Thermodynamic bounds on coherent transport in periodically driven conductors

Elina Potanina1, Christian Flindt1, Michael Moskalets2, and Kay Brandner1

1Department of Applied Physics, Aalto University, 00076 Aalto, Finland2Department of Metal and Semiconductor Physics,

NTU Kharkiv Polytechnic Institute, 61002 Kharkiv, Ukraine

We establish a family of new thermodynamic constraints on heat and particle transport in coherentmulti-terminal conductors subject to slowly oscillating driving fields as well as moderate electricaland thermal biases. These bounds depend only on the number of terminals of the conductor andthe base temperature of the system. Going beyond the second law of thermodynamics, they implythat every local current puts a lower limit on the mean dissipation caused by the overall transportprocess. As a key application of this result, we derive two novel trade-off relations restricting theperformance of adiabatic quantum pumps and isothermal engines. On the technical level, our workcombines Floquet scattering and linear-adiabatic-response theory with recent techniques from small-scale thermodynamics. Using this framework, we illustrate our general findings by working out twospecific models describing either a quantum pump or an isothermal engine. These case studies showthat our bounds are tight and provide valuable benchmarks for realistic devices.

I. INTRODUCTION

In macroscopic systems at finite temperature, trans-port is typically governed by the irreversible dynamics ofparticles constantly undergoing collisions, in which theyrandomly change their energy and direction of motion.When the size of the sample becomes comparable to themean free path of the carriers, however, the situationchanges drastically1. In this regime, which is realizedin mesoscopic conductors at low temperatures and canbe probed accurately in experiments2–12, the carrier dy-namics is no longer stochastic but is dominated by the re-versible laws of quantum mechanics. The transfer of mat-ter and energy thus becomes a coherent process, whichcan be elegantly described as the elastic scattering of ef-fectively non-interacting particles originating from ther-mal reservoirs. This picture applies even to systems thatare subject to oscillating electric or magnetic fields.

In such dynamic conductors, however, the microscopicscattering events are no longer elastic, since carriers canabsorb or emit discrete quanta of energy while passingthrough the sample13. This mechanism makes it possibleto realize cyclic nano-scale machines like adiabatic quan-tum pumps13, motors14 or heat engines15, which use slowperiodic control fields to generate either a directed flowof particles or mechanical work, see Fig. 1.

What are the fundamental performance limits of suchdevices? Finding answers to this question is one of themain purposes of the present paper. In fact, as our cen-tral result, we derive the universal relation

σ ≥m

m − 1Kxx

(Jxα)2, (1)

which bounds the total dissipation σ caused by a coherenttransport process in an m-terminal conductor in terms ofevery period-averaged particle (x = ρ) and heat (x = q)

current. The factor Kxx thereby effectively depends onlyon the equilibrium temperature of the system.

Following ultimately from the fundamental laws of par-ticle and energy conservation, our new bound (1) is sig-

FIG. 1. Coherent quantum pumps. (a) Magnetic-flux pumpconsisting of a mesoscopic ring connected to two leads viaideal cyclic beam splitters. All incoming carriers are reflectedafter passing through the loop. A directed current from leftto right can be generated by applying a time-dependent mag-netic flux φ, which accelerates counterclockwise-moving par-ticles (red path) and decelerates clockwise-moving ones (bluepath). (b) Tunable-barrier pump driven by periodic variationsof two control parameters λ1 and λ2 determining the strengthsof the barriers. The device operates in a four-stroke cycle, inwhich incoming carriers from the left are first absorbed, thencaptured between the barriers and finally ejected to the right.

nificantly stronger than the second law of thermodynam-ics, which only requires σ ≥ 0. It therefore restricts not

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only the efficiency but also the practically often moreimportant output of mesoscopic transport devices16. Forexample, Eq. (1) implies a universal maximum for thecurrent that a quantum pump can generate with givenenergy input. This result could provide a valuable bench-mark for future technologies involving adiabatic quantumpumps, which, owing to their ability to transfer particlesone-by-one with high accuracy, can be used as dynamicalsingle-electron source2–12.

In a broader perspective, the quest for universal con-straints on the figures of merit that govern the perfor-mance of coherent nano-scale machines based on meso-scopic conductors has emerged as an active research topicover the last years17–29. As their main new feature, thebounds derived in this paper cover both mechanical driv-ing exerted through oscillating control fields and steady-state driving induced by thermochemical gradients. Infact, we will show that our general scheme makes it pos-sible to recover and, to some extent, unify various earlierresults by considering specific limits.

On the technical level, the basis for our work is pro-vided by the Floquet scattering formalism13, a dynami-cal extension of the conventional scattering approach toquantum transport in stationary systems; going back tothe pioneering work of Landauer30, the idea to use scat-tering theory for the description of transport phenom-ena developed into a powerful and well-tested tool ofmesoscopic physics during the following decades31. Asa second ingredient for our analysis, we use a combina-tion of adiabatic and linear response theory15. Leadingto a thermodynamically consistent perturbation scheme,these methods allow us to capture the intricate inter-play between coherent transport dynamics and mechan-ical driving in concise and physically transparent equa-tions. This mathematical framework enables a generalapproach, which covers a wide range of systems and ap-plications and does not rely on specific models.

Our paper is organized as follows. In Sec. II, we pro-vide a short review of Floquet scattering theory, showhow this framework can be furnished with a consis-tent thermodynamic structure and outline the linear-adiabatic response scheme. We then derive our newbounds on coherent transport in dynamic mesoscopicconductors in Sec. III A. We also discuss how our mainresults are related to time-reversal symmetry and geo-metric aspects and argue that they are robust againstphase-breaking perturbations. Moving on to practicalapplications, in Sec. IV, we establish our thermodynamictrade-off relations for quantum pumps and isothermal en-gines. Furthermore, by analyzing two specific models,we show that our bounds are generally tight and demon-strate their practical applicability. Finally, in Sec. V, webriefly summarize our work and discuss its relation withearlier studies. We conclude with a short outlook.

II. FRAMEWORK

A. Scattering Theory

We begin with a brief review of the scattering ap-proach to quantum transport in dynamic conductors. Tothis end, we consider a mesoscopic sample, whose energylandscape is periodically modulated by external drivingfields λ = {λk}. The terminals of this conductor arecoupled to thermochemical reservoirs with different tem-peratures Tα and chemical potentials µα, see Fig. 2. Inthe coherent regime, the mean free path of the carri-ers injected by these reservoirs exceeds the dimensionsof the conductor. The emerging transport process canthen be described as a continuous series of coherent scat-tering events involving non-interacting particles, whichexchange a quantized amount of energy with the exter-nal fields while passing through the sample13.

