Many-bodyeﬀectsonthethermodynamicsofclosedquantumsystems · 2019. 5. 2. · Thermodynamics of quantum systems out-of-equilibrium is very important for the progress of quantum technologies,

Many-body effects on the thermodynamics of closed quantum systems

A. H. Skelt,1 K. Zawadzki,2 and I. D’Amico1, 3

1Department of Physics, University of York, UK2Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA

3International Institute of Physics, Federal University of Rio Grande do Norte, Natal, Brazil(Dated: May 2, 2019)

Thermodynamics of quantum systems out-of-equilibrium is very important for the progress ofquantum technologies, however, the effects of many body interactions and their interplay withtemperature, different drives and dynamical regimes is still largely unknown. Here we present asystematic study of these interplays: we consider a variety of interaction (from non-interacting tostrongly correlated) and dynamical (from sudden quench to quasi-adiabatic) regimes, and drawsome general conclusions in relation to work extraction and entropy production. As treatment ofmany-body interacting systems is highly challenging, we introduce a simple approximation whichincludes, for the average quantum work, many-body interactions only via the initial state, while thedynamics is fully non-interacting. We demonstrate that this simple approximation is surprisinglygood for estimating both the average quantum work and the related entropy variation, even whenmany-body correlations are significant.

I. INTRODUCTION

Progress on applications of quantum technologies islinked to acquiring deeper understanding of the out-of-equilibrium thermodynamic properties of small quantumsystems. Small quantum systems operating at finite tem-perature constitute the hardware for most of these tech-nologies, so that out-of-equilibrium quantum thermody-namics guides, and ultimately may limit, these technolo-gies [1–5]. Quantum thermodynamics has then become arapidly expanding field, also supported by the advancesin experimental techniques. Nonetheless, a remainingchallenge is properly accounting for the effects of many-body interactions on quantum thermodynamics proper-ties. Many-body interactions in quantum systems giverise to complex phenomena, such as collective behavioursand quantum phase transitions. From a practical per-spective, these effects are challenging to calculate andoften require approximations. In this respect, there havebeen recent works studying out-of-equilibrium thermody-namics of many-body systems such as quantum harmonicoscillator chains and spin chains [6–15], and a proposalfor a density-functional-theory-based set of approxima-tions which is in principle applicable to systems of highcomplexity [16, 17].

Here we present a systematic study of the out-of equi-librium thermodynamics of many-body quantum systemssubject to a set of qualitatively different driving poten-tials. For each type of driving potential, we consider dy-namical regimes from sudden quench, to finite times, toquasi-adiabatic; for each dynamical regime we considerdifferent interaction strengths, from non-interacting tostrongly correlated systems. For each driving potential,dynamical regime, and interaction strength, we considerdifferent temperatures: low, intermediate and high tem-perature. For all cases considered we calculate and dis-cuss the average quantum work extracted and entropyproduced in the dynamical trajectory. Our systematicstudy allows to uncover some important dependencies of

work and entropy on the systems’ correlation and dy-namical regimes, which cut across the different applieddrives.

Afterwards we consider two quite drastic approxima-tions and compare their estimates with the exact results.The first is the completely non-interacting approxima-tion, where many-body interactions are set to zero in allphases of the thermodynamic processes considered. Thesecond approximation assumes knowledge of the initialinteracting many-body state, but completely neglects in-teractions afterwards, during the driven dynamics. Ourresults show that including interactions just within theinitial state provides surprisingly good accuracy. We pro-vide an analytical analysis that explains this accuracy inthe sudden quench and adiabatic regimes for a generalHamiltonian and driving potential.

II. THEORY

A. Hubbard model

The one-dimensional Hubbard model can depict sys-tems from weakly to strongly correlated and model nu-merous phases of matter and related phase transitions,such as metallic, antiferromagnetic, Mott-insulator, su-perconductivity, and FFLO transition [18–21]. It is be-ing widely used to study many physical systems, fromcoupled quantum dots, to molecules, to chains of atoms[22–27]. These are systems of importance, as hardware,to quantum technologies. For small chains, the Hubbardmodel is numerically exactly solvable, yet still displaysnon-trivial behaviours, including, for repulsive interac-tions, the precursor to the metal-Mott insulator phasetransition. Hence it is often the system of choice for ex-ploring approximations to interacting quantum systems[16, 17, 28, 29].

For a fermionic system of N sites, the Hamiltonian of

arX

iv:1

905.

0031

8v1

[qu

ant-

ph]

1 M

ay 2

019

2

the Hubbard model can be written as (h = 1)

H(t) = −JN∑i,σ

(c†i,σ ci+1,σ + c†i+1,σ ci,σ

)

+ U

N∑i

ni,↑ni,↓ +

N∑i

vi(t)ni, (1)

where J is the hopping parameter, c†i,σ (ci,σ) is the cre-ation (annihilation) operator for a fermion with spin σ(σ =↑ or ↓) in site i, U is the strength of the on-siteCoulomb interaction, ni,σ = c†i,σ ci,σ is the spin σ, i-sitenumber operator, ni = ni,↑ + ni,↓, and vi is the on-sitepotential.

