# Many-body enhancement in a spin-chain quantum heat engine

L. A. Williamson  ARC Centre of Excellence for Engineered Quantum Systems,
School of Mathematics and Physics, University of Queensland, St Lucia,
Queensland 4072, Australia  Matthew J. Davis  ARC Centre of Excellence for
Engineered Quantum Systems, School of Mathematics and Physics, University of
Queensland, St Lucia, Queensland 4072, Australia

###### Abstract

We show that ferromagnetic interactions can enhance the adiabatic performance
of a quantum spin chain engine at low temperatures. The enhancement in work
output is particular pronounced, increasing exponentially with interaction
strength. The performance enhancement occurs in the paramagnetic phase and is
qualitatively explained by considering just the ground and first excited
state, in which case the system exhibits bipartite entanglement. As the
temperature is increased, thermal occupation of higher energy levels
diminishes performance. We find that these thermal fluctuations are smallest
for long-range interactions, resulting in the highest efficiency. Diabatic
work extraction degrades performance due to quantum friction. We identify an
approximate, experimentally realisable counterdiabatic drive that can mitigate
friction for weak interactions.

Quantum heat engines convert heat into work utilising some distinctly quantum
effect in the reservoir or working substance [1]. Reservoirs possessing
coherence [2, 3, 4], squeezing [5, 6, 7, 8, 9, 10, 11] or entanglement [12,
13] have been shown to improve engine performance. Coherence in a working
substance can be utilised as a resource [14, 15, 16, 17] and can improve power
output for rapid engine cycles [18, 19]. In the many-body regime, interactions
in a Bose gas can enhance engine performance compared to a non-interacting gas
[20, 21, 22, 23]. Interactions in a many-body quantum system can also be tuned
to change the energy of a working substance, hence providing a means to
extract work [24, 25].

One of the simplest quantum working substances is an ensemble of two-level
systems (“spins”) [26, 27, 28, 19, 29, 30, 31, 32, 33, 34, 35]. Work can be
extracted by tuning the level spacing $\hbar\omega(t)$ via control of an
external field, see Fig. 1(a). Including interactions between spins opens up
the possibility to explore many-body quantum effects. While considerable work
has explored engines with two interacting spins [36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47], much less is known about higher numbers of spins. For
nearest-neighbour interactions, a spin chain can function as both a heat
engine and a refrigerator [48] with critical scaling of performance close to
the critical point [49]. However, the question of whether many-body effects
can improve the performance of a spin-chain quantum heat engine has not been
explored.

In this work we characterise the performance of an Otto cycle with a
ferromagnetic spin chain as the working substance. In addition to displaying
rich many-body physics, this system may be realised in experiments with a
remarkable degree of control [50, 51, 52, 53, 54]. We show that both short and
long-range interactions improve the adiabatic work output and efficiency in
the paramagnetic phase at low temperatures $k_{B}T\lesssim\hbar\omega$. The
performance enhancement is qualitatively explained by an analytic model
considering just the ground and first excited state, in which case the thermal
state exhibits bipartite entanglement. For temperatures $k_{B}T>\hbar\omega$,
higher energy eigenstates are occupied and the performance degrades,
approaching the non-interacting performance. This effect is reduced as the
range of interactions is increased, and hence high efficiency is most robust
for long-range interactions. For diabatic work extraction, decreasing the
engine cycle time decreases performance due to quantum friction [36, 37, 55].
We demonstrate an approximate, experimentally realisable counterdiabatic drive
that can mitigate friction for weak interactions, and hence a performance
enhancement is possible at finite power output.

