差分進化を用いた最適なナノ熱機関の探索
東大理
蘆田祐人
Ref: YA and T. Sagawa, Comm. Phys. 4, 45 (2021).

Motivation: What is the “best” nanoscale heat engines?
Remarkable developments in the ability to control nanoscale heat engines:
Single trapped ion

J. Rossnagel et al.,Science 352, 6283 (2016).

Quantum dot

Colloidal particle

M. Josefsson et al.,Nat. Nanotech. 13, 920 (2018).

S. Toyabe et al.,Nat. Phys. 6, 988 (2010).

So far, single-particle or noninteracting regimes are well explored.
Toward realizing high power, one has to assemble a large number of microscopic engines,
in which interaction effects become essential.

Motivation: What is the “best” nanoscale heat engines?
Remarkable developments in the ability to control nanoscale heat engines:
Trapped ions

Quantum-dot array

Colloidal particles

Univ. Innsbruck

I. Piquero-Zulaica et al.,Nat. Commun. 8, 787 (2017).

P. J. Lu et al.,Annu. Rev. Cond. Matt. 4, 217 (2013).

So far, single-particle or noninteracting regimes are well explored.
Toward realizing high power, one has to assemble a large number of microscopic engines,
in which interaction effects become essential.

What is the “best” nanoscale heat engines with interactions?

Motivation: What is the “best” nanoscale heat engines?

Stochastic Thermodynamics

Search for thebest nanoscaleheat engines.

Machine Learning/Optimization

So far, single-particle or noninteracting regimes are well explored.
Toward realizing high power, one has to assemble a large number of microscopic engines,
in which interaction effects become essential.

What is the “best” nanoscale heat engines with interactions?

The “best” heat engines as Pareto-optimal solutions
Two conflicting objectives: thermodynamic efficiency and power
Carnot efficiency:⌘C = 1

Efficiency<latexit sha1_base64="y82IqJUjteQ/PQFogvaU7DBHluM=">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</latexit>

Tc
Th

Both of objectives cannot be optimized
simultaneously in general.
The best heat engines =A set of engines whose efficiency andpower cannot be further improved
without comprising the other.
=“Pareto front”
cf. Sawaragi et al.,Theory of multiobjective optimization. (1985).

Specific examples: Carnot machine, heatengine operating at the maximum power.

Power

Optimize the model parameters to findthe best thermodynamic tradeoff

The “best” thermoelectric heat engines in linear-response regime
Thermoelectric system as a steady-state heat engine:
Convert heat flows into work in the form of electrical power.
ColdreservoirsystemHot
reservoir

Heat flow

The “best” thermoelectric heat engines in linear-response regime
Thermoelectric system as a steady-state heat engine:
Convert heat flows into work in the form of electrical power.
Linear-response formula(cf. Benenti, Casati, Saito and Whitney, Phys. Rep. 694 (2017)):<latexit sha1_base64="CG1vj+6Y7hFIPB4z7G1IqFwl8Nw=">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</latexit>⌘(P )P/(Q T 2 /4)
i
= h
p
⌘C
2 1 + 2/ZT ⌥ 1 P/(Q T 2 /4)

ZT : figure of meritp⌘
ZT + 1 1
p⌘C
ZT + 1 + 1

Q : power factor<latexit sha1_base64="Is/khSMSR12ZhMS4KUgmPzWp0SE=">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</latexit><latexit sha1_base64="lqLwlrDKJaIlSeWKtUctff2CCJI=">AAACpXichVHLShxBFD12YtSJxjHZCNkMGXxAcLgtgiGrQTduBB8zo8SWprtTMzb2y+qaAW36B/IDLlwpaAj5jGyyyNaAnyAuDWSTRe70NPgi5nZX1b3n3nPrVJUdeW6siC76tCdP+58NDA4Vng+PvBgtjr1sxGFbOqLuhF4oN20rFp4biLpylSc2Iyks3/bEhr272M1vdISM3TCoqf1IbPtWK3CbrmMphsxi1WhKy0kMoaw0m83EkH5pMU0NT+zlyXhPquRD7a2ezujp7ZB/s1imCmVWeujouVNGbith8TMMfEQIB234EAig2PdgIeZvCzoIEWPbSBiT7LlZXiBFgbltrhJcYTG6y3OLo60cDTju9owztsO7eDwkM0uYoHP6Qtf0nb7SJf35Z68k69HVss+r3eOKyBz9NL7++78sn1eFnRvWo5oVmniXaXVZe5Qh3VM4PX7n4PB6/f3aRDJJJ3TF+o/pgr7xCYLOL+d0VawdocAPoN+/7odOY7aiU0VfnStXF/KnGMRrvME03/c8qljCCuq87xl+4Bw/tSltWatpjV6p1pdzXuGOaeZfBV6jwQ==</latexit><latexit sha1_base64="6K+Ams7AaoELaw9nycgNx6B4kwc=">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</latexit>