We shall use the time-dependent, inhomogeneous, one-dimensional Hubbard model (i) to calculate the exactaverage quantum work and the entropy production forthe set of system sizes and regimes described in sectionIII and (ii) within the approximations for these quantitiesdescribed in section IV.

B. Average quantum work and entropy variation

The average quantum work, much like its classicalcounter part, is described as the usable energy in a quan-tum system [2]. In a closed system at temperature T , itcan be calculated as [2]

〈W 〉 = Tr[ρf Hf

]− Tr

[ρ0H0

], (2)

with ρ0(f) and H0(f) the initial (final) system state andHamiltonian, respectively.

For a given dynamic process, the variation in thermo-dynamic entropy is defined using the average work andthe change in the free energy of the system [1, 30],

∆S = β (〈W 〉 −∆F ) (3)

where β = 1/kBT , and the free energy variation is

∆F = − 1

βln

(ZfZ0

), (4)

with Z0(f) the partition function at the beginning (end)

of the dynamics, Z0(f) = Tr[exp

(−βH0(f)

)]. This ther-

modynamic entropy can be considered a measure of thedegree of irreversibility of the system dynamics: in factit captures an uncompensated heat which would need tobe dispersed to the environment for the system to returnto thermodynamic equilibrium at the end of the drivenprocess [16, 30].

III. SYSTEMS, DRIVING POTENTIALS,TEMPERATURE RANGE AND DYNAMICAL

REGIMES

We shall calculate the work extracted and entropy pro-duced for different systems, temperatures, driving poten-tials, and dynamical regimes.

We will consider Hubbard chains of 2, 4, and 6 sites,at half-filling and under open boundary conditions, andexplore low (T = 0.2J/kB), medium (T = 2.5J/kB), andhigh (T = 20J/kB) temperatures.

For each system size and temperature, we will exploreregimes from non-interacting (U = 0J) all the way tostrongly correlated (U = 10), and dynamics from suddenquench (τ = 0.5/J , τ the overall driving time) all theway to quasi-adiabatic (τ = 10/J).

For each parameter combination, we will consider threetypes of driving potentials [31], where each potential hasa linear time dependency via vi(t) = µ0

i + µτi t/τ , withµ0i and µτi the time-independent coefficients for site i at

time 0 and τ respectively. With this choice, the char-acter of the dynamics will depend on τ , while the finalHamiltonian Hf will be independent of it. These drivingpotentials are:

• “Comb”: for each site i, µ0i = µ0(−1)i at t = 0

and µτi = µτ (−1)i at t = τ , where µ0 = 0.5J andµτ = 4.5J .

• “Middle Island (MI)”: the inhomogeneity µi isdriven only for the middle two sites of the chain;µ0i = 0 for i 6= L/2, (L/2) + 1 where i goes from 1

to L, and L is the chain length. For the middle twosites, i = L/2, (L/2) + 1, µ0

i = 0.5J and µτi = 10J .

• “Applied Electric Field (AEF)”: this potential mim-ics the application of a potential difference betweenthe extremes of the chain. The sites form a linearslope from i = 1 to i = L and are described us-ing µ0

i = 2µ0/L × i − µ0 where µ0 = 0.5J , andµτi = 2µτ/L× i− µτ with µτ = 10J .

The t = 0 and t = τ form of the driving potentials for asix-site Hubbard chain are illustrated in figure 1.

IV. APPROXIMATIONS

A. Average quantum work

The type of approximations we consider for the averagequantum work are of the form:

〈W is+evo〉 = Tr[ρis+evof Hevo

f

]− Tr

[ρis0 H

evo0

]. (5)

Here is (initial system) refers to the approximationused to derive the system state at t = 0, ρis0 =

exp(−βHis

0

)/Tr

[exp

(−βHis

0

)], and evo is the ap-

proximation used for the evolution operator Uevo =

3

0.00.5

-0.5

5.0

-5.0

Site number1 2 3 4 5 6

1.0

2.0

3.0

4.0

-1.0

-2.0

-3.0

-4.0

Pot

entia

l in

units

of J

(a) Comb dynamics.

5.0

0.5

10.0

0.0

6.0

7.0

8.0

9.0

4.0

3.0

2.0

1.0

10.5


Pot

entia

l in

units

of J

(b) MI dynamics.

0.00.5

-0.5

10.5

-10.5

2.0

4.0

6.0

8.0

-2.0

-4.0

-6.0

-8.0


Pot

entia

l in

units

of J

(c) AEF dynamics.

FIG. 1: On-site driving potentials versus site number for a 6 site chain; red dashed lines show the potentials att = 0, and blue solid lines show the potentials at t = τ .

