2$ we have

$\displaystyle
J\big{[}h_{j}^{(r-1)}-h_{j+1}^{(r-1)}+h_{j}^{(r+1)}-h_{j-1}^{(r+1)}\big{]}+(\epsilon_{j}-\epsilon_{j+r-1})h_{j}^{(r)}+\hbar\Big{[}\gamma+\frac{\Gamma}{4}(\delta_{j,1}+\delta_{j+r-1,L})\Big{]}b^{(r)}_{j}=0\;,$
(19) $\displaystyle
J\big{[}b_{j}^{(r-1)}-b_{j+1}^{(r-1)}+b_{j}^{(r+1)}-b_{j-1}^{(r+1)}\big{]}+(\epsilon_{j}-\epsilon_{j+r-1})b_{j}^{(r)}-\hbar\Big{[}\gamma+\frac{\Gamma}{4}(\delta_{j,1}+\delta_{j+r-1,L})\Big{]}h^{(r)}_{j}=0\;.$
(20)

In total, Eqs. (16)–(20) are a closed set of $L^{2}$ coupled equations and
will be the starting point of our study. This set of equations is equivalent
to a Lyapunov-type equation for the correlation matrix in the presence of
dephasing Žnidarič and Horvat (2013). Instead of the exponential scaling with
the system size, the quadratic scaling makes the problem computationally
tractable. Moreover, as we show below, it also allow us to find analytical
expressions for the current under some mild assumptions.

##  5 Dephasing-assisted transport

We now discuss separately the effect of the tilt and dephasing on the
transport properties of the model, while their interplay is addressed in the
next section. The bias $f$ of each reservoir at the boundaries induces
transport along the chain. Intuitively, a similar effect is expected in the
presence of an electric field which increases/decreases at each site at a rate
$U$, or tilt between neighbour sites. If the tilt decreases from left to
right, the local potentials are

$\displaystyle\epsilon_{j}=-Uj\;.$ (21)

Setting the bias to $f=1$ drives excitations in the same direction (see Fig.
1). Note that changing the sign of $f$ inverts the sign of the current and
magnetization, while changing the sign of $U$ affects neither Mendoza-Arenas
and Clark (2024).

Since the tilt $U$ represents the energy difference between nearest neighbours
in the chain, the total tilt $V=U(L-1)$ represents the difference in energy
across the whole chain. This quantity diverges in the thermodynamic limit, so
that a physically meaningful analysis requires fixing the total tilt $V$ and
allowing $U$ to decrease with the system size $L$. As discussed in Secs. 5.2
and 5.3, this procedure introduces a non-trivial dependence of the current
with system size in the presence of dephasing: while for fixed $U$ the current
is expected to decrease with the system size $L$, this will no longer be the
case for fixed $V$.

Figure 2: (a) Steady-state magnetization profile
$\\{\langle\sigma^{z}_{j}\rangle_{\infty}\\}_{j=1}^{L}$ for system size $L=50$
and different values of total tilt $V$. The site positions are normalized
$x=j/L$. (b) Steady-state current $\langle I\rangle_{\infty}$ as a function of
system size for different values of total tilt $V$. The dashed lines represent
exponential fits $\langle I\rangle_{\infty}\propto e^{-L/L_{0}}$ to the data
with $L_{0}=4.06,1.75,1.20,0.82,0.67$ respectively for $V=5,7.5,10,15,20$.
Parameters: $\hbar=f=J=1$, $\Gamma=0.01$ and $\gamma=0$.

Such a linear potential is known to induce Wannier-Stark localization in the
absence of dephasing, with $U/\hbar$ being the frequency of Bloch oscillations
Zener (1934); Wannier (1962); Hartmann et al. (2004); Guo et al. (2021). In
the absence of dephasing $\gamma=0$ and for non-vanishing tilt $V>0$, the
steady state magnetization profile shows the formation of domain walls
Mendoza-Arenas and Clark (2024) – regions of constant magnetization
$\langle\sigma^{z}_{j}\rangle_{\infty}\simeq\pm f$ starting at the boundaries
and extending inwards as shown in Fig. 2(a). According to Eqs. (8) or (10),
this implies a vanishing current, which characterizes the localized system.
Indeed we observe in Fig. 2(b) that, as $V$ increases, the current displays an
exponential decay with the system size $\langle I\rangle_{\infty}\sim
e^{-L/L_{0}}$ with $L_{0}$ being the localization length. Note that $L_{0}$
here is different from the localization region of a single particle in a
Wannier-Stark localized system given by $J/U$ Hartmann et al. (2004). We did
not find any direct relationship between the two.

Figure 3: Steady-state magnetization profile
$\\{\langle\sigma^{z}_{j}\rangle_{\infty}\\}_{j=1}^{L}$ for system size $L=50$
and different values of dephasing $\gamma$. The site positions are normalized
$x=j/L$. The inset shows a zoom-in of the results for $\gamma=0$ and
$\gamma=0.001$. Parameters: $\hbar=f=J=1$, $V=0$.