## System setup.

A chain of $N$ ferromagnetic interacting two-level spins is described by the
Hamiltonian (hereon $\hbar\equiv 1$),

$\hat{H}(\omega(t))=-\omega(t)\sum_{i=1}^{N}\hat{\sigma}_{z}^{(i)}-g\sum_{\begin{subarray}{c}i,j=1\\\
(j\neq
i)\end{subarray}}^{N}J_{ij}\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)},$ (1)

with $\hat{\sigma}_{\mu}^{(i)}$ ($\mu=x,y,z$) the Pauli spin-1/2 matrices for
spins $i=1,...,N$. The interaction strength between spins $i$ and $j$ is
$gJ_{ij}$ with $J_{ij}=1/|i-j|^{p}$, $g\geq 0$ the nearest-neighbour
interaction strength and $p>0$ determining the range of interactions, with
both $g$ and $p$ tuneable in experiments [51, 53]. For $N\rightarrow\infty$,
the system may be ferromagnetic $(g\lesssim\omega)$ or paramagnetic
$(g\gtrsim\omega)$ with the precise cross-over $g_{c}(p)$ dependent on $p$
[56, 57, 58, 59, 60, 61, 62]. We denote nearest-neighbour interactions by
$p=\infty$.

We consider an Otto engine cycle with the following steps, as shown in Fig.
1(a). (1) We begin with a hot thermal state
$\rho^{\mathrm{th}}_{H}=e^{-\beta_{H}\hat{H}(r\omega_{0})}/Z(\beta_{H},r\omega_{0})$
at level spacing $\omega=r\omega_{0}$, with $r>1$ the “compression ratio”
111In analogy with the ideal gas Otto cycle, where the compression ratio is
given by the ratio of maximum and minimum gas volumes.,
$Z(\beta,\omega)=\operatorname{Tr}e^{-\beta\hat{H}(\omega)}$ the partition
function, and $\beta=(k_{B}T)^{-1}$ the inverse temperature. (1 $\rightarrow$
2) The system is then thermally isolated and work is extracted by decreasing
$\omega$ from $r\omega_{0}$ to $\omega_{0}$, via the protocol
$\omega(t)/\omega_{0}=f(t)=r+(1-r)\sin^{2}(\pi t/2\tau)$ ($0\leq t\leq\tau$).
(2 $\rightarrow$ 3) Next, we cool the system at fixed $\omega=\omega_{0}$,
leaving the system in a cold thermal state
$\rho^{\mathrm{th}}_{C}=e^{-\beta_{C}\hat{H}(\omega_{0})}/Z(\beta_{C},\omega_{0})$.
(3 $\rightarrow$ 4) We thermally isolate the system again and increase
$\omega$ from $\omega_{0}$ back to $r\omega_{0}$, with the protocol
$\omega(t)/\omega_{0}=f(\tau-t)$ ($0\leq t\leq\tau$). (4 $\rightarrow$ 1)
Finally we heat the system at fixed $\omega=r\omega_{0}$ back to the initial
state. The work output $W$ and efficiency $\eta$ of the engine cycle are

$W=Q_{H}-Q_{C},\hskip 17.07182pt\eta=\frac{W}{Q_{H}}.$ (2)

Here
$Q_{H}=\operatorname{Tr}\left[\hat{H}(r\omega_{0})(\rho^{\mathrm{th}}_{H}-\rho_{4})\right]$
is the heat input from the hot reservoir and
$Q_{C}=-\operatorname{Tr}\left[\hat{H}(\omega_{0})(\rho^{\mathrm{th}}_{C}-\rho_{2})\right]$
is the heat output to the cold reservoir, with $\rho_{4}$ the density matrix
prior to coupling to the hot reservoir and $\rho_{2}$ the density matrix prior
to coupling to the cold reservoir. The density matrix at points 2 and 4 are
obtained by time-evolving the von Neumann equation
$\dot{\rho}(t)=-i[H(t),\rho(t)]$ with initial conditions
$\rho^{\mathrm{th}}_{H}$ and $\rho^{\mathrm{th}}_{C}$ respectively, using
Runge-Kutta integration.

