# Enhancing the performance of coupled quantum Otto thermal machines without
entanglement and quantum correlations

Abdelkader El Makouri  LPHE-Modeling and Simulation, Faculty of Sciences,
Mohammed V University in Rabat, Morocco.  Abdallah Slaoui  LPHE-Modeling and
Simulation, Faculty of Sciences, Mohammed V University in Rabat, Morocco.
Centre of Physics and Mathematics, CPM, Faculty of Sciences, Mohammed V
University in Rabat, Rabat, Morocco.  Mohammed Daoud  Department of Physics,
Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco.

###### Abstract

We start with a revision study of two coupled spin-$1/2$ under the influence
of Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interaction and a magnetic
field. We first show the role of idle levels, i.e., levels that do not couple
to the external magnetic field, when the system is working as a heat engine as
well as when it is a refrigerator. Then we extend the results reported in
[journal PRE. 92, (2015) 022142NoStop] by showing that it is not necessary to
change both the magnetic field as well as the coupling parameters to break the
extensive property of the work extracted globally from two coupled spin-$1/2$
as has been demonstrated there. Then we study the role of increasing the
number of coupled spins on efficiency, extractable work, and coefficient of
performance (COP). First, we consider two- and three-coupled spin-$1/2$
Heisenberg $\mathrm{XXX}$-chain. We prove that the latter can outperform the
former in terms of efficiency, extractable work, and COP. Then we consider the
Ising model, where the number of interacting spins ranges from two to six. We
show that only when the number of interacting spins is odd the system can work
as a heat engine in the strong coupling regime. The enhancements in efficiency
and COP are explored in detail. Finally, this model confirms the idea that
entanglement and quantum correlations are not the reasons behind the
enhancements observed in efficiency, extracatable work, and COP, but only due
to the structure of the energy levels of the Hamiltonian of the working
substance. In addition to this, the extensive property of global work as well,
is not affected by entanglement and quantum correlations.

##  I Introduction

Classical thermodynamics Prigogine  __(2015) was developed in the
$\mathrm{19^{th}}$ century to enhance the efficiency of heat engines Carnot
__(1824). On the other hand, quantum thermodynamics Kosloff1984  _et al._
(1984); Rezek  _et al._ (2017); Deffner  __(2019); Gemmer  _et al._ (2009);
Binder  __(2018); Goold  _et al._ (2016); Kosloff  _et al._ (2013); Anders
_et al._ (2016); Alicki  _et al._ (1979); Millen  _et al._ (2016); Quan  _et
al._ (2007); Quan1  _et al._ (2009); Allahverdyan  _et al._ (2015); Levy  _et
al._ (2013); Skrzypczyk  _et al._ (2012); Mitchison  __(2019), is an emergent
field that aims to extend thermodynamics to the regime where the working
substance is quantum in nature and quantum effects are no longer negligible.
In this field, particular attention has been devoted to the study of thermal
machines fabricated from tiny quantum systems to improve their performance.
This upsurge of interest started with the first work of Scovil and Schulz-
DuBois when they showed that a three-level system is equivalent to a quantum
heat engine Scovil  __(1959). Since then, a lot of work has been developed to
understand thermal machines at the quantum scale. Furthermore, in 2016, the
first experimental realization of a quantum heat engine was demonstrated using
a trapped ion as a working fluid Rossnagel  __(2012); Rossnagel2  __(2014);
Rossnagel  __(2016). In 2017, the same platform was used to fabricate a
quantum absorption refrigerator made of only three trapped ions Maslennikov
__(2017). Other platforms to realize quantum thermal machines experimentally
could be found in Bariani  __(2014); Sothmann  __(2012); QuanZhang  __(2006);
AltintasHardal  __(2015); Peterson  __(2019); Uzdin  __(2019); Fazio
__(2013); DVon  __(2019), and more recently, all of these platforms have been
collected in an article review MayersDeffner  __(2022). When the baths are not
thermal ScullyZubairy  __(2003); Dillenschneider  __(2009); HuangWang
__(2012); AbahLutz  __(2014); RossnagelAbah  __(2014); HardalM  __(2015);
Niedenzu  __(2016); Manzano  __(2016); Klaers  __(2017); Agarwalla  __(2017);
ScullyMarlan  __(2001), it has been shown that the efficiency of these quantum
thermal machines could surpass the thermodynamic Carnot bound. However, there
is no violation of the laws of thermodynamics since in this case we have used
resources, e.g., coherence, squeezing, and quantum correlations.

Many studies have been done since the first work done by Kieu Kieu  _et al._
(2004); Kieu2  _et al._ (2006) on a quantum heat engine (QHE) composed of one
qubit undergoing an Otto cycle composed of two adiabatic and two isochoric
stages in which he clarifies some important aspects of the second law of
thermodynamics. First, Zhang et al. have extended the same work to two coupled
spin-$1/2$ Heisenberg XXX chain Zhang  _et al._ (2007) where the exchange
coupling $\mathrm{J}$ is the parameter changed in the adiabatic stages, and
then in Ref.Johal  __(2011) where the external magnetic field $\mathrm{h}$ is
the parameter changed in the adiabatic stages. The studies in this direction
could be stated as follows: in Ref.Altintas  _et al._ (2015) the authors
investigated the case when one of the composite system is a spin-$1/2$
particle and the other is a qudit, in Ref.Ukraine  _et al._ (2015) when both
interacting spins are qudits where the role of negative temperature on the
efficiency has been investigated in detail, in Ref.Mehta  _et al._ (2017) the
role of degeneracy, in Ref.Talkner  _et al._ (2017); Anka  _et al._ (2021);
DasGosh  _et al._ (2019) when a thermal bath being replaced by a quantum
measurement as a new source of heat, in Ref.GuoZhang  _et al._ (2008); Zhao
_et al._ (2017); Sodeif  _et al._ (2021) the role of Dzyaloshinski-Moriya
interaction, in Ref.Ramandeep  _et al._ (2021) the authors examine the
performance of a two-coupled spins of arbitrary magnitudes (where majorization
Marshall  _et al._ (2011) has been used to find the upper bound of efficiency,
which was an open question to be answered), and last but not least, the role
of non-adiabaticity (i.e., when the unitary strokes are done in finite time)
for one spin and two coupled spin-$1/2$ particles Johall  _et al._ (2022);
Cakmak  _et al._ (2019); CampisiM  _et al._ (2020); Cakmak2  _et al._ (2021);
Thomas  _et al._ (2013).

Recently, it has been shown in Johal  __(2011) that the efficiency of two
coupled spin-$1/2$ particles is greater than the one of the Otto system. And
since quantum systems are known for strange phenomena, e.g., quantum
coherence, entanglement and quantum correlations, special attention has been
devoted to understanding the role of entanglement and quantum correlations in
efficiency and extractable work. An investigation of the role of entanglement
and quantum correlations has been done in Chang  _et al._ ; Altintas1  _et
al._ (2014); Hewgill  _et al._ (2018); Zhang  _et al._ (2007). However, even
with these works, there was no sufficient and clear answer until recently,
when Oliveira and Jonathan Oliveira2020  _et al._ (2021) provided an
explanation by demonstrating that the enhancement observed in efficiency is
solely due to the presence of idle levels. Moreover, in Ref. Anka  _et al._
(2021) the same enhancement was proved even for a three-level system, a system
that cannot be divided into subsystems. However, in our opinion, a revisited
study and more clarifications are needed to gain more insights in this
direction.

Motivated by these works, we use the Ising model (with and without KSEA
interaction) and the Heisenberg model to strengthen and demonstrate the
correctness of their reasoning Anka  _et al._ (2021); Oliveira2020  _et al._
(2021); Johall  _et al._ (2022). First, we use as a working fluid two coupled
spin-$1/2$ particles under the influence of KSEA interaction Yurischev  _et
al._ (2020); Yildirim  _et al._ (1995); Kaplan  _et al._ (1983); Aharony  _et
al._ (1993); Shekhtman  _et al._ (1992) and a magnetic field along the
$\mathrm{z}$-direction. The KSEA interaction arises from spin-orbit coupling.
We will see that when we have only quantum entanglement and working levels
(i.e., levels that do couple to the external magnetic field), there is no
enhancement in efficiency as it is expected. However, when the
$\mathrm{z}$-component of $\mathrm{J}$ is not equal to zero, two idle levels
emerge, and we see that there is an enhancement in the efficiency which is due
to heat passing through these levels from the cold to the hot bath. Then we
show how idle levels affect the COP when the system is working as a
refrigerator. We show that they shift the enhancement in COP to large values
of $\mathrm{\Gamma_{z}}$. When the system is working as a refrigerator, it has
been studied in Sodeif  _et al._ (2021); Deniz  _et al._ (2020), but in
contrast to them, we explore the role of idle levels as well as the role of
increasing the number of interacting spins on the COP, two situations that
have not been explored there. Moreover, taking advantage of this model, we
generalize the results reported in Altintas  _et al._ (2015), by showing that
it is not necessary to change both the external magnetic field and the
coupling parameters to break the extensive property of the global extractable
work.

Secondly, we will increase the number of interacting systems to more than two
coupled spin-$1/2$ up to six. We first compare the efficiency, extractable
work, and COP of two- and three-coupled Heisenberg XXX chain. Our results show
that three coupled spins could harvest more work than two coupled spin-$1/2$.
Even more, the former could work as a heat engine in the strong coupling
regime, which is not the case for the latter. When it comes to the COP, three
coupled spins could surpass that of the Otto, which is not the case with two-
coupled spin-$1/2$. Then we use the Ising model in which the number of
interacting spins ranges from two to six spin-$1/2$ particles. We show that,
in contrast to Altintas  _et al._ (2015), when we increase the number of
coupled spins, we see a remarkable enhancement in the extractable work. The
efficiency will be enhanced only for small values of the coupling parameter
$\mathrm{J}$. Even more, a new conclusion has been drawn, which is that only
when the number of interacting spins is odd, the system can work as a heat
engine in the strong coupling regime. When it comes to COP, we found that when
the system is ferromagnetic, there is a remarkable enhancement in the COP even
though for three- to six-coupled spin-$1/2$ they are nearly coincidable. When
the number of interacting spins ranges from two to six, we only consider the
$\mathrm{z}$-component of $\mathrm{J}$ to ensure that no entanglement or
quantum correlations will build up along the cycle. More precisely, the
coupled system will be only in a statistical mixture of factorized states.
Therefore, the enhancements in efficiency, extractable work, and COP are only
due to the structure of the energy levels of the system. Note that for all
models chosen and studied in this work, we have degenerate eigenvalues, where
in Ref.DasGosh  _et al._ (2019) the role of degneracy on efficiency and
extractable work has been explored in detail.

The rest of the article is organized as follows: In section II a brief review
of the quantum Otto cycle and the relevant thermodynamic quantities to
characterize the Otto heat engine and refrigerator is given. In section III we
study the role of KSEA interaction on the efficiency, local and global
extractable works and COP. The comparison between the efficiency, COP and
extractable work from two and three up to six-coupled spin-$1/2$ is done in
the section IV. Then in section V, we clarify some very important points. And
finally, in section VI we give a summary of our results with future
directions.

##  II Quantum Otto cycle

Before we start presenting our results, we should provide some necessary
definitions and expressions used in this paper. Suppose we have a quantum
system $\mathrm{S}$ described by the state $\mathrm{\rho}$ and a Hamiltonian
$\mathrm{H}$. The expectation value of the measured energy of $\mathrm{S}$ is

$\mathrm{U=\langle E\rangle=Tr(\rho H)=\sum_{i}p_{i}E_{i}},$ (1)

where $\mathrm{E_{i}}$ are the energy levels and $\mathrm{p_{i}}$ are the
corresponding occupation probabilities. The derivative of $\mathrm{U}$ gives

$\mathrm{dU=\sum_{i}E_{i}dp_{i}+p_{i}dE_{i},}$ (2)

which can be divided into heat and work given, respectively, as follows;
$\mathrm{\delta Q=\sum_{i}E_{i}dp_{i}}$ and $\mathrm{\delta
W=\sum_{i}p_{i}dE_{i}}$ Quan  _et al._ (2007); Kieu  _et al._ (2004); Alicki
_et al._ (1979). From these definitions, the average heat and work are:
$\mathrm{Q=\sum_{i}\int E_{i}dp_{i}}$ and $\mathrm{W=\sum_{i}\int
p_{i}dE_{i}}$. Therefore, the first law of thermodynamics reads

$\mathrm{dU=\delta Q+\delta W.}$ (3)

Mathematically speaking, $\mathrm{dU}$ is an exact differential, however,
$\mathrm{\delta Q}$ and $\mathrm{\delta W}$ are not total differentials but
are path dependent.