Once the system has reached a periodic state, the av-erage particle and energy currents flowing in terminal αtowards the scattering region are given by the Landauer-Buttiker-type formulas13

Jρα =1

h∫

∞

0dE∑β

(δαβ −∑n∣SαβEn,E ∣

2)fβE and (2)

JEα =1

h∫

∞

0dE∑β

(Eδαβ −∑nEn∣S

αβEn,E

∣2)fβE

respectively. Here, we have used the shorthand notationEn ≡ E+nhω, where h = h/2π denotes the reduced Planckconstant and ω the driving frequency. Thermodynamicsenters the expressions (2) via the Fermi functions of thereservoirs,

fαE ≡ 1/(1 + exp[(E − µα)/Tα]), (3)

with Boltzmann’s constant being set to 1 throughout.The properties of the sample are encoded in the Floquet

scattering amplitudes SαβEn,E . These objects describe thetransmission of incoming particles with energy E fromthe terminal β to the terminal α under the absorption(n > 0) or emission (n < 0) of n energy quanta of size hω.

Though generally dependent on the specific architec-ture of the system, the Floquet scattering amplitudes stillobey two universal relations, which arise from fundamen-tal principles. First, the sum rules

∑α∑n∣SαβEn,E ∣

2=∑α∑n

∣SβαE,En∣2= 1, (4)

which follow from the unitarity of the Floquet scatter-ing matrix13, ensure the conservation of probabilities insingle-particle scattering events. The double sum therebyruns over all terminals α and all integers n, for whichEn > 0. Note that, throughout this article, we focuson single-channel conductors for simplicity. Second, thetime-reversal symmetry of Schrodinger’s equation implies

SαβEn,E = TBTλSβαE,En

, (5)

3

FIG. 2. Dynamic multi-terminal conductor. A central scat-tering region subject to periodic driving fields λ is connectedto m reservoirs with chemical potentials µ1, . . . , µm and tem-peratures T1, . . . , Tm. Each reservoir injects a constant meancurrent of particles (x = ρ) and heat (x = q) into the conduc-tor. Additionally, the external driving provides a continuousinflow of energy proportional to the photon flux Jω.

where the symbolic operators TB and Tλ indicate thereversal of external magnetic fields and driving protocols,respectively. Hence, while the sum rules (4) constrain thescattering amplitudes in any given setup, the symmetry(5) connects the scattering amplitudes of two differentsystems that are T-symmetric counterparts of each other.

Since neither particles nor energy are accumulated inthe conductor over a full driving cycle, the mean currents(2) have to obey the conservation laws

∑αJρα = 0 and Pac +∑α

JEα = 0. (6)

The first of these relations can be easily verified usingEq. (2) and Eq. (4). Along the same lines, the secondone leads to a microscopic expression for the average me-chanical power that is absorbed by the system,

Pac = −∑αJEα =

1

h∫

∞

0dE∑αβ∑n

nhω∣SαβEn,E ∣2fβE (7)

≡ hωJω.

Here, Jω corresponds to the average current of energyquanta, henceforth called photons, that is injected intothe conductor. Note that this quantity can in principle bedefined without reference to the microscopic dynamics ofthe system. Its physical interpretation as a photon flux,however, relies on the fact that particles and driving fieldscan exchange only discrete units of energy in elementaryscattering events. This phenomenon is a genuine featureof quantum conductors13. On the technical level, it arisesas a consequence of the Floquet theorem, which has nocounterpart in classical mechanics32.

B. Thermodynamics

The non-equilibrium thermodynamics of dynamic co-herent conductors can be developed starting from the

first and the second law,

Pac + Pel +∑αJqα = 0 and σ ≡ −∑α

Jqα/Tα ≥ 0. (8)

Here, the average electrical power absorbed by the sys-tem and the mean heat current flowing in the terminal αtowards the scattering region are given by

Pel ≡∑αµαJ

ρα, and Jqα ≡ JEα − µαJ

ρα, (9)

respectively, and σ denotes the total rate of entropy pro-duction accompanying the transport process. As we showin App. A, the relations (8) can be verified explicitly us-ing the microscopic expressions (2) and (7) for the meanparticle and energy currents and the sum rules (4). Thus,the dynamical scattering approach, which provides thebasis for our work, is inherently consistent with the lawsof thermodynamics.

Rewriting these laws in terms of thermodynamic fluxesand forces reveals a universal structure, which resem-bles the standard irreversible thermodynamics of non-equilibrium steady states16. Specifically, upon introduc-ing the affinities

Fω ≡ hω/T, F ρα ≡ (µα − µ)/T, F qα ≡ 1/T − 1/Tα, (10)

where µ and T are the reference chemical potential andtemperature, the first law becomes

∑αJqα + TJ

ωFω +∑αTJραF

ρα = 0 (11)

and the second law assumes the canonical bilinear form

σ = JωFω +∑αJραF

ρα + J

qαF

qα ≥ 0. (12)

This result shows that, on the level of mean fluxes, thethermodynamic properties of periodic states resemblethose of non-equilibrium steady states as has been ob-served earlier for classical stochastic systems33–35. Quiteremarkably, Eqs. (11) and (12) put the conventional ther-mochemical forces F ρα and F qα, which are determinedby the external reservoirs, on equal footing with thenew affinity Fω, which describes the effect of periodicdriving fields on charge carriers inside the conductor.This unification of thermal and mechanical driving is,as opposed to earlier works33,36–38, achieved here byidentifying the frequency rather than the amplitude ofthe time-dependent perturbation as a thermodynamicforce. Though similar in their formal structure, these twoschemes describe quite different physical scenarios. Inparticular, the frequency-based approach, which was firstsuggested in Ref.15, yields a natural generalization of On-sager’s kinetic flux-force relations in the linear-adiabaticregime, which we will discuss in the next section. Bycontrast, a perturbation theory in the driving strengthalways leads to a trivial decoupling of thermochemicalcurrents and mechanical driving for coherent conductorsas we show in App. B.

4

C. Linear-Adiabatic Response

1. Approximation Scheme

Using the formulas (2) and (7), the thermodynamicfluxes can in principle be determined for any given setup.However, these expressions are notoriously hard to applyin practice and the Floquet scattering amplitudes can beobtained exactly only for a few simple models. In thefollowing we show how linear-adiabatic response theorymakes it possible to simplify the situation by invokingtwo key assumptions, which are typically well justifiedunder realistic conditions; coherent quantum pumps, forexample, which we will discuss further in Sec. IV, can beaccurately described within the adiabatic approximation.