## Adiabatic low-temperature performance.

Figure 1:  (a) An Otto cycle can be realised in an ensemble of two-level
spins, as described in the main text. Area of blue and red spots indicate
relative ground and excited state occupations, respectively. (b),(c) Engine
performance for a 10-spin chain operating adiabatically at low temperature
($\beta_{H}=10\omega_{0}^{-1}$, $\beta_{C}=2\beta_{H}$). Increasing $g$
increases both (b,i) work output and (b,ii) efficiency, with maximum
performance at $g\approx g_{c}(p)$ (results for
$r=r_{\mathrm{NI}}^{\mathrm{max}}$). Above this, the system transitions to the
ferromagnetic phase and no longer functions as an engine. For $p=\infty$, the
behaviour is captured by Eq. (3) (black dashed line) and the approximation Eq.
(Adiabatic low-temperature performance.) (gray dot-dashed line). (c,i) Work
output and (c,ii) efficiency as a function of compression ratio for
$g=g_{c}(p)$, with dotted lines indicating the non-interacting results
$W_{\mathrm{NI}}$ and $\eta_{\mathrm{NI}}$ ($W_{\mathrm{NI}}$ is scaled by a
factor of $100$). Gray horizontal lines in (b,ii),(c,ii) is the Carnot
efficiency.

We first examine the quantum adiabatic limit $\tau\gg\omega_{0}^{-1},g^{-1}$
(we set $\tau=100\omega_{0}^{-1}$) such that transitions between eigenstates
during the work steps are suppressed [64, 65, 66, 67]. For zero interactions
and fixed $\beta_{H}\omega_{0}\gg 1$, the maximum work output occurs at a
compression ratio $r_{\mathrm{NI}}^{\mathrm{max}}\approx
1+(\beta_{H}\omega_{0})^{-1}$, which gives a small efficiency
$\eta_{\mathrm{NI}}\approx(\beta_{H}\omega_{0})^{-1}$ that decreases with
decreasing temperature. We find that interactions drastically improve both
work output and efficiency in the paramagnetic phase for temperatures
$\beta_{H}^{-1}\ll\omega_{0}$, see Fig. 1(b). The improvement in work output
is particularly pronounced, with a maximum work output $\sim 10^{2}$ times
larger than the non-interacting ensemble.

The behaviour is qualitatively similar in all cases $p=1,2,3,\infty$ after
rescaling interactions by $g_{c}(p)$, which we define to be the point at which
$\partial^{2}\Delta/\partial g^{2}\big{|}_{\omega=\omega_{0}}$ has a maximum
[68]. Here $\Delta$ is the energy gap to the first excited state. The
improvement increases monotonically up to $g\approx g_{c}(p)$, before dropping
abruptly. For $p=2,3,\infty$ the performance scales approximately extensively
with increasing $N$ [68]. For $p=1$ the performance increases non-extensively
[68] due to the non-extensive thermodynamics of long-range interactions [69,
70]. This is removed after scaling $g$ by $g_{c}$ [68].

In the limit of large $N$, a chain with nearest neighbour interactions has a
solvable spectrum [71]. For $\beta_{H}^{-1}\ll\omega_{0},g$, only the first
$N$ excited states are appreciably occupied. The dimensionless free energy is
then 222In a frame where ground-state energy is zero. The dimensionless free
energy is the free energy divided by $k_{B}T$.

$\ln
Z\approx\frac{N}{\pi}\int_{0}^{\pi}e^{-\beta\sqrt{\omega^{2}+g^{2}-2\omega
g\cos\theta}}\,d\theta.$ (3)

The performance computed from Eq. (3) (see [68]) is plotted alongside the full
numerical results in Fig. 1(b), and agrees well with the $p=\infty$ results
for $g<g_{c}$. Transforming the spin operators using a Holstein-Primakoff
transformation and expanding to quadratic order in the bosonic operators gives
an analytically tractable theory even for long-range interactions [73]. To
lowest order in $g/\omega$ and for large $N$ and $\beta$, $\ln Z\approx
N\mathcal{G}_{p}(\beta g)e^{-\beta\Delta}$, where $Ne^{-\beta\Delta}$ is the
low-temperature free energy of $N$ two-level systems with level splitting
$\Delta(\omega)=\omega(1-\omega_{0}g/\omega g_{c})$. The factor
$\mathcal{G}_{p}(\beta g)$ arises from thermal fluctuations and depends on $p$
[68]. To lowest order in $\beta^{-1}$, this gives,