The Otto cycle is composed of four stages: two adiabatic stages in which an
external controlled parameter is varied, and two isochoric stages in which the
system is in contact with a heat bath. These stages are given as follows.
$\mathbf{Stage\ 1:}$ The occupation probabilities of each level are
$\mathrm{p^{\prime}_{i}}$. The system is put in contact with a hot bath at
tempereture $\mathrm{T_{h}}$ until it reaches thermal equilibrium. Then it is
decribed with the new occupation probabilities $\mathrm{p_{i}}$. Since at this
stage we have only a change in the occupation probabilities, only heat is
exchanged between the working substance and the hot bath. $\mathbf{Stage\ 2:}$
The system is isolated from the hot bath and the magnetic field is changed
from $\mathrm{h}$ to $\mathrm{h^{\prime}}$ (with $\mathrm{h>h^{\prime}}$).
This transformation will be done slowly to ensure the holding of the quantum
adiabatic theorem Kieu  _et al._ (2004). At this stage, only work is performed
since the occupation probabilities stay the same. $\mathbf{Stage\ 3:}$ The
system is put in contact with a cold bath at temperature
$\mathrm{T_{c}}$($\mathrm{T_{h}>T_{c}}$) until it reaches equilibrium with it.
In this case we have a change in the occupation probabilities from
$\mathrm{p_{i}}$ to $\mathrm{p^{\prime}_{i}}$. At this stage, only heat is
exchanged. $\mathbf{Stage\ 4:}$ The system is again isolated from the cold
bath and the magnetic field is changed back from $\mathrm{h^{\prime}}$ to
$\mathrm{h}$. At this stage, only work is performed. After this stage, the
system is again attached to the hot bath at temperature $\mathrm{T_{h}}$ to
complete the cycle and return the system to its initial state. Furthermore,
note that the Otto cycle is the most commonly used cycle to study quantum
thermal machines. The reason is that in this cycle, the system at each stage
exchanges either heat or work, not both of them, which leaves no ambiguity in
the identification of them correctly. In addition, the definitions of heat and
work used in this paper are only valid in the weak coupling regime.

When the system is working as a heat engine (for more details see, Kieu  _et
al._ (2004); Zhang  _et al._ (2007); Kieu2  _et al._ (2006)), we must have
$\mathrm{Q_{h}}>0$, $\mathrm{Q_{c}}<0$ and $\mathrm{W}>0$. In this case, in
$\mathbf{Stage\ 1}$ the system will absorb heat from the hot bath. Its
expression is given by

$\mathrm{Q_{h}=\sum_{i}E_{i}(p_{i}-p_{i}^{\prime}),}$ (4)

where $\mathrm{E_{i}}$ are the eigenvalues of the system at the hot bath side,
$\mathrm{p_{i}}$ and $\mathrm{p_{i}^{\prime}}$ are the populations of the
system when it is in contact with the hot bath and the cold bath,
respectively. In $\mathbf{Stage\ 3}$ the system will release heat to the cold
bath, which is given as follows

$\mathrm{Q_{c}=\sum_{i}E_{i}^{\prime}(p_{i}^{\prime}-p_{i}).}$ (5)

From the first law of thermodynamics, we have

$\mathrm{W=Q_{h}+Q_{c}=\sum_{i}(E_{i}-E_{i}^{\prime})(p_{i}-p_{i}^{\prime}).}$
(6)

Therefore, the expression of efficiency is

$\mathrm{\eta=\frac{W}{Q_{h}}=1+\frac{Q_{c}}{Q_{h}}.}$ (7)

When the system is working as a refrigerator, we must have $\mathrm{Q_{h}}<0$,
$\mathrm{Q_{c}}>0$ and $\mathrm{W}<0$, in other words, the system is running
in the reverse of the Otto heat engine cycle. In this case, the system is
characterized by its COP, which is given as follows

$\mathrm{COP=\frac{Q_{c}}{|W|}=\frac{Q_{c}}{|Q_{h}+Q_{c}|}.}$ (8)

We should note that the efficiency of the Otto cycle when the working medium
is either a single or (multiple but uncoupled) spin(s) or harmonic
oscillator(s) is given by $\mathrm{\eta_{o}=1-h^{\prime}/h}$. In the same
manner, when the system is working as a refrigerator
$\mathrm{COP_{o}=\frac{h^{\prime}}{h-h^{\prime}}}$, see Refs.Quan  _et al._
(2007); Quan1  _et al._ (2009). In this paper, the units are chosen such that
$\mathrm{k_{B}=\hbar=1}$.

##  III Revisited study of a coupled quantum Otto heat engine and refigerator

Our working fluid in this section is a two-coupled spin-$1/2$ $\mathrm{1D}$
Ising model with a $\mathrm{z}$-component KSEA interaction parameter
$\mathrm{\Gamma_{z}}$ under the influence of a magnetic field $\mathrm{h}$ in
the $\mathrm{z}$-direction. The expression of the Hamiltonian describing this
system is Moriya  _et al._ (1960)

$\mathrm{H=J_{z}\sigma_{z}^{1}\sigma_{z}^{2}+\Gamma_{z}(\sigma_{x}^{1}\sigma_{y}^{2}+\sigma_{y}^{1}\sigma_{x}^{2})+h(\sigma_{z}^{1}+\sigma_{z}^{2}),}$
(9)

where $\mathrm{\sigma_{x,y,z}^{i}}$ are the standard Pauli matrices acting on
the site $i\in[1,2]$. The eigenvalues of $\mathrm{H}$ are:
$\mathrm{E_{1}=E_{2}=-J_{z}}$,
$\mathrm{E_{3}=J_{z}-2\sqrt{h^{2}+\Gamma_{z}^{2}}}$ and
$\mathrm{E_{4}=J_{z}+2\sqrt{h^{2}+\Gamma_{z}^{2}}}$. Their corresponding
eigenstates in the standard basis
$\mathrm{\\{|11\rangle,|10\rangle,|01\rangle,|00\rangle\\}}$ are:
$\mathrm{|\psi_{1}\rangle=|10\rangle}$,
$\mathrm{|\psi_{2}\rangle=|01\rangle}$,
$|\psi_{3}\rangle=\frac{1}{\sqrt{|\alpha_{1}|^{2}+1}}(\alpha_{1}|11\rangle+|00\rangle)$
and
$|\psi_{4}\rangle=\frac{1}{\sqrt{|\alpha_{2}|^{2}+1}}(\alpha_{2}|11\rangle+|00\rangle)$,
with $\mathrm{\alpha_{1}=i(-h+\sqrt{h^{2}+\Gamma_{z}^{2}})/\Gamma_{z}}$ and
$\mathrm{\alpha_{2}=-i(h+\sqrt{h^{2}+\Gamma_{z}^{2}})/\Gamma_{z}}$. The KSEA
interaction has been chosen for two reasons. First, when the
$\mathrm{z}$-component of $\mathrm{J}$ is equal to zero, we have
entanglement-(between the two coupled spins described by Eq.9 when they are in
thermal equilibrium with either the hot bath or the cold bath, see Eq.10)-but
no idle levels. More precisely, here we want to test if the presence of only
entanglement or quantum correlations could boost the efficiency beyond that of
the Otto. Second, for this Hamiltonian, we see that the eigenstates are
dependent on the magnetic field $\mathrm{h}$ and $\mathrm{\Gamma_{z}}$ and
that the eigenvalues are non-linear in $\mathrm{h}$. This situation is
different from the cases considered in Zhang  _et al._ (2007); Johal
__(2011); Ukraine  _et al._ (2015); DasGosh  _et al._ (2019). Below, we see
that this influences the extensive property of the work extracted globally.
Moreover, as far as we know this interaction has not been considered in the
previous works to the authors’ knowledge.

When the system in a thermal equilibrium with a heat bath at inverse
temperature $\mathrm{\beta}$, it can be described by the density operator
$\mathrm{\rho_{th}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|}$. In the
computational basis $\\{|11\rangle,|10\rangle,|01\rangle,|00\rangle\\}$, this
matrix is given as follows

$\begin{split}\mathrm{\rho_{th}}=\begin{pmatrix}\frac{|\alpha_{1}|^{2}}{|\alpha_{1}|^{2}+1}\mathrm{p_{3}}+\frac{|\alpha_{2}|^{2}}{|\alpha_{2}|^{2}+1}\mathrm{p_{4}}&0&0&\frac{\alpha_{1}^{\star}}{|\alpha_{1}|^{2}+1}\mathrm{p_{3}}+\frac{\alpha_{2}^{\star}}{|\alpha_{2}|^{2}+1}\mathrm{p_{4}}\\\
0&p_{1}&0&0\\\ 0&0&p_{1}&0\\\
\frac{\alpha_{1}}{|\alpha_{1}|^{2}+1}\mathrm{p_{3}}+\frac{\alpha_{2}}{|\alpha_{2}|^{2}+1}\mathrm{p_{4}}&0&0&\frac{1}{|\alpha_{1}|^{2}+1}\mathrm{p_{3}}+\frac{1}{|\alpha_{2}|^{2}+1}\mathrm{p_{4}}\\\
\end{pmatrix},\end{split}$ (10)

with $\mathrm{p_{i}=e^{-\beta E_{i}}/Z}$. The partition function is
$\mathrm{Z=2(e^{\beta J_{z}}+e^{-\beta
J_{z}}\cosh(2\beta\sqrt{h^{2}+\Gamma_{z}^{2}}))}$, where
$\mathrm{\beta=1/k_{B}T}$, $\mathrm{k_{B}}$ is the Boltzmann constant and $T$
is the temperature. Therefore to get the expression of the density matrix of
the system at the $\mathbf{Stages}$ $1$ and $3$ we only have to replace
$\mathrm{\beta}$ by $\mathrm{\beta_{h}}$ and $\mathrm{\beta_{c}}$,
respectively.

To arrive to the expression of this state, we have to follow the next steps.
First, we take the expression of
$\rm\rho_{th}=p_{1}|\psi_{1}\rangle\langle\psi_{1}|+p_{2}|\psi_{2}\rangle\langle\psi_{2}|+p_{3}|\psi_{3}\rangle\langle\psi_{3}|+p_{4}|\psi_{4}\rangle\langle\psi_{4}|$,
then we replace the eigenstates $|\psi_{i}\rangle$ by their expressions, we
get

$\begin{split}&\hskip 5.69046pt\rm\rho_{th}=\rm p_{1}|10\rangle\langle
10|+p_{2}|01\rangle\langle 01|\\\ &+\rm
p_{3}\left(\frac{|\alpha_{1}|^{2}|11\rangle\langle
11|+\alpha_{1}|11\rangle\langle 00|+\alpha_{1}^{\star}|00\rangle\langle
11|+|00\rangle\langle 00|}{|\alpha_{1}^{2}|+1}\right)\\\
&\rm+p_{4}\left(\frac{|\alpha_{2}|^{2}|11\rangle\langle
11|+\alpha_{2}|11\rangle\langle 00|+\alpha_{2}^{\star}|00\rangle\langle
11|+|00\rangle\langle 00|}{|\alpha_{2}^{2}|+1}\right).\end{split}$ (11)

Then, rearranging this expression, one obtains equation 10. Furthermore, note
that the Hamiltonian given in equation 9 it only describes the coupling
between the spins, i.e., it does not include the properties of the implicitly
assumed heat bath, since we are interested in the thermodynamics of the cycle
in the long time limit, when the system completely gets thermalized. However,
if the bath is included explicitly, the total Hamiltonian will be given as
follows: $\mathrm{H_{T}=H_{S}+H_{B}+H_{SB}}$, with $\mathrm{H_{B}}$ is the
Hamiltonian of the bath and $\mathrm{H_{BS}}$ is the Hamiltonian describing
the interaction between the S and B. If the interaction between the bath and
the system is assumed to be weak and markovian then the system would
thermalizes to the Gibbs state $\mathrm{\rho_{th}=\frac{e^{-\beta
H}}{Tr(e^{-\beta H})}}$. In general, the evolution of the state of the system
is given by the next equation,

$\mathrm{\frac{\partial\rho}{\partial t}=-i[H,\rho]+\mathcal{L}(\rho),}$ (12)

with the first term describing the unitary evolution of the system when there
is no interaction with the heat bath and the second describes the effect of
the heat bath on the system. The thermal state given in Eq. 10 is the steady
state of this equation. Furthermore, since it is not trivial to assume
thermalization, one can suppose that each one of the coupled spins is coupled
to a heat bath and assuming that both the reservoirs are at the same
temperature T and lead to the same decohering rates $\mathrm{\gamma}$, then
one will compute the distance between the state of the system as a result of
this interaction $\mathrm{\rho}$ (this state will be obtained from Eq. 12) and
$\mathrm{\rho_{th}}$ (given in Eq. 10) given by the fidelity
$\mathrm{F(\rho_{th},\rho)=[\mathrm{Tr}(\sqrt{\sqrt{\rho_{th}}\rho\sqrt{\rho_{th}}})]^{2}}$,
and when $\mathrm{F(\rho_{th},\rho)}\rightarrow$ 1 (depending on the chosen
values the parameters $\mathrm{T,J_{z},\Gamma_{z},hand\gamma}$), one would be
sure that the system is now thermalized (see, Ref. Altintas  _et al._ (2015)).