First, the temperature and chemical potential gradi-ents are usually small compared to their respective ther-modynamic references. Specifically, we can assume that

F ρα ≪ µ/T and F qα ≪ 1/T (13)

The Fermi functions in (2) and (7) can thus be expandedto first order in the thermodynamic forces, i.e., we have

fβE ≃ fE + hgE ⋅ (Fρβ + (E − µ)F qβ), where (14)

fE ≡ fβE ∣Fρβ=F q

β=0 and gE ≡ 1/4h cosh2

[(E − µ)/2T ]

denotes the negative derivative of the Fermi function upto a scaling factor h/T , which has been introduced tosimplify the notation in the following.

The second corner stone of the adiabatic approach isthe slow-driving criterion13

ω ≪ δE/h. (15)

Here, δE is the typical energy scale, over which the frozen

scattering amplitudes SαβE,λ vary significantly; these ob-

jects describe the transmission of particles with energyE at fixed values of the control parameters λ, i.e., in thequasi-static limit. The quantity δE/h can be regarded asa measure for the inverse dwell time of particles in thescattering region39,40. The criterion (15) thus essentiallyrequires that the energy landscape of the sample changesonly slightly during the passage of individual carriers.Under this condition, the Floquet scattering amplitudesare approximately given by

SαβEn,E ≃1

T∫

T

0dt (S

αβE,λ + nhω∂ES

αβE,λ + hωA

αβE,t) e

inωt

with T ≡ 2π/ω (16)

and the corrections AαβE,t being required to ensure thatthe adiabatic approximation does not spoil the unitar-ity of the Floquet scattering matrix, for details see41.Note that, at low temperatures, the function gE in (14)is sharply peaked around µ. The transmission of carriersthen occurs only at energies close to the Fermi edge. Itis therefore typically sufficient to require (15) for E ≃ µ.

Upon inserting the approximations (14) and (16), intoEqs. (2) and (7), the thermodynamic fluxes become lin-ear functions of the corresponding affinities given by

Jxα = Lxωα Fω +∑β∑yLxyαβF

yβ and (17)

Jω = LωωFω +∑α∑xLωxα F xα with x, y = ρ, q.

The kinetic coefficients appearing in these relations can,after some algebra, be expressed in the compact form

Lxyαβ = ∫∞

0dE gE ⋅ ξ

xEξ

yE(δαβ − ⟪∣S

αβE,λ∣

2⟫), (18)

Lxωα = ∫

∞

0dE gE ⋅ ξ

xEξ

ω∑β

Im[⟪SαβE,λS

αβ∗E,λ⟫],

Lωxα = ∫

∞

0dE gE ⋅ ξ

xEξ

ω∑β

Im[⟪SβαE,λS

βα∗E,λ⟫],

Lωω = ∫

∞

0dE gE ⋅ (ξ

ω)2∑αβ

⟪∣SαβE,λ∣

2⟫/2 with

ξρE ≡ 1, ξqE ≡ E − µ, ξω ≡ 1/ω,

dots indicating time derivatives and double brackets de-noting the time average over one period of the driving.This result shows that, under linear-adiabatic-responseconditions, the thermodynamic fluxes depend only on

the frozen scattering amplitudes SαβE,λ, which are signif-

icantly easier accessible than the full Floquet scatteringamplitudes entering the non-linear expressions (2) and

(7). Note in particular that the corrections AαβE,t appear-

ing in (16) do not contribute to the response coefficients(18) as a consequence of the unitarity requirement15,42.

2. Thermodynamic Consistency & Reciprocity Relations

After introducing the linear-adiabatic-responsescheme, we are now ready to discuss the connectionbetween thermodynamics and microdynamics in slowlydriven coherent conductors. To this end, we first notethat the frozen scattering amplitudes and their timederivatives have to satisfy the sum rules

∑α∣SαβE,λ∣

2=∑α

∣SβαE,λ∣

2= 1, ∑αβ

SαβE,λS

αβ∗E,λ = 0. (19)

The first condition thereby accounts for probability con-servation in single-particle scattering events at fixed con-trol parameters. The second one fixes the global phaseof the frozen scattering amplitudes such that the particleand heat currents (17) obey the conservation laws

∑αJρα = 0 and ∑α

Jqα = 0, (20)

respectively. Note that, while the particle currents sat-isfy this constraint also in the non-linear regime, the heatcurrents are conserved only in linear order with respectto the thermodynamic forces (10). This result is well inagreement with the first law (11), since the mechanicaland the electrical power are both quadratic in the affini-ties; to recover (11) exactly, we would have to includesecond-order corrections to the heat currents.

5

To validate the second law (12), we observe that, uponinserting the kinetic equations (17), the total rate of en-tropy production becomes a quadratic form in the affini-ties (10), which can be written as

σ = ∫

∞

0dE gE∑αβ

⟪∣XαβE S

αβE,λ +ZS

αβE,λ∣

2⟫/2 ≥ 0 (21)

with XαβE ≡∑x

ξxE(F xα − Fxβ ) and Z ≡ iξωFω.

This expression follows from (18) and the sum rules (19);it shows that the rate of entropy production is an inher-ently positive quantity in the linear-adiabatic-responsescheme, which is thus consistent with the second law.

Finally, owing to time-reversal symmetry, the frozenscattering amplitudes obey the relation

SαβE,λ = TBS

βαE,λ. (22)

This constraint is substantially stronger than (5), whichrequires the reversal of driving protocols. It implies thereciprocity relations

TBLxyαβ = L

yxβα, TBL

ωxα = −Lxωα , TBTλL

ωxα = Lxωα (23)

for the adiabatic response coefficients (18), which canbe regarded as extensions of the conventional Onsager-Casimir relations43–45. As we will discuss in Sec. III B,these symmetries have profound consequences for thethermodynamics of slowly driven coherent conductors.

III. THERMODYNAMIC CONSTRAINTS

A. New Bounds

We are now ready to derive our new bounds on co-herent transport in slowly driven mesoscopic conduc-tors. To this end, we employ a mathematical tech-nique that has proven effective to unveil thermodynamicconstraints in systems subject to steady-state19,20,24,25,46

and periodic27,33,37 driving. The key idea of this ap-proach is to introduce a generating function for the quan-tities of interest, which is then bounded from below andafterwards contracted to a given set of variables.