T e−i∫ τ0Hevot (t)dt where T is the time-ordered operator.

The final state is then ρis+evof = Uevoρ

is0 U†evo. We note

that Hevo0 = Hevo

t (t = 0). In the approximation where isand evo are the same, only one acronym shall be written.As (5) indicates, the Hamiltonians Hf and H0 explicitlyentering (2) and the evolution Hamiltonian Hevo

t are tobe taken in the same approximation: if this is not thecase, we found that the mismatch in eigenstates leads tospurious oscillations in the work production (not shown).

We will consider two approximations, as described inTable I. The first, 〈WNI〉, corresponds to a completelynon-interacting system, the one obtained by setting U =0 in the Hubbard Hamiltonian. The second approxi-mation, 〈W exact+NI〉, uses the exact many-body initialstate, but the approximated (non-interacting) Hamilto-nian for the evolution of the system, according to thenotation previously introduced.

B. Entropy variation

When approximating the entropy, we use 〈W is+evo〉,while the free energy is estimated in the same way asρis. In the NI approximation, this implies that the freeenergy is constant. For ∆Sexact+NI , the exact free en-ergy is then used as this uses the same assumption madefor calculating the initial thermal state, that the systemHamiltonian can be exactly – or very accurately – diag-onalised.

V. WORK EXTRACTION

The system is considered at equilibrium at time t = 0−,when the coupling with the thermal bath is switched off.Then the closed system is driven by a time-dependentexternal potential from the initial Hamiltonian H0 to thefinal Hamiltonian Hf in a time τ , and the extracted work

〈Wext〉 from this dynamics is calculated according to (2),with 〈Wext〉 = −〈W 〉.

We stress that with each of the driven dynamics de-scribed in section III, the final Hamiltonian Hf (U) is thesame for all τ ’s, so that the latter controls the rate ofdriving. Therefore, the larger τ is, the slower the systemhas evolved and hence more adiabatic the evolution.

For each of the many-body systems, their approxima-tions, temperatures, and driving potentials described insection IV, we will consider the parameter space 0.5 ≤τ × J ≤ 10 and 0 ≤ U/J ≤ 10. Due to the sheer num-ber of results from all the combination of parameters, wewill only explicitly show results for 6 site chains (all 6site chains’ exact results, and part of them for the ap-proximations), and comment on the rest.

A. Exact results

Figure 2 shows the exact average quantum work ex-tracted from a 6 site chain driven via “applied electricfield (AEF)” (right column), “comb” (middle column),and “middle island (MI)” (left column) potentials at tem-peratures of T = 0.2J/kB (first row), T = 2.5J/kB (sec-ond row), and T = 20J/kB (third row). Each panelshows a wide range of regimes: from non-interacting tostrongly correlated systems as U increases along the y-axis; and from sudden quench towards adiabaticity asτ increases along the x-axis. A lighter shade of colourcorresponds to higher extracted work.

Figure 2 presents a variety of behaviours, with the ex-tracted work varying over a wide range of values, andeven from positive to negative. This confirms that thechosen dynamics are a good test-bed for understandingwork extraction in systems representable via Hubbardchains, and hence a good test bed for related approxima-tions.

At all temperatures, the largest work can be extractedvia the AEF dynamics, while work needs to be done on

4

Acronym Approximation Hamiltonian Initial State

NI 〈WNI〉 HNI = −JN∑iσ


)+

N∑i

vini ρNI0 = exp(−βHNI

0

)/ZNI

exact + NI 〈W exact+NI〉 HNI = −JN∑iσ


)+

N∑i

vini ρexact0 = exp

(−βHexact

0

)/Zexact

TABLE I: Types of approximations with their Hamiltonians and initial states.

the system in order to perform the MI dynamics. Forall driving potentials, increasing temperatures decreasesboth the range of extracted/provided work and the max-imum extractable work (the minimum work to be per-formed on the system in the case of MI). The maximumapplied potential difference at t = τ is comparable to thehighest temperature, so, as the temperature rises, thesystems become less sensitive to the applied field. For alltemperatures and dynamics the highest work that canbe extracted (the lowest work performed on the systemin the case of MI) is reached for large τ ’s, as the sys-tem gets closer to adiabaticity: here the dynamical statebetter adjusts to the driving force and, compatible withtemperature and many-body interactions, the energeticsfavour low-potential chain sites.

In the case of AEF, at all temperatures, increasing Uhampers the transient current dynamics, so the highestwork is achieved for zero to weak correlations. The max-imum potential step between nearby sites is about 4J so,as U increases, the AEF dynamics tends to freeze andlesser and lesser work can be extracted from the system.At the highest temperature, the thermal energy is al-most equal to the potential difference between the chainextremes at the end of the dynamics.