Below, when the system is working as a heat engine, the parameter values for
the plot are $\mathrm{T_{h}=4}$, $\mathrm{T_{c}=1}$, $\mathrm{h=4}$ and
$\mathrm{h^{\prime}=3}$, on the other hand, when it is working as a
refrigerator, the parameter values for the plot are $\mathrm{T_{h}=2}$,
$\mathrm{T_{c}=1}$, $\mathrm{h=5}$ and $\mathrm{h^{\prime}=2}$. These
parameters have been chosen to ensure that $\mathrm{h>h^{\prime}}$ and
$\mathrm{h/T_{h}<h^{\prime}/T_{c}}$ for the system to function as a heat
engine Kieu  _et al._ (2004); Kieu2  _et al._ (2006), $\mathrm{h>h^{\prime}}$
and $\mathrm{h/T_{h}>h^{\prime}/T_{c}}$ for the system to function as a
refrigerator. This is for uncoupled systems, however, for coupled ones, the
conditions become more involved because of the coupling.

###  III.1 Global description

Shifting the eigenvalues of the Hamiltonian $\mathrm{H}$ will not alter heats
and work Quan  _et al._ (2007). They become: $\mathrm{E_{1}=E_{2}=-2J_{z}}$,
$\mathrm{E_{3}=-2\sqrt{h^{2}+\Gamma_{z}^{2}}}$ and
$\mathrm{E_{4}=2\sqrt{h^{2}+\Gamma_{z}^{2}}}$. Therefore, the expressions of
the global thermodynamical quantities, i.e., heats and work, are

$\mathrm{Q_{h}=\frac{2\sqrt{h^{2}+\Gamma_{z}^{2}}\sinh(2\beta_{c}\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})+2J_{z}e^{2\beta_{c}J_{z}}}{e^{2\beta_{c}J_{z}}+\cosh(2\beta_{c}\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})}-\frac{2\sqrt{h^{2}+\Gamma_{z}^{2}}\sinh(2\beta_{h}\sqrt{h^{2}+\Gamma_{z}^{2}})+2J_{z}e^{2\beta_{h}J_{z}}}{e^{2\beta_{h}J_{z}}+\cosh(2\beta_{h}\sqrt{h^{2}+\Gamma_{z}^{2}})},}$
(13) $\mathrm{Q_{c}=\frac{-2\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}\sinh(2\beta_{c}\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})-2J_{z}e^{2\beta_{c}J_{z}}}{e^{2\beta_{c}J_{z}}+\cosh(2\beta_{c}\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})}+\frac{2\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}\sinh(2\beta_{h}\sqrt{h^{2}+\Gamma_{z}^{2}})+2J_{z}e^{2\beta_{h}J_{z}}}{e^{2\beta_{h}J_{z}}+\cosh(2\beta_{h}\sqrt{h^{2}+\Gamma_{z}^{2}})},}$
(14) $\mathrm{W=2(\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})(\frac{\sinh(2\beta_{c}\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})}{e^{2\beta_{c}J_{z}}+\cosh(2\beta_{c}\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})}-\frac{\sinh(2\beta_{h}\sqrt{h^{2}+\Gamma_{z}^{2}})}{e^{2\beta_{h}J_{z}}+\cosh(2\beta_{h}\sqrt{h^{2}+\Gamma_{z}^{2}})}).}$
(15)

From these formulas and Eqs. (7) and (8), it is straightforward to get the
formulas of the efficiency as well as the COP. However, since their
expressions are too long, we will not write them here. Moreover, since here
the thermodynamical quantities are unchanged when substituting
$\mathrm{\Gamma_{z}}$ by -$\mathrm{\Gamma_{z}}$ we only plot them for positive
values of $\mathrm{\Gamma_{z}}$. For negative values of $\mathrm{J_{z}}$ we
found no enhancement, so we plotted them only for positive values of
$\mathrm{J_{z}}$.

Figure 1: (Color online). Plot of (a) efficiency versus $\mathrm{\Gamma_{z}}$
for three different values of $\mathrm{J_{z}=0}$ (blue dashed line),
$\mathrm{J_{z}=0.5}$ (orange dotted line) and $\mathrm{J_{z}=2.6}$ (green
solid line) (b) the heat absorbed by idle levels $\mathrm{q_{h}}$ from the hot
bath as a function of $\mathrm{\Gamma_{z}}$ for $\mathrm{J_{z}=0}$, 0.5 and
2.6. Note that $\mathrm{q_{h}}$ here is the heat absorbed globally from the
hot bath by idle levels and should be distinguished from the local heats given
by, $\mathrm{q_{h1}}$ and $\mathrm{q_{h2}}$. The parameter values for the
plots are $\mathrm{T_{h}=4}$, $\mathrm{T_{c}=1}$, $\mathrm{h=4}$ and
$\mathrm{h^{\prime}=3}$. Note that if one plot $\mathrm{\eta}$ and
$\mathrm{q_{h}}$ versus $\mathrm{J_{z}}$ for different values of
$\mathrm{\Gamma_{z}}$ will get the same plots. Figure 2: (Color online).
Contour plot of $\rm{q_{h}}$ as a function of $\rm{J_{z}}$ and
$\rm{\Gamma_{z}}$, where the parameter values are the same as in Fig.1. We see
that for an arbitrary value of $\rm{\Gamma_{z}}$, there is a maximal value of
$\rm{J_{z}}$ above which $\rm{q_{h}}$ will become positive. The blue dashed
line represents Eq.17. For example, at $\rm{\Gamma_{z}}=0$ and when
$\rm{J_{z}}$ exceeds 2.65, then $\rm{q_{h}}$ becomes positive. In this case,
the efficiency would be less than $\eta_{o}$. Figure 3: (Color online). COP
versus $\mathrm{\Gamma_{z}}$ for three different values of $\mathrm{J_{z}=0}$
(blue dotted line), $\mathrm{J_{z}=0.5}$ (orange dashed line), and
$\mathrm{J_{z}=2.6}$ (green large-dashing line). The parameter values for the
plot are $\mathrm{T_{h}=2}$, $\mathrm{T_{c}=1}$, $\mathrm{h=5}$, and
$\mathrm{h^{\prime}=2}$. The red solid line corresponds to that of Carnot
$\mathrm{COP_{c}=T_{c}/(T_{h}-T_{c})}$, and the purple dot-dashed line
corresponds to that of Otto $\mathrm{COP_{o}=h^{\prime}/(h-h^{\prime})}$. When
the COP of the system gets above that of $\mathrm{COP_{c}}$ it is no longer
working as a refrigerator. Note that if one plots the COP versus
$\mathrm{J_{z}}$ for different values of $\mathrm{\Gamma_{z}}$ will get the
same plot.

In Fig.1(a), the efficiency is plotted as a function of $\mathrm{\Gamma_{z}}$
for three different values of $\mathrm{J_{z}=0}$, $0.5$, and $2.6$. For
$\mathrm{J_{z}=0}$, we see that the efficiency could not surpass that of the
Otto $\mathrm{\eta_{o}=1-h^{\prime}/h}$, since in this case $\rm{q_{h}}=0$.
However, for $\mathrm{J_{z}>0}$, e.g., $\mathrm{J_{z}=0.5}$ and $2.6$, we see
that the efficiency could surpass $\mathrm{\eta_{o}}$. This could be explained
from Fig. 1(b). When $\rm{J_{z}}\neq 0$, we can have $\rm{q_{h}}<0$ depending
on $\rm{\Gamma_{z}}$. This is why efficiency can be enhanced. For example,
when $\rm{\Gamma_{z}}=0$, $|\rm{q_{h}}|$ is a monotonically increasing
function as we increase $\rm{J_{z}}$ and it reaches its maximal
value($\approx$0.435) when $\rm{J_{z}}\approx 1.788$. However, for
$\rm{\Gamma_{z}}=0$ and after $\rm{J_{z}}$ exceeding 1.788, $|\rm{q_{h}}|$
starts decreasing again, and if $\rm{J_{z}}$ is kept increasing, then
$\rm{q_{h}}$ can even become positive. This is why we see that $|\rm{q_{h}}|$
for $\mathrm{J_{z}=2.6}$ is less than $|\rm{q_{h}}|$ for $\mathrm{J_{z}=0.5}$
for $\Gamma_{z}\lesssim 0.61$. The explicit expression of $\rm{q_{h}}$ in
terms of $\rm{J_{z}}$ and $\rm{\Gamma_{z}}$ is given as follows,

$\begin{split}\mathrm{q_{h}}=&\mathrm{-2J_{z}(\frac{e^{-J_{z}/4}}{2(e^{J_{z}/4}+e^{-J_{z}/4}\cosh(\sqrt{16+\Gamma_{z}^{2}}/2))}}\\\
&\mathrm{-\frac{e^{-J_{z}}}{2(e^{J_{z}}+e^{-J_{z}}\cosh(2\sqrt{9+\Gamma_{z}^{2}}))}).}\end{split}$
(16)

Fig.2 displays the contour plot of $\rm{q_{h}}$ as a function of $\rm{J_{z}}$
and $\rm{\Gamma_{z}}$. For an arbitrary value of $\rm{\Gamma_{z}}$ there is a
critical value of $\rm{J_{z}}$ above which $\rm{q_{h}}$ would become positive,
and it is given by,

$\mathrm{J_{z}\leq
J_{z}^{\star}=\frac{2}{3}\log\left(\frac{\cosh\left(2\sqrt{9+\Gamma_{z}^{2}}\right)}{\cosh\left(\sqrt{16+\Gamma_{z}^{2}}/2\right)}\right).}$
(17)

For example, when $\rm{\Gamma_{z}=0}$, we have $\rm{J_{z}^{\star}}\approx
2.65.$

In Fig.3, we plot COP as a function of $\mathrm{\Gamma_{z}}$ for different
values of $\mathrm{J_{z}}$. We see that, as it is expected, idle levels
diminish the COP of the system when it is working as a refrigerator. More
precisely, for $\mathrm{J_{z}=0}$ we have no idle levels which makes the COP
surpass that of the Otto when $\mathrm{\Gamma_{z}\neq 0}$. However, when
$\mathrm{J_{z}\neq 0}$ the COP get enhanced only after certain value of
$\mathrm{\Gamma_{z}}$, i.e., the enhancement of COP shift to big values of
$\mathrm{\Gamma_{z}}$. This is because for small values of
$\mathrm{\Gamma_{z}}$ and when $\mathrm{J_{z}\neq 0}$ we found that idle
levels are taking heat from the hot to the cold bath which diminishes the COP.
This is due to the fact that they will heat the cold bath we want to cool. For
example, when $\mathrm{J_{z}=2.6}$ the COP gets above that of the Otto only
when $\mathrm{\Gamma_{z}>3.45}$. This shows that idle levels shift the
enhancement in the COP to high values of $\mathrm{\Gamma_{z}}$, more
precisely, the enhancement in the performance is present only when idle levels
stop taking heat to the cold bath. Therefore, we see that our system without
idle levels is a good refrigerator but a bad heat engine. However, this is not
the general conclusion, below we give the conditions on the heat absorbed or
released by idle levels such that the condition $\mathrm{COP>COP_{o}}$ is met.
More precisely, we show that if idle levels channel heat in the same direction
as working levels, then the COP can be enhanced. We do not add the plot of
entanglement and quantum correlations here since when we plotted entanglement
we found no clear relationship between it and the efficiency, the extractable
work, or the COP. Moreover, this enhancement will also be seen for coupled but
un-correlated spins in the section VI.