The results in the absence of tilt $U=V=0$ are shown in Figs. 3 and 5(a),
respectively for the magnetization and the current. We distinguish two
regimes: $\gamma=0$ and $\gamma>0$. The first case corresponds to ballistic
transport, characterized by a small (but non-zero) magnetization at the
boundaries and zero bulk magnetization (inset of Fig. 3), together with a
size-independent current (see later Fig. 5(a)). The second case corresponds to
diffusive transport, characterized by an emerging linear magnetization profile
as the dephasing rate increases, accompanied by a decreasing current in
accordance with Eqs. (8) or (10). For very large dephasing or system sizes the
current scales as $\langle I\rangle_{\infty}\sim 1/(\gamma L)$ signaling
diffusive transport. The exact expression for the current in the absence of
tilt is given by Eq. (30) (with $U=V=0$) as we discuss in Sec. 5.2.

###  5.1 Exact current for small system sizes

Having shown the degrading effect of the tilt and dephasing separately on the
spin transport across the system, we now study their interplay which gives
rise to dephasing-enhanced transport. In this section, we focus on transport
for small system sizes $L=2,3$. An exact and compact solution for the current
can be found from Eqs. (16)–(20) in these cases. For $L=2$ we obtain

$\displaystyle\langle
I\rangle_{\infty}=f~{}\frac{16J^{2}\Gamma(2\gamma+\Gamma)}{16J^{2}(2\gamma+\Gamma)+4\Gamma
U^{2}+\hbar^{2}\Gamma(2\gamma+\Gamma)^{2}}\;.$ (22)

Dephasing-assisted transport can be characterised through the maximum of Eq.
(22) with respect to the dephasing rate $\gamma$, yielding an optimal
dephasing rate

$\displaystyle\gamma_{\mathrm{max}}=\frac{U}{\hbar}-\frac{\Gamma}{2}\geq 0\;.$
(23)

The behaviour of Eq. (22) is shown in Fig. 4. Note that since the dephasing is
always positive, the maximum only occurs if $U/\hbar\geq\Gamma/2$.

Figure 4: Steady-state current $\langle I\rangle_{\infty}$ for $L=2$ as a
function of dephasing $\gamma$ for different values of tilt $U=V$, computed
from Eq. (22). The large filled markers represent the maxima computed from Eq.
(23). Parameters: $\hbar=f=J=1$, $\Gamma=0.01$.

For $L=3$ we have a similar expression, namely

$\displaystyle\langle
I\rangle_{\infty}=f~{}\frac{16J^{2}\Gamma(4\gamma+\Gamma)}{16J^{2}(4\gamma+\Gamma)+16\Gamma
U^{2}+\hbar^{2}\Gamma(4\gamma+\Gamma)^{2}}\;,$ (24)

where the maximum occurs at

$\displaystyle\gamma_{\mathrm{max}}=\frac{U}{\hbar}-\frac{\Gamma}{4}\geq 0\;.$
(25)

The complexity of these expressions increases rapidly for $L>3$, so the exact
evaluation of the current and its maximum is too involved to be displayed here
for larger systems. In the following section, we resort to an approximate
solution for the current.

###  5.2 Tridiagonal approximation

In order to find an exact solution for systems of arbitrary size, we perform
the tridiagonal approximation which consists of retaining strings of Pauli
matrices up to $r=2$, that is, ignoring higher order coefficients in Eqs.
(16)–(20) so that $h^{(r)}_{j},b^{(r)}_{j}=0$ for $r\geq 3$. In the
correlation matrix approach, this is equivalent to retaining only the diagonal
and first off-diagonal elements Žnidarič and Horvat (2013). Physically, this
implies that spin and current correlations can be ignored within first order
in $f$, which allows the derivation of an exact expression for the current as
a function of all parameters as we now show. We only have to consider Eqs.
(19) and (20) for $r=2$. Summing Eqs. (19) from $j=1$ to $j=L-1$ we obtain

$\displaystyle J(h^{(1)}_{1}-h^{(1)}_{L})$
$\displaystyle+\sum_{j=1}^{L-1}(\epsilon_{j}-\epsilon_{j+1})h^{(2)}_{j}+\hbar
b[\gamma(L-1)+\Gamma/4]=0\;,$ (26)

where $b\equiv b^{(2)}_{j}$ is the coefficient associated with the current,
which is independent of $j$ as a consequence of Eq. (18). First, we can write
$h^{(1)}_{1}-h^{(1)}_{L}$ as a function of $b$ by using Eqs. (16) and (17),
which yields

$\displaystyle h^{(1)}_{1}-h^{(1)}_{L}=\frac{8Jb}{\hbar\Gamma}-2\;.$ (27)

Second, we can also express $h^{(2)}_{j}$ as a function of $b$ by using Eq.
(20), which leads to

$\displaystyle
h^{(2)}_{j}=\frac{(\epsilon_{j}-\epsilon_{j+1})b}{\hbar[\gamma+(\delta_{j,1}+\delta_{j,L-1})\Gamma/4]}\;.$
(28)

We can now substitute the previous two equations into Eq. (26) and obtain an
expression for $b$, namely

$\displaystyle
b=\frac{4J}{2\hbar\gamma(L-1)+\hbar\Gamma+\frac{16J^{2}}{\hbar\Gamma}+\frac{2}{\hbar}\sum_{j=1}^{L-1}\frac{(\epsilon_{j}-\epsilon_{j+1})^{2}}{\gamma+(\delta_{j,1}+\delta_{j,L-1})\Gamma/4}}\;.$
(29)

The current is now obtained from Eq. (15). For the local tilted potential in
Eq. (21) it becomes