P  Q T 2 /4
<latexit sha1_base64="8TulVv832Pf13e4jZqRmOiIp4Ck=">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</latexit>

Relation to transport coefficients:

ZT =<latexit sha1_base64="gdciADArTnxkRoK+6PxKvODnR+c=">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</latexit>

S2T
QT
=



: electrical conductance<latexit sha1_base64="0zDWTdT162F4Jr2x7h1FajH93WY=">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</latexit> : thermal conductivity<latexit sha1_base64="r9Yg7l1ZE9HmnAzttqz05czw2k0=">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</latexit><latexit

S : Seebeck coefficient<latexit sha1_base64="9yNzImxJvprXGiaRkxpDcM2l+GI=">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</latexit><latexit

The “best” thermoelectric heat engines in linear-response regime
Thermoelectric system as a steady-state heat engine:
Convert heat flows into work in the form of electrical power.
Linear-response formula(cf. Benenti, Casati, Saito and Whitney, Phys. Rep. 694 (2017)):⌘(P )P/(Q T 2 /4)
i
= h
p
⌘C
2 /4)
In the linear-responsethe⌥search
the T
best
heat
2 1regime,+ 2/ZT
1 for
P/(Qengines reduces to finding the Pareto front on the Q-ZT plane.
<latexit sha1_base64="CG1vj+6Y7hFIPB4z7G1IqFwl8Nw=">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</latexit>

Pareto front on Q-ZT plane

Envelope of ”lasso” curves= Pareto front on P-η plane.

Model: interacting fermions in the sequential regime
Prototypical and generic model for interacting fermions:H=<latexit sha1_base64="PVCkudNwZi533pG2E1WeJnLvzPE=">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</latexit>

X1Xvg )nl +wlm nl nm
2

(✏l

l

l6=m

vg : ground voltage✏l : single-electron modes<latexit sha1_base64="C1Woyjnz5UNFvSlLO5qM25MmVtM=">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</latexit><latexit sha1_base64="HGQw38cm/QpRMMI90QUvGDm73Uk=">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</latexit>

nl : electron occupancy<latexit sha1_base64="56wkSazY9clATPcE/jH2nBXbPBE=">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</latexit>

wlm : repulsive interaction<latexit sha1_base64="+4CwmWafsrIqJoGYt4PemWwHOBQ=">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</latexit>

Graphical
representation“interaction network”✏m✏n<latexit sha1_base64="dihjtdhIdC+K3LiEeEkhrag20Z0=">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</latexit><latexit sha1_base64="NmjzUziG7Y/VWRmiFYdY8ilqGXs=">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</latexit>

Node: single-electron level
Edge: interaction between nodes
Our aim: finding a set of the best heat engines in the presence of interaction= optimization of “network topology” in the graphical representation.

Calculations of ZT and Q
We neglect quantum coherence; the dynamics is described by
the classical master equation (=Stochastic Thermodynamics)
cf. Seifert, Rep Prog. Phys. 75 126001 (2012)<latexit sha1_base64="K0ASx0SZrnOywLtqdY7yJ9jixNw=">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</latexit>X
dpa=
Wab pb , Wab =
dt
b

2

X

db ,

=

ab

ab

Nf

d
Nf transfer matrix<latexit sha1_base64="0YvNojZjmhQ4HDoJKMqjjLvEbUo=">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</latexit>

⇥2

h

+

c

,

For the matrix elements between many-body states a and b whose particlenumber difference is one, we set the detailed balance conditions:<latexit sha1_base64="JStRQMNFeThufU0zIK4gBiBhmfM=">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</latexit>

Entropy production
Fermi distribution
✓
◆
1
E
E
µ
a
b
i
i
i
i
,
s
=
+
(N
N
)
,
a
b
ab = i f ( sab ), f (x) =ab
1 + exk B Tik B Ti

Reservoir: i = h, c<latexit sha1_base64="meRaoXbbbMLlr/H0hzu2pBchb9g=">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</latexit>

