# Heat current rectification in segmented $XXZ$ chains

Vinitha Balachandran  Science and Math Cluster and EPD Pillar, Singapore
University of Technology and Design, 8 Somapah Road, 487372 Singapore
Giuliano Benenti  Center for Nonlinear and Complex Systems, Dipartimento di
Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via Valleggio
11, 22100 Como, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di
Milano, via Celoria 16, 20133 Milano, Italy  NEST, Istituto Nanoscienze-CNR,
I-56126 Pisa, Italy  Emmanuel Pereira  Departamento de Física–Instituto de
Ciências Exatas, Universidade Federal de Minas Gerais, CP 702, 30.161-970 Belo
Horizonte MG, Brazil  Giulio Casati  Center for Nonlinear and Complex Systems,
Dipartimento di Scienza e Alta Tecnologia, Università degli Studi
dell’Insubria, via Valleggio 11, 22100 Como, Italy  International Institute of
Physics, Federal University of Rio Grande do Norte, 1613 Natal, Brazil  Dario
Poletti  Science and Math Cluster and EPD Pillar, Singapore University of
Technology and Design, 8 Somapah Road, 487372 Singapore

###### Abstract

We study the rectification of heat current in an $XXZ$ chain segmented in two
parts. We model the effect of the environment with Lindblad heat baths. We
show that, in our system, rectification is large for strong interactions in
half of the chain and if one bath is at cold enough temperature. For the
numerically accessible chain lengths, we observe that the rectification
increases with the system size. We gain insight in the rectification mechanism
by studying two-time correlations in the steady state. The presence of
interactions also induces a strong nonlinear response to the temperature
difference, resulting in superlinear and negative differential conductance
regimes.

##  I Introduction

Control of nanoscale heat transport can lead to promising applications in
thermal waste management. An important step in this direction is devising heat
rectifiers (diodes) that allow heat to flow in a preferential direction.
Possible pathways to thermal rectification were proposed in classical systems
exploiting tailored transport in asymmetric non-linear chains
TerraneoCasati2002 ; LiCasati2004 ; Baowen2012 ; Benenti2016 ; LeiLi2011 ;
Bagchi2013 ; KaushikMarathe2018 . In such systems, the rectification mechanism
stems from a temperature dependent mismatch in the power spectrum of portions
of the nonlinear chain. These theoretical results were confirmed in pioneering
experimental investigations of thermal rectification of phonon transport Chang
; Kobayashi .

Investigations of rectifiers have been done also at the quantum level using
other energy carriers such as photons Ruokola , electrons Kuo ; Ren ; Perez
and spins Segal2005 ; Segal ; Segal2009 ; ArracheaAligia2009 ;
LandiKarevski2014 ; Yan2011 ; Flores ; Chen . In Segal ; Segal2009 ,
asymmetric quantum structures and hybrid material junctions are identified as
the main ingredients for thermal rectification. From this perspective, spin-
boson models (spin chains connected to bosonic baths) have been extensively
studied Yan ; LifaLi2009 ; WerlangValente2014 ; MotzAnkerhold2018 . In
MotzAnkerhold2018 , weak anharmonicities were used to generate some heat
current rectification. In WerlangValente2014 , it was shown that two spins
coupled via Ising interaction can work as a perfect heat rectifier in the
regime of strong spin-spin coupling. However, studies in longer chains
observed moderate rectification. In short, an effective way to construct a
well-performing heat diode for larger systems is an open challenge. This quest
is of utmost practical relevance, since it is difficult to apply large
temperature biases on systems of small size size_note .