$\displaystyle\langle
I\rangle_{\infty}=f~{}\frac{16J^{2}\Gamma}{16J^{2}+\hbar^{2}\Gamma[\Gamma+2\gamma(L-1)]+4\Gamma
U^{2}\begin{cases}1/(2\gamma+\Gamma)&L=2\\\
(L-3)/(2\gamma)+4/(4\gamma+\Gamma)&L\geq 3\end{cases}}\;.$ (30)

We note that this expression is exact for $L=2$ and $L=3$, in which case we
recover Eqs. (22) and (24), respectively. It is also exact in the absence of
tilt $U=V=0$ Žnidarič (2010); Žnidarič and Horvat (2013); Turkeshi and Schiró
(2021), which is verified in Fig. 5(a). In last three panels of Fig. 5 we plot
the exact solution and the tridiagonal approximation for non-zero values of
total tilt $V>0$. We observe that the approximation works remarkably well,
deviating from the exact solution only for very small dephasing rates and
system sizes; when $\gamma=0$, the approximation fails dramatically and only
the exact solution is shown. The latter case corresponds to a Wannier-Stark
localized system which we already discussed. Moreover, we see that for fixed
$\gamma$ the current can now increase with the system size up to a critical
size $L_{\mathrm{max}}$ where it is maximal, and only then decreases. In the
same way, we see that for fixed $L$ there is a critical dephasing rate
$\gamma_{\mathrm{max}}$ at which the current is maximal.

Figure 5: Steady-state current $\langle I\rangle_{\infty}$ as a function of
system size $L$ for different values of dephasing rate $\gamma$ and total tilt
$V=U(L-1)$. (a) $V=0$, (b) $V=5$, (c) $V=10$ and (d) $V=20$. The unfilled
markers represent the exact values computed from Eqs. (16)–(20); the lines
represent the tridiagonal approximation in Eq. (30). The large, filled markers
represent the maximum of the current as a function of the system size
occurring at $L_{\mathrm{max}}\simeq V/(\hbar\gamma)$. Parameters:
$\hbar=f=J=1$, $\Gamma=0.01$.

###  5.3 Interplay between dephasing and tilt

Since the validity of Eq. (30) is established, we now use it to determine both
$L_{\mathrm{max}}$ and $\gamma_{\mathrm{max}}$. The maxima of Eq. (30) as a
function of $L$ (for fixed $V$ and $\gamma$) and $\gamma$ (for fixed $L$ and
$V$) can be determined exactly. However, as before, the expressions are too
lengthy to be displayed. More physically intuitive expressions are obtained by
assuming that $L\gg 1$ and $L\gg 8\gamma/(4\gamma+\Gamma)$ in the denominator
of Eq. (30) which leads to the approximate maxima

$\displaystyle L_{\mathrm{max}}$ $\displaystyle\simeq\frac{V}{\hbar\gamma}\;,$
(31) $\displaystyle\gamma_{\mathrm{max}}$ $\displaystyle\simeq\frac{V}{\hbar
L}=\frac{U}{\hbar}\;.$ (32)

These maxima characterize dephasing-assisted transport as we show in Fig.
6(a).

Figure 6: (a) Exact steady-state current $\langle I\rangle_{\infty}$ as a
function of system size $L$ and $\gamma$ for total tilt $V=20$. The line
represents the maximum current, obtained for dephasing values dictated by Eq.
(32). The colour scheme is a visual aid representing the value of the current
increasing from blue to red. (b) Diffusion coefficient as a function of
dephasing rate $\gamma$ for different values of total tilt $V$. The large,
filled markers represent the maximum occurring at $\gamma_{\mathrm{max}}\simeq
V/(\hbar L)$. The remaining parameters are the same as in previous figures.

Below the critical length ($L_{\mathrm{max}}\gg L\gg 1$), dephasing is not yet
effective and an increase in $L$ decreases the local tilt $U=V/(L-1)$ and thus
localization, which increases the current according to Eq. (30). Above this
length scale ($L\gg L_{\mathrm{max}}\gg 1$) dephasing becomes effective and
diffusive transport clearly emerges $\langle I\rangle_{\infty}\sim 1/(\gamma
L)$. Analogously, below the critical dephasing rate
($\gamma_{\mathrm{max}}\gg\gamma$) dephasing jump events happen at a much
lower frequency than Bloch oscillations, so the current flowing remains
hampered by localization. Increasing the rate of dephasing destroys
localization and increases the current. When the dephasing rate passes this
threshold ($\gamma\gg\gamma_{\mathrm{max}}$) diffusive behaviour emerges. We
illustrate this in Fig. 6(b) where we plot the (finite size) diffusion
coefficient defined by $\langle I\rangle_{\infty}\equiv fD/L$ as a function of
dephasing rate for the largest system size we reached $L=1000$.

The imprint of dephasing-enhanced transport is also evident in the
magnetization profile, as shown in Fig. 7. Here, varying the dephasing
strength towards its critical value (32) decreases the magnetization gradient,
rendering the magnetization profile almost uniform. This is in contrast with
the results obtained in the absence of tilt, where an increasing dephasing
rate tends to increase the magnetization gradient and decrease uniformity as
shown in Fig. 3(a). Uniformity as a feature of environment-assisted transport
has been pointed out for quantum systems subject to on-site disorder Zerah-
Harush and Dubi (2020); our results confirm that the same phenomenon appears
in the presence of a linear potential inducing Wannier-Stark localization.