At low temperatures, ‘comb’ dynamics presents a workextraction pattern similar to AEF; however, as the tem-perature increases, maximum work extraction can beachieved for higher many-body interactions, and at highT it is achieved only for relatively strong many-body in-teractions (4 <∼ U/J <∼ 8). This can be understood byrealising that in this case the thermal energy kBT = 20Jis twice the potential barrier between even and odd chainsites, and hence a certain degree of repulsion is necessaryto depopulate the high-energy sites completely and max-imise work extraction.

Extracted work under MI dynamics is negative, mean-ing that work must be performed on the systems toachieve the final states. Indeed in this case the driveraises the potential of the central sites and, under thedynamics, the (closed) system cannot decrease its overallenergy. As even for τ = 10 the system is not fully adi-abatic, a finite amount of repulsion helps to completelydeplete the central island, even at low temperature, andmore so as temperature increases. The 10.5J maximumbarrier between high and low potential sites is about halfof the maximum thermal energy considered.

1. Adiabatic regime

From figure 2 we see that, as τ increases and we ap-proach the adiabatic regime, the average work becomesstrongly dependent on U and weakly dependent on τ .This behaviour can be easily explained. For a closedsystem, 〈Wext〉 can be also calculated from the work dis-tribution function, and hence written as [2]

〈Wext〉 =∑n,m

pτn,mEfm −

∑n

p0nE0n, (6)

where Efm (E0n) is the m-th (n-th) eigenvalue of the final

(initial) Hamiltonian, and pτn,m = p0npτm|n is the joint

probability distribution of arriving to the eigenstate |Ψfm〉

of Hf , given the probability p0n of initially being in theeigenstate |Ψ0

n〉 of H0.In this formalism, the effect of the dynamics driven

by the external potential (and hence the dependenceon τ) is contained in the conditional probabilitiespτm|n = |〈Ψf

m|Uevo(τ)|Ψ0n〉|2. In the adiabatic regime

Uevo(τ)|Ψ0n〉 = |Ψf

n〉, so that we obtain

〈Wext〉adiabatic =∑n,m

Efmp0n|〈Ψf

m|Ψf,NIn 〉|2 −

∑n

p0nE0n

=∑n

p0n(Efn − E0n), (7)

which indeed depends on the interaction coupling U butnot on τ . By looking at figure 2 we can then note that thevalue of τ at which the different systems enter a nearlyadiabatic behaviour depends both on the driving poten-tial and on the temperature, with the ‘comb’ driving po-tential getting closer to adiabaticity for smaller values ofτ , and increasing temperature seemingly decreasing adi-abaticity. The latter can be understood by realising thatwhen temperature increases to the point of dominatingby far inter-particle interactions, the system behaves asnon-interacting, and hence the dependence of any quan-tity over U will be lost (see also discussion in sectionVB1).

B. Approximated results

We will firstly approximate the many-body systemwith a non-interacting system and use this to estimatequantum thermodynamic quantities. Afterwards we will

5

2 4 6 8 10τ × J

0

2

4

6

8

10U/J

Extracted work 〈W exactext 〉

−19.51−18.07−16.63−15.18−13.74−12.30−10.86−9.42 −7.98 −6.54 −5.10

(a) T = 0.2J/kB with MI dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


1.22 3.39 5.55 7.71 9.88 12.04 14.20 16.37 18.53 20.69 22.86

(b) T = 0.2J/kB with comb dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.22 2.76 5.30 7.84 10.38 12.92 15.46 18.00 20.54 23.08 25.62

(c) T = 0.2J/kB with AEF dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


−19.53−19.01−18.48−17.96−17.44−16.91−16.39−15.86−15.34−14.81−14.29

(d) T = 2.5J/kB with MI dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


0.96 1.53 2.10 2.68 3.25 3.82 4.39 4.97 5.54 6.11 6.68

(e) T = 2.5J/kB with comb dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.48 1.36 2.23 3.11 3.98 4.86 5.74 6.61 7.49 8.36 9.24

(f) T = 2.5J/kB with AEF dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


−19.76−19.70−19.64−19.58−19.52−19.46−19.41−19.35−19.29−19.23−19.17

(g) T = 20J/kB with MI dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.49 0.55 0.60 0.66 0.72 0.78 0.83 0.89 0.95 1.00 1.06

(h) T = 20J/kB with comb dynamics.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.39 0.47 0.55 0.64 0.72 0.80 0.88 0.97 1.05 1.13 1.21

(i) T = 20J/kB with AEF dynamics.

FIG. 2: (a)-(i) Exact extracted average quantum work versus total dynamics time τ (x-axis)and interaction strengthU (y-axis). Data are presented for 6 site Hubbard chains driven by MI, comb, and AEF potentials, and at low,

medium, and high temperatures, as indicated. The lighter the colour shade, the more work is extracted, compatiblewith the respective work range indicated over each panel.

extend this approach to include some memory of theelectron-electron interaction through the system initialstate.