Recently, it was shown in BalachandranPoletti2018  that large interactions in
a segmented $XXZ$ chain can produce strong spin current rectification when the
system is coupled to magnetization baths (and not heat baths). In particular,
the authors showed that when the system was biased in one direction a spin
current could flow diffusively, but when the system was biased in the opposite
direction, and the interactions strong enough, the current was strongly
suppressed and the system became an insulator. The mechanism at the basis of
such dynamics was recognized to be the emergence of an excitation gap only in
reverse bias and for strong enough interactions. If a tunnelling between the
two halves of a segmented chain is possible at no energy cost, then a current
is generated. If however a tunnelling between the two halves can occur only
after overcoming an energy gap, then the excitation generated at the interface
is localized and no current is generated. The rectification improves with the
system size, and both numerical results and theoretical arguments suggest a
perfect spin diode in the thermodynamic limit. It was also noticed, in
BalachandranPoletti2018 , that rectification is significantly stronger when
one bath tends to polarize the chain, so that all spins point down (or up).
The translation of the above results to thermal rectification is not trivial
as (i) one should consider heat instead of spin baths and (ii) it is unclear
whether the heat rectification effect remains sizeable when the temperature of
the cold heat bath is not close to zero temperature.

In this work, we show that a segmented $XXZ$ chain can be a well-performing
heat diode. We find that large rectification can be observed when the
interaction is strong enough, even for not excessively cold temperatures
(i.e., comparable with the other energy scales in the model). We characterize
the dependence of rectification on the interactions, interface coupling and
baths temperatures, and discuss in which scenarios the rectification is
stronger. For the chain lengths accessible to numerical investigations, we
observe that the rectification increases with the system size. To gain a
deeper insight of the physical mechanism behind thermal rectification, we also
analyse the two-time correlations of a tunneling excitation in the middle of
the chain. We observe that only in reverse bias the frequency response of the
system depends significantly on the magnitude of the interaction. Finally, we
characterize the nonlinear response of the system to temperature differences.
In this respect we show that it is possible to observe both superlinear and
negative differential conductance, the latter being a key feature for building
up a thermal transistor.

This paper is organized as follows: in Sec.II we introduce the model, in
Sec.III we present our results and in Sec.IV we draw our conclusions.

##  II Model

We study a spin-$1/2$ chain segmented in two parts, described by the $XXZ$
Hamiltonian

$\displaystyle\hat{H}=\hat{H}_{L}+\hat{H}_{R}+\hat{h}_{N/2}(J_{N/2},0),$ (1)
$\displaystyle\hat{H}_{L}=\\!\\!\sum_{n=1}^{N/2-1}\hat{h}_{n}(J,\Delta),\;\;\;\;\hat{H}_{R}=\\!\\!\sum_{n=N/2+1}^{N}\hat{h}_{n}(J,0),$
(2)
$\displaystyle\hat{h}_{n}(J,\Delta)=J(\hat{\sigma}_{n}^{x}\hat{\sigma}_{n+1}^{x}+\hat{\sigma}_{n}^{y}\hat{\sigma}_{n+1}^{y})+\Delta\hat{\sigma}_{n}^{z}\hat{\sigma}_{n+1}^{z},$
(3)

where $J$ and $\Delta$ are the magnitudes of the $XX$ tunneling and the $ZZ$
coupling, $\hat{\sigma}_{n}^{\alpha}$, with $\alpha=x,y,z$ are the Pauli
matrices for the $n$-th spin and $N$ is the (even) total number of sites in
the chain. The ratio $\Delta/J$ signals the strength of the interaction and it
is also referred to as the anisotropy parameter. $J_{N/2}$ is the magnitude of
the coupling between the two half-chains, and, as we will show later, plays an
important role in determining the system’s response.

The chain is coupled to heat baths at different temperatures at its edges. The
effect of the heat baths on the system is modeled by a Lindblad master
equation Lindblad ; GoriniSudarshan ,

$\displaystyle\frac{d\hat{\rho}}{dt}=-\frac{{\rm
i}}{\hbar}[\hat{H},\hat{\rho}]+\mathcal{D}_{1}(\hat{\rho})+\mathcal{D}_{N}(\hat{\rho})=\mathcal{L}(\hat{\rho}),$
(4)

where $\hat{\rho}$ is the density operator of the system, $\hbar$ is the
(reduced) Planck constant, and $\mathcal{D}_{1}$ ($\mathcal{D}_{N}$) is the
dissipator due to the coupling of the first (last) site with the left (right)
bath. Each dissipator is given by _global_ operators, that is operators that,
in general, act on the full system (PetruccioneBreuer2002, ). More
specifically,