Figure 7: Steady-state magnetization profile
$\\{\langle\sigma^{z}_{j}\rangle_{\infty}\\}_{j=1}^{L}$ as a function of
dephasing $\gamma$ for system size $L=50$ and total tilt $V=20$. The site
positions are normalized $x=j/L$. The colour scheme is a visual aid
representing the value of the magnetization increasing from blue to red. The
remaining parameters are the same as in previous figures.

##  6 Conclusion and Outlook

In this paper, we have exploited an exact ansatz to obtain the steady state of
a non-interacting spin chain subject to bulk dephasing, a linear potential and
boundary-driving. This description has allowed us to unravel the delicate
interplay between Wannier-Stark localization and dephasing-induced diffusive
transport. We applied the tridiagonal approximation, where only on-site and
nearest-neighbour correlation matrix elements are retained, to derive an
approximate expression for the steady-state current as a function of both
dephasing and tilt. This expression is found to closely match the exact
solution for a wide range of parameters, and reveals that the maximum current
occurs for a dephasing rate equal to the period of Bloch oscillations in the
Wannier-Stark localized system. Fixing the total tilt across the system then
revealed a maximum in the current as a function of system size. This evidences
a critical system size beyond which Bloch oscillations are suppressed by
dephasing.

Our work motivates future analytical and numerical analysis on the interplay
between dephasing and a tilted potential in more complex and richer systems,
e.g. those including interactions between neighboring sites Mendoza-Arenas et
al. (2013b, a, 2014), time-periodic driving De et al. (2023, 2024), or more
realistic (non-Markovian) boundary-driving schemes Prior et al. (2010);
Tamascelli et al. (2018); Strathearn et al. (2018); Brenes et al. (2020);
Purkayastha et al. (2021); Anto-Sztrikacs et al. (2023). Furthermore, our
results can be readily verified in current photonic Ma et al. (2019), ion-trap
Morong et al. (2021) and cold-atom Krinner et al. (2017); Scherg et al.
(2021); Amico et al. (2022) experimental platforms.

## Funding

The authors declare financial support was received for the research,
authorship, and/or publication of this article. S.L.J. acknowledges the
financial support from a Marie Skłodowska-Curie Fellowship (Grant No.
101103884). S.R.C. gratefully acknowledge financial support from UK’s
Engineering and Physical Sciences Research Council (EPSRC) under grant
EP/T028424/1 which is part of an EPSRC-SFI joint project QuamNESS funded also
by the SFI under the Frontier For the Future Program. J.G. is supported by a
SFI Royal Society University Research Fellowship.

## Acknowledgements

The authors want to thank Gabriel T. Landi for useful discussions.

##  7 Supplementary Material

###  7.1 Heisenberg equation

For completeness, we show here how to obtain Eq. (6). Differentiating
$\langle\sigma^{z}_{j}\rangle(t)\equiv\mathrm{Tr}[\sigma_{j}\rho(t)]$ with
respect to time and using Eq. (2) yields

$\displaystyle\frac{d\langle\sigma^{z}_{j}\rangle(t)}{dt}=\frac{i}{\hbar}~{}\mathrm{Tr}\big{[}[H,\sigma^{z}_{j}]\rho(t)\big{]}+\sum_{\alpha=l,r,d}\mathrm{Tr}\big{[}\sigma^{z}_{j}\mathcal{L}^{\alpha}[\rho(t)]\big{]}\;.$
(33)

Throughout this section, we make use of several relations for the Pauli
matrices, which can all be deduced from their commutation and anti-commutation
relations
$[\sigma^{\alpha},\sigma^{\beta}]=2i\varepsilon_{\alpha\beta\gamma}\sigma^{\gamma}$
and
$\\{\sigma^{\alpha},\sigma^{\beta}\\}=2\delta_{\alpha\beta}\mathbb{I}_{2}$,
respectively. Here $\alpha,\beta,\gamma\in\\{x,y,z\\}$, $\mathbb{I}_{2}$ is
the identity operator for a two-level system and
$\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita symbol with the Einstein
summation notation being implicit. The commutator in Eq. (33) can then be
easily computed

$\displaystyle\frac{i}{\hbar}[H,\sigma^{z}_{j}]$
$\displaystyle=\frac{iJ}{\hbar}\sum_{k=1}^{L-1}[\sigma_{k}^{x}\sigma^{x}_{k+1}+\sigma_{k}^{y}\sigma^{y}_{k+1},\sigma^{z}_{j}]$
$\displaystyle=\frac{2J}{\hbar}(\sigma_{j-1}^{x}\sigma_{j}^{y}-\sigma_{j-1}^{y}\sigma_{j}^{x})-\frac{2J}{\hbar}(\sigma_{j}^{x}\sigma_{j+1}^{y}-\sigma_{j}^{y}\sigma_{j+1}^{x})$
$\displaystyle=I_{j-1}-I_{j}\;,$ (34)