1. Non-interacting approximation

For non-interacting (NI) systems, U = 0, the averagework extracted 〈WNI

ext 〉 has no dependence on U . Thisis shown in the upper panels of figure 3, where we plot〈WNI

ext 〉 for ‘comb’ dynamics, with 0.5 ≤ τ × J ≤ 10(x-axis) and 0 ≤ U/J ≤ 10 (y-axis), and low (left), in-termediate (middle), and high (right) temperatures. The

range of variation in 〈WNIext 〉 decreases with temperature

(see colour bar above each panel), as the maximum driv-ing potential becomes comparable to the thermal energy.

The lower panels of figure 3 show the correspond-ing relative error of the NI approximation, where thedarker the blue, the more accurate the approximationis in that regime. This approximation accurately cap-tures the work extraction only in the parameters regionswhere interactions are weak compared to the other energyscales. These regions include higher values of U as thetemperature increases, see related discussion in sectionVA. However it is worth noting that there is very littlework extracted at high temperatures, where the thermal

6

energy is comparable to the driving potential which isthen less effective.

As the lowest values of the average extracted work aredriven by the freezing of the system dynamics due tostrong many-body repulsion (dark areas in figure 2), theNI approximation strongly overestimates the minimumwork that can be extracted by a system. To see thiscompare the work range indicated over each panel forthe mid column of figure 2 to the corresponding upperpanels of figure 3. At the other end, the value of themaximum average work extracted is quantitatively wellcaptured by the NI approximation, but the correspondingparameter regions are qualitatively wrong, compare theshape of the light-shade areas of the mid-column panelsof figure 2 to the corresponding areas of the upper panelsof figure 3.

We find a similar accuracy pattern with increasing tem-perature for 2 [32] and 4 site chains.

The ‘comb’ driving potential corresponds to an accu-racy of the results consistently in-between those of theAEF and MI potentials.

For the MI driving potential, the NI approximationworks better than for the ‘comb’ potential at all temper-atures. As discussed in section VA, for these dynamicsmany-body interactions become comparatively dominantat higher values of U . As a consequence, for 6 site chains,the MI driving potential results in 10% accuracy (or bet-ter) for U <∼ 3J at low temperatures, for U <∼ 7J atintermediate temperatures, and for all regimes at hightemperatures: here thermal energy dominates so thatthe average extracted-work range is very narrow and soweakly sensitive to parameter changes.

For the AEF driving potential, the results in the NIapproximation are in general worse than with the ‘comb’driving potential, and comparatively worsen as the tem-perature increases. At difference with the MI and combdriving potentials, the maximum AEF potential differ-ence between nearby sites becomes at most of 4J , so thateven a Coulomb repulsion of just U∼1J will remain rel-evant. The NI approximation is then bound to fail, evenat high temperatures, where, for 6 site chains, we get anaccuracy of 10% for U <∼ 3J only.

2. Exact initial system with non-interacting evolutionoperator

To try and improve the estimate of the work extracted,we shall consider to still implement a non-interacting evo-lution, but this time starting from the exact many-bodyinitial state. The rationale is that a many-body evolutionis in general a more challenging part of the calculation(and hence here it is approximated), while an accurateestimate for the initial state would be more readily avail-able. This approximation is referred to as 〈W exact+NI〉in table I.

Indeed this simple approximation leads to strikinglyimproved accuracy. Results are presented in figure 4:

〈W exact+NIext 〉 in the upper three panels, and its relative

error with respect to the exact results in the lower panels.Parameters are the same as in figure 3.

By comparing 〈W exact+NIext 〉 to the corresponding ex-

act results in panels (b), (e), and (h) of figure 2 we seethat including interactions just through the initial stateis sufficient to recover the qualitative (and in great partquantitative) behaviour at low and intermediate temper-atures. The greatest improvement is seen in the low tem-perature, where 〈W exact+NI〉 recaptures the correct workto 10-20% for most regimes up to U ≈ 9J .

At high temperature the qualitative behaviour is notrecovered, but, as the work extracted varies only slightlyat this temperature, quantitatively the approximation re-mains overall good, as it reproduces rather well the over-all work variation range (compare the colour bar limitsof figures 2 and 4). In general the ‘exact + NI’ approxi-mation significantly improves for the minimum extractedaverage work over the NI value and at all temperatures.

A similar pattern is seen in all the other systems con-sidered, i.e. for 2 and 4 site chains, and for the MIand AEF evolutions (see appendix), demonstrating thescalability and versatility of this approximation. The〈W exact+NI〉 approximation handles weak to mediumcorrelated systems well in all regimes and temperatures,from adiabatic to sudden quench, and from low to hightemperatures.

3. Adiabatic behaviour and the ‘exact + NI’ approximation

The most striking improvement of the ‘exact + NI’approximation with respect to the NI one, is the recov-ery of the qualitative behaviour of the average quantumwork in the quasi-adiabatic regime for low to interme-diate temperatures. In this regime the exact 〈Wext〉 isbecoming independent from τ – as to be expected in theadiabatic regime – but strongly depends on U , see fig-ure 2. The recovery of this behaviour by the ‘exact +NI’ approximation can be explained as follows.