Steady-state solution: W pss = 0<latexit sha1_base64="+8YOs8idc/ru/wblnO+Ab0dJcUo=">AAACdHichVHLSsNAFD2Nr1ofrboRdFEsFVflRgRFEIpuXPqqFWwtSRw1NC+StFBDf8AfcNGVooj4GW78ARd+grhUdOPC2zQgKuoNkzlz5p47Z+aqjqF7PtFDTOrq7unti/cnBgaHhpOpkdFtz665mihotmG7O6riCUO3RMHXfUPsOK5QTNUQRbW60t4v1oXr6ba15TccUTaVQ0s/0DXFZ6qSShadvSAouWba85rNJaqkMpSjMNI/gRyBDKJYs1NXKGEfNjTUYELAgs/YgAKPv13IIDjMlREw5zLSw32BJhKsrXGW4AyF2Sr/D3m1G7EWr9s1vVCt8SkGD5eVaWTpnq7pme7ohh7p/ddaQVij7aXBs9rRCqeSPBnffPtXZfLs4+hT9adnHwdYCL3q7N0JmfYttI6+fnz6vLm4kQ2m6Zye2P8ZPdAt38Cqv2iX62KjhQQ3QP7+3D/B9mxOppy8PpfJL0etiGMCU5jh955HHqtYQyHsSQsXuIy9SpNSRsp2UqVYpBnDl5ByH0n+kAA=</latexit>✏m<latexit sha1_base64="NmjzUziG7Y/VWRmiFYdY8ilqGXs=">AAACbXichVG7SgNBFD1Z3/EVFUFQRAxRq3BXBMUqaGMZH1HxQdhdJ3FwX+xuAhr8AWvBQhQURMTPsPEHLPwEsbBQsLHwZrMgGtQ7zMyZM/fcOTOju6b0A6LHmNLQ2NTc0toWb+/o7OpO9PSu+k7JM0TOcEzHW9c1X5jSFrlABqZYdz2hWbop1vS9+er+Wll4vnTslWDfFduWVrRlQRpawNTGlnB9aTp23sonkpSmMEbqgRqBJKLIOolrbGEHDgyUYEHARsDYhAaf2yZUEFzmtlFhzmMkw32BQ8RZW+IswRkas3s8Fnm1GbE2r6s1/VBt8Ckmd4+VI0jRA93QK93TLT3Rx6+1KmGNqpd9nvWaVrj57qOB5fd/VRbPAXa/VH96DlDATOhVsnc3ZKq3MGr68sHJ6/LsUqoyRpf0zP4v6JHu+AZ2+c24WhRLp4jzB6g/n7serE6mVUqri1PJzFz0Fa0YxCgm+L2nkcECssjxuTaOcYbz2IvSrwwpw7VUJRZp+vAtlPFPreaN/A==</latexit>✏n<latexit sha1_base64="dihjtdhIdC+K3LiEeEkhrag20Z0=">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</latexit>Calculations of ZT and Q
We work in the linear-response regime:<latexit sha1_base64="jiZWUJR1XTkdudIcdpbWh7SmCpc=">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</latexit>

T = ThTc ⌧ Th| µ| = |µhµ c | ⌧ k B Th

<latexit sha1_base64="a9pHCUyaJQQIqztdbt9lDs8+geo=">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</latexit>

Using the steady-state values of the currents (J ss , Jqss )T for<latexit sha1_base64="8vADSJEnCMwhoc11rRKdg97CDdk=">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</latexit><latexit sha1_base64="9Uaw4P7cQFe4/OI52V69cU4lxz8=">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</latexit><latexit sha1_base64="JTSV979i6C+xDp/JrANQrPmAc20=">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</latexit>

T = 0, µ 6= 0,
T 6= 0, µ = 0

we calculate the Onsager matrix and figure of merit ZT and power factor Q accordingly

ZT =<latexit sha1_base64="exaovUNyuJVUB8MLGAbnVgwabHU=">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</latexit>

S2T
L12 L21
L212
2=, Q= S = 3,
det(L)
T L11

✓

h

J
Jqh

◆

=L

✓

µ/T
T /T 2◆,

Jqh : heat current out of the hot reservoir<latexit sha1_base64="KocHmOmzda2bZkwrUrcD33UXUNg=">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</latexit>