In the first line only the kinetic energy part of the Hamiltonian is non-
vanishing. In the second line, only the pairs $(k,k+1)$ which overlap with $j$
survive. In the last line, we identified the current operator in Eq. (7).
Regarding the last term in Eq. (33), we can make progress by considering the
dual superoperator acting on $\sigma^{z}_{j}$ defined by
$\mathrm{Tr}\big{[}\sigma^{z}_{j}\mathcal{L}^{\alpha}[\rho(t)]\big{]}=\mathrm{Tr}\big{[}\mathcal{L}^{\alpha\dagger}[\sigma^{z}_{j}]\rho(t)\big{]}$
with

$\displaystyle\mathcal{L}^{\alpha\dagger}[\cdot]=\sum_{j,\beta}L^{\alpha\beta\dagger}_{j}\cdot
L_{j}^{\alpha\beta}-\frac{1}{2}\big{\\{}L_{j}^{\alpha\beta\dagger}L^{\alpha\beta}_{j},\cdot\big{\\}}\;.$
(35)

The dephasing superoperator is equal to its dual and reads explicitly

$\displaystyle\mathcal{L}^{d}[\cdot]=\frac{\gamma}{2}\sum_{j=1}^{L}(\sigma_{j}^{z}\cdot\sigma_{j}^{z}-~{}\cdot)=\mathcal{L}^{d\dagger}[\cdot]\;.$
(36)

It is straightforward to see that its action vanishes when applied to any
Pauli $z$-matrix. Therefore dephasing does not contribute to Eq. (33). For the
reservoir at the left boundary, we have the dual superoperator

$\displaystyle\mathcal{L}^{l\dagger}[\cdot]$
$\displaystyle=\frac{\Gamma(1+f)}{2}O_{1}^{l+\dagger}[\cdot]~{}+~{}\frac{\Gamma(1-f)}{2}O_{1}^{l-\dagger}[\cdot]$
(37) $\displaystyle O_{1}^{l\pm\dagger}[\cdot]$
$\displaystyle\equiv\sigma_{1}^{\mp}\cdot\sigma_{1}^{\pm}-\frac{1}{2}\\{\sigma_{1}^{\mp}\sigma_{1}^{\pm},\cdot\\}\;,$
(38)

and a similar expression holds for the right reservoir in Eq. (4). Applying
these superoperators to the Pauli z-matrix yields after some manipulation

$\displaystyle\mathcal{L}^{l\dagger}[\sigma^{z}_{j}]$
$\displaystyle=\delta_{j,1}\Gamma(f-\sigma^{z}_{1})\;,$ (39)
$\displaystyle\mathcal{L}^{r\dagger}[\sigma^{z}_{j}]$
$\displaystyle=-\delta_{j,L}\Gamma(f+\sigma^{z}_{L})\;.$ (40)

Using the last two expressions and Eq. (7.1) in Eq. (33) immediately leads to
Eq. (6).

###  7.2 Steady-state ansatz

In order to derive Eqs. (16)–(20), we insert Eq. (11) into Eq. (2) and set the
left-hand side to zero as demanded at steady state. This gives

$\displaystyle 0$
$\displaystyle=\frac{1}{2^{L}}\sum_{\alpha=l,r,d,0}\big{[}\mathcal{L}^{\mathrm{\alpha}}[\mathbb{I}]+f(\mathcal{L}^{\mathrm{\alpha}}[H]+\mathcal{L}^{\mathrm{\alpha}}[B])]\;,$
(41) $\displaystyle\mathcal{L}^{\mathrm{\alpha}}[H]$
$\displaystyle=\sum_{r=1}^{L}\sum_{j=1}^{L+1-r}h^{(r)}_{j}\mathcal{L}^{\mathrm{\alpha}}[H^{(r)}_{j}]\;,$
(42) $\displaystyle\mathcal{L}^{\mathrm{\alpha}}[B]$
$\displaystyle=\sum_{r=2}^{L}\sum_{j=1}^{L+1-r}b^{(r)}_{j}\mathcal{L}^{\mathrm{\alpha}}[B^{(r)}_{j}]\;.$
(43)

In the last expression we ignore all terms beyond first order in $f$ and
defined the superoperator for free evolution

$\displaystyle\mathcal{L}^{0}[\cdot]\equiv-\frac{i}{\hbar}[H,~{}\cdot~{}]\;.$
(44)

####  7.2.1 Action of the dephasing reservoir

We start by computing the action of the dephasing superoperator which reads
explicitly

$\displaystyle\mathcal{L}^{d}[\cdot]=\frac{\gamma}{2}\sum_{j=1}^{L}(\sigma_{j}^{z}\cdot\sigma_{j}^{z}-~{}\cdot)\equiv\frac{\gamma}{2}\sum_{j}O^{d}_{j}[\cdot]\;.$
(45)

It is straightforward to see that the action of this superoperator vanishes
when applied to $\mathbb{I}$ and $\sigma^{z}$. For $r\geq 2$, we have to
compute terms of the following form

$\displaystyle\mathcal{L}^{d}\Big{[}\sigma^{\alpha}_{j}Z^{(r-2)}_{j+1}\sigma^{\beta}_{j+r-1}\Big{]}$
$\displaystyle=\frac{\gamma}{2}\Big{[}O^{d}_{j}[\sigma^{\alpha}_{j}]~{}Z^{(r-2)}_{j+1}\sigma_{j+r-1}^{\beta}+\sigma^{\alpha}_{j}Z^{(r-2)}_{j+1}O^{d}_{j+r-1}[\sigma_{j+r-1}^{\beta}]\Big{]}\;,$
(46)