$\displaystyle\mathcal{D}_{n}(\hat{\rho})=$
$\displaystyle\sum_{\omega>0}J(\omega)\left\\{\left[1+n_{n}(\omega)\right]\left[\hat{A}_{n}(\omega)\hat{\rho}\hat{A}_{n}^{\dagger}(\omega)\right.\right.$
$\displaystyle\left.-1/2\left(\hat{A}_{n}^{\dagger}(\omega)\hat{A}_{n}(\omega)\hat{\rho}+\hat{\rho}\hat{A}_{n}^{\dagger}(\omega)\hat{A}_{n}(\omega)\right)\right]$
$\displaystyle+n_{n}(\omega)\left[\hat{A}_{n}^{\dagger}(\omega)\hat{\rho}\hat{A}_{n}(\omega)\right.$
$\displaystyle\left.\left.-1/2\left(\hat{A}_{n}(\omega)\hat{A}_{n}^{\dagger}(\omega)\hat{\rho}+\hat{\rho}\hat{A}_{n}(\omega)\hat{A}_{n}^{\dagger}(\omega)\right)\right]\right\\}.$
(5)

Here, $\omega=\epsilon_{j}-\epsilon_{i}$ is the energy difference of two
eigenstates, $|\epsilon_{i}\rangle$ and $|\epsilon_{j}\rangle$, of $\hat{H}$,
$n_{n}(\omega)=[\exp(\hbar\omega/k_{B}T_{n})-1]^{-1}$ is the Bose-Einstein
distribution characterizing the heat baths ($n=1$ or $N$) and $k_{B}$ is the
Boltzmann constant. The Lindblad operators
$\hat{A}_{n}(\omega)=\sum_{\omega}|\epsilon_{i}\rangle|\langle\epsilon_{i}|\hat{\sigma}^{x}_{n}|\epsilon_{j}\rangle\langle\epsilon_{j}|$
describe the transitions induced by the bath. In the following we use an Ohmic
bath, with the bath spectral function $J(\omega)=\gamma\omega$. This master
equation can be justified in weak coupling limit and it has been shown that it
can predict accurately the heat current exchanged between the system and the
baths WichterichMichel2007 ; Alicki1979 ; LevyKosloff2014 . To lighten the
notations, henceforth we will work in units in which $J=k_{B}=\hbar=1$.

The heat current $\mathcal{J}$ is calculated from the continuity equation,

$\displaystyle\frac{d\langle\hat{H}\rangle}{dt}=\mathcal{J}_{L}-\mathcal{J}_{R}.$
(6)

It follows from the master equation in Eq. (4) that

$\displaystyle\frac{d\langle\hat{H}\rangle}{dt}=\mathrm{Tr}\big{[}\hat{H}\mathcal{D}_{1}(\hat{\rho})\big{]}+\mathrm{Tr}\big{[}\hat{H}\mathcal{D}_{N}(\hat{\rho})\big{]}.$
(7)

In the steady state, $\frac{d\langle\hat{H}\rangle}{dt}=0$. Thus, the steady
state heat current is

$\displaystyle\mathcal{J}=$
$\displaystyle\mathcal{J}_{L}=\mathcal{J}_{R}=\mathrm{Tr}\big{[}\hat{H}\mathcal{D}_{1}(\hat{\rho}_{s})\big{]}=-\mathrm{Tr}\big{[}\hat{H}\mathcal{D}_{N}(\hat{\rho}_{s})\big{]},$
(8)

where $\hat{\rho}_{s}=\hat{\rho}(t=\infty)$ is the steady state. We refer to
forward bias the case in which the hot bath is coupled to the first (left-
most) site and the cold one to the last (right-most) site, while we use
reverse bias for the opposite case. The magnitude of the rectification is
signalled by the rectification coefficient, which is the ratio between the
current in forward bias $\mathcal{J}_{f}$ and that in reverse bias
$\mathcal{J}_{r}$:

$\mathcal{R}=-\frac{\mathcal{J}_{f}}{\mathcal{J}_{r}}.$ (9)

The rectification coefficient $\mathcal{R}$ is much larger, or much smaller,
than $1$ for good rectifiers fn1 , while $\mathcal{R}=1$ for no rectification
indgamma .