$\displaystyle W=$ $\displaystyle
N(r-1)\omega_{0}\mathcal{G}_{p}(\beta_{H}g)\left(e^{-\beta_{H}\Delta(r\omega_{0})}-e^{-\beta_{C}\Delta(\omega_{0})}\right),$
$\displaystyle\eta=$ $\displaystyle 1-\frac{\Delta_{2}}{\Delta_{1}}.$ (4)

For $g<g_{c}(p)$, increasing $g$ decreases $\Delta(\omega)$. From examination
of Eq. (Adiabatic low-temperature performance.), this increases low-
temperature work output as $W\sim
W_{\mathrm{NI}}e^{\beta_{H}\omega_{0}g/g_{c}}$, consistent with the
exponential increase in Fig. 1(b,i), and efficiency as
$\eta\sim\eta_{\mathrm{NI}}/(1-g/rg_{c})$. Above $g_{c}$ the system
transitions to the ferromagnetic state and $\Delta$, and therefore
$\partial^{2}\ln Z/\partial\beta\partial\omega$, changes sign. The cycle
therefore no longer functions as a heat engine [48], resulting in the abrupt
drop in performance in Fig. 1(b) above $g_{c}$.

The quadratic approximation above permits a calculation of the bipartite
entanglement of the spin chain, which allows us to determine the “quantum”
nature of the performance enhancement. For low temperatures, a thermal state
can be approximated by
$\rho_{T}^{01}\approx(\ket{0}\bra{0}+e^{-\beta\Delta}\ket{1}\bra{1})/(1+e^{-\beta\Delta})$,
with $\ket{0}$ the ground state,
$\ket{1}=\sum_{i=1}^{N}\hat{\sigma}_{+}^{(i)}\ket{0}/\sqrt{N}$ the approximate
first-excited state (independent of $p$) and
$\hat{\sigma}_{+}^{(i)}=\hat{\sigma}_{x}^{(i)}+i\hat{\sigma}_{y}^{(i)}$ [68].
In [68], we show that $\rho_{T}^{01}$ is entangled according to the Peres-
Horodecki criterion [74, 75, 76]. The performance enhancement, Eq. (Adiabatic
low-temperature performance.), requires access to an entangled thermal state,
and so is a many-body quantum effect. The entanglement of $\ket{1}$ is also
directly evident from the entanglement entropy, which is
$\mathcal{S}=\ln(N/2)$ for a partition dividing the spin chain in half [68].
In contrast, in a mean-field approximation, the interaction of spin $i$ with
the remaining spins is replaced by $-g\Omega_{i}\hat{\sigma}_{x}^{(i)}$. Here
$\Omega_{i}=\sum_{j\neq i}J_{ij}s_{j}$ is an effective transverse drive and
$s_{j}=\langle\hat{\sigma}_{x}^{(j)}\rangle_{\mathrm{mf}}$ is a mean-field
approximation for spin $j$. The energy gap of spin $i$ then increases with $g$
as $\sqrt{\omega^{2}+g^{2}\Omega_{i}^{2}}$ and interactions degrade
performance.