Note that the right-hand side has only two non-vanishing terms on the
boundaries since the action on $\sigma^{z}$ vanishes. We then have

$\displaystyle
O^{d}_{j}[\sigma^{\alpha}_{j}]=\sigma_{j}^{z}\sigma_{j}^{\alpha}\sigma_{j}^{z}-\sigma_{j}^{\alpha}=-2\sigma_{j}^{\alpha}(\delta_{\alpha,y}+\delta_{\alpha,x})\;.$
(47)

Using this last expression and carrying through simple algebraic manipulation,
the action of the dephasing is written as

$\displaystyle\mathcal{L}^{d}[\mathbb{I}]$
$\displaystyle=\mathcal{L}^{d}[H_{j}^{(1)}]=0\;,$ (48)
$\displaystyle\mathcal{L}^{d}[H_{j}^{(r)}]$ $\displaystyle=-2\gamma
H_{j}^{(r)}\hskip 7.11317pt,\hskip 7.11317ptr\geq 2$ (49)
$\displaystyle\mathcal{L}^{d}[B_{j}^{(r)}]$ $\displaystyle=-2\gamma
B_{j}^{(r)}\hskip 7.11317pt,\hskip 7.11317ptr\geq 2\;.$ (50)

####  7.2.2 Action of the boundary reservoirs

We first illustrate the action for the left boundary reservoir. The
superoperator is given by

$\displaystyle\mathcal{L}^{l}[\cdot]$
$\displaystyle=\frac{\Gamma(1+f)}{2}O_{1}^{l+}[\cdot]~{}+~{}\frac{\Gamma(1-f)}{2}O_{1}^{l-}[\cdot]\;,$
(51) $\displaystyle O_{1}^{l\pm}[\cdot]$
$\displaystyle\equiv\sigma_{1}^{\pm}\cdot\sigma_{1}^{\mp}-\frac{1}{2}\\{\sigma_{1}^{\mp}\sigma_{1}^{\pm},\cdot\\}\;.$
(52)

In opposition to dephasing, this superoperator does not preserve the identity

$\displaystyle O_{1}^{l\pm}[\mathbb{I}]=\pm\sigma_{1}^{z}=\mp H^{(1)}_{1}\;.$
(53)

As a consequence, its action on the Pauli matrices $\sigma_{j}^{z}$ depends on
$j$. Namely, we obtain

$\displaystyle
O_{1}^{l\pm}[\sigma_{j}^{z}]=\begin{cases}-\sigma^{z}_{1}\;,&j=1\\\
\pm\sigma_{1}^{z}\sigma^{z}_{j}\;,&j=2,3...L\;.\end{cases}$ (54)

Inserting these results into Eq. (51) we obtain the actions

$\displaystyle\mathcal{L}^{l}[\mathbb{I}]$ $\displaystyle=-\Gamma
fH^{(1)}_{1}\;,$ (55) $\displaystyle\mathcal{L}^{l}[H^{(1)}_{j}]$
$\displaystyle=\begin{cases}-\Gamma H_{1}^{(1)}&j=1\\\ f\Gamma
H_{1}^{(1)}H_{j}^{(1)}&j=2,3,\dots,L\;.\end{cases}$ (56)

Inspecting the last expression together with Eq. (41), we see that for
$j=2,3,..L$ the reservoir creates terms in the expansion which are
proportional to $f^{2}$ instead of $f$ – they represent higher order spin
correlations which do not affect the steady state values of magnetization and
current. We henceforth ignore these second order terms by considering the
action of the reservoir on Pauli matrices located at the reservoir site.
Proceeding with the calculations for $r\geq 2$, we have to compute terms of
the form

$\displaystyle\mathcal{L}^{l}\Big{[}\sigma^{\alpha}_{j}Z^{(r-2)}_{j+1}$
$\displaystyle\sigma^{\beta}_{j+r-1}\Big{]}=\delta_{j,1}\Bigg{\\{}\frac{\Gamma(1+f)}{2}O^{l+}_{1}[\sigma^{\alpha}_{1}]+\frac{\Gamma(1-f)}{2}O^{l-}_{1}[\sigma^{\alpha}_{1}]\Bigg{\\}}~{}Z^{(r-2)}_{j+1}\sigma^{\beta}_{j+r-1}\;.$
(57)

It is easy to verify that

$\displaystyle
O^{l\pm}_{1}[\sigma^{\alpha}_{1}]=-\frac{\sigma_{1}^{\alpha}}{2}\hskip
7.11317pt,\hskip 7.11317pt\alpha=x,y\;.$ (58)

Using the last two expressions and carrying out the algebra, the results for
the left reservoir to first order in $f$ are

$\displaystyle\mathcal{L}^{l}[\mathbb{I}]$ $\displaystyle=-\Gamma
fH^{(1)}_{1}\;,$ (59) $\displaystyle\mathcal{L}^{l}[H^{(1)}_{j}]$
$\displaystyle=-\delta_{j,1}\Gamma H^{(1)}_{j}\;,$ (60)
$\displaystyle\mathcal{L}^{l}[H^{(r)}_{j}]$
$\displaystyle=-\delta_{j,1}\frac{\Gamma}{2}H_{j}^{(r)}\hskip 7.11317pt,\hskip
7.11317ptr\geq 2\;,$ (61) $\displaystyle\mathcal{L}^{l}[B^{(r)}_{j}]$
$\displaystyle=-\delta_{j,1}\frac{\Gamma}{2}B_{j}^{(r)}\hskip 7.11317pt,\hskip
7.11317ptr\geq 2\;.$ (62)