##  III Results

We now study how the magnitude of the anisotropy $\Delta$, the temperature
$T_{C}$ ($T_{H}$) of the cold (hot) bath, and the interface coupling $J_{N/2}$
affect the rectification obtained in the system. The steady state is
calculated from the stationary solution of the master equation (4),

$\mathcal{L}(\hat{\rho}_{s})=0,$ (10)

i.e., the zero$-$th mode of the Lindblad super operator $\mathcal{L}$ which in
our case is unique. The memory cost of finding the eigenvector for the
zero$-$th eigenvalue of $\mathcal{L}$ using exact diagonalization is $4^{2N}$.
Thus, it is highly non-trival to go beyond $N>8$. So for larger chains, we
employ Runge-Kutta method to numerically integrate the master equation and
find the steady state. Although the memory cost is reduced to $2^{2N}$, long
integration times are required for convergence to the steady state, and this
limits the accessible chain lengths to $N\approx 10$.

###  III.1 Role of interaction

Figure 1: Role of interaction.  (a) Rectification coefficient $\mathcal{R}$ as
a function of anisotropy parameter $\Delta$ for different chain lengths $N$.
Cold bath temperature is $T_{C}=0.1$ and hot bath temperature is
$T_{H}=10+T_{C}$. (b) Dependence of rectification coefficient $\mathcal{R}$ on
anisotropy parameter $\Delta$ with different cold bath temperature $T_{C}$ for
a chain of length $N=8$ and $T_{H}=10+T_{C}$. In both panels we used
$J_{N/2}=1$.

First, we investigate the effect of interactions on the rectification. Fig.1
shows the variation of rectification $\mathcal{R}$ with the interaction
strength $\Delta$. Similar to the spin current in BalachandranPoletti2018 ,
rectification increases significantly when $\Delta\gtrsim 2$. In Fig.1(a) in
particular, we show how the rectification changes with the chain length $N$.
For the system sizes analyzed, we find that significant rectification can be
observed even for small $N$. Moreover we observe an increase of the
rectification for larger $N$. We are however unable to directly explore large
system sizes and cannot predict on analytical grounds what would be the
behavior in the thermodynamic limit.

However we can conjecture what would happen in the thermodynamic limit by
reverting to results for spin currents and spin baths as in
BalachandranPoletti2018 . In that system, in fact, we observed that when the
fully polarizing bath was not pushing the system exactly in the state
$|\downarrow\downarrow\dots\downarrow\rangle$, not only the rectification was
reduced, but $\mathcal{R}$ would decrease with the system size. Hence, we
expect that also in our set-up, as soon as $T_{C}>0$, the system will never be
perfectly insulating in reverse bias, hence the temperature bias per unit-
length will be reduced as the system size increases, thus bringing the system
towards linear response where rectification is not possible. This argument
implies that for a given bath temperature there is an optimal system size for
heat current rectification.

In Fig.1(b) we shift our focus on the importance of the temperature of the
cold bath $T_{C}$. Here we show that going to low values of $T_{C}$ can induce
a significant increase in the observed rectification. This is consistent with
BalachandranPoletti2018  in which, when one of the baths was trying to fully
polarize the chain, the rectification was stronger.

###  III.2 Role of $\Delta T$

Figure 2: Role of $\Delta T$. (a) Variation of rectification coefficient
$\mathcal{R}$ with temperature bias $\Delta T$ for different anisotropy
parameter $\Delta$. Here, cold bath temperature is $T_{C}=0.1$. (b) Dependence
of rectification coefficient $\mathcal{R}$ on temperature bias $\Delta T$ for
different cold bath temperature $T_{C}$. Here the anisotropy $\Delta=4$. In
both panels the chain length $N=8$, $T_{H}=T_{C}+\Delta T$ and $J_{N/2}=1$.