Increasing the compression ratio increases performance until $r=r^{\prime}$,
with $r^{\prime}\sim 1.1$ at $g=g_{c}(p)$, at which point the performance
abruptly drops, see Fig. 1(c). Unlike the non-interacting case, the peak work
output and efficiency can both occur at a comparable compression ratio.
Equation (Adiabatic low-temperature performance.) describes this behaviour:
within this approximation, performance increases until
$1-\Delta_{2}/\Delta_{1}\sim\eta_{C}$, with $\eta_{C}=1-\beta_{H}/\beta_{C}$
the Carnot efficiency. Hence
$r^{\prime}\approx(1-g\eta_{C}/g_{c}(p))/(1-\eta_{C})+O(g^{2})$ diminishes
with increasing $g$. As a result, for $g\sim g_{c}(p)$ we can have high
efficiency at small compression ratios $r\sim r_{\mathrm{NI}}^{\mathrm{max}}$.
Without interactions, the maximum compression ratio satisfies
$1-1/r^{\prime}=\eta_{C}$ and therefore $r^{\prime}=1/(1-\eta_{C})\gg
r_{\mathrm{NI}}^{\mathrm{max}}$.

## Effect of increasing temperature.

As the temperature increases, thermal fluctuations render Eq. (Adiabatic low-
temperature performance.) invalid and we find that the performance enhancement
relative to the non-interacting system is diminished, see Fig. 2(a). A
performance enhancement is present as long as $\beta_{H}\gtrsim
4\omega_{0}^{-1}$ (the precise cross-over point is dependent on $g$),
coinciding with the regime where only the ground and first $N$ excited levels
are appreciably occupied, see Fig. 2(b). The transverse Ising model gives a
qualitative understanding of this diminished performance enhancement. The
first $N$ excited energy levels of this model are
$\sqrt{\omega^{2}+g^{2}-2\omega g\cos\theta_{k}}$ with $\theta_{k}=2\pi k/N$
($k=0,...,N-1$) [71]. Interactions diminish the energy of only the lowest
$N/2$ excited levels, with the most pronounced reduction occurring for the
first excited level ($k=0$). Hence, the enhancement is largest when only the
first excited level is occupied, and diminishes as more excited levels are
occupied. The efficiency enhancement is most robust to increasing temperature
for long-range interactions, see Fig. 2(a,ii). At a given temperature, the
occupation $\sum_{i=2}^{N}n_{i}$ decreases as the range of interactions
increases, see Fig. 2(b). Hence long-range interactions are most effective at
suppressing fluctuations beyond the approximation (Adiabatic low-temperature
performance.). For $\beta_{H}<4\omega_{0}^{-1}$, interactions degrade
performance, see Fig. 2(a). Expanding the dimensionless free energy in powers
of $\beta$, we obtain

$\ln Z=\ln
Z_{\infty}+\frac{\beta^{2}\omega^{2}}{4}+\frac{\beta^{2}g^{2}\sum_{i}(\Omega_{i}^{\prime})^{2}}{4}+O(\beta^{4}),$
(5)

with $Z_{\infty}=2^{N}$ the infinite temperature partition function and
$\Omega_{i}^{\prime}=(1/2)\sum_{j\neq i}J_{ij}$ is $\Omega_{i}$ with
$s_{j}=1/2$. At order $\beta^{2}$, the free energy is indistinguishable from
the mean-field free energy
$\sum_{i}\ln\operatorname{Tr}e^{-\beta(\omega\hat{\sigma}_{z}^{(i)}-\Omega_{i}^{\prime}\hat{\sigma}_{x}^{(i)})}$,
in which case interactions degrade performance. The scaling
$\ln(Z/Z_{\infty})\propto\beta^{2}$ is clearly present for temperatures
$\beta_{H}\lesssim\omega_{0}^{-1}$, see Fig. 2(c).