Here, we use the quadratic generating function

G ≡ σ +∑α∑xJxαG

xα +∑α∑xy

KxyGxαGyα/2

with Kxy≡ ∫

∞

0dE gE ⋅ ξ

xEξ

yE (24)

and the Gxα being real but otherwise arbitrary variables.Upon expressing the rate of entropy production σ andthe currents Jxα in terms of the kinetic coefficients (18)and using the sum rules (19), this object can be shownto be manifestly non-negative. That is, we have

G = ∫

∞

0dE gE∑αβ

⟪∣Y αβE SαβE,λ +ZS

αβE,λ∣

2⟫/2 ≥ 0

with Y αβE ≡∑xξxE(F yα − F

yβ +G

xα). (25)

This result implies a whole family of bounds, each ofwhich can be extracted by contracting (24) in a specificway. In particular, taking the minimum of G with respectto the auxiliary variables Gxα yields

σ ≥∑α∑xyKxyJxαJ

yα/2, (26)

where the parameters Kxy are defined by the condition

∑yKxyKyz

≡ δxz. (27)

To obtain a bound that involves only a single current,we further minimize the right-hand side of (26) with re-spect to all but one of the flux variables Jxα, thereby tak-ing into account the conservation laws (20) as additionalconstraints. This procedure yields our central result,

σ ≥m

m − 1Kxx

(Jxα)2/2, (28)

where m is the number of terminals of the conductor.This inequality provides a significantly stronger con-

straint than the bare second law, which requires onlyσ ≥ 0. Specifically, (28) shows that the total rate of en-tropy production is bounded by the square of every indi-vidual heat and particle current times a proportionalityfactor, which depends only on the number of terminalsm, the equilibrium temperature T of the system and, for-mally, its base chemical potential µ. In practice, however,the parameters Kxx can be estimated as

Kρρ≥ h/T and Kqq

≥ 3h/π2T 3, (29)

where equality holds in the limit µ/T → ∞, which iseffectively realized in mesoscopic systems. Notably, nouniversal relation of the form (28) exists for the photonflux Jω as we will demonstrate explicitly in Sec. IV B byworking out a specific example.

B. Time-Reversal Symmetry

The bound (28) is universal in that it holds for anarbitrary potential and magnetic field landscape insidethe conductor and any sufficiently slow driving protocols.As we will show in Sec. IV B, it is generally also tight. Forsystems with additional symmetries, however, strongerbounds can be established, as we will see in the following.

For time-symmetric driving protocols, the reciprocityrelations (23) imply Lxωα = Lωxα = 0, that is, the thermo-chemical variables, Jxα and F xα , decouple from the me-chanical ones, Jω and Fω. The total entropy productioncan thus be divided into two non-negative contributions,

σth ≡∑α∑xJxαF

xα ≥ 0 and σω = JωFω ≥ 0, (30)

which arise solely from thermal gradients and periodicdriving, respectively. Repeating essentially the steps thatlead to (28), and using a convexity argument, which weprovide in App. C, then leads to the refined bound

σth ≥m

m − 1Lxxαα(J

xα)

2/2 with ∑yLxyααL

yzαα ≡ δxz, (31)

6

which is stronger than (28) in two respects. First, itinvolves only the thermal rather than the total entropyproduction; these two quantities are identical if and onlyif the external control parameters are frozen, that is, if notime-dependent driving is applied to the system. Second,the factors Kxx, which are independent of the structuralproperties of the conductor, are now replaced with theinverse kinetic coefficients, Lxxαα ≥ Kxx.

The latter feature leads to an interesting physicalinterpretation of (31). According to the fluctuation-dissipation theorem47, the diagonal response coefficientsLxxαα are proportional to the period averaged equilibriumnoise of the current Jxα. Specifically, we have

Dxα ≡ ⟪Dx

α,λ⟫ = 2Lxxαα, (32)

where Dxα,λ denotes the zero-frequency noise of Jxα at

constant temperature and chemical potential, T and µ,and fixed external parameters λ, for details see Ref.13.Thus, upon noting that Lxxαα ≥ 1/Lxxαα, the bound (31)implies

σεxα ≥ σthεxα ≥m/(m−1), where εxα ≡ (Jxα)

2/Dx

α (33)

denotes the relative uncertainty of the current Jxα48,49.

This figure can be regarded as a measure for the accu-racy, at which either particles or heat are extracted fromthe reservoir α. Hence, (33) entails a universal trade-offbetween the thermodynamic cost, i.e., the entropy thatmust be generated to maintain the fluxes Jxα, and theprecision of adiabatically driven coherent transport. Wewill further discuss this concept and the relation of (33)with previous results at the end of this paper50.

For now, we move on to fully time-reversal symmet-ric systems, i.e., we now assume that the driving proto-cols are time-symmetric and no magnetic field is appliedto the conductor. According to the reciprocity relations(23), the thermochemical response coefficients are thensubject to the constraint Lxyαβ = L

yxβα. Using this symme-

try, the bounds (31) and (33) can be further tightened byreplacing the m-dependent factor on the right-hand sidewith 2, that is, we have

σth ≥ Lxxαα(Jxα)

2 and σεxα ≥ σthεxα ≥ 2. (34)

These relations can either be derived using the methodexplained in Sec. III A or inferred directly from theCauchy-Schwarz inequality28. They apply to any system,whose adiabatic-response coefficients obey Lxωα = Lωxα = 0and Lxyαβ = L

yxβα. To this end, full time-reversal symmetry

is a sufficient but not a necessary condition as we willlearn in the next part of this section.

C. Geometry

Adiabatic-response theory is closely related to geomet-ric concepts51. In the framework discussed here, this con-nection is concealed in the coupling coefficients between

thermochemical and mechanical variables. In fact, theadiabatic transmission coefficients, which enter the ex-pressions (18) for Lxωα and Lωxα , have the same formalstructure as Berry’s geometric phase52. They can there-fore be rewritten as

Im[⟪SαβE,λS

αβ∗E,λ⟫] = −Im[⟪S

αβE,λS

αβ∗E,λ⟫] =

1

T∮γdλ ⋅Aαβ

E,λ,

where AαβE,λ ≡ Im[S

αβE,λ∇λS

αβ∗E,λ ] (35)

plays the role of a generalized vector potential and γ isthe closed path in the space of control parameters that isencircled by the driving protocols. Upon applying Stokes’theorem, this expression can be converted into

Im[⟪SαβE,λS

αβ∗E,λ⟫] =

1

T∫

ΣγdS ⋅BαβE,λ. (36)

The integral thereby extends over an arbitrary surface

Σγ that is bounded by the curve γ and BαβE,λ denotes

the Berry curvature. For a three-dimensional parameterspace, this quantity is given by

BαβE,λ = ∇λ ×AαβE,λ. (37)

In higher dimensions, a proper generalization of this for-mula is provided by the theory of differential forms52.