Now we focus on the physics behind the enhancements in the efficiencies. The
expressions of heats and work in terms of the eigenvalues and populations are,

$\mathrm{Q_{h}=-4J_{z}(p_{2}-p_{2}^{\prime})+2\sqrt{h^{2}+\Gamma_{z}^{2}}\left(p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3}\right),}$
(18) $\mathrm{Q_{c}=4J_{z}(p_{2}-p_{2}^{\prime})-2\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}\left(p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3}\right),}$
(19)

and,

$\mathrm{W=2(\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})\left(p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3}\right).}$
(20)

These equations could be expressed as well as follows:
$\mathrm{Q_{h}=q_{I}+q_{Wh}}$, $\mathrm{Q_{c}=-q_{I}-q_{Wc}}$ and
$\mathrm{W=q_{Wh}-q_{Wc}}$. Here $\mathrm{q_{I}}$ and $\mathrm{q_{Wh}}$ are
the heats absorbed from the hot bath by idle and working levels respectively.
The efficiency is,

$\mathrm{\eta=\left(1-\frac{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}{\sqrt{h^{2}+\Gamma_{z}^{2}}}\right)\left(\frac{1}{1-\frac{4J_{z}}{2\sqrt{h^{2}+\Gamma_{z}^{2}}}\frac{p_{2}-p_{2}^{\prime}}{p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3}}}\right),}$
(21)

which could be written as well as follows,
$\mathrm{\eta=\eta_{\Gamma_{z}}\frac{1}{1+\frac{q_{I}}{q_{Wh}}}}$, with
$\mathrm{\eta_{\Gamma_{z}}=1-\frac{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}{\sqrt{h^{2}+\Gamma_{z}^{2}}}}$.
One can see that for $\mathrm{\Gamma_{z}=0}$, $\mathrm{\eta}$ will be
$\mathrm{\eta_{o}}$ times another term (which has to be bigger than one for
$\mathrm{\eta>\eta_{o}}$). Thus, the condition for $\mathrm{\eta>\eta_{o}}$ is
that,
$\mathrm{0<\frac{4J_{z}}{2h}\frac{p_{2}-p_{2}^{\prime}}{p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3}}<1}$,
or equivalently
$\mathrm{-2h(p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3})<q_{I}<0}$. This
condition shows that as long as the heat channeled from the hot to the cold
bath by idle levels is not positive and not less than $\mathrm{-q_{Wh}}$ then
efficiency could be improved. However, if $\mathrm{q_{I}<-q_{Wh}}$ then
$\mathrm{Q_{h}}$ will be negative, thus the system will not work as a heat
engine. The same thing could be said when $\mathrm{\Gamma_{z}\neq 0}$ even
though things get complicated because the eigenvalues are not linear in the
magnetic field. Thus, different from $\mathrm{\Gamma_{z}=0}$, in this case we
must have
$\mathrm{1+\frac{q_{I}}{q_{Wh}}<\frac{\eta_{\Gamma_{z}}}{\eta_{o}}}$. Adding
the next condition, $\mathrm{\eta_{o}<\eta<\eta_{c}}$ (even though
$\mathrm{\eta}$ may not be able to reach the Carnot bound, but let’s take the
biggest upper bound allowed by the second law fo thermodynamics), in this case
the condition on heat absorbed by idle level from the hot bath is,

$\mathrm{-q_{Wh}\frac{\eta_{c}-\eta_{\Gamma_{z}}}{\eta_{c}}<q_{I}<-q_{Wh}\frac{\eta_{o}-\eta_{\Gamma_{z}}}{\eta_{o}}<0.}$
(22)

We see that $\mathrm{q_{I}}$ not only has to be less than 0 as in the case
when $\mathrm{\Gamma_{z}=0}$, but only when it is less than
$\mathrm{-q_{Wh}\frac{\eta_{o}-\eta_{\Gamma_{z}}}{\eta_{o}}}$, the efficiency
can be enhanced, and this is due to the fact that the eigenvalues are
nonlinear in the magnetic field.

The expression of the COP is,

$\begin{split}&\rm COP=\frac{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}{\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}\\\
&\rm-\frac{4J_{z}(p_{2}-p_{2}^{\prime})}{2(\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})(p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3})}.\end{split}$
(23)

From this equation, for $\mathrm{J_{z}=0}$ we have,
$\mathrm{COP=\frac{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}{\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}}$, this term is always bigger than $\mathrm{COP_{o}}$ as
long as $\mathrm{\Gamma_{z}\neq 0}$. Moreover, if we continue increasing
$\mathrm{\Gamma_{z}}$ one can see that this term could reach the Carnot bound.
In this case ($\mathrm{J_{z}=0}$) we have no idle levels, but we have two
working levels and two degenerate levels with energy 0, that neither
contribute to heats nor work. Thus, in this case, our system is equivalent to
a two-level system with an energy gap $\mathrm{4\sqrt{h^{2}+\Gamma_{z}^{2}}}$.
However, when $\mathrm{J_{z}}$ is not equal to zero things will be different
and complicated. For example, when $\mathrm{J_{z}>0}$, the enhancement in this
case is dependent on the sign of $\mathrm{p_{2}-p^{\prime}_{2}}$. More
precisely, if $\mathrm{p_{2}-p^{\prime}_{2}<0}$, in this case we see that the
second term will be negative thus it will diminish the COP. However, if
$\mathrm{p_{2}-p^{\prime}_{2}>0}$, in this case the COP will be enhanced.
Therefore, in this case, for $\mathrm{COP>COP_{o}}$ the term
$\mathrm{q_{c}=4J_{z}(p_{2}-p_{2}^{\prime})}$ has to be positive, i.e. idle
levels has to take heat in the right direction from the cold bath to the hot
bath. When $\mathrm{J_{z}=0}$ it does not matter if
$\mathrm{p_{2}-p^{\prime}_{2}<0}$ or $\mathrm{p_{2}-p_{2}^{\prime}>0}$, since
in this case idle levels will channel no heat. However, when
$\mathrm{J_{z}>0}$ then in this case we must have
$\mathrm{p_{2}-p^{\prime}_{2}\geq 0}$, i.e. the probability of occupation of
the degenerate idle level $\mathrm{-2J_{z}}$ at the hot bath side has to be
equal or bigger than the one at the cold side.

Now we give the theory used to describe the thermodynamics of the global and
the local cycles. Let’s give the expressions of the global as well as the
local work and heats. We start with the global ones.

$\mathrm{-W_{1}=Tr\left[\rho(H_{g}-H_{g}^{\prime})\right],}$ (24)

this is the work performed during the adiabatic stage 2. $\mathrm{\rho}$ is
the state of the system when it is in equilibrium with the hot bath,
$\mathrm{H_{g}}$ and $\mathrm{H_{g}}^{\prime}$ are the global Hamiltonians
when the magnetic field is equal to $\mathrm{h}$ and $\mathrm{h}^{\prime}$
respectively.

$\mathrm{Q_{h}=Tr\left[H_{g}(\rho-\rho^{\prime})\right],}$ (25)

this is the heat absorbed from the hot bath, $\mathrm{\rho^{\prime}}$ is the
state of the system when it is in equilibrium with the cold bath.

$\mathrm{-W_{2}=-Tr\left[\rho^{\prime}(H_{g}-H_{g}^{\prime})\right],}$ (26)

this is the work performed during the adiabatic stage 4. Finally,

$\mathrm{Q_{c}=-Tr\left[H_{g}^{\prime}(\rho-\rho^{\prime})\right],}$ (27)

this is the heat released to the cold bath. The first law of thermodynamics
reads as $\mathrm{W_{1}+W_{2}+Q_{1}+Q_{2}=0}$, from which we have
$\mathrm{W=-(W_{1}+W_{2})=T((\rho-\rho^{\prime})(H_{g}-H_{g}^{\prime}))}$. The
local ones are given in the same way as follows,

$\mathrm{-w_{\alpha}=Tr\left[\rho_{\alpha}(H_{l}-H_{l}^{\prime})\right],}$
(28)
$\mathrm{q_{h\alpha}=Tr\left[H_{l}(\rho_{\alpha}-\rho_{\alpha}^{\prime})\right],}$
(29)
$\mathrm{-w_{\alpha}^{\prime}=-Tr\left[\rho_{\alpha}^{\prime}(H_{l}-H_{l}^{\prime})\right],}$
(30)
$\mathrm{q_{c\alpha}=-Tr\left[H_{l}^{\prime}(\rho_{\alpha}-\rho_{\alpha}^{\prime})\right],}$
(31)

with $\mathrm{\alpha=1,\ 2}$, $\mathrm{H_{l}=diag\\{h,-h\\}}$ is the local
Hamiltonian of the spins. The definitions of the global and local works will
be compared below for the Ising+KSEA model. Since the reduced state of both
spins is the same $\mathrm{\rho_{1}=\rho_{2}}$, the total work extracted
locally is,

$\mathrm{w=2Tr\left[(\rho_{1}-\rho_{1}^{\prime})(H_{l}-H_{l}^{\prime})\right],}$
(32)

and if $\mathrm{\rho_{1}\neq\rho_{2}}$,
$\mathrm{w=\sum_{\alpha}Tr((\rho_{\alpha}-\rho_{\alpha}^{\prime})(H_{l}-H_{l}^{\prime}))}$.
Another definition of local heats and work which one can use as well is the
case when the local heats and work are defined with respect to the global
Hamiltonian $\mathrm{H_{g}}$ and local state $\mathrm{\rho_{\alpha}}$. In this
case we have,

$\mathrm{-w_{\alpha}=Tr\left[\rho_{\alpha}(H_{g}-H_{g}^{\prime})\right],}$
(33)
$\mathrm{q_{h\alpha}=Tr\left[H_{g}(\rho_{\alpha}-\rho_{\alpha}^{\prime})\right],}$
(34)
$\mathrm{-w_{\alpha}^{\prime}=-Tr\left[\rho_{\alpha}^{\prime}(H_{g}-H_{g}^{\prime})\right],}$
(35)
$\mathrm{q_{c\alpha}=-Tr\left[H_{g}^{\prime}(\rho_{\alpha}-\rho_{\alpha}^{\prime})\right].}$
(36)

If $\mathrm{\rho_{1}=\rho_{2}}$, the total work extracted locally is,

$\mathrm{w=2Tr\left[(\rho_{1}-\rho_{1}^{\prime})(H_{g}-H^{\prime}_{g})\right].}$
(37)

In general, we have four cases with respect to which we can compute the
average of work and heats: 1) with respect to the global state and the global
Hamiltonian, 2) with respect to the local state and the global Hamiltonian, 3)
with respect to the global state and the local Hamiltonian, and finally 4)
with respect to the local state and the local Hamiltonian. 3 and 4 are
equivalent. Furthermore, 1, 2 and 3 may be equal for the total work extracted
depending on the model. However, in terms of heats will not be equal. Thus, we
would obtain different efficiencies with these definitions.

###  III.2 Local description

Now let us see how the coupled spin-$1/2$ particles are undergoing the cycle
locally. The reduced density matrix of the sub-systems $1$ and $2$ in the
standard basis $\\{|1\rangle,|0\rangle\\}$ when they are in thermal
equilibrium with the hot bath is

$\mathrm{\rho_{1}=\rho_{2}}=\begin{pmatrix}\frac{1}{2}(1-\frac{\mathrm{h}(\mathrm{p_{3}-p_{4}})}{\mathrm{\sqrt{h^{2}+\Gamma_{z}^{2}}}})&0\\\
0&\frac{1}{2}(1+\frac{\mathrm{h}(\mathrm{p_{3}-p_{4}})}{\mathrm{\sqrt{h^{2}+\Gamma_{z}^{2}}}})\\\
\end{pmatrix}.$ (38)

The reduced density matrix of the sub-systems $1$ and $2$ in the standard
basis $\\{|1\rangle,|0\rangle\\}$ when they are in thermal equilibrium with
the cold bath is

$\mathrm{\rho^{\prime}_{1}=\rho_{2}^{\prime}=}\begin{pmatrix}\frac{1}{2}(1-\frac{\mathrm{h^{\prime}}(\mathrm{p_{3}^{\prime}}-\mathrm{p_{4}^{\prime}})}{\mathrm{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}})&0\\\
0&\frac{1}{2}(1+\frac{\mathrm{h^{\prime}(p_{3}^{\prime}-p_{4}^{\prime})}}{\mathrm{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}})\\\ \end{pmatrix}.$ (39)

Note that Eqs. 38 and 39, has been obtained using the normalisation constraint
$\mathrm{\sum_{i}p_{i}=\sum_{i}p^{\prime}_{i}=1}$. Following Johal  __(2011),
the local Hamiltonians for the spins are $\mathrm{H={\rm
diag}\left(h,-h\right)}$ and $\mathrm{H^{\prime}={\rm
diag}\left(h^{\prime},-h^{\prime}\right)}$ at $\mathrm{\mathbf{Stages}}$ $1$
and $3$, respectively. And using the formulas given in the section above, the
local heat absorbed from the hot bath $\mathrm{q_{h1,2}}$, the local heat
released to the cold bath $\mathrm{q_{c1,2}}$, and the amount of work
extracted locally $\mathrm{w_{1,2}}$ for each spin are expressed as follows

$\mathrm{q_{h1}=q_{h2}=h\left(\frac{h(p_{4}-p_{3})}{\sqrt{h^{2}+\Gamma_{z}^{2}}}+\frac{h^{\prime}(p_{3}^{\prime}-p_{4}^{\prime})}{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}\right),}$ (40)
$\mathrm{q_{c1}=q_{c2}=-h^{\prime}\left(\frac{h(p_{4}-p_{3})}{\sqrt{h^{2}+\Gamma_{z}^{2}}}+\frac{h^{\prime}(p_{3}^{\prime}-p_{4}^{\prime})}{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}\right),}$ (41)

and

$\mathrm{w_{1}=w_{2}=(h-h^{\prime})\left(\frac{h(p_{4}-p_{3})}{\sqrt{h^{2}+\Gamma_{z}^{2}}}+\frac{h^{\prime}(p_{3}^{\prime}-p_{4}^{\prime})}{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}\right).}$ (42)

The total work extracted locally by both subsystems $1$ and $2$ is given by

$\mathrm{w=2(h-h^{\prime})\left(\frac{h(p_{4}-p_{3})}{\sqrt{h^{2}+\Gamma_{z}^{2}}}+\frac{h^{\prime}(p_{3}^{\prime}-p_{4}^{\prime})}{\sqrt{h^{\prime
2}+\Gamma_{z}^{2}}}\right).}$ (43)

Figure 4: (Color online). Eq. 15 and Eq. 43 are plotted versus
$\mathrm{\Gamma_{z}}$ for different values of $\mathrm{J_{z}=0}$, 0.5, and
2.6. The parameter values are the same as in Fig. 1. $\mathrm{w_{J_{z}=0}}$
(blue dotted line), $\mathrm{W_{J_{z}=0}}$ (orange dashed line),
$\mathrm{w_{J_{z}=0.5}}$ (green large-dashing line), $\mathrm{W_{J_{z}=0.5}}$
(red medium-dashing line), $\mathrm{w_{J_{z}=2.6}}$ (purple solid line) and
$\mathrm{W_{J_{z}=2.6}}$(brown dot-dashed line).