Proceeding the same way for the right boundary reservoir we obtain the result

$\displaystyle\mathcal{L}^{r}[\mathbb{I}]$ $\displaystyle=\Gamma
fH^{(1)}_{L}\;,$ (63) $\displaystyle\mathcal{L}^{r}[H^{(1)}_{j}]$
$\displaystyle=-\delta_{j,L}\Gamma H^{(1)}_{j}\;,$ (64)
$\displaystyle\mathcal{L}^{r}[H^{(r)}_{j}]$
$\displaystyle=-\delta_{j+r-1,L}\frac{\Gamma}{2}H_{j}^{(r)}\hskip
7.11317pt,\hskip 7.11317ptr\geq 2\;,$ (65)
$\displaystyle\mathcal{L}^{r}[B^{(r)}_{j}]$
$\displaystyle=-\delta_{j+r-1,L}\frac{\Gamma}{2}B_{j}^{(r)}\hskip
7.11317pt,\hskip 7.11317ptr\geq 2\;.$ (66)

####  7.2.3 Action of Hamiltonian

Regarding the Hamiltonian superoperator in Eq. (44), it is easy to see that it
preserves the identity $\mathcal{L}^{0}[\mathbb{I}]=0$ while
$\mathcal{L}^{0}[H_{j}^{(1)}]$ yields the well-known result

$\displaystyle\mathcal{L}^{0}[H_{j}^{(1)}]=-\frac{2J}{\hbar}(B^{(2)}_{j}-B^{(2)}_{j-1})\;.$
(67)

This result was already implicitly used in Eq. (7.1) and simply expresses that
the change in magnetization at a given site is due to differences between
incoming and outgoing currents at that site. For $r\geq 2$, it is useful to
write the superoperator as a sum of two contributions
$\mathcal{L}^{0}\equiv\mathcal{L}^{0_{1}}+\mathcal{L}^{0_{2}}$, where

$\displaystyle\mathcal{L}^{0_{1}}\cdot$
$\displaystyle\equiv-\frac{iJ}{\hbar}\sum_{j=1}^{L-1}~{}\Big{\\{}[\sigma_{j}^{x}\sigma_{j+1}^{x},\cdot]+[\sigma_{j}^{y}\sigma_{j+1}^{y},\cdot]\Big{\\}}\;,$
(68) $\displaystyle\mathcal{L}^{0_{2}}\cdot$
$\displaystyle\equiv-\frac{i}{\hbar}\sum_{j=1}^{L}~{}\epsilon_{j}[\sigma_{j}^{z},\cdot]\;.$
(69)

We compute only the action of $\mathcal{L}^{0_{1}}$ and the one for
$\mathcal{L}^{0_{2}}$ is obtained in a similar manner. We first evaluate
operators for $r=2$ which contain no Pauli $z$-strings (see Eq. (4.2) for the
exact expressions). It involves computing terms of the form

$\displaystyle\sum_{j=1}^{L-1}[\sigma_{j}^{\gamma}\sigma_{j+1}^{\gamma},\sigma_{k}^{\alpha}\sigma_{k+1}^{\beta}]$
$\displaystyle=\sigma^{\gamma}_{k-1}[\sigma_{k}^{\gamma},\sigma_{k}^{\alpha}]\sigma_{k+1}^{\beta}+[\sigma_{k}^{\gamma}\sigma_{k+1}^{\gamma},\sigma_{k}^{\alpha}\sigma_{k+1}^{\beta}]+\sigma^{\alpha}_{k}[\sigma_{k+1}^{\gamma},\sigma_{k+1}^{\beta}]\sigma_{k+2}^{\gamma}$
$\displaystyle=2i~{}[\varepsilon_{\gamma\alpha\omega}\sigma_{k-1}^{\gamma}\sigma_{k}^{\omega}\sigma_{k+1}^{\beta}+\varepsilon_{\alpha\beta\omega}(\delta_{\gamma\alpha}\sigma_{k+1}^{\omega}-\delta_{\gamma\beta}\sigma_{k}^{\omega})+\varepsilon_{\gamma\beta\omega}\sigma_{k}^{\alpha}\sigma_{k+1}^{\omega}\sigma_{k+2}^{\gamma}]\;.$
(70)

In the first line, the only non-vanishing terms are for those pairs $(j,j+1)$
that partially or completely overlap with $(k,k+1)$. In the second line, we
used the standard properties of the Pauli matrices. The last expression is
enough to compute the action of $\mathcal{L}^{0_{1}}$ on $H^{(2)}_{k}$ and
$B^{(2)}_{k}$ which reads

$\displaystyle\mathcal{L}^{0_{1}}[H^{(2)}_{j}]$
$\displaystyle=-\frac{2J}{\hbar}(B_{j}^{(3)}-B_{j-1}^{(3)})\;,$ (71)
$\displaystyle\mathcal{L}^{0_{1}}[B^{(2)}_{j}]$
$\displaystyle=-\frac{4J}{\hbar}(H_{j+1}^{(1)}-H_{j}^{(1)})-\frac{2J}{\hbar}(H_{j}^{(3)}-H_{j-1}^{(3)})\;.$
(72)