This result shows that the adiabatic transmission co-efficients are independent of the specific parameteriza-tion of γ and vanish if the mechanical driving is ex-erted through a single control parameter. We then haveLωxα = Lxωα = 0 and the currents Jxα obey the bounds(31), regardless of whether or not the driving protocolis time-symmetric. If, in addition, the thermochemicalresponse coefficients are symmetric, the stronger bounds(34) apply. We conclude this discussion by noting thatthe expression (36) is closely connected to a geometricquantization principle and plays an important role in thetheory of adiabatic quantum pumps53–59, which we willrevisit in Sec. IV.

D. Dephasing

Ideal coherent transport, which we have consideredso far, is characterized by a fixed phase relation be-tween incoming and outgoing carriers. Under realisticconditions, however, phase-breaking mechanisms such ascarrier-carrier or carrier-phonon interactions are difficultto suppress completely. An elegant method to accountfor such effects within the scattering formalism goes backto Buttiker60. In this approach, a number of virtualreservoirs is attached to the conductor, whose temper-ature and chemical potential are adjusted such that theydo not draw or supply any particles or heat on average.Hence, these probe terminals do not contribute to the ac-tual transport process but rather only mimic incoherentscattering events.

7

On the technical level, the virtual reservoirs differ fromphysical ones only in that their affinities are not free pa-rameters but implicitly fixed by the condition of zeromean currents. Since the relation (28) and (31) were de-rived without making any assumptions on the affinities,it is clear that they remain valid for models with arbi-trarily many probe terminals. The parameter m therebya priori refers to the total number of physical and virtualterminals.

However, since only non-vanishing currents have to beconsidered in the contraction leading from (26) to (28)61,the constraint (28) even holds in a stronger version, wherem counts only the physical terminals. Hence, our generalnew bound is robust against dephasing and applies evenin the limit of completely incoherent transmission60. Thisobservation further underlines the universality of our re-sults. As we will discuss in Sec. V B 2, it also allows usto recover an earlier bound on stationary transport inmulti-terminal systems. Note, however, that the sameargument can not be used for the symmetric bound (31),which can be proven only with m being the total numberof terminals, see App. C.

IV. ADIABATIC QUANTUM DEVICES

A. Setup

We now show how our general theory can be appliedto practical problems. To this end, we use our key result(28) to derive two universal trade-off relations restrict-ing the performance of adiabatic quantum pumps andisothermal engines, respectively. To explore the qualityof these bounds, we work out two specific models.

Coherent quantum pumps use a periodically modu-lated scattering potential to generate a directed flowof particles between two reservoirs with equal temper-atures and chemical potentials53, see Fig. 1. In linear-adiabatic response, such devices are subject to the power-flux trade-off relation

QP ≡ h(Jρ)2/Pac ≤ 1, (38)

which provides a lower bound on the mean energy inputPac that is required to sustain a given pump current Jρ.This result follows directly from (28) upon using (29) andnoting that, for vanishing thermochemical gradients, theentropy production (11) is proportional to the mechanicalpower (9), i.e., σ = hωJω/T = Pac/T .

In order to extract useful power from a periodicallydriven conductor, an external bias voltage ∆µ = TF ρ

must be applied such that the pumped particle currentflows uphill. The device then operates as an isothermalengine converting mechanical into electrical energy. Thesecond law puts a universal upper limit on the thermo-dynamic efficiency of this process,

η ≡ −Pel/Pac ≤ 1. (39)

However, the laws of thermodynamics do not constrainthe bare power output, −Pel = −∆µJρ, due to their inher-ent lack of a fundamental time scale. This gap is closed byour theory as the bound (28) implies the power-efficiencytrade-off relation33,62

QE ≡ h(−Pel/∆µ2)/(1/η − 1) ≤ 1. (40)

This result shows in particular that the ideal efficiency 1can be approached only at the price of vanishing power,for further discussions of this phenomenon see Sec. V B 2.

B. Magnetic-Flux Device

As a first benchmark for our trade-off relations (38)and (40), we consider a technically simple model, whichwas introduced in17 as an example of an optimal quantumpump. This system consists of an Aharonov-Bohm loopthreaded by a linearly increasing magnetic flux φ = ωt, seeFig. 1a. The corresponding frozen scattering amplitudesand driving protocols are given by

SαβE,λ = δαβ(λ1 − imαλ2)e

iχ and (41)

λ1 = cos[ωt], λ2 = sin[ωt],

respectively, where m1 = 1, m2 = −1 and the global phaseχ is determined by the circumference of the loop. Insert-ing these scattering amplitudes into (18) and (17) yieldsthe mean particle and photon flux,

Jρ = 1/T and Jω = 1/T −∆µ/h. (42)

Note that, from here onwards, we focus on the low-temperature limit, µ/T ≫ 1, where the function gE de-fined in (14) is strongly peaked around E = µ such thatthe energy integrals in (18) can be carried out explicitly.

Upon evaluating the mechanical and the electricalpower generated by the currents (42),

Pac = (h/T −∆µ)/T and − Pel = −∆µ/T , (43)

we find that the trade-off relations (38) and (40) are bothsaturated, for ∆µ = 0 and ∆µ < 0, respectively. Thisresults show that our bounds are tight. Furthermore,from (42), we find that the mean entropy production ofthe magnetic-flux engine,

σ = F ρJρ + FωJω = (h/T )/T2, (44)

is independent of the applied bias. Thus, changing ∆µ,in principle, makes it possible to tune Jω to any givenvalue without altering σ. This observation proves thatthe photon flux can indeed not be constrained by a uni-versal bound of the form (28).