This equation and Eq. 15 are plotted in Fig. 4 as a function of
$\mathrm{\Gamma_{z}}$ for three different values of $\mathrm{J_{z}=0}$, $0.5$
and $2.6$. We see that, in general, the total work extracted locally
$\mathrm{w}$ is not equal to the one extracted globally $\mathrm{W}$, and only
for very small values of $\mathrm{\Gamma_{z}}$ when they become equal to each
other. It is seen that as $\Gamma_{z}$ is increasing, the difference between
the local and global works become more pronounced. This is in contrast to
Altintas  _et al._ (2015) in which it was suggested that to violate the
extensive property of the work extracted globally, we have to vary the
coupling parameters next to the magnetic field $\mathrm{h}$ in the adiabatic
stages. This is not the case as our results show clearly. The reason is that
the eigenstates of their Hamiltonian were not dependent on the coupling
parameters as in the case under consideration and the eigenvalues were linear
in the magnetic field. More precisely, it was stated there that when
$\mathrm{dH_{int}=0}$ then we can automatically conclude that the global work
will be extensive. This is not true in general, since the total work is also
dependent on the state of the total system. That is, if $\mathrm{dW={\rm
Tr}(\rho\sum dH_{\rm loc})}$ does not mean the work will be extensive. Put it
in another way this does not mean that dW=$\mathrm{\sum dw_{\rm loc}}$ with
$\mathrm{dw_{\rm loc}={\rm Tr}_{\rm loc}(\rho_{\rm loc}dH_{\rm loc})}$. The
extensive property depends as well on the different parameters entering the
eigenstates of the Hamiltonian describing the system. However, when
$\mathrm{\Gamma_{z}=0}$ the results are in agreement with each other.

At this point we want to give some clarifications about $\mathrm{W-w\geq 0}$.
Our study shows that the extensive property is not related to any type of
classical or quantum correlations. To make it clear, let’s take the above
example and divide the expression of the global work into two terms. The
expression of the total work extracted globally is given as follows:

$\mathrm{W=\sum_{i}(E_{i}-E_{i}^{\prime})(p_{i}-p_{i}^{\prime}).}$ (44)

From this expression we have,

$\mathrm{W=\sum_{i}(E_{i}-E_{i}^{\prime})p_{i}+\sum_{i}(E_{i}^{\prime}-E_{i})p_{i}^{\prime}=-(W_{1}+W_{2}).}$
(45)

The first work ($\mathrm{-W_{1}}$) is the work performed when we change the
magnetic field from $\mathrm{h}$ to $\mathrm{h}^{\prime}$ with respect to the
state $\mathrm{\rho}$. The second term ($\mathrm{-W_{2}}$) is the work
performed when we change back $\mathrm{h}^{\prime}$ to $\mathrm{h}$. Now let’s
consider these two works separately and examine the extensive property. We
want to compare $\mathrm{-W_{1}}$ and $\mathrm{-W_{2}}$ with their
corresponding works invested $\mathrm{-w_{1}}$ and extracted $\mathrm{-w_{2}}$
locally. In the same way we have,

$\mathrm{w=2\sum_{i}((E_{i1}-E_{i1}^{\prime})p_{i1}+(E_{i1}^{\prime}-E_{i1})p_{i1}^{\prime})=-2(w_{1}+w_{2}).}$
(46)

with $\mathrm{E_{i1}=\\{h,-h\\}}$, and $\mathrm{p_{i\alpha}}$ are the local
probabilities. Let’s now compare $\mathrm{-W_{1}}$ with $\mathrm{-w_{1}}$.
More precisely, we want to compare the works performed globally and locally
with each other during stage 2. In this case we have,

$\mathrm{-W_{1}=2(\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})(p_{4}-p_{3}),}$ (47)

and,

$\mathrm{-2w_{1}=2(h-h^{\prime})(\frac{h(p_{4}-p_{3})}{\sqrt{h^{2}+\Gamma_{z}^{2}}}).}$
(48)

Since $\mathrm{(p_{4}-p_{3})}<0$, thus $\mathrm{-W_{1}}$ and
$\mathrm{-2w_{1}}$ are both negative, therefore they are the works invested
globally and locally, respectively. Their difference is given as follows:

$\begin{split}\rm-(W_{1}-2w_{1})=&\rm
2((\sqrt{h^{2}+\Gamma_{z}^{2}}-\sqrt{h^{\prime
2}+\Gamma_{z}^{2}})-\frac{(h-h^{\prime})h}{\sqrt{h^{2}+\Gamma_{z}^{2}}})\\\
&\rm(p_{4}-p_{3}).\end{split}$ (49)

One can see clearly that if $\mathrm{\Gamma_{z}=0}$ we get
$\mathrm{-W_{1}=-2w_{1}}$, in this case the model reduces to the Ising model.
Since $\mathrm{p_{4}-p_{3}}<0$ and that the first term is as well negative
when $\mathrm{\Gamma_{z}}\neq 0$, thus $\mathrm{-(W_{1}-2w_{1})>0}$. Thus, the
total work invested locally during the adiabatic stage 2 is not equal to the
one invested globally. In the same way, one can show that
$\mathrm{-(W_{2}-2w_{2})>0}$. Note that because $\mathrm{-W_{2}}$ and
$\mathrm{-2w_{2}}$ are positive, thus they are the works extracted during
stage 4. And because the total works invested (extracted) globally and locally
are not equal in stage 2 (stage 4), therefore, the global work will not be
extensive. The explanation for this gap is that the eigenvalues of Ising+KSEA
model are not linear in the magnetic field and that the eigentates are
parameter dependent.

Let’s now explain why $\mathrm{W-w\geq 0}$ has nothing to do with quantum
correlations. If one studies the work extracted by two coupled spins in the
Heisenberg, Ising and Ising+KSEA models, one will find that only the first and
second models preserve the extensive property, even though both Heisenberg and
Ising+KSEA models are quantum mechanically correlated. Thus, the extensive
property of the work has nothing to do with quantum correlations, it only
depends on the structure of the Hamiltonian. For example, in Ref. Chang  _et
al._ the eigenvalues were non-linear in the magnetic field and the eigenstates
were parameter dependent. However, there they explain the gap $\mathrm{W-w>0}$
by quantum correlations. In contrast to their conclusion, ours show that
quantum correlations are not responsible for this gap. And thus it is
difficult to give an interpretation of the extracted work in terms of quantum
correlations, since there is no clear relationship between them in our study.
However, note that if one includes measurement and feedback in the protocol,
in this case, quantum correlations will affect the work output. More
precisely, suppose we apply a measurement on one of the correlated spins and
then apply some operations on the unmeasured one depending on the result of
the measurement(feedback). In this case, the correlations between the spins
will be important. However, in our study we were only interested in running
the Otto cycle on each spin locally.

Another point that should be clarified is the one concerning the local and
global descriptions. More precisely, one may think that the reason behind
$\mathrm{W-w>0}$ is that the global cycle is Otto and the local one is not.
Let’s explain this in detail. This is because from Eqs. 38 and 39 one may
notice that the local probabilities at both sides are magnetic field
dependent. Then, one may directly conclude that the cycle is globally Otto and
that the quantum adiabatic theorem is valid. However, the adiabatic stages of
the local cycle may not be adiabatic and, therefore, the quantum adiabatic
theorem is no longer satisfied locally. However, this conclusion is not true.
Because according to this conclusion, even the global probabilities will not
stay the same in the adiabatic stages. And this is because the global
probabilities as well are magnetic field dependent either through $\rm
e^{-\beta E_{i}}$ (if $\rm E_{i}$ is magnetic field dependent) or through $\rm
Z=2(e^{\beta J_{z}}+e^{-\beta J_{z}}\cos(2\beta\sqrt{h^{2}+\Gamma_{z}^{2}}))$,
however, if the the transformation is done slowly there will be no transitions
between the states and thus the cycle would satisfy the quantum adiabatic
theorem, i.e. there will be no changes in the populations. Therefore, using
the same reasoning, the local adiabatic stages would be adiabatic if the
transformation is done slowly, thus the system will only exchange work and no
heat $\mathrm{q_{ad}=0}$, and the adiabatic theorem is still satisfied.
Therefore, the local cycle is also Otto. Furthermore, this model makes it
difficult to show that correlations are not necessary to surpass the
efficiency of the Otto. To show this, below we use the Ising model, keeping
only the z-components which exhibit no quantum correlations at all. The system
would be along the cycle in a separable state.

##  IV Comparison between multi-coupled spin-$1/2$ particles

Here we first compare the efficiency, the extractable work, and the COP of
two- and three-coupled spin-$1/2$ $\mathrm{1D}$ Heisenberg
$\mathrm{XXX}$-chain. Then we do the same comparison between two coupled and
the systems corresponding up to six spins in the Ising model. The latter has
been chosen for two reasons: first to ensure that no entanglement and quantum
correlations will be created during the cycle between the interacting spins,
as our purpose is to show and confirm that the enhancement is only due to the
structure of the energy levels of the system. Secondly, we want to study the
role of increasing the number of interacting spins on efficiency, extractable
work and COP.

###  IV.1 Heisenberg model

Here, the working fluid is a three-coupled spin-$1/2$ Heisenberg
$\mathrm{XXX}$-chain under the influence of a magnetic field $\mathrm{h}$
along the $\mathrm{z}$-axis. We call that the previous works Zhang  _et al._
(2007); Johal  __(2011); Ukraine  _et al._ (2015); Altintas  _et al._ (2015);
Ramandeep  _et al._ (2021) have considered only two coupled spin-$1/2$. The
expression of the Hamiltonian is given by

$\mathrm{H=J\sum_{i}^{N}(\sigma_{x}^{i}\sigma_{x}^{i+1}+\sigma_{y}^{i}\sigma_{y}^{i+1}+\sigma_{z}^{i}\sigma_{z}^{i+1})+h\sum_{i}^{N}\sigma_{z}^{i},}$
(50)

where $\mathrm{\sigma_{x,y,z}^{i}}$ are the standard Pauli matrices acting on
the site $\mathrm{i=1,2,3}$. Note that the periodicity is presumed.
$\mathrm{J}$ and $\mathrm{h}$ are the exchange coupling and the strength of
the external magnetic field, respectively. When $\mathrm{J<0}$ the model is
ferromagnetic, while $\mathrm{J>0}$ corresponds to an antiferromagnetic
system. Here, we have the isotropic situation $\mathrm{J_{x}=J_{y}=J_{z}=J}$.
The eigenvalues of $\mathrm{H}$ are given by: $\rm
E=\\{-(h+3J),-(h+3J),h-3J,h-3J,-3(h-J),3(h+J),-h+3J,h+3J\\}$. Their associated
eigenstates will not be reported here, since we will not use them.

Figure 5: (Color online). Plot of (a) the extracted work when $\mathrm{N=2}$
(orange dashed line) and $\mathrm{N=3}$ (blue solid line) as a function of
$\mathrm{J}$ (b) the efficiency for both $\mathrm{N=2}$ and $\mathrm{N=3}$ as
a function of $\mathrm{J}$. The parameter values are the same as in Fig. 1.
Figure 6: (Color online). The plot of COP for $\mathrm{N=2}$ (red large-
dashing line) and $\mathrm{N=3}$ (blue dotted line) as a function of
$\mathrm{J}$. The parameter values are the same as in Fig. 3. The green dot-
dashed line and the solid orange line correspond to the COP of Otto and
Carnot, respectively.