As discussed in section VA1, the dependency of〈Wext〉 over τ is contained in the conditional probabil-ities pτm|n = |〈Ψf

m|Uevo(τ)|Ψ0n〉|2. In the NI approxima-

tion these would read pτ,NIm|n = |〈Ψf,NIm |UNIevo(τ)|Ψ0,NI

n 〉|2which, would lead, as expected, to a result potentiallydependent on τ but completely independent from U .However in the ‘exact + NI’ approximation we havepτ,exact+NIm|n = |〈Ψf,NI

m |UNIevo(τ)|Ψ0n〉|2, which could lead

to dependency on U (through |Ψ0n〉) as well as on τ [33].

Let us name |ΨNIm (t)〉 the m-th eigenstate of HNI(t):

then |ΨNIm (τ)〉 = |Ψf,NI

m 〉, independent of τ , and |Ψ0n〉 =

7

2 4 6 8 10τ × J

0

2

4

6

8

10U/J

Extracted work 〈WNIext 〉

16.32 16.97 17.63 18.28 18.93 19.59 20.24 20.89 21.55 22.20 22.86

(a) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


4.56 4.77 4.98 5.20 5.41 5.62 5.83 6.04 6.26 6.47 6.68

(b) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.58 0.61 0.64 0.67 0.69 0.72 0.75 0.78 0.80 0.83 0.86

(c) T = 20J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J

|〈WNIext 〉 − 〈W exact

ext 〉|/〈W exactext 〉

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(d) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(e) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(f) T = 20J/kB .

FIG. 3: Panels (a) to (c): Work extracted in the NI approximation for 6 site chains driven by the ‘comb’ potential.Considered regimes go from non-interacting to strongly coupled along the y-axis, and from sudden quench to nearlyadiabatic along the x-axis. The lighter the colour shade, the more work is extracted, compatible with the respective

value ranges indicated above each panel. Temperature increases from left to right panel, as indicated.Panels (d) to (f): Relative error for 〈WNI

ext 〉 with respect to the exact results for the same parameters as the upperpanels. The darker the colour, the more accurate the approximation is in that regime.

∑j aj,n|ΨNI

j (0)〉, with aj,n = aj,n(U). Consider

∑n,m

pτn,mEfm

∣∣∣∣∣exact+NI

=∑n,m

Efmp0n|〈Ψf,NI

m |UNIevo(τ)|Ψ0n〉|2

=∑n,m

Efmp0n|〈Ψf,NI

m |∑j

aj,nUNIevo(τ)|ΨNIj (0)〉|2. (8)

In the adiabatic regime UNIevo(τ)|ΨNIj (0)〉 = |Ψf,NI

j 〉, sothat (8) becomes

∑n,m

pτn,mEfm

∣∣∣∣∣exact+NI

adiab.=

∑n,m

Efmp0n|〈Ψf,NI

m |∑j

aj,n|Ψf,NIj 〉|2

=∑n,m

Efmp0n|am,n(U)|2, (9)

which indeed depends on U but not on τ , as observed.At high temperatures, inclusion of many-body inter-

actions just via the initial state is a too-weak correc-tion to counter the high thermal energy, which becomes

even more dominant than in the exact case: at high tem-peratures for the ‘exact+NI’ approximation a behaviourcloser to the fully NI approximation is recovered (com-pare figure 4 panel (c) to figure 3 panel (c)). We observethis for all driving potentials (see appendix, panel (c) offigures 8 and 9).

4. Sudden quench and the ‘exact + NI’ approximation

With respect to the NI approximation, the ‘exact + NI’approximation recovers the qualitative and, in great part,quantitative exact behaviour for the average quantumwork also in the sudden quench limit. This can be seenby comparing results for the small-τ parameter region ofthe upper panels of figure 4 with the corresponding panelsin the central column of figure 2. We give an explanationfor this below.

In the quasi-sudden-quench regime τ << 1/J , ρf ≈ρ0 + δρ(τ), with the second term a small correction. Byusing this and the fact that ∆H = Hf−H0 is a constant,

8

2 4 6 8 10τ × J

0

2

4

6

8

10U/J

Extracted work 〈W exact+NIext 〉

2.23 4.29 6.36 8.42 10.48 12.54 14.61 16.67 18.73 20.79 22.86

(a) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


1.41 1.94 2.46 2.99 3.52 4.04 4.57 5.10 5.63 6.15 6.68

(b) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.53 0.56 0.60 0.63 0.66 0.70 0.73 0.76 0.79 0.83 0.86

(c) T = 20J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J

|〈W exact+NIext 〉 − 〈W exact


0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(d) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(e) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(f) T = 20J/kB .