C. Tunable-Barrier Device

We now move on to a second type of periodic quan-tum device, which, operating in a four-stroke cycle, uses

8

FIG. 3. Tunable-barrier system as adiabatic quantum device.(a) Contour plot of the Berry curvature defined in (46) for thescattering amplitudes (45) and χ = π/10. Black lines indicatethe two symmetries of this function. The circles correspondto the path encircled by the driving protocols (48) for v =1/4, . . . ,5/4. (b) Performance coefficients QP (blue) and QE

(red) as functions of the dimensionless current T Jρ and theisothermal engine efficiency η, respectively. For both plots,we have set χ = π/10 and tuned v continuously from 1/10 to100. To evaluate QE, the bias ∆µ has been determined bymaximizing the efficiency (39).

a periodically modulated confinement potential to moveparticles one-by-one through a mesoscopic conductor,see Fig. 1b. Owing to their application as a metrolog-ical current standard such systems have attracted signif-icant interest. In fact, it has become clear that single-electron pumps implemented with tunable-barrier quan-tum dots provide an experimentally accessible realizationof the quantum ampere63–65 with close-to-metrologicalaccuracy66,67.

The behavior of these devices is determined by the con-ductance G of the point contacts or metallic gates sepa-rating the dot from the leads. In the Coulomb blockaderegime, where G ≪ e2/h with e being the elementarycharge, strong interactions lead to incoherent but quan-

tized single-particle transport. In the coherent regime,G ≫ e2/h, which is realized in large open quantum dotswith nearly vanishing level spacings, quantization is con-nected to topological properties of the underlying scat-tering matrix55. Furthermore, even for almost open dots,i.e., G ≃ e2/h , current quantization can be achieved byapproaching the resonant transmission regime57, as wewill show in the following.

To demonstrate the main features of adiabatic single-electron pumping in the coherent regime, we here con-sider a one-dimensional model, where the confinementpotential consists of two delta-type barriers with dimen-sionless strengths λ1 and λ2. These control parametersdetermine the conductances of the individual barriers ac-cording to G ∼ e2/hλ2. The frozen scattering amplitudesfor this system are given by41

S12E,λ = S

21µ,λ = χ2eiχ/Z, (45)

S11E,λ = λ1(λ2 − iχ)/Z − λ2(λ1 + iχ)e

2iχ/Z,

S22E,λ = λ2(λ1 − iχ)/Z − λ2(λ1 + iχ)e

2iχ/Z, where

Z ≡ λ1λ2e2iχ

− (λ1 − iχ)(λ2 − iχ) and χ ≡ L√

2ME/h.

Here, L denotes the spatial distance between the twobarriers and M the effective carrier mass.

The complicated structure of the scattering amplitudes(45) makes it difficult to find proper driving protocolsfor the tunable-barrier system. Here, we approach thisproblem from a geometric perspective using the frame-work discussed in Sec. III C. We first rewrite the particlecurrent at zero bias and low temperatures as

Jρ = LρωFω =1

2πT∫

ΣγdS ⋅Bµ,λ, (46)

where the total Berry curvature, Bµ,λ ≡ B11µ,λ +B

12µ,λ, can

be calculated by inserting the scattering amplitudes (45)into (35) and (37). The resulting expression is, however,quite involved. Rather than spelling it out explicitly, wehere discuss the general properties of this quantity withthe help of Fig. 3a.

This plot reveals three key features of Bµ,λ as a func-tion of the control parameters λ1 and λ2. First, it issymmetric with respect to the line λ2 = λ1 and antisym-metric with respect to λ2 = −λ1 + 2Λ. Second, we have

Bµ,λ ≥ 0 for λ2 ≥ −λ1 + 2Λ and (47)

Bµ,λ ≤ 0 for λ2 ≤ −λ1 + 2Λ with Λ ≡ −χ cot[χ]/2.

Third, the function Bµ,λ shows two isolated peaks alongits symmetry axis and decays rapidly for λ1, λ2 → ∞;this behavior is ultimately a consequence of the resonantstructure of the transmission coefficients correspondingto the scattering amplitudes (45). Thus, to generate asignificant current with moderate driving amplitudes, thecontrol path γ must lie in the half plane λ2 > −λ1+2Λ andencircle the positive peak of the Berry curvature. Thesetwo requirements can be met with the driving protocols

λ1 = Λ+v/√

2−v cos[ωt], λ2 = Λ+v/√

2−v sin[ωt], (48)

9

which correspond to harmonically modulated gate volt-ages. The tuning parameter v, i.e., the radius of the circleparameterized by (48), thereby determines the amplitudeof the driving, see Fig. 3a.

We are now ready to explore the performance of thetunable-barrier system as a coherent quantum pump andisothermal engine. To this end, we first numerically eval-uate the response coefficients (18) in the low-temperaturelimit using the scattering amplitudes (45) and the proto-cols (48) for increasing values of v. We then calculate theparticle flux Jρ, the mechanical power Pac and, for theisothermal engine, the electrical power Pel with the bias∆µ being fixed such that the efficiency (39) is maximized.

The results of this analysis are summarized in Fig. 3b,which can be understood in our geometric picture as fol-lows. For ∆µ = 0, increasing v first leads to a rapidlygrowing particle current Jρ as the contour γ expands intothe positive peak of the Berry curvature. Once the peakis fully encircled, the generated current settles to a finitevalue close to 1/T . Further increasing v then only leadsto successively larger driving amplitudes requiring moreand more power input. Thus, the coefficient QP, whichdescribes the performance of the system as an adiabaticquantum pump, goes to zero. A similar argument appliesif the device is operated as an isothermal engine. Af-ter the pumped current is saturated, increasing v makesit necessary to apply a successively larger bias ∆µ tokeep the efficiency at its maximum, that is, close to 1/4.The rescaled output −Pel/∆µ

2 thus becomes smaller andsmaller and the performance coefficient QE drops to zero.

Figure 3b shows that the tunable-barrier system canindeed produce considerable currents as a quantum pumpand operate at viable efficiencies as an isothermal engine.The corresponding performance coefficients, QP and QE,however, do not even come close to their upper boundof 1. In fact, these figures reach their respective maximaat Q∗P ≃ 0.12 and Q∗E ≃ 0.04. On the microscopic level,this result is a consequence of idle scattering events, inwhich carriers pick up energy from the external drivingwithout contributing to the pumped current. This typeof process is fully suppressed in the magnetic-flux device,where ideal beam splitters force each particle that hasbeen accelerated inside the loop to escape in the directionof the mean current. In contrast to the tunable-barriersetup, which is well within reach of current experimentaltechniques65, this idealized system can, however, hardlybe implemented in practice. Finding strategies to test ournew bounds experimentally is an interesting challenge forfuture investigations.