J h : particle current out of the hot reservoir<latexit sha1_base64="tAhtWzaRaDYXF62VX4hIPv5F1KM=">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</latexit>(*System works as a heat engine when we set)<latexit sha1_base64="W8gdXua8EGC8UTrTkAi3bjUlgak=">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</latexit><latexit sha1_base64="zqb1zA27gaSJuASdoYnXkutUysE=">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</latexit>µ < 0 if S > 0µ > 0 if S < 0Previous studies in noninteracting regimes
Noninteracting case: wlm = 0<latexit sha1_base64="2j3zITPmQgyJxS4n72FQ9OX6bFU=">AAACa3ichVHLSsNAFD2N7/qqdiPqQiwVV+VGBEUQim5c+qotVClJnNZgXiRpRYM/4NKNC90oiIif4cYfcOEniLhScOPCmzQgKuoNkzlz5p47Z+6ojqF7PtFDQmppbWvv6OxKdvf09vWnBgY3PLvuaqKg2YbtllTFE4ZuiYKv+4YoOa5QTNUQRXV3MdwvNoTr6ba17u87YstUapZe1TXFZ6q0VwkM83CeKqkM5SiKsZ9AjkEGcSzbqStsYhs2NNRhQsCCz9iAAo+/MmQQHOa2EDDnMtKjfYFDJFlb5yzBGQqzu/yv8aocsxavw5pepNb4FIOHy8oxZOmerumF7uiGHun911pBVCP0ss+z2tQKp9J/NLT29q/K5NnHzqfqT88+qpiNvOrs3YmY8BZaU984OHlZm1vNBhN0QU/s/5we6JZvYDVetcsVsXqKJD+A/L3dP8HGVE6mnLwynckvxE/RiRGMY5L7PYM8lrCMQtTnY5ziLPEspaVhabSZKiViTRpfQsp+AECPjNo=</latexit>

The best engine in the ideal situation:
perfectly degenerate single-electron levels✏ 1 = ✏ 2 = · · · = ✏ Nf
<latexit sha1_base64="jtyh4qJ/9Pl5ZlcMgHcEuEX2lN8=">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</latexit>DOS

cf. Mahan & Sofo, PNAS 93, 7436 (1996).

ZT ! 1<latexit sha1_base64="2PQgwqF89Q7gXK/LcBa4IOlIAno=">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</latexit>ideal

Q / Nf<latexit sha1_base64="OZ0TqgyvVz8HvEy+8oaTQlp1lRw=">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</latexit>∵) tight-coupling condition satisfied:

J / Jq<latexit sha1_base64="ILaaM3Romg781jxDjKPkt6mByr8=">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</latexit>

Interaction?

The best engine in a realistic situation:DOS(imposing upperbound on DOS)boundnondegenerate single-electron levels✏ 1 < ✏2 < · · · < ✏Nf<latexit sha1_base64="9sljJMN+efsjBWV1P1eRAzXUnrk=">AAACkHichVG7SgNBFD2u7/iK2gg2YlAEIdwVQRHBVyMW4isqGAm76yQObnaX3UlAgz9gZ2VhpSAifoCtYuMPWPgJYqlgY+HNZlFU1DvMzJkz99w5M2N6tgwU0UOVVl1TW1ff0Bhram5pbYu3d6wGbsG3RMpybddfN41A2NIRKSWVLdY9Xxh50xZr5s5MeX+tKPxAus6K2vXEZt7IOTIrLUMxlYkPpoUXSJuhPv4Bh8bT1pargk+mNJ/J7mfiCUpSGD0/gR6BBKJYcOPnSGMLLiwUkIeAA8XYhoGA2wZ0EDzmNlFizmckw32BfcRYW+AswRkGszs85ni1EbEOr8s1g1Bt8Sk2d5+VPeije7qgZ7qjS3qkt19rlcIaZS+7PJsVrfAybQddy6//qvI8K2x/qv70rJDFaOhVsncvZMq3sCr64t7R8/LYUl+pn07pif2f0APd8g2c4ot1tiiWjhHjD9C/P/dPsDqU1CmpLw4nJqejr2hAN3oxwO89gknMYgEpPvcQV7jGjdahjWoT2lQlVauKNJ34EtrcO2OimxA=</latexit>

cf. Whitney, PRB 91, 115425 (2015).