Figure 2:  (a) Adiabatic engine performance for varying temperature ($r=1.1$,
$\beta_{C}=2\beta_{H}$). (a,i) Work output and (a,ii) efficiency exceed the
non-interacting values (dotted lines) for cold temperatures $\beta_{H}\gtrsim
4\omega_{0}^{-1}$, whereas interactions degrade performance for
$\beta_{H}<4\omega_{0}^{-1}$. The gray line in (a,ii) is the Carnot
efficiency. (b) Thermal energy-level occupations $n_{0}$ (dotted lines),
$n_{1}$ (solid lines), $\sum_{i=2}^{N}n_{i}$ (dashed lines) and
$\sum_{i=N+1}^{2^{N}-1}n_{i}$ (dash-dotted lines). Here $n_{i}=e^{-\beta
E_{i}}/Z(\beta,\omega)$ is the thermal occupation of energy level $i$, indexed
in order of increasing energy $E_{i}$. The performance enhancement occurs when
occupation is predominantly in the ground and first $N$ excited states. Long-
range interactions suppress occupation beyond the first excited state,
resulting in the highest efficiency in (a,ii). (c) Dimensionless free energy
$\ln Z$ showing the $\beta^{2}$ scaling for $\beta\lesssim\omega^{-1}$,
coinciding with the regime of low performance. All results are for a 10-spin
chain at $g=g_{c}(p)$.

## Diabatic work extraction.

Figure 3:  (a) Diabatic work output $W_{\mathrm{D}}$ and efficiency
$\eta_{\mathrm{D}}$ for $p=\infty$ with weak interactions $g/g_{c}=0.2$. The
maximum power output $P_{\mathrm{D}}$ (see inset) occurs at $\tau\approx
4\omega_{0}^{-1}$ (star), with performance rapidly declining for smaller
$\tau$. The approximate counterdiabatic driving (Eq. (6)) results in work
output ($W_{\mathrm{CD}}$) and efficiency ($\eta_{\mathrm{CD}}$) close to the
adiabatic performance even for rapid engine cycles. The counterdiabatic power
output ($P_{\mathrm{CD}}$) grows as $\tau^{-1}$ (inset). (b) The effectiveness
of Eq. (6) diminishes for larger $g/g_{c}(p)$ or smaller $p$ (results for
$\tau=\omega_{0}^{-1}$; for $p=\infty$, we use the exact $\chi_{ij}$, whereas
for $p=1,2,3$, we set $\chi_{ij}=1$). All results are for $N=10$ and
$r=r_{\mathrm{NI}}^{\mathrm{max}}$.

For diabatic (finite-time) work extraction, interactions generally degrade
engine performance due to “quantum friction” [36, 37, 55]. This friction
arises when the interaction component of the Hamiltonian does not commute with
the driving component, and hence the density matrix develops off-diagonal
elements in the energy eigenbasis. The diabatic performance of a $p=\infty$
engine with weak interactions is shown in Fig. 3(a). The peak power output
occurs for a time step $\tau\approx 4\omega_{0}^{-1}$ (the precise value is
dependent on $g$), at which point the efficiency is close to the adiabatic
efficiency. For faster cycles, the performance rapidly decreases.

In principle, quantum friction can be mitigated completely using a
counterdiabatic driving field $\hat{H}_{\mathrm{cd}}$ [77, 78]. In practice,
exact counterdiabatic driving in a many-body system requires unrealistic
interactions between all particles [79, 80, 81, 82, 83, 84, 85], and
approximate protocols are required. A powerful approximation method is to find
$\hat{H}_{\mathrm{cd}}$ variationally by minimising the action
$S=\operatorname{Tr}[G(\hat{H}_{\mathrm{cd}})^{2}]$, with
$G(\hat{H}_{\mathrm{cd}})=\partial\hat{H}/\partial
t+i[\hat{H}_{\mathrm{cd}},\hat{H}]$ and $\hat{H}_{\mathrm{cd}}$ expanded in
some truncated set of operators [86, 87]. We use
$\hat{H}_{\mathrm{cd}}=\sum_{ij(j\neq
i)}C_{ij}\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{y}^{(j)}$, which is the optimal
counterdiabatic drive over all one-body and two-body operators [68] (c.f.
[88]). For large $N$ in the paramagnetic phase, we obtain [68],