In Fig.5(a), we plot the work as a function of $\mathrm{J}$ for $\mathrm{N=2}$
and $\mathrm{N=3}$. We see that when $\mathrm{N=3}$, the work extracted
globally is bigger than the one from two-coupled spin-$1/2$. Even more, the
three coupled spins could still harvest work in the strong coupling regime in
contrast to the two-coupled spin-$1/2$. In Fig.5(b), the efficiency
$\mathrm{\eta}$ is plotted as a function of $\mathrm{J}$. We see that
$\mathrm{N=3}$ outperforms $\mathrm{N=2}$ in terms of the highest value riched
of the efficiency. More precisely, for the considered value of the parameters,
the maximum value riched when N=3 is $\approx$0.36, and when N=2 it is
$\approx$0.34. We should add that for J$\leq$0.52, the efficiency of two
coupled spins is higher than that of N=3, as Fig.5(b) shows. In Fig.6 we plot
their COPs as well as the ones of Otto $\mathrm{COP_{o}}$ and Carnot
$\mathrm{COP_{c}}$. We see that only three-coupled spin-$1/2$ COP could
surpass the COP of the Otto $\mathrm{COP_{o}}$, however, the COP of two-
coupled spin-$1/2$ is always equal or less than it. We can conclude from those
figures that increasing the number of spins enhances the amount of extractable
work as well as the maximal values of both the efficiency and the COP. Please,
note that here we do not mean that the COP of three coupled spins is always
bigger than that of N=2 since, as figure 6 shows, the latter can be higher
than that of N=3 for broad values of J, but its maximum value does not exceed
the one of N=3. In addition to this, notice that when the system is working as
a heat engine for three-coupled spin-$1/2$, its efficiency could surpass that
of the Otto $\mathrm{\eta_{o}}$ only when it is anti-ferromagnetic. And when
it is a refrigerator, the enhancement is seen only when the system is
ferromagnetic. We have also calculated the work extracted locally by the three
coupled spins, and we found that it is equal to the one extracted globally. As
we said before, this is due to the fact that the eigenstates of the system are
parameter independent and that the eigenvalues are linear in the magnetic
field Johal  __(2011). However, we didn’t include its plot here for brevity.

Now let’s explain the reasons behind the enhancement in the work extracted.
First, we expect that more levels will help extract more work, e.g. a harmonic
oscillator could extract more work than a single spin. However, their
efficiencies are equal if all energies are shifted by the same amount of
energy. More precisely, we see from the figure 5 that when the coupling is
turned off, i.e. $\mathrm{J=0}$, then as we increase the number of spins (i.e.
when $\mathrm{N=2}$ and $\mathrm{N=3}$ for $\mathrm{J=0}$) more work will be
extracted. When turning on the interaction, we see that the work extracted by
coupled spins enhances the one from uncoupled ones. Thus we see that both the
number of spins N as well as the interaction $\mathrm{J}$ between them
enhances the work extracted globally W.

When we compared the efficiency of 3 and 2 coupled Heisenberg spins we found
that the latter cannot work as a heat engine in the strong coupling regime,
and the reason is that the positive work condition is not satisfied, since in
this regime the system consumes work instead of outputing it, thus the system
no longer works as a heat engine. More precisely, the work extracted
$\mathrm{-W_{2}}$ during the adiabatic stage 4 is less than the one invested
$\mathrm{-W_{1}}$ in the adiabatic stage 2, thus $\mathrm{W=-(W_{1}+W_{2})\leq
0}$. When it comes to the three coupled spins in the strong coupling regime,
the eigenvalues of the system become equidistant, thus with efficiency equal
to the one of the Otto $\mathrm{\eta_{o}=1-\frac{h^{\prime}}{h}}$.

Concerning the local and global thermodynamics of the three spins Heisenberg
model, we have the preservation of the extensive property of the work, i.e.,

$\mathrm{W=w_{1}+w_{2}+w_{3}},$ (51)

with the local work extracted by each spin is
$\mathrm{w_{\alpha}=Tr((H_{l}-H^{\prime}_{l})(\rho_{\alpha}-\rho_{\alpha}^{\prime}))}$,
with $\mathrm{\alpha=1}$, 2, and 3. The local states are,
$\mathrm{\rho_{1}=(a,1-a)}$, $\mathrm{\rho_{2}=(b,1-b)}$ and
$\mathrm{\rho_{3}=(c,1-c)}$. The parameters a, b and c are functions of the
populations and they are different, not like the case of two coupled spins,
because in this case the state is no longer symmetric with respect to the
partial trace. Therefore, the local extractable work is,

$\mathrm{w_{1}+w_{2}+w_{3}=2(h-h^{\prime})(a-a^{\prime}+b-b^{\prime}+c-c^{\prime})}.$
(52)

If we replace a, a’, b, b’, c, and c’ with their expressions in terms of
populations, we get that the global work W is equal to the total local one. It
would be very important to compute the upper bound of the efficiency when we
are no longer interested in two interacting systems but N s-spins, and to
investigate if the number of coupled spins could affect the upper bound of the
efficiency in addition to the coupling and the spin of the particles (see,
Ramandeep  _et al._ (2021)), i.e. if $\mathrm{\eta_{up}}$ is a function of N.

###  IV.2 Ising model

The Hamiltonian of the $\mathrm{N}$ spin-$\frac{1}{2}$ Ising model under the
influence of a magnetic field $\mathrm{h}$ is

$\mathrm{H=J\sum_{i=1}^{N}\sigma_{z}^{i}\sigma_{z}^{i+1}+h\sum_{i=1}^{N}\sigma_{z}^{i},}$
(53)

where $\mathrm{\sigma_{z}^{i}}$ is the $\mathrm{z}$-component of the Pauli
spin matrices acting on the site $\mathrm{i\in[1,N]}$. The periodicity is here
as well as assumed, i.e., $\mathrm{\sigma_{z}^{N+1}=\sigma_{z}^{1}}$.
$\mathrm{N}$ is the total number of sites, and here it will range from $2$ to
$6$. The eigenvalues of this Hamiltonian are given in the appendix. Note that
this Hamiltonian has been used in Ref.Oliveira2020  _et al._ (2021), to show
that the enhancement is only a matter of the structure of the system and it is
not quantum correlations. Here, we also study the case when the system is
working as a refrigerator which was not considered in Oliveira2020  _et al._
(2021).

Figure 7: (Color online). Plot of (a) the extracted work from $\mathrm{N}$
coupled spins as a function of $\mathrm{J}$, where $\mathrm{N}$ ranges from
$2$ to $6$ (b) the efficiency of coupled spin-$1/2$ as a function of
$\mathrm{J}$. The parameter values are the same as in Fig.1. It is worth
noting that the efficiencies of three to six coupled spins are nearly
coincidable. $\mathrm{N=6}$ (blue dashed line), $\mathrm{N=5}$ (orange dotted
line), $\mathrm{N=4}$ (green dot-dashed line), $\mathrm{N=3}$ (red large-
dashing line), $\mathrm{N=2}$ (purple solid line). Figure 8: (Color online).
Plot of (a) the extracted work from $\mathrm{N}$ coupled spin-$1/2$ as a
function of $\mathrm{J}$, where $\mathrm{N}$ ranges from $2$ to $6$ (b) the
efficiency of coupled spin-$1/2$ as a function of $\mathrm{J}$. Here, in
contrast to Fig.7 the work and efficiency are plotted in the weak as well as
in the strong coupling regimes. The parameter values are the same as in Fig.
1. $\mathrm{N=6}$ (blue dashed line), $\mathrm{N=5}$ (orange dotted line),
$\mathrm{N=4}$ (green dot-dashed line), $\mathrm{N=3}$ (red large-dashing
line), $\mathrm{N=2}$ (purple solid line). Figure 9: (Color online). The work
extracted versus the efficiency, where $\mathrm{J}$ is in the interval
$[0,10]$, and $\mathrm{N}$ ranges from $2$ to $6$. The parameter values are
the same as in Fig.1. $\mathrm{N=6}$ (blue dashed line), $\mathrm{N=5}$
(orange dotted line), $\mathrm{N=4}$ (green dot-dashed line), $\mathrm{N=3}$
(red large-dashing line), $\mathrm{N=2}$ (purple solid line). Figure 10:
(Color online). The COP of $\mathrm{N}$ coupled spin-$1/2$ as a function of
$\mathrm{J}$. It is only in the interval $[-0.5,0]$ that we have an
enhancement in the COP for three to six-coupled spin-$1/2$. The parameter
values are the same as in Fig.3. $\mathrm{N=6}$ (blue dashed line),
$\mathrm{N=5}$ (orange dotted line), $\mathrm{N=4}$ (green dot-dashed line),
$\mathrm{N=3}$ (red large-dashing line), $\mathrm{N=2}$ (purple solid line).

In Fig. 7(a), we plot the work extracted from two coupled spins up to six. We
see that as we increase their number, more work can be extracted. As explained
before, it is the number of spins N and the interaction between them J which
enhances the work extracted. In Fig.7(b), we plot the efficiency as a function
of $\mathrm{J}$ for $\mathrm{N=2}$ up to $\mathrm{N=6}$. We see that
increasing the number of coupled spins enhances the efficiency, but only for
small values of $\mathrm{J}$. However, the efficiency of three up to six
coupled spins is nearly coincidable. In Altintas  _et al._ (2015) the authors
studied a coupled spin-$1/2$ with another $s$ spin where $s$ ranges from $1$
to $3$, and they found that all of these interacting spins can work as heat
engines in the strong coupling regime. However, in our case, we see from
Fig.8, that only three and five interacting spins could work as heat engines
in the strong regime. Furthermore, in our case, in contrast to them, we see a
remarkable enhancement in the extractable work, as in Ref.DasGosh  _et al._
(2019), since in our case as well, we have degeneracy. In Fig.9 we plot work
versus efficiency for $\mathrm{J}$ in the interval $[0,10]$. We can see that
the efficiency at maximum work is not altered by the number of interacting
spins. However, in contrast to Altintas  _et al._ (2015), the work at maximum
efficiency is affected by the number of interacting spins.

Now let us study the case when our coupled spins are working as a
refrigerator. For this, the COP is plotted in Fig.10. It is seen that when the
system is ferromagnetic, the COP of $3$ to $6$ coupled spins is above that of
two-coupled spin-$1/2$. However, all of them surpass the COP of the Otto
engine. In addition to this, notice that when the system is working as a heat
engine (Figs.7 and 8), its efficiency can surpass Otto one only when the
system is anti-ferromagnetic. And when it is a refrigerator, the enhancement
is seen when the system is ferromagnetic (Fig.10). We now briefly compare our
results with the ones reported in GuoZhang  _et al._ (2008); Zhang  _et al._
(2007); Altintas  _et al._ (2015). It was shown in GuoZhang  _et al._ (2008);
Zhang  _et al._ (2007) that entanglement between the two-coupled spins at the
end of the hot isochoric stage is not necessary for harvesting work. They show
that we only need to have entanglement at the end of the cold isochotic stage.
In contrast, in Altintas  _et al._ (2015) , it was shown that we can harvest
work even if we have no entanglement on the cold isochoric side and the hot
isochoric side, but quantum discord was present. In our case, we have neither
entanglement nor quantum correlations, and even more, the system could harvest
work. To resume, from Figs. 7, 8, 9 and 10 we see that the extractable work,
the efficiency, as well as the COP of these coupled but not correlated systems
could surpass the corresponding ones of the Otto without exploiting
entanglement and quantum correlations.

Here, when the system is working as a heat engine, we plot work and efficiency
only for positive values of $\mathrm{J}$. This is because only when the system
is anti-ferromagnetic do we have an enhancement in work as well as efficiency.
When the system is working as a refrigerator, the enhancement is observed in
the COP only when the system is ferromagnetic. One can explain this just from
the eigenvalues and the populations of the interacting spins. For
$\mathrm{N=2}$, the enhancement in the COP is approximately in the interval
$[-2,0]$. For $\mathrm{N=3}$ to $6$, the enhancement in COP is approximately
in the interval $[-0.5,0]$.

As for the Ising model, the eigenvalues are linear in the magnetic field and
the eigenstates are parameter independent, thus one can easily demonstrate
that we have the preservation of the extensive property of the work, i.e.
$\mathrm{W_{g}=Nw_{local}}$ which means that the work extracted from N coupled
spins globally will be equal to N times the one extracted locally from one
spin $\mathrm{w}$. This is due to the fact that, for the Ising model
considered here, all the local spins have the same reduced state independently
on N, thus $\mathrm{W=Nw}$. We conclude that as long as the eigenvalues are
linear in the magnetic field and the eigenstates are parameter independent,
the global work would be extensive.