ZT finite<latexit sha1_base64="Is/khSMSR12ZhMS4KUgmPzWp0SE=">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</latexit>

Q / Nf0<latexit sha1_base64="ubjdpeIh4TMmEowHmnyiXjSjdn4=">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</latexit>

realistic/ Nf0<latexit sha1_base64="OPVi0mNklHyyyeO/cpBu7WfPn1I=">AAACcHichVFNLwNBGH66vuuruEgclIaIQ/OuSIhTw8VJtFUkVLO7pmxsd9futEk1/oA/4OCCRET8DBd/wKE/QdyQuDh4u91EELyTmXnmmfd555kZ3bVMXxLVI0pLa1t7R2dXtLunt68/NjC47jtlzxA5w7Ecb1PXfGGZtshJU1pi0/WEVtItsaEfLDX2NyrC803HXpNVV+RL2p5tFk1Dk0zlt13PcaUTXykUd6gQS1CSgoj/BGoIEghj1YldYxu7cGCgjBIEbEjGFjT43LagguAyl0eNOY+RGewLHCPK2jJnCc7QmD3gcY9XWyFr87pR0w/UBp9icfdYGccEPdANvdA93dIjvf9aqxbUaHip8qw3tcIt9J8MZ9/+VZV4ltj/VP3pWaKI+cCryd7dgGncwmjqK0enL9mFzERtki7pif1fUJ3u+AZ25dW4SovMGaL8Aer35/4J1meSKiXV9GwitRh+RSdGMI4pfu85pLCMVeT43EOc4hwXkWdlWBlVxpqpSiTUDOFLKNMfaNWOuQ==</latexit>/ Nf<latexit sha1_base64="vnCPseA/RGA2ZSDdIDI+1owuXW8=">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</latexit>

Can interaction push up the boundclose to the ideal case?

Problem setting and optimization algorithm
Problem:

Given generic (nondegenerate) single-electron levels ✏1 < ✏2 < · · · < ✏Nf ,<latexit sha1_base64="9sljJMN+efsjBWV1P1eRAzXUnrk=">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</latexit>

find a set of the optimal parameters W = {vg , {wlm }l>m },<latexit sha1_base64="+u7GJNsSvSEJiP1khbBE2FGbGS4=">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</latexit>

which provide the Pareto front on the Q-ZT plane.

Strategy:
1. Maximize Q with respect to W via solving the single-objective optimization problem.
→ Identifying an unambiguous element for the Pareto front.
<latexit sha1_base64="wUvuxW63ypUMhsCzAvJ9g5O79FM=">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</latexit><latexit

2
2. Search the Pareto front starting from the above solution.
The iterative alternate method:
Custodio et al., SIAM J. Optim. 21, 1109 (2011).

Optimizing (training) Wto maximize efficiency and power= “Reinforcement learning” of the underlying topology andweights of the interaction network.
<latexit sha1_base64="wUvuxW63ypUMhsCzAvJ9g5O79FM=">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</latexit><latexit

1

Learning the best heat engines via global search-Q

The optimization problem is challenging: local (gradient-based) algorithms failed.
local optimum

<latexit sha1_base64="wb94oGq+i7iFwn3pb0ho4snvAxc=">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</latexit>

Q
Globaloptimum

cf. Goodfellow et al., ICLR 2015.

L(↵, ) =<latexit sha1_base64="nMi78lnUwhs+tiVbE0mY4yinWdg=">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</latexit>

⇤
Q(WQ
+↵ +

Randomly generate vectors:<latexit sha1_base64="cIu5F6iaro86/lU8w6k2kB1JRQI=">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</latexit>

Optimal solution:

,

⇤
WQ

2 Rd

)

d = 1 + Nf (Nf<latexit sha1_base64="KD8va5S8qBHBKjekFmr4EaRB4zQ=">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</latexit><latexit sha1_base64="Ow1D0u2ZFC1YmU5jSmyZqw0EtYM=">AAACb3ichVHLSsNAFD2Nr1pfVRcKghRLRV2UGxEUV6Ibl1btA1otSRw1mCYhSQu1+AN+gC5c+AAR8TPc+AMu/ARxJRXcuPA2DYgW9Q4zc+bMPXfOzKi2obse0VNIamvv6OwKd0d6evv6B6KDQxnXKjuaSGuWYTk5VXGFoZsi7emeIXK2I5SSaoiserDS2M9WhOPqlrnpVW2xVVL2TH1X1xSPqUKtoClGLHtUTG3PFKNxSpIfsVYgByCOINas6A0K2IEFDWWUIGDCY2xAgcstDxkEm7kt1JhzGOn+vsARIqwtc5bgDIXZAx73eJUPWJPXjZqur9b4FIO7w8oYEvRIt1SnB7qjZ/r4tVbNr9HwUuVZbWqFXRw4Ht14/1dV4tnD/pfqT88edrHge9XZu+0zjVtoTX3l8LS+sbieqE3SFb2w/0t6onu+gVl5065TYv0MEf4A+edzt4LMbFKmpJyaiy8tB18RxhgmMMXvPY8lrGINaT7XxgnOcRF6lUakcSnWTJVCgWYY30Ka/gQhTI4h</latexit>

We then plot L as a function of α and β, which gives
the 2D projection of the d-dimensional optimization
landscape.