$\hat{H}_{\mathrm{cd}}=-\sum_{\begin{subarray}{c}i,j=1\\\ (j\neq
i)\end{subarray}}^{N}\frac{g\omega^{\prime}(t)J_{ij}}{2\omega(t)^{2}}\chi_{ij}(t)\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{y}^{(j)},$
(6)

with $\chi_{ij}(t)=1+O(g^{2}/\omega^{2})$ given in [68]. The work protocols
$f(t)$ and $f(\tau-t)$ satisfy $f^{\prime}(0)=f^{\prime}(\tau)=0$, and hence
the net power transferred to the counterdiabatic drive field is zero 333this
can be shown explicitly by integrating by parts the instantaneous power
$\langle\partial(\hat{H}+\hat{H}_{\mathrm{cd}})/\partial t\rangle$ and noting
that only the boundary term remains..

For nearest neighbour interactions, $\chi_{ij}=1/(1+g^{2}/\omega(t)^{2})$ and
Eq. (6) drastically improves the diabatic engine operation for $g\lesssim
0.3g_{c}$, see Fig. 3. For rapid cycles, the work output approaches a constant
with little cost in efficiency, and hence the power $P$ increases as
$\tau^{-1}$. In practice, the time scale of the thermalization steps will
limit the engine to finite power [90, 26]. Note $\eta\propto W$ irrespective
of counterdiabatic driving (Fig. 3(a)), hence $Q_{H}$ depends only weakly on
$\tau$.

For increasing $g/g_{c}$ there is a trade-off in the performance gained from
interactions and the performance lost from quantum friction, with peak
performance occurring for $g/g_{c}\approx 0.3$ for $\tau=\omega_{0}^{-1}$.
Here, the work output from a chain with nearest neighbour interactions is
about $50\%$ larger than the non-interacting chain and both show comparable
efficiency, see Fig. 3(b). For $p=1,2,3$, $\chi_{ij}(t)$ is difficult to
engineer since the interactions must be reconfigured at different times. To
simplify, we expand to lowest order in $g/\omega$ and set $\chi_{ij}(t)=1$.
While this is somewhat effective at mitigating diabatic degradation for weak
interactions, the performance enhancement diminishes as the range of
interactions increases. Hence a chain with $p=1$, $g/g_{c}\lesssim 0.3$ and
$\tau=\omega_{0}^{-1}$ has approximately the same performance as a non-
interacting chain. Interestingly, we find that Eq. (6) is most effective for
$\beta_{H}\lesssim 10\omega_{0}^{-1}$, with reduced performance for colder
temperatures. This may be due to thermal fluctuations countering quantum
friction [91].

## Conclusion.

We have shown that an engine of interacting spins outperforms a non-
interacting engine in the paramagnetic phase for low temperatures and
adiabatic operation, due to a lowering of the first excited state energy gap.
The enhancement in work output is particular pronounced, with
$W/W_{\mathrm{NI}}$ increasing exponentially with increasing interactions. The
efficiency enhancement is largest for long-range interactions, which suppress
occupation of energy levels beyond the first excited state. A performance
enhancement due to long-range interactions has also been identified in Kitaev
chains [92, 93]. For diabatic engine operation, quantum friction degrades
performance. We have presented one counterdiabatic method that mitigates
friction for weak interactions, however other methods could be explored [91,
79, 94, 85, 95, 96, 97, 98, 99]. Modulating the phase and detuning of the
drive profile may better isolate the two lowest energy eigenstates [100, 101,
102, 103, 104], limiting degradation due to thermal fluctuations and quantum
friction. The low-temperature performance enhancement is a many-body quantum
effect due to bipartite entanglement arising from the first excited state. A
more thorough investigation of the entanglement properties of the thermal spin
chain could reveal how entanglement changes for higher temperatures [105, 106,
107, 108] or diabatic operation.

## Acknowledgements.

We thank C. Woffinden and M. Edmonds for useful comments on the manuscript.
This research was supported by the Australian Research Council Centre of
Excellence for Engineered Quantum Systems (EQUS, CE170100009).

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