In this section, we will detail the importance of the temperature bias in the
rectification mechanism. An increase in $\Delta T=T_{H}-T_{C}$ allows a strong
nonlinear response and hence the possibility of strong rectification. In
Fig.2(a) we plot the rectification versus $\Delta T$ for different magnitudes
of the anisotropy, while keeping the cold temperature at $T_{C}=0.1$. For
small $\Delta T$, the current increases both in the forward and reverse
biases, but in a very similar manner and the rectification remains always
$\mathcal{R}\approx 1$. When $\Delta T\gtrsim 1$ strong rectification appears,
however, it is only for large anisotropy $\Delta$ that $\mathcal{R}$ become
particularly prominent. We also observe a decrease of rectification with
larger $\Delta T$, and eventually a large bias favours reverse current over
the forward current, thus resulting in a rectification coefficient
$\mathcal{R}<1$. Since we numerically observed that the value of $\Delta T$
where rectification reversal takes place increases with the system size, we
interpret this phenomenon as a proximity effect related to the small chain
length (see Appendix A for more details).

In Fig.2(b) we instead show the rectification versus the temperature
difference $\Delta T$ but for different values of the cold temperature
$T_{C}$, while keeping $\Delta=4$. Fig.2(b) confirms that, even for large
anisotropy, it is only when the cold temperature $T_{C}$ is low enough that
one observes strong rectification.

###  III.3 Role of interface tunneling

Figure 3: Interface tunneling. Rectification ratio versus anisotropy $\Delta$
for different magnitudes of the tunneling at the interface between the
interacting and the non-interacting halves of the chain (purple circles for
$J_{N/2}=0.1$, blue squares for $J_{N/2}=0.2$, red triangles for $J_{N/2}=0.5$
and green diamonds for $J_{N/2}=1.0$). The smaller the interface tunneling
$J_{N/2}$ the stronger the rectification effect, however larger are the
resonance effects due to finite size. Other parameters are $N=8$, $T_{C}=0.1$
and $T_{H}=10$.

The tunnelling at the interface $J_{N/2}$ plays a very important role in
determining the amount of current in the system and the magnitude of the
rectification. Smaller $J_{N/2}$ implies smaller currents, however it also
allows to recognize more clearly the differences in the spectrum of the halves
of the spin chain, by separating/distinguishing them more strongly. This
results in stronger rectifications for smaller interface tunnelling $J_{N/2}$,
but also in sharper responses close to resonances, where there is no energy
gap for the propagation of a magnon excitation created at the interface
between the two halves of the segmented chain, and therefore
$\mathcal{R}\approx 1$ BalachandranPoletti2018 . This is clearly depicted in
Fig.3 in which we show $\mathcal{R}$ versus the anisotropy $\Delta$ for
different values of the interface tunneling $J_{N/2}$.

###  III.4 Frequency response

Figure 4: Frequency response. ${\rm Im}[\mathcal{C}(\omega)]$, i.e. the
imaginary part of the Fourier transform of the two-time correlation $C(t)$ in
Eq.(11) as a function of the frequency $\omega$. We consider different values
of the anisotropy (red diamonds for $\Delta=1$, yellow circles for $\Delta=2$,
purple squares for $\Delta=3$, green downward triangles for $\Delta=4$, and
blue upward triangles for $\Delta=5$) for forward, panel (a), and reverse
bias, panel (b). Other parameters are $N=8$, $T_{C}=1$, $T_{H}=100$,
$\gamma=1$ and $J_{N/2}=1$.