1)/2

Learning the best heat engines via global search-Q

The optimization problem is challenging: local (gradient-based) algorithms failed.
local optimum

Global optimum

Visualizing the optimization landscape:highly nonconvex<latexit sha1_base64="wb94oGq+i7iFwn3pb0ho4snvAxc=">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</latexit>

Differential evolution:
one of the most competitive global search algorithm
Stron & Price Tech. Rep. 95-012 (1995).
Wang et al., IEEE Trans. Evol. 15, (2011).

Q
Globaloptimumhttps://pablormier.github.io

Results: representative examples for Nf = 5 levels<latexit sha1_base64="oxmwzVa/4QmYY6myoPQUOykDFxA=">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</latexit>

Noninteracting caseInteraction network

State-transfer network✏ 1 < ✏2 < · · · < ✏Nf<latexit sha1_base64="9sljJMN+efsjBWV1P1eRAzXUnrk=">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</latexit>

Only a few edges activated → low power output
Modest ZT~10 at the highest power∵) nonzero heat flow at zero particle current(= bipolar effect)*Qualitatively consistent with previous studies.
cf. Whitney, PRB 91, 115425 (2015).

Results: representative examples for Nf = 5 levels<latexit sha1_base64="oxmwzVa/4QmYY6myoPQUOykDFxA=">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</latexit>

Interacting case, highest power factor Q
Interaction network

State-transfer network

Maximum power achieved by a sparse interaction:
1. Degeneracy of single-hole excitation energies(activating many transfer edges→high power)
2. Suppressing hole-hole interactions(nondiverging, but still large ZT)

Results: representative examples for Nf = 5 levels<latexit sha1_base64="oxmwzVa/4QmYY6myoPQUOykDFxA=">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</latexit>

Interacting case, intermediate Q and ZTInteraction network

State-transfer network

Making interaction stronger and denser, ZT is
improved at the expense of compromising Q.
∵) Strong and dense interaction isolates aparticular energy manifold, realizing theapproximate tight-coupling condition: J / Jq<latexit sha1_base64="ILaaM3Romg781jxDjKPkt6mByr8=">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</latexit>Results: representative examples for Nf = 5 levels<latexit sha1_base64="oxmwzVa/4QmYY6myoPQUOykDFxA=">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</latexit>

Interacting case, highest ZT
Interaction network

State-transfer network

Further increasing interaction, one can isolate twoparticular levels far from other many-body levels.
The divergence of ZT originates from almost perfectunicyclic structure in the probability flow, ensuring
the tight-coupling condition J / Jq<latexit sha1_base64="ILaaM3Romg781jxDjKPkt6mByr8=">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</latexit>Results at larger highest-power machines and finite-size scaling analysis(interacting)<latexit sha1_base64="3DFjKIacMFkiMoX5/1S230rhIoE=">AAAC2XichVFNSxxBEK2ZxERXjZN4CeQyZFEEYakRISEnMZecxFVXhR0Zetre3Wbni5neRTPMIbfg1YOHnBIIIeRn5JI/kIP5B+LRgBcP1s4M+LGo1XR31Xv1qqu73ciTiUI81vRHj0eePB0dq4xPTD6bMp6/2EzCXsxFg4deGG+7LBGeDERDSeWJ7SgWzHc9seV23w/4rb6IExkGG2o/Ejs+aweyJTlTBDmGslsx42ndSe3YN322l2XpitPKbBWa9p40C7pb0MtEbmQFZLeZ77MC72TXI05ZN9n5W7RjVLGGuZnDjlU6VShtNTR+gA27EAKHHvggIABFvgcMEhpNsAAhImwHUsJi8mTOC8igQtoeZQnKYIR2aW1T1CzRgOJBzSRXczrFoxmT0oQZ/Is/8Qz/4C88wYs7a6V5jUEv+7S7hVZEztTBy/XzB1U+7Qo6V6p7e1bQgrd5r5J6j3JkcAte6Psfj87W363NpLP4DU+p/694jL/pBkH/P/9eF2tfoEIfYN1+7mFnc6FmYc2qL1aXlsuvGIVX8Brm6L3fwBJ8gFVo0Ln/NNDGtIre1D/pn/WDIlXXSs003DD98BJERLe7</latexit>

Qmax
kB
!⇠
Nf
T

h c
h+

⇠ ' 0.439<latexit sha1_base64="LDm/xweXT7PlrKrOU8BZIE9Y0gg=">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</latexit>c

Fundamental bound on Q per level
cf. Esposito et al., EPL 85, 60010 (2009).