V. CONCLUDING DISCUSSION

A. Main Results

The new bounds (28) and (31) constitute the first ma-jor result of this paper. Being derived in a universalframework, they hold for any coherent multi-terminal

system in adiabatic response with only (31) requiringan additional symmetry condition. In particular, bothconstraints (28) and (31) apply in the stationary limit,which is included in our general theory as the special caseof frozen, i.e., time-independent, control parameters.

From a practical perspective, the relations (28) and(31) open a new avenue to experimentally estimate theotherwise hardly accessible total dissipation caused bya coherent transport process, a strategy known as ther-modynamic interference68. Specifically, to determine thetotal rate of entropy production, it would in principlebe necessary to measure all thermodynamic fluxes in thesystem. Heat currents in mesoscopic system are, howevernotoriously difficult to access in experiments and the pho-ton flux is virtually out of reach for direct observation.Our bound (28) now makes it possible to obtain at leasta lower bound on the entropy production through prac-tically feasible measurements of the particle currents inthe individual terminals.

Another key application of our bounds is the deriva-tion of universal performance constraints for mesosopicmachines, as we have shown in Sec. IV A for adiabaticquantum pumps and isothermal engines. These thermo-dynamic trade-off relations, are our second main result.By applying them to two specific system in Secs. IV Band IV C, we have demonstrated that, first, our boundsare technically tight and, second, that they make it pos-sible to quantitatively assess the performance of realisticmesoscopic devices. These combined insights constitutethe third achievement of our work. In the following, wewill briefly discuss how our results relate to ongoing de-velopments and previous studies before concluding thispaper with a brief outlook.

B. Recent Developments and Earlier Work

1. Thermodynamic Uncertainty Relations

Thermodynamic uncertainty relations are inequalitiesof the general form

σε ≥ ψ with ε ≡ J2/D. (49)

Here, σ corresponds to the total dissipation caused bya stationary non-equilibrium process, ε denotes the rel-ative uncertainty of a generated current with mean Jand fluctuations D and ψ is a numerical constant48,49.This bound was first discovered as a universal featureof Markovian biomolecular processes, for which ψ = 2.It has since then triggered considerable research effortsseeking to extend its applicability and to explore itsbroader implications, see for example27,28,69–76. Forsteady-state coherent transport in linear response, threedifferent relations of the type (49) are now established;the corresponding values of ψ are

ψP =m/(m− 1), ψB ≃ 0.89612, ψM = 2 sin2[π/m], (50)

10

where m is the number of terminals of the conductor.The strongest of these bounds, ψP, follows from our newresult (31) by freezing the external control parameters.The second one, ψB, was derived only for particle cur-rents, under isothermal conditions and in a classical set-ting but without a linear-response assumption. It can besaturated far from equilibrium but covers the quantumregime only in the limit of small biases, where the cur-rent fluctuations become quasi-classical27. Finally, ψM

provides the weakest but also the most general availableconstraint, which applies also to non-elementary fluxes,i.e., arbitrary linear combinations of the heat and parti-cle currents flowing in the individual terminals; for n > 3this bound cannot be saturated by a single currents28.

2. Power-Efficiency Trade-off Relations

The first and the second law put universal limits onthe efficiency of thermodynamic machines like heat en-gines, refrigerators or isothermal energy converters. Theydo, however, not constrain the output power of such sys-tems, which in contrast to efficiency, is generally nota dimensionless figure. This observation has recentlytriggered an active debate on whether or not it is, atleast in principle, possible to realize devices that oper-ate reversibly while still delivering finite output, see forexample19,20,25,33,36,62,72,77,78. These investigations haveuncovered a variety of thermodynamic trade-off relationsthat rule out the option of finite power at ideal efficiencyfor broad classes of systems. In Sec. IV A, we have de-rived such a constraint for adiabatic isothermal engines.Our bound (28) is, however not limited to this specificsetup. It can, in fact, be applied to any type of deviceoperating in linear-adiabatic response. In particular, forsteady-state thermoelectric generators, it allows us to re-cover the power-efficiency trade-off relation

η(ηC − η) ≥ (3h/π2T 2)P, where ηC ≡ TF q (51)

is the Carnot efficiency. This bound still holds when amagnetic field is applied to the system and dephasing isincluded through an arbitrary number of probe terminals.In this general form, it was first conjectured in25 basedon numerical evidence and later proven in79.

3. Optimal Quantum Pumps

Adiabatic quantum pumps can be elegantly describedin terms of geometric concepts, as we have seen inSec. IV C for a tunable-barrier device. In general, thegeometric approach has proven remarkably useful to un-derstand the physical principles that govern the prop-erties of such systems, most importantly the origins ofquantized charge currents, see for example53–59. In17,the adiabatic approximation was used to derive a uni-versal criterion for optimal coherent pumps operating at

zero temperature. This bound,

µJρα − JEα ≥ h(Jρα)

2/2, (52)

which involves the particle and the energy currents, Jραand JEα , flowing in the terminal α = 1,2, is quite similarto our trade-off relation (38). In fact, (38) can be derivedfrom (52) upon using the conservation laws Jρ1 + J

ρ2 = 0

and Pac +JE1 +JE2 = 0. However, (38) does not imply the

stronger relation (52). This observation also explains,why our bound (38) is saturated by the magnetic-fluxdevice discussed in Sec. IV B, which was introduced in17

as an example of an optimal quantum pump.

C. Outlook

The foregoing discussion underlines the versatile appli-cability of the theoretical results obtained in this paper.In fact, it shows that our general approach makes it pos-sible to connect a whole variety of different topics rangingfrom fundamental questions of quantum and stochasticthermodynamics to practical problems of mesoscopic de-vice engineering. Our work thus paves the way for fur-ther studies seeking to develop a broader unifying frame-work for the thermodynamic description of periodicallydriven mesoscopic conductors. This ambitious perspec-tive leads to two immediate challenges for future inves-tigations. First, it is yet an open problem to developviable strategies that make it possible to test our boundsin experiments. Second, it remains to be understood howour results can be extended beyond the regimes of linear-adiabatic response and coherent transport.

ACKNOWLEDGMENTS

E.P. acknowledges helpful comments from V.Kashcheyevs. K.P. thanks K. Saito for insightfuldiscussions. E.P. acknowledges support from the Vilho,Yrjo and Kalle Foundation of the Finnish Academyof Science and Letters through the grant for doctoralstudies. K.B. acknowledges support from Academyof Finland (Contract No. 296073). This work wassupported by Academy of Finland (Projects No. 308515and No. 312299). All authors from Aalto Universityare associated with the local Centre for QuantumEngineering.