In order to characterize further the rectification system here studied, we
consider the frequency response of a tunnelling excitation at the interface. A
tunnelling excitation corresponds to a spin exchange between two nearest
neighbour spins. More precisely, we consider the two-time correlation function

$\displaystyle
C(t)=\mathrm{Tr}\left[\sigma^{-}_{N/2}\sigma^{+}_{N/2+1}\;e^{\mathcal{L}t}\left(\sigma^{+}_{N/2}\sigma^{-}_{N/2+1}\hat{\rho}_{s}\right)\right].$
(11)

This correlation function corresponds to imposing a tunneling on the steady
state, evolving it for a time $t$ with the Lindbladian $\mathcal{L}$ and then
tunneling back. The Fourier transform of this two-time correlation,
$\mathcal{C}(\omega)$, gives the frequency response of the system. We show the
dependence of the imaginary part of ${\rm Im}[\mathcal{C}(\omega)]$ on the
frequency $\omega$ for different values of the anisotropy $\Delta$ in Fig.4.
In forward bias, Fig.4(a), the two-time correlation is almost unaffected by a
change in the anisotropy $\Delta$ fn3 . In reverse bias instead, for small
values of $\Delta$ we observe that there is a sizeable response at all
frequencies, including low frequencies. However, for large enough
anisotropies, the low frequency response is significantly suppressed, and only
at large enough frequencies (i.e., energies in the system), it is possible to
have a sizeable response. This is in agreement with our previous work
BalachandranPoletti2018 , where we showed that in reverse bias there is an
excitation gap which hinders spin current, while in forward bias no excitation
gap is present. See Appendix B for further insights.

###  III.5 Nonlinear response and negative differential conductance

Figure 5: Negative differential conductance. Reverse, panel (a), and forward
bias current, panel (b), vs $T_{C}$ for different anisotropies $\Delta=5$
(green triangles), $\Delta=4$ (purple squares), $\Delta=3$ (yellow circles)
and $\Delta=2$ (red diamonds). The hot bath temperature is set to $T_{H}=10$.
Other parameters are $N=8$, $\gamma=1$ and $J_{N/2}=1$. In the inset of panel
(a), the range for the $y$-axis is modified, so that for $\Delta=2$ the
monotonic behavior of the heat current can be seen down to low values of
$T_{C}$.

Due to the presence of strong interactions, our system offers the opportunity
to have nonlinear responses and, in particular, negative differential
conductance (see BenentiRossini2009 ; BenentiZnidaric2009 ; GuoPoletti2017  in
the context of spins and hard-core bosons transport). Negative differential
conductance, i.e. the counter-intuitive phenomenon of decreasing the heat flow
despite an increase of thermal bias plays a vital role in the performance of
various thermal devices, such as the thermal transistors LiCasati2006  and
thermal memories WangLi2008 . It is a very important subject of thermal
transport research, and targeting the future possibility of building
experimental devices, we want to stress that it is present in our proposal of
thermal diode. In order to show this, we consider again a chain with $N=8$
spins and the first four of them interacting with anisotropy $\Delta$. Current
in the system vs the cold temperature is plotted in Fig.5(a,b), respectively
for reverse and forward biases. We first consider the reverse bias where the
first site is set to the cold temperature $T_{C}$ and the last site to the hot
temperature $T_{H}=10$. For $T_{C}=10$, the cold bath is at the same
temperature as the hot one, so the current is clearly zero. As the cold
temperature decreases the current increases until $T_{C}$ is cold enough and,
after that, the current decreases despite the increase in temperature
difference $T_{H}-T_{C}$. The change of derivative of the current vs
temperature curve occurs at higher temperatures for larger anisotropies. At
low anisotropy $\Delta$, the current increases super-linearly for small
$\Delta T$, i.e. the current increases more than linearly with the change in
bias, however, it may show negative differential conductance only for $T_{C}$
cold enough (e.g. for $\Delta=3$) or not at all (e.g. for $\Delta=2$, not
shown in the main panel but highlighted in the inset). In forward bias
instead, no negative differential conductance is observed, on the contrary,
for larger anisotropy we observe superlinear conductance, as clearly shown in
Fig.5(b).