Let’s now focus on the local and the global thermodynamics of the two coupled
spin-1/2 Ising model. We recall that the eigenvalues of this system are
$\rm\\{E_{1}=2h,E_{2}=E_{3}=-2J,E_{4}=-2h\\}$ and their corresponding
probabilities are, $\rm\rho=\\{p_{1},p_{2},p_{2},p_{4}\\}$. The local reduced
states of the subsystems are equal,
$\rm\rho_{1}=\rho_{2}=diag\\{p_{1}+p_{2},p_{4}+p_{2}\\}$. The expression of
the global work in terms of the energies and the populations is,

$\mathrm{W=(E_{1}-E^{\prime}_{1})(p_{1}-p^{\prime}_{1}+p^{\prime}_{4}-p_{4}).}$
(54)

The local work extracted from each subsystem is,

$\mathrm{w_{1}=(E_{1}-E^{\prime}_{1})(p_{1}-p^{\prime}_{1}+p^{\prime}_{4}-p_{4})/2.}$
(55)

Thus, the global work is extensive. In addition, one should note that even
though the state is only classically correlated, the local effective
temperatures are not equal to the ones of the baths, and this is because of
the coupling J. The effective temperatures Johal  __(2011) are given by,

$\mathrm{T_{eff,1}=T_{eff,2}=2h(\log(\frac{p_{2}+p_{4}}{p_{1}+p_{2}}))^{-1},}$
(56)

at the hot bath side and,

$\mathrm{T_{eff,1}^{\prime}=T_{eff,2}^{\prime}=2h^{\prime}(\log(\frac{p_{2}^{\prime}+p_{4}^{\prime}}{p_{1}^{\prime}+p_{2}^{\prime}}))^{-1},}$
(57)

at the cold bath side. This tells us that even though the global extracted
work is the same as the local one and that the spins are only classically
correlated, the heats absorbed from and released to the heat baths are not
equal. Therefore, the operation modes will not be the same. This is because
the local temperatures are not equal to the ones of the heat baths. The global
heats are,

$\mathrm{Q_{h}=E_{1}(p_{1}-p^{\prime}_{1}+p^{\prime}_{4}-p_{4})+2E_{2}(p_{2}-p^{\prime}_{2}),}$
(58)
$\mathrm{Q_{c}=-E^{\prime}_{1}(p_{1}-p^{\prime}_{1}+p^{\prime}_{4}-p_{4})-2E_{2}(p_{2}-p^{\prime}_{2}),}$
(59)

and the total local ones are,

$\mathrm{q_{h,12}=E_{1}(p_{1}-p^{\prime}_{1}+p^{\prime}_{4}-p_{4}),}$ (60)
$\mathrm{q_{c,12}=-E^{\prime}_{1}(p_{1}-p^{\prime}_{1}+p^{\prime}_{4}-p_{4}).}$
(61)

We see that they are not equal, and are different by the quantity
$\mathrm{2E_{2}(p_{2}-p^{\prime}_{2})}$, which is the responsible for th
enhancement of the efficiency. More precisely, this term, even though makes no
difference between the work extracted by a global agent and a local one, the
global $\mathrm{\eta_{g}}$ and $\mathrm{COP_{g}}$ can exceed the ones of the
Otto $\mathrm{\eta_{o}}$ and $\mathrm{COP_{o}}$, which is not the case for the
local efficiency ($\mathrm{\eta_{l}=1-h^{\prime}/h}$) and local COP
($\mathrm{COP_{l}=h^{\prime}/h-h^{\prime}}$) which are equal to the ones of
the Otto. This means that a global agent can extract the same work as a local
one, but by absorbing less amount of heat from the hot bath. And for
refrigeration, for the same amount of consumed work, the global agent can
absorb a bigger amount of heat than the local one. Thus, the global agent will
cool more efficiently than a local one, and this is independently if the
system is classically or quantum mechanically correlated.

Finally, let’s explain the enhancement observed in the COP of the Ising model.
The COP is given as follows:

$\mathrm{COP=\frac{h^{\prime}}{h-h^{\prime}}+\frac{-4J_{z}(p_{2}-p_{2}^{\prime})}{2(h-h^{\prime})(p_{4}-p^{\prime}_{4}+p^{\prime}_{3}-p_{3})}}.$
(62)

$\mathrm{q_{c}=4J_{z}(p_{2}-p_{2}^{\prime})}$ this is the heat absorbed from
the cold bath by the degenerate idle level $\mathrm{-2J_{z}}$. In this case,
one sees that if $\mathrm{J_{z}=0}$ the COP is equal to the one of the Otto.
And to exceed it, the second term has to be positive, which is equivalent to
$\mathrm{-4J_{z}(p_{2}-p_{2}^{\prime})<0}$, since the denominator is negative.
This means that idle levels have to absorb heat from the cold bath in addition
to working ones, otherwise the COP will not be degraded. Thus, if
$\mathrm{J_{z}>0}$ then $\mathrm{p_{2}-p^{\prime}_{2}}$ has to be positive.
However, for $\mathrm{J_{z}>0}$, we have $\mathrm{p_{2}-p^{\prime}_{2}<0}$,
thus the second term will degrade the COP. This expalins why the enhancement
in the COP is observed only when the system is ferromagnetic. Note that in
contrast to Ising+KSEA model in which idle levels are not needed to enhance
the COP, here we need them, but they have to channel heat in the same
direction as working levels. Finally, following the same arguments, we can
explain the enhancement in the efficiency.

##  V Clarifying some important points

At this point, before we conclude this paper, we should clarify some important
points:

First, in the previous works Johal  __(2011); Ukraine  _et al._ (2015);
Altintas  _et al._ (2015) we have seen that the eigenvalues were linear in
$\mathrm{h}$ for the local as well as for the global Hamiltonian. However, we
see here (Sec. III) that the eigenvalues of the global Hamiltonian are not
linear in the $\mathrm{h}$ in contrast to the local one. Therefore, the
extensive property of the work is dependent on the eigenvalues as well as on
the eigenstates of the Hamiltonian. Moreover, we can safely say that the
extensive property of the work is not related to any type of quantum or
classical correlations. It is only related to the coupling parameters. More
precisely, if the chosen interaction between the spins does still preserve the
linearity of the eigenvalues in the magnetic field, then in this case the
extensive property of the work would still be preserved. However, if it
violates the linearity, then we can no longer ensure that the global work is
equal to the total local one. This is because the local Hamiltonian is linear
in h.
Note that one may think that the preservation of the extensive property has to
do with the assumption that the system is in equilibrium with a heat bath
apositive temperature $\mathrm{\beta>0}$, thus the state is passive, i.e.
higher levels of energy are occupied with less probabilities (for
$\mathrm{E_{i}\geq E_{j}}$ we have $\mathrm{p_{i}\leq p_{j}}$, see
ref.Ergotropy  __(2004)), which means that state ergotropy is zero. However,
this is not the case. More precisely, we can let the system thermalizes with a
heat bath at negative temperature $\mathrm{\beta}<0$, thus the state in this
case is not passive because higher levels will be occupied with high
populations, i.e. for $\mathrm{E_{i}\geq E_{j}}$ we have $\mathrm{p_{i}\geq
p_{j}}$, thus the state would have non-zero ergotropy. However, even in this
case, the extensive property is still preserved for the Heisneberg model but
not for the Ising+KSEA model. And this is because the temperature of the bath
will only affect the value of populations, but not the expression of the
global as well as the local works in terms of energies and populations.
Therefore, $\mathrm{W-w\geq 0}$ is neither related to quantum correlations nor
to ergotropy, i.e. is the state being passive or not, it is only dependent on
the Hamiltonian $\mathrm{H}$.

Second, actually one may ask that even though if entanglement is not present,
however there will be quantum discord which could affect the thermodynamical
quantities, i.e. heats, work and efficiency. However, the Ising model shows us
that this enhancement in efficiencies, i.e. $\mathrm{\eta>\eta_{o}}$ and
$\mathrm{COP>COP_{o}}$ could be observed as well in the absence of discord.

Third, note that when one sees the eigenvalues of the interacting spins,
either for the Heisenberg or the Ising model, one may still not be convinced
if it is the presence of idle levels or something else that is the reason
behind this enhancement. In this case, one can simply shift the eigenvalues
and will get two sets of levels, i.e. working and idle levels. Let’s take two
examples: the first is the case of three coupled spins XXX chain. In this
case, shifting the eigenvalues $\rm
E=\\{-(h+3J),-(h+3J),h-3J,h-3J,-3(h-J),3(h+J),-h+3J,h+3J\\}$ by +h one gets
the new eigenvalues: $\rm E=\\{-3J,-3J,2h-3J,2h-3J,-2h+3J,4h+3J,-3J,2h+3J\\}$.
Thus, we obtain three idle levels. These levels would not contribute to work,
but only to heat. The other example is the Ising model of two interacting
spins which are, $\rm\\{-2h,-2J,-2J,2h\\}$. And if one plots the heat absorbed
by the idle levels, one sees that they are channeling heat in the wrong
direction of the one by the working levels. However, it is not necessary that
all idle levels take heat in the opposite direction, but only the average of
heat should be in the opposite direction of the working levels. Furthermore,
one should note that even though there is a part of working levels that does
not depend on the magnetic field, i.e. J, this part only contributes to
$\mathrm{Q_{h}}$ and $\mathrm{Q_{c}}$, but work is only dependent on the
working part, i.e. the part that depends on the magnetic field. Thus, it is
not necessary to divide the levels into two sets: idle and working levels to
explain the enhancements.
Let’s take the case when the eigenvalues are linear in the h and J, i.e.
$\mathrm{E_{i}=a_{i}h+b_{i}J}$ and
$\mathrm{E^{\prime}_{i}=a_{i}h^{\prime}+b_{i}J}$. $\mathrm{a_{i}}$ and
$\mathrm{b_{i}}$ are just numbers, they can be positive or negative. Then the
heats and work expressions are,

$\mathrm{Q_{h}=\sum_{i}(a_{i}h+b_{i}J)(\frac{e^{-\beta_{h}(a_{i}h+b_{i}J)}}{Z_{h}}-\frac{e^{-\beta_{c}(a_{i}h^{\prime}+b_{i}J)}}{Z_{c}})},$
(63)
$\mathrm{\mathrm{Q_{c}=-\sum_{i}(a_{i}h^{\prime}+b_{i}J)(\frac{e^{-\beta_{h}(a_{i}h+b_{i}J)}}{Z_{h}}-\frac{e^{-\beta_{c}(a_{i}h^{\prime}+b_{i}J)}}{Z_{c}}})},$
(64)

and work is,

$\mathrm{W=\sum_{i}a_{i}(h-h^{\prime})(\frac{e^{-\beta_{h}(a_{i}h+b_{i}J)}}{Z_{h}}-\frac{e^{-\beta_{c}(a_{i}h^{\prime}+b_{i}J)}}{Z_{c}})}.$
(65)

With $\mathrm{Z=\sum_{i}e^{-\beta E_{i}}}$. We see that the heats are affected
by both, the part of eigenvalues that depend oh h and the one of J. However,
work is affected by J only through the populations, i.e. the idle part of
$\mathrm{E_{i}}$ disappears because of the difference
$\mathrm{E_{i}-E^{\prime}_{i}}$. Thus, if J is tuned in the right way, more
work will be extracted than in the case $\mathrm{J=0}$. The same thing can be
said when the eigenvalues are linear in h but not in J. However, when
$\mathrm{E_{i}}$ are nonlinear in h things become more complicated.

The above expressions of heats and work can be rewritten as,

$\mathrm{Q_{h}=ah+bJ},\ \mathrm{Q_{c}=-ah^{\prime}-bJ,\ and\
W=a(h-h^{\prime})}.$ (66)

With
$\mathrm{a=\sum_{i}a_{i}(e^{-\beta_{h}(a_{i}h+b_{i}J)}/Z_{h}-e^{-\beta_{c}(a_{i}h^{\prime}+b_{i}J)}/Z_{c})}$
and
$\mathrm{b=\sum_{i}b_{i}(e^{-\beta_{h}(a_{i}h+b_{i}J)}/Z_{h}-e^{-\beta_{c}(a_{i}h^{\prime}+b_{i}J)}/Z_{c})}$.
In this case the efficiency will be,

$\mathrm{\eta=\eta_{o}\frac{1}{1+\frac{bJ}{ah}}}.$ (67)

For $\mathrm{\eta>\eta_{o}}$ the second term has to be bigger than one. From
$\mathrm{W>0}$ we have $\mathrm{a>0}$ (we have assumed that
$\mathrm{h>h^{\prime}}$), thus depending on the sign of $\mathrm{bJ}$,
$\mathrm{\eta}$ either can be enhanced or degraded. More precisely, if
$\mathrm{bJ>0}$ then $\mathrm{\eta<\eta_{o}}$ and if $\mathrm{bJ<0}$ then
$\mathrm{\eta>\eta_{o}}$. Thus, when the term of global heat $\mathrm{Q_{h}}$
absorbed from the hot bath multiplied by h takes heat from the hot bath to the
cold bath, the heat multiplied by J has to take it in the other direction.
Now, let’s explain the enhancement in the COP. The expression of the COP is,

$\mathrm{COP=COP_{o}+\frac{bJ}{a(h-h^{\prime})}.}$ (68)

Since $\mathrm{a(h-h^{\prime})<0}$, in this case bJ has to be negative as
well. And because -bJ is the amount of heat absorbed from the cold bath, thus
it will enhance the COP. Therefore, it is necessary in this case that idle
levels take heat from the bath we want to cool otherwise the COP will be
degraded.