Appendix A: Thermodynamic Consistency

Here, we show that the Floquet scattering formalismdiscussed in Sec. II A and Sec. II B is consistent withthe laws of thermodynamics (8). For the first law, itis sufficient to insert Eq. (9) into Eq. (8) and recall thatEq. (4) implies the conservation laws (6). The second lawcan be verified using a general argument put forward in80.

11

To this end, we first rewrite the total entropy productiondefined in Eq. (8) as

σ =1

h∫

∞

0dE∑αβ∑n

(uβEn − uαE)∣SβαEn,E ∣

2f[uαE] (A1)

with uαE ≡ (E − µα)/Tα, f[u] ≡ 1/(1 + exp[u]).

Next, we note that, since f[u] is monotonically decreas-ing, an arbitrary antiderivative F [u] of this functionmust be concave. Hence, we have

(uβEn − uαE)f[uαE] ≥ F [uβEn] − F [uαE]. (A2)

Upon inserting this relation into (A1), we arrive at

σ ≥1

h∫

∞

0dE∑αβ∑n

(F [uβEn] − F [uαE])∣SβαEn,E ∣2

(A3)

=1

h∫

∞

0dE (∑β

F [uβE] −∑αF [uαE]) = 0,

where the second line follows by shifting the integrationvariable in the first term and using that the Floquet scat-tering amplitudes vanish if one of their energy argumentsbecomes negative.

Appendix B: Weak-Driving Theory

It is instructive to compare the linear-adiabatic re-sponse theory of Sec. II C with the more common weak-driving scheme, which can be formally developed for co-herent conductors as follows. We first divide the single-particle Hamiltonian that describes the dynamics insidethe sample into two contributions,

Ht ≡H0+ εVt. (B1)

The free Hamiltonian H0 is thereby time-independent,the dynamical scattering potential Vt = Vt+T accounts forthe external driving fields and the parameter ε controlsthe strength of this perturbation. The Floquet scatteringamplitudes can thus be formally expanded as

SαβEn,E ≃ δn0SαβE + εAαβEn,E

up to second-order corrections in ε. The first contribu-tion in this series describes elastic transmission and re-flection events and the ε-dependent corrections arise frominelastic processes induced by the periodic driving.

The decomposition (B1) determines the dynamicalperturbation only up to time-independent shift. There-fore, Vt can alway be chosen such that the inelastic termin (B) obeys

AαβEn,E ∣n=0= 0. (B2)

Consequently, we have

∣SαβEn,E ∣2≃ δn0∣S

αβE ∣ (B3)

up to second-order corrections in the driving strength.Inserting this result into Eq. (2) and Eq. (7) shows thatthe thermochemical variables, Jxα and F xα , and the me-chanical ones, Jω and Fw, become independent of eachother in the weak-driving limit, that is, in linear orderwith respect to ε. By contrast, these quantities are gen-erally not decoupled in the linear-adiabatic scheme, sincethe slow-driving scattering amplitudes (16) remain finitefor n ≠ 0, even in lowest order with respect to the effectiveexpansion parameter hω.

Appendix C: Symmetric New Bound

We derive the bound (31). To this end, we proceed infour steps. First, we observe that (19) implies

∑α⟪∣S

αβE,λ∣

2⟫ =∑α

⟪∣SβαE,λ∣

2⟫ = 1. (C1)

Hence, the period averaged transmission coefficients forma bistochastic matrix. According to the Birkhoff-vonNeumann theorem, they can thus be expressed as

⟪∣SαβE,λ∣

2⟫ =∑ν

ρνEPαβν , (C2)

where the positive coefficients ρνE add up to 1 and thePαβν are the elements of a permutation matrix81. Con-sequently, we can decompose the thermal entropy pro-duction, the thermochemical currents and the diagonalthermochemical response coefficients as

σth =∑ν∑xy∑αβ(δαβ − P

αβν )Mxy

ν F xαFyβ ≡∑ν

σνth,

Jxα =∑ν∑y∑β(δαβ − P

αβν )Mxy

ν F yβ ≡∑νJx,να ,

Lxyαα =∑ν(1 − Pααν )Mxy

ν ≡∑νLxy,ναα with

Mxyν ≡ ∫

∞

0dE gE ⋅ ρ

νEξ

xEξ

yE , (C3)

respectively. Note that, in the second line, we have usedthat the coupling coefficients Lxωα and Lωxα vanish fortime-reversal symmetric driving.

Second, we define the detailed generating function

Gν≡ σνth +∑α∑x

Jx,να Gxα +∑α∑xyLxy,ναα GxαG

yα/2

= ∫

∞

0dE gE∑αβ

ρνE(Y αβE )2Pαβν /2 ≥ 0, (C4)

where we have used the definition (25) for the variables

Y αβE . We now observe that the detailed currents Jx,ναvanish if Pααν = 1 for given α. Therefore, (C4) can berewritten as

Gν= σνth +∑α

(1 − Pααν )∑xJx,να Gxα (C5)

+∑α(1 − Pααν )∑xy

Mxyν GxαG

yα.

Taking the minimum of this expression with respect tothe auxiliary variables Gxα and recalling that Gν ≥ 0 yields

σνth ≥∑α(1 − Pααν )Mxy

ν Jx,να Jy,να /2 with (C6)

∑yMxyMyz

≡ δxz.

12

Third, we note that the detailed currents satisfy theconservation laws

∑αJx,να =∑α

(1 − Pααν )Jx,να = 0. (C7)

Minimizing (C6) with respect to all detailed currents ex-cept for Jx,να while taking into account these constraintsleads to the bound

σνth ≥m −mν

m − 1 −mν(1 − Pααν )Mxx

ν (Jx,να )2/2 (C8)

≥m

m − 1(1 − Pααν )Mxx

ν (Jx,να )2/2,

where mν denotes the number of fixed points of the per-mutation corresponding to Pαβν .

Fourth, summing (C8) over ν and applying Jensen’sinequality leaves us with

σth ≥m

m − 1Nα∑n

(1 − Pααν )Mxxν (Jx,ν)2

/2Nα (C9)

≥m

m − 1Nα(∑ν

(1 − Pααν )Mxxν /Nα)(J

xα/Nα)

2

/2,

where Nα ≡∑ν

(1 − Pααν ).

Finally, by convexity of the matrix inverse82, we have

∑ν(1 − Pααν )Mxx

ν /Nα ≥ NαLxxαα. (C10)

Inserting this inequality into (C9) completes our proof.

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