##  IV Conclusions

Motivated by our findings in BalachandranPoletti2018 , in which it was shown
that a segmented $XXZ$ chain can become a perfect diode for spin currents, we
investigate the heat current rectification in the same setup. Using global
Lindblad heat baths, we show that a segmented chain works as a good diode
provided one half of the chain is strongly interacting and the cold bath
temperature is low enough. We have shown that the spectral response is
significantly suppressed at low frequencies when the anisotropy is large
enough. Our simulations show that, up to the numerically accessible chain
lengths, the rectification improves with the system size. Unlike the diode for
spin currents in BalachandranPoletti2018 , we find that the heat diode can
work over a wide range of bath parameters, although the peak performance of
the latter is, in general, worse.

The presence of strong interactions can, in general, induce a nonlinear
response to an external bias. Here we show that, depending on the magnitude of
the anisotropy and whether the system is in forward or reverse bias, both
superlinear and negative differential conductance can emerge, the latter being
a key ingredient for building up a thermal transistor.

The presence of a nonlinear response and of strong rectification can be used
to control the heat flow and could have applications also in energy conversion
from heat to work BenentiWhitney2017 ; XuPoletti2018 .

Acknowledgments: D.P. and V. B. acknowledge support from the Singapore
Ministry of Education, Singapore Academic Research Fund Tier-II (project
MOE2016-T2-1-065). E.P. was partially supported by CNPq (Brazil). G.B.
acknowledges the financial support of the INFN through the project “QUANTUM”.

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Figure 6: Scaling of rectification reversal. Variation of rectification
coefficient $\mathcal{R}$ with temperature bias $\Delta T$ for different chain
lengths $N$. Dotted line corresponds to $\mathcal{R}=1.$ Here, the anisotropy
is $\Delta=5$, cold bath temperature $T_{C}=0.1$, hot bath temperature
$T_{H}=T_{C}+\Delta T$ and $J_{N/2}=1$.

##  Appendix A Reversal of rectification

To demonstrate the proximity effect, we study the rectification for different
temperature bias $\Delta T$ at different chain lengths $N$ in Fig. 6. For a
chain of spins $N=4$ with anisotropy $\Delta=5$, the rectification $R$ is
below $1$ when $\Delta T=74$ whereas for $N=6$, it is after $\Delta T=5000$.
The temperature required to reverse the rectification increases to values
larger than $10^{4}$ for $N=8$ and $10^{5}$ for $N=10$. From our numerical
data we can conclude that, for a longer chain, the increase in the reverse
current takes place at larger bias.

Figure 7: Frequency response for a half-chain. $|\mathcal{C}_{1}(\omega)|$,
i.e. the absolute value of the Fourier transform of the two-time correlation
$C_{1}(t)$ as a function of the frequency $\omega$ with $T=1$, for different
anistropies $\Delta$. The length of the chain is $N=4$ and $\gamma=1$.

##  Appendix B Spectral response for two-time correlation

Here, we further show the suppression of the low frequency response in the
reverse bias at larger anisotropy. For this, we take the half-chain with
uniform anisotropy $\Delta$ at a temperature $T$. The corresponding state of
the chain is  $[\rho_{s}]_{i,j}=e^{-E_{i}/T}\delta_{i,j}/\mathcal{Z}$, where
$E_{i}$ are the eigenenergies of the half chain and
$\mathcal{Z}=\sum_{i}e^{-E_{i}/T}$. Next, we attach this chain to a single
thermal bath with temperature $T$ and calculate the two-time correlation
$C_{1}(t)=\mathrm{Tr}\left[\sigma^{+}_{1}e^{\mathcal{L}t}\left(\sigma^{-}_{1}\rho_{s}\right)\right]$
and its Fourier transform $\mathcal{C}_{1}(\omega)$. The frequency response of
the absolute value of $\mathcal{C}_{1}(\omega)$ for different anisotropies is
plotted in Fig. 7 at temperatures $T=1$. This figure shows that, at low
temperature, the zero frequency response of the chain is highly suppressed
when the anisotropy is large. This is a clear indication of opening of an
excitation gap, implying small reverse current and large rectification for
large anisotropy.