Third, in Ref.Chang  _et al._ the authors found a nice relation between the
work extracted and entropies. This relationship is given as follows:

$\mathrm{W=(T_{h}-T_{c})(S(\rho)-S(\rho^{\prime}))-T_{h}H[p^{\prime}_{i}|p_{i}]-T_{c}H[p_{i}|p^{\prime}_{i}]}.$
(69)

H is the relative entropy and it is positive. $\rho$ and $\rho^{\prime}$ are
the equilibrium states of the system at the hot and cold baths, and p and p’
are the corresponding populations, respectively. This relationship can be
easily proven starting from,

$\mathrm{W=\sum_{i}(E_{i}-E^{\prime}_{i})(p_{i}-p^{\prime}_{i})}.$ (70)

Then using $\mathrm{E_{i}=-T_{h}\log(p_{i}Z_{h})}$ and
$\mathrm{E^{\prime}_{i}=-T_{c}\log(p^{\prime}_{i}Z_{c})}$, we get this
relation. Therefore, for $\mathrm{W>0}$ we have to ensure not only that
$\mathrm{T_{h}>T_{c}}$ but as well $\mathrm{S(\rho)>S(\rho^{\prime})}$, and an
addition to this
$\mathrm{(T_{h}-T_{c})(S(\rho)-S(\rho^{\prime}))>T_{h}H[p^{\prime}_{i}|p_{i}]+T_{c}H[p_{i}|p^{\prime}_{i}}$.
More precisely, this relationship tells us that for the positive work
condition to be satisfied the entropy of the system at the hot bath side has
to be greater than the one at the cold bath side. And according to the second
law, this is true, as heat will flow spontaneously from a high entropy system
to a less entropy one. Moreover, this relation is useful either for a coupled
or only one system, and it’s difficult to relate it to quantum correlations,
since it tells us for $\mathrm{W>0}$ we have only to compare the entropies of
the system at the hot and cold bath sides independtly of the system being
correlated or just a one system. In the same manner, if one is interested in
the local thermodynamics of correlated systems, one can have an expression of
the local work in terms of entropies and relative entropies. In this case, the
expression of the local work in terms of entropies is,

$\begin{split}\rm
w_{\alpha}=&\mathrm{(T_{h}-T_{c})(S(\rho_{\alpha})-S(\rho^{\prime}_{\alpha}))-T_{h}H[p^{\prime}_{i,\alpha}|p_{i,\alpha}]}\\\
&\mathrm{-T_{c}H[p_{i,\alpha}|p^{\prime}_{i,\alpha}].}\end{split}$ (71)

with $\mathrm{\alpha=1}$ and 2. $\mathrm{\rho_{\alpha}}$ is the local reduced
state of the system, $\mathrm{p_{i,\alpha}}$ and
$\mathrm{p^{\prime}_{i,\alpha}}$ are the local populations of the system when
it is in thermal equilibrium with a hot bath and cold bath, respectively.
Therefore, according to our study, the equations 69 and 71 tell us that the
condition for a positive work either globally or locally can be explained only
from the entropies of the system at the hot and cold bath sides without
mentioning quantum correlations and entanglement. Furthermore, one should note
that Eqs. 69 and 71 are not used to explain efficiency enhancement, but they
are only related to the positive work condition.

Fourth, our results do not claim that quantum correlations should be excluded
at all to enhance the performance of thermal machines. But it shows that
surpassing the Otto efficiencies could be possible even without using them.
Therefore, one may wonder if the Carnot efficiency could be surpassed using
idle levels, and our answer would be no. And the reason is that even though
some levels take heat in the opposite direction to the average of heat flow,
the effective temperature associated with each of the two levels of the system
is equal to the heat baths, i.e. $\mathrm{T_{h}}$ and $\mathrm{T_{c}}$, thus
$\mathrm{\eta\leq\eta_{C}=1-\frac{T_{c}}{T_{h}}}$. In addition to this, one
should note that the enhancement in efficiency comes at a cost which is due to
creating such non-uniform eigenvalues.

Finally, our studies show that neither the enhancement in work, efficiency,
COP nor the extensive property of work are related to the presence of quantum
correlations. We have seen as well that idle levels are needed to take heat in
the wrong direction to enhance efficiency. However, when the system is working
as a refrigerator, we saw that depending on the model, they may not be
necessary to contribute to the heats to enhance the COP or if their
contribution is necessary they have to take heat in the same direction as the
working levels, otherwise they will degrade the heat absorbed from the cold
bath, thus $\mathrm{COP\leq COP_{o}}$.

##  VI Conclusions

By studying two coupled spin-$1/2$ Ising model under the influence of KSEA
interaction in a magnetic field along the $\mathrm{z}$-axis, we show that the
enhancement observed in the efficiency is only due to the structure of the
energy levels of the system and not entanglement or quantum correlations.
Furthermore, we reexamined the results reported in Altintas  _et al._ (2015)
according to which to break the extensive property of the work extracted
globally, we have to change the coupling parameters next to the magnetic
field. In our case, we show that this is not necessary and we can break it
even when only the magnetic field is changed in the adiabatic stages. Then we
study the effect of increasing the number of interacting spin-$1/2$ on
efficiency, extractable work, and COP. In contrast to Ref.Altintas  _et al._
(2015), when they extend the dimension of one spin, they have a little
enhancement in efficiency and the amount of extractable work, in our case we
see a remarkable enhancement in the extractable work, though the maximum of
the efficiencies is still bounded by the one of two-coupled spin-$1/2$. In
addition, we see that only when the number of the interacting spins is odd,
the system could work as a heat engine in the strong coupling regime. The
enhancement in COP is seen as well when the system is working as a
refrigerator. Moreover, note here that we did not consider the $\mathrm{XY}$
components of $\mathrm{J}$, only the $\mathrm{z}$-component of $\mathrm{J}$ to
ensure that no entanglement or quantum correlations will build up between the
interacting spins along the cycle. Our results support the expectation, put
forward in Refs.Anka  _et al._ (2021); Oliveira2020  _et al._ (2021); Johall
_et al._ (2022), that the structure of the Hamiltonian is a resource and that
this is the reason behind the enhancement observed in the efficiency and the
extractable work, as well as the COP.

It is worth noting that the enhancement observed in efficiency and COP was
also observed in refs.Mendes  _et al._ (2021); Sourav  _et al._ . In
Ref.Sourav  _et al._ the role of anharmonicity on COP has been investigated.
The authors there showed that the COP of a quantum harmonic oscillator after
introducing anharmonicity could surpass that of the Otto. Moreover, this was
shown even for a two-level system, as in our case, when we have two coupled
spin-$1/2$ under the influence of KSEA interaction in the absence of idle
levels, i.e., when $\mathrm{J_{z}=0}$. In Ref. Mendes  _et al._ (2021), the
role of Kerr-nonlinearity on efficiency and COP has been studied in detail,
showing that Kerr-nonlinearity could boost them above the ones of Otto. To
resume, our work supports the idea that the enhancement in the performance of
coupled spins is only a matter of the structure of the Hamiltonian. It would
be useful to look at the role of thermal fluctuations Jarzynski1995  __(1995);
Campisi  __(2011); Crooks  __(1999), since here and in the previous works
Zhang  _et al._ (2007); Johal  __(2011); Altintas  _et al._ (2015); Sodeif
_et al._ (2021); Deniz  _et al._ (2020); Ukraine  _et al._ (2015); Mehta  _et
al._ (2017); Anka  _et al._ (2021); Zhao  _et al._ (2017); Ramandeep  _et al._
(2021) the interest was only in the average of the thermodynamical quantities,
i.e., heat, work, efficiency, and COP. Furthermore, our work and the previous
ones will pave the way for efficient quantum thermal machines in which the
structure of the system is used as a resource. Inspired by other quantum
resource theories, Brandao  __(2013); Brandao2  __(2015); Winter  __(2014);
Baumgratz  __(2014); Gour  __(2019); Spekkens  __(2008), a resource theory of
the structure of the energy levels of quantum systems, will also be
interesting to study Mauro  __(2016).

Our results go in the same way as in Kurizki23  __(2016); GWatanabe  __(2016);
Latune  __(2016); Kamimura  __(2016); Souza  __(2016), which show the effect
of the coupling on the performance of quantum heat engines. We hope to
investigate further our study concerning the global and the local
thermodynamics more thoroughly, or in general, when both the unitary and the
thermalization strokes are in finite time.

Acknowledgment: We gratefully thank anonymous reviewers for their constructive
remarks and for their insightful comments and suggestions, which have improved
the quality of the paper.

##  Appendix A EIGENVALUES OF THE WORKING MEDIUM

Here we give the eigenvalues of the Hamiltonian 53. Their corresponding
eigenstates are the elements of the computational basis. The eigenvalues of
the Hamiltonian when $\mathrm{N=2}$ are $\rm\\{2h,-2J,-2J,-2h\\}$ Johal
__(2011); Anka  _et al._ (2021).

The eigenvalues of the Hamiltonian when $\mathrm{N=3}$ are
$\rm\\{3h+3J,h-J,h-J,-h-J,h-J,-h-J,-h-J,-3h+3J\\}$. The partition function of
the system when it is in equilibrium with a heat bath at inverse temperature
$\mathrm{\beta}$ is: $\rm Z=2e^{-3\beta J}\cosh(3\beta h)+6e^{\beta
J}\cosh(\beta h)$.

The eigenvalues of the Hamiltonian when $\mathrm{N=4}$ are
$\rm\\{4h+4J,2h,2h,0,2h,-4J,0,-2h,2h,0,-4J,-2h,0,-2h,-2h,\\\ -4h+4J\\}$. The
partition function of the system when it is in equilibrium with a heat bath at
inverse temperature $\mathrm{\beta}$ is: $\rm Z=2(e^{4\beta J}+e^{-4\beta
J}\cosh(4\beta h))+4(1+2\cosh(2\beta h))$.

The eigenvalues of the Hamiltonian when $\mathrm{N=5}$ are
$\rm\\{5h+5J,3h+J,3h+J,h+J,3h+J,h-3J,h+J,-h+J,3h+J,h-3J,h-3J,-h-3J,h+J,-h-3J,-h+J,-3h+J,3h+J,h+J,-h-3J,-3h+J,h+J,-h+J,-h-3J,-3h+J,h+J,-h+J,-h-3J,-3h+J,-h+J,-3h+J,-3h+J,-5h+5J\\}$.
The partition function of the system when it is in equilibrium with a heat
bath at inverse temperature $\mathrm{\beta}$ is: $\rm Z=2e^{-5\beta
J}\cosh(5\beta h)+10e^{-\beta J}(\cosh(\beta h)+\cosh(3\beta h))+10e^{3\beta
J}\cosh(\beta h)$.

The eigenvalues of the Hamiltonian when $\mathrm{N=6}$ are
$\rm\\{6h+6J,4h+2J,4h+2J,4h+2J,4h+2J,4h+2J,4h+2J,2h+2J,2h+2J,2h+2J,2h+2J,2h+2J,2h+2J,2h-2J,2h-2J,2h-2J,2h-2J,2h-2J,2h-2J,2h-2J,2h-2J,2h-2J,2J,2J,2J,2J,2J,2J,-2J,-2J,-2J,-2J,-2J,-2J,\\\
-2J,-2J,-2J,-2J,-2J,-2J,-2h+2J,-2h+2J,-2h+2J,-2h+2J,-2h+2J,-2h+2J,-2h-2J,-2h-2J,-2h-2J,-2h-2J,-2h-2J,-2h-2J,-2h-2J,-2h-2J,-2h-2J,-4h+2J,-4h+2J,-4h+2J,-4h+2J,-4h+2J,-4h+2J,-6J,-6J,-6h+6J\\}$.
The partition function when the system is in thermal equilibrium with a heat
bath at inverse temperature $\beta$ is: $\rm Z=2(e^{-6\beta J}\cosh(6\beta
h)+e^{6\beta J})+12e^{-2\beta J}(\cosh(4\beta h)+\cosh(2\beta h))+e^{2\beta
J}(18\cosh(2\beta h)+6)+12\cosh(2\beta J)$.

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