FIG. 4: Panels (a) to (c): Work extracted in the ‘exact + NI’ approximation for 6 site chains driven by the ‘comb’potential. Considered regimes go from non-interacting to strongly coupled along the y-axis, and from sudden quenchto nearly adiabatic along the x-axis. The lighter the colour shade, the more work is extracted, compatible with therespective value ranges indicated above each panel. Temperature increases from left to right panel, as indicated.Panels (d) to (f): Relative error for 〈W exact+NI

ext 〉 with respect to the exact results for the same parameters as theupper panels. The darker the colour, the more accurate the approximation is in that regime.

we can approximate (2) as

〈W 〉 ≈ Tr[(ρ0 + δρ(τ))Hf − ρ0H0

]= Tr

[ρ0∆H + δρ(τ)Hf

](10)

τ=0= Tr [ρ0∆H] . (11)

In the ‘exact + NI’ approximation, (10) and (11) be-come

〈W exact+NI〉 ≈ Tr[ρexact0 ∆HNI + δρ(τ)exact+NIHNI

f

](12)

τ→0= Tr

[ρexact0 ∆HNI

], (13)

where we have explicitly stressed which terms on ther.h.s. are to be computed exactly and which ones ina specific approximation. In contrast to a fully NI ap-proximation, (13) is now U -dependent through ρexact0 ,which means that this dependence for τ = 0 is recovered.Similarly, the correction for small (but finite) τ valuesin (12) contains a U -dependence through δρ(τ)exact+NI ,and will then improve both qualitatively and quantita-tively over NI results, as comparison of figures 2, 3, and4 demonstrates.

VI. ENTROPY VARIATION

A. Exact results

Let us now examine the exact results for the entropy,∆S from (3). This quantity corresponds to the heatwhich the system would have to disperse in the envi-ronment to return to thermodynamic equilibrium at theend of the dynamics. Apart from the average quantumwork, the other key ingredient for ∆S is the free-energyvariation, shown by (4). Since our final Hamiltonians areindependent of τ , the free energy only depends on U andβ.

Figure 5, panels (a)-(c), shows the variation of freeenergy as U varies at each of the temperatures considered(green line for low, blue for medium, and red for hightemperature) and for each of the driving potentials. ∆Fis weakly dependent on U for high temperatures, while,at intermediate and, even more, at low temperatures, itsignificantly changes with the interaction strength. Thisconfirms that the system behaves more and more like anon-interacting system as the temperature increases.

The exact entropy variation for all driving potentials

9

and temperatures is shown in figure 5, second to fourthrows: left column for MI, middle for ‘comb’, and rightfor AEF driving potential. ∆S increases as the colourshade becomes lighter; note however that the same shadecorresponds to different values in different panels, as theoverall entropy range significantly changes according toboth temperature and type of driving potential.

The temperature affects the entropy production dras-tically, compare the extent of ∆S ranges between the sec-ond and last row of figure 5. This can be understood bycomparing energy scales. By the end of the dynamics, ourdriving potentials reach the maximum energy differenceof 10J for comb, 10.5J for MI, and 21J for AEF poten-tial. For the low temperature kBT = 0.2J and the rangeof parameters explored, both the interaction strength Uand the driving potentials can be much bigger than thethermal energy, and so they have a large impact on thesystem evolution. The system can change quite drasti-cally leading to the possibility of large work extractionand large entropy production. However at high temper-ature, kBT = 20J , the interaction strength and drivingpotential are, at most, comparable to the thermal energy:the system is not as receptive to being driven, it will re-main closer to its thermal state, and hence the energyrequired to be dispersed to return to equilibrium (i.e.the entropy that we are here considering) will be muchless.

In general at the lowest temperature the suddenquench with weak-medium strength coupling parameterregion corresponds to very high ∆S values, while a dra-matic reduction in entropy is seen moving towards theadiabatic regime. Given the correspondence between ∆Sand the heat to be dispersed at the end to recover equi-librium, it stands to reason that a sudden quench wouldrequire a larger dissipation of energy to return to an equi-librium state compared to an adiabatic evolution, whichremains closer to an equilibrium state at all times.

We note that systems subject to AEF potential showat low T a relatively larger entropy production in thestrongly-coupled regime and τ ∼ 10/J than systems sub-ject to ‘comb’ and MI potentials. As discussed in sec-tion VA1, the level of adiabaticity reached for the samevalue of τ differs with driving potential. Indeed the dy-namic induced by AEF at U >∼ 6J and τ ∼ 10/J is lessadiabatic than the ones by MI or ‘comb’, as can be ob-served by comparing panel (c) to panels (a) and (b) offigure 2. This leads to a larger amount of entropy pro-duction occurring with AEF even in this strongly-coupledlarge-τ region.

B. Non-interacting approximation

We extend the approximations for the extracted workto the entropy production. ∆SNI will be calculated from〈WNI〉 = −〈WNI

ext 〉 and by setting U = 0 in the calcula-tion of the free energy (U = 0 values in figure 5, upperpanels).