For $r\geq 3$, the operators in Eq. (4.2) have Pauli-$z$ strings. We need to
compute terms of the form

$\displaystyle\sum_{j=1}^{L-1}$
$\displaystyle[\sigma_{j}^{\gamma}\sigma_{j+1}^{\gamma},\sigma_{k}^{\alpha}Z^{(r-2)}_{k+1}\sigma_{k+r-1}^{\beta}]=\sigma_{k-1}^{\gamma}[\sigma_{k}^{\gamma},\sigma_{k}^{\alpha}]Z^{(r-2)}_{k+1}\sigma_{k+r-1}^{\beta}+[\sigma_{k}^{\gamma}\sigma_{k+1}^{\gamma},\sigma_{k}^{\alpha}\sigma_{k+1}^{z}]Z^{(r-2)}_{k+2}\sigma_{k+r-1}^{\beta}$
$\displaystyle+\sigma_{k}^{\alpha}Z^{(r-2)}_{k+1}[\sigma^{\gamma}_{k+r-1},\sigma^{\beta}_{k+r-1}]\sigma^{\gamma}_{k+r}+\sigma_{k}^{\alpha}Z^{(r-3)}_{k+1}[\sigma_{k+r-2}^{\gamma}\sigma_{k+r-1}^{\gamma},\sigma_{k+r-2}^{z}\sigma_{k+r-1}^{\beta}]\;,$
$\displaystyle=2i~{}\big{[}\varepsilon_{\gamma\alpha\omega}\sigma_{k-1}^{\gamma}\sigma_{k}^{\omega}Z^{(r-2)}_{k+1}\sigma_{k+r-1}^{\beta}+\varepsilon_{\gamma\beta\omega}\sigma_{k}^{\alpha}Z^{(r-2)}_{k+1}\sigma_{k+r-1}^{\omega}\sigma_{k+r}^{\gamma}$
$\displaystyle+(\varepsilon_{\gamma\alpha\omega}\delta_{\gamma
z}\sigma_{k}^{\omega}+\varepsilon_{\gamma
z\omega}\delta_{\gamma\alpha}\sigma_{k+1}^{\omega})Z^{(r-3)}_{k+2}\sigma_{k+r-1}^{\beta}+\sigma_{k}^{\alpha}Z^{(r-3)}_{k+1}(\varepsilon_{\gamma
z\omega}\delta_{\gamma\beta}\sigma_{k+r-2}^{\omega}+\varepsilon_{\gamma\beta
z}\delta_{\gamma z}\sigma_{k+r-1}^{\omega})\big{]}\;.$ (73)

Again, the only non-vanishing terms are for those pairs $(j,j+1)$ that
partially or completely overlap with the string $(k,k+1,...k+r-1)$. No
commutator appears containing pairs of $\sigma_{z}$ since these also vanish.
The last expression suffices to compute the action of $\mathcal{L}^{0_{1}}$ on
$H^{(r)}_{k}$ and $B^{(r)}_{k}$ for $r\geq 3$ and a similar reasoning holds
for $\mathcal{L}^{0_{2}}$. We summarize the results below:

$\displaystyle\mathcal{L}^{0}[\mathbb{I}]$ $\displaystyle=0\;,$ (74)
$\displaystyle\mathcal{L}^{0}[H_{j}^{(1)}]$
$\displaystyle=-\frac{2J}{\hbar}(B^{(2)}_{j}-B^{(2)}_{j-1})\;,$ (75)
$\displaystyle\mathcal{L}^{0}[B_{j}^{(2)}]$
$\displaystyle=\frac{4J}{\hbar}(H_{j}^{(1)}-H_{j+1}^{(1)})+\frac{2J}{\hbar}(H_{j}^{(3)}-H_{j-1}^{(3)})+\frac{2(\epsilon_{j}-\epsilon_{j+1})}{\hbar}H_{j}^{(2)}\;,$
(76) $\displaystyle\mathcal{L}^{0}[H_{j}^{(r)}]$
$\displaystyle=-\frac{2J}{\hbar}\big{[}(B_{j}^{(r-1)}-B_{j+1}^{(r-1)})+(B_{j}^{(r+1)}-B_{j-1}^{(r+1)})\big{]}-\frac{2(\epsilon_{j}-\epsilon_{j+r-1})}{\hbar}B_{j}^{(r)}\hskip
7.11317pt,\hskip 7.11317ptr\geq 3\;,$ (77)
$\displaystyle\mathcal{L}^{0}[B_{j}^{(r)}]$
$\displaystyle=\frac{2J}{\hbar}\big{[}(H_{j}^{(r-1)}-H_{j+1}^{(r-1)})+(H_{j}^{(r+1)}-H_{j-1}^{(r+1)})\big{]}+\frac{2(\epsilon_{j}-\epsilon_{j+r-1})}{\hbar}H_{j}^{(r)}\hskip
7.11317pt,\hskip 7.11317ptr\geq 3\;.$ (78)

With the action of all superoperators computed and using Eq. (41) we obtain
the set of Eqs. (16)–(20) for the coefficients used in the main text.

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