(noninteracting) Qmax / Nf0<latexit sha1_base64="gzAqUwNbIjPKcZNxNZbkQ9IF0qg=">AAACfXichVHLSiNBFD3p0VGjM4m6EdwEQ2QWEm6LEHElzsaVGDVRMNp091S0Sb/orgQ1OB8wP+DClaKIiP6EG39gFvmEwaWCG0FvOg3iiHqLqjp16p5bp6oM37ZCSdRKKF+6ur/29PYl+we+fU+lB4fKoVcPTFEyPdsL1gw9FLblipK0pC3W/EDojmGLVaP2s72/2hBBaHnuitz1xYajb7lW1TJ1yZSWHilqzUrgZBx9Z7/iB54vvcyCVt0kLZ2lPEWReQvUGGQRx6KXPkMFv+DBRB0OBFxIxjZ0hNzWoYLgM7eBJnMBIyvaF9hHkrV1zhKcoTNb43GLV+sx6/K6XTOM1CafYnMPWJlBjv7SOd3RDV3QP3p8t1YzqtH2ssuz0dEKX0v9GVl++FTl8Cyx/aL60LNEFdORV4u9+xHTvoXZ0Tf2Du6WZ5ZyzXE6plv2f0QtuuYbuI1787Qolg6R5A9Q/3/ut6A8mVcprxansrNz8Vf0YhRj+MHvXcAs5rGIEp/7Gye4xFXiSckpE0q+k6okYs0wXoVSeAZsQJOh</latexit><latexit sha1_base64="KlZLZi03E+NZRwmt6ON3rqfZTx4=">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</latexit>

h,c : tunneling rates

Results at larger highest-power machines and finite-size scaling analysis

idealInteraction!
realistic
noninteracting/ Nf0<latexit sha1_base64="OPVi0mNklHyyyeO/cpBu7WfPn1I=">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</latexit>/ Nf<latexit sha1_base64="vnCPseA/RGA2ZSDdIDI+1owuXW8=">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</latexit>(interacting)<latexit sha1_base64="3DFjKIacMFkiMoX5/1S230rhIoE=">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</latexit>

Qmax
kB
!⇠
Nf
T

h c
h+

⇠ ' 0.439<latexit sha1_base64="LDm/xweXT7PlrKrOU8BZIE9Y0gg=">AAACdHichVHLLgRBFD3T3uMxg42ExcRkxKpzG4nHasLGcjwGiZFJdytU9Et3zwQTP+AHLKwIEfEZNn7AwieIJWFj4U5PJ4Lgdqrr1Kl7bp2qa3iWDEKih4TS1NzS2tbekezs6u5JpXv7VgK34puiaLqW668ZeiAs6YhiKENLrHm+0G3DEqvG7lx9f7Uq/EC6znJ44IkNW9925JY09ZCpcjpV2pelQNpiL0PqxPh0OZ0llaLI/ARaDLKIo+Cmr1DCJlyYqMCGgIOQsQUdAX/r0EDwmNtAjTmfkYz2BY6QZG2FswRn6Mzu8n+bV+sx6/C6XjOI1CafYvHwWZlBju7pmp7pjm7okd5/rVWLatS9HPBsNLTCK6eOB5be/lXZPIfY+VT96TnEFqYir5K9exFTv4XZ0FcPT56XZhZztRE6pyf2f0YPdMs3cKov5uWCWDxFkhugfX/un2BlTNVI1RYmsvnZuBXtGMQwRvm9J5HHPAooRj05xQUuE6/KkJJVco1UJRFr+vElFPUDbTSPEw==</latexit>c

Fundamental bound on Q per level
cf. Esposito et al., EPL 85, 60010 (2009).