Figure 6 presents the ∆SNI results for 6 sites, ‘comb’driving potential, and increasing temperature (left toright). The upper panels show the non-interacting en-tropy variation, and the lower ones the relative differencewith the exact entropy variation (the darker the purple,the more accurate the approximation is in that region).

For each given temperature, the entropy ∆SNI is just〈WNI

ext 〉 with an added constant, so, much like the non-interacting work, for all driving potentials this approxi-mation is unable to qualitatively describe the exact en-tropy produced.

Comparing with the NI-work approximation accuracy,the overall quantitative accuracy of the entropy is in gen-eral reduced for all three temperatures, which is to be ex-pected since, on top of the work, we are severely approx-imating the free energy as well. As U increases, we areexploring highly correlated systems, including systemssubject to a precursor to the Mott-insulator transition,but these Coulomb correlations are completely neglectedby the NI approximation.

For the AEF driving potential, the NI approximationworks better then for ‘comb’ drive at all temperatures,giving a 10% accuracy (or better) for U <∼ 1.5J at low andintermediate temperatures and for almost all regimes athigh temperatures. For the MI driving potential, the ac-curacy of the results is comparable to the ‘comb’ drivingpotential.

C. Exact initial state with non-interactingevolution

Let us now see how considering the exact initial stateaffects the estimate of the entropy production. Initiallythe entropy ∆Sexact+NI is calculated from (3) using〈W exact+NI〉 = −〈W exact+NI

ext 〉 and the exact free energyvariation: in this approximation, we are already assum-ing that we can diagonalise the initial Hamiltonian toget the exact initial state, we then make the same as-sumption for Hf , as this operation would have the samecalculation costs/difficulties. This leads to the exact freeenergy variation. However we note that, with the im-plementation described above, this approximation couldlead to the nonphysical occurrence of negative entropy:in fact the two contributions to the entropy have op-posite sign, and one of them (the work) has been ap-proximated, so the occurrence of a negative sign can-not apriori be excluded. We then further impose that∆Sexact+NI = max{∆Sexact+NI , 0} to correct for it.

Related results are plotted in figure 7 for 6 sites, ‘comb’driving potential, and increasing temperature (left toright). By comparing the upper panels of figure 7 tothe upper panels of figure 6 and to the correspondingones in the mid column of figure 5, we note a markedimprovement in the qualitative behaviour of the approx-imation. As with the work, we can see an improvementalso in the quantitative results. The high temperatureaccuracy is very much akin to that of the work, and is

10

0 2 4 6 8 10U/J

2

4

6

8

10

12

14

16

18

20∆F

Free Energy change ∆Fexact

T = 20/kBJ

T = 2.5/kBJ

T = 0.2/kBJ

(a) MI driving potential.

0 2 4 6 8 10U/J

−25

−20

−15

−10

−5

0

∆F


T = 20/kBJ

T = 2.5/kBJ

T = 0.2/kBJ

(b) Comb driving potential.

0 2 4 6 8 10U/J

−35

−30

−25

−20

−15

−10

−5

0

∆F


T = 20/kBJ

T = 2.5/kBJ

T = 0.2/kBJ

(c) AEF driving potential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J

Entropy production ∆Sexact1.45 7.88 14.31 20.74 27.17 33.60 40.03 46.46 52.89 59.31 65.74

(d) T = 0.2J/kB with MI drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(e) T = 0.2J/kB with comb drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(f) T = 0.2J/kB with AEF drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(g) T = 2.5J/kB with MI drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(h) T = 2.5J/kB with comb drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(i) T = 2.5J/kB with AEF drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(j) T = 20J/kB with MI drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(k) T = 20J/kB with comb drivingpotential.

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(l) T = 20J/kB with AEF drivingpotential.

FIG. 5: (a)-(c) Variation of free energy ∆F versus U for 6-site chains at low (green), medium (blue), and high (red)temperatures and for the three driving potential (as indicated). (d)-(l) Exact entropy variation ∆S versus τ (x-axis)and U (y-axis), for 6 site chains, with MI (left column), ‘comb’ (middle) and AEF (right column) driving potential;temperatures as indicated. Darker colour shades correspond to lower entropy production, whilst lighter to higher

entropy production.

11

2 4 6 8 10τ × J

0

2

4

6

8

10U/J

Entropy production ∆SNI1.76 5.03 8.30 11.57 14.84 18.11 21.38 24.65 27.92 31.18 34.45

(a) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(b) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(c) T = 20J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J

|∆SNI −∆Sexact|/∆Sexact0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(d) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(e) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(f) T = 20J/kB

FIG. 6: Upper panels: Non-interacting entropy production versus τ (x-axis) and U (y-axis) for 6-site chains withcomb driving potential. Lower panels: Non-interacting entropy production relative difference for the same

parameters as the upper panels.

accurate