(noninteracting) Qmax / Nf0<latexit sha1_base64="gzAqUwNbIjPKcZNxNZbkQ9IF0qg=">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</latexit><latexit sha1_base64="KlZLZi03E+NZRwmt6ON3rqfZTx4=">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</latexit>

h,c : tunneling rates

Conditions for the highest-power heat engines
Given generic single-electron levels {✏l } , we conjecture thatthe highest-power machine is achieved by satisfying the following conditions:<latexit sha1_base64="sDOK0UojYToQIXlV3Gpw1ohlOiY=">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</latexit>(i) Single-hole excitation energies are degenerate (NfXwlm|el el+1 | ⌧ kB T , el = ✏l +1 constraints):<latexit sha1_base64="2J/RmOfgT2Ks6RQinccj2ywLObE=">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</latexit><latexit sha1_base64="+mhcfv4BuZcrIqfJapgNY46+0tA=">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</latexit>

m6=l<latexit sha1_base64="oS+Ws6hYFubkHvqVrLBJK9neG50=">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</latexit>(ii) At most Nf<latexit sha1_base64="2J/RmOfgT2Ks6RQinccj2ywLObE=">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</latexit>

1 variables of {wlm }l>m can be nonzero (sparse interaction).
<latexit sha1_base64="MH7LnP8fC4Arv3stOKeCDHUWnmY=">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</latexit>(iii) The ground voltage is set to be vg = eh + ↵kB T<latexit sha1_base64="Lp6nNIiNKzA4G09FYsMTPHEMTHE=">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</latexit>cf. Murphy et al., PRB 78, 161406 (2008).
Esposito et al., EPL 85, 60010 (2009).

↵ ' 2.40(ii)<latexit sha1_base64="z6364Fzr8EIPiEQSq14p4UOxYCk=">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</latexit>(i)

For any {✏l, }there in general exist an excessive number ofsolutions for {wlm }l>m
, which allow for the highest power→ flexible design of optimal nanoscale engines.
<latexit sha1_base64="sDOK0UojYToQIXlV3Gpw1ohlOiY=">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</latexit><latexit sha1_base64="MH7LnP8fC4Arv3stOKeCDHUWnmY=">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</latexit>*In noninteracting case, the highest power is possible only if ✏1 = ✏2 = · · · = ✏Nf<latexit sha1_base64="jtyh4qJ/9Pl5ZlcMgHcEuEX2lN8=">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</latexit>

Conditions for the highest-power heat engines

Conditions for the highest-power heat engines
Nf = 7

Nf = 6<latexit sha1_base64="t1K1fjM3hwpPU7N2LZEMeoiCx5I=">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</latexit><latexit sha1_base64="YS0K+uFaNvTrBOVJetN6SycWO84=">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</latexit>

Conjecture numerically confirmed up toNf =9levels
Nf = 9<latexit sha1_base64="Vd3LwHTyJaPguUdQKUAD+bllzfI=">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</latexit>

Nf = 8<latexit sha1_base64="J+EyGTOOOi+nukFEPGBGYMXI96k=">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</latexit><latexit sha1_base64="Vd3LwHTyJaPguUdQKUAD+bllzfI=">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</latexit>

Possible experimental relevance
Quantum-dot array

Trapped ions

・Coulomb interactions in dense regimes.
・Two prototypical configurations forthe highest power with ✏1 < ✏2 < · · · < ✏Nf<latexit sha1_base64="9sljJMN+efsjBWV1P1eRAzXUnrk=">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</latexit>

Bylinskii et al.,Science 348, 6239 (2015).

• Noise of electric fields act as
equilibrium baths.
• Inevitable Coulomb interactions.
• Single-particle manipulation
realized.

Summary• We develop a global-optimization framework to identify the best tradeoff
relation in the multiple objectives for interacting nanoscale machines.
• We apply it to optimizing power and efficiency in nanothermoelectrics tofind a set of the best heat engines.
• For generic single-electron levels, thermoelectric figure of merit and power factor canin principle be enhanced by orders of magnitudes in the presence of interaction.
• Our findings could be of relevance to quantum-dot array and trapped ions.
Outlook:✔ Application to other nanosystems described by the master equation(such as solar photovoltaics, molecular motors and biophysics networks).
✔ Multiobjective optimization with other objectives; finding a way to maximize solar power,
molecular mobility and biophysical-reaction yield while keeping high efficiency.
✔ Role of nonlinear effects, quantum many-body effects (e.g., Kondo physics),
and time-reversal symmetry breaking in interacting nano-heat engines.
Ref: YA and T. Sagawa, Commun. Phys. 4, 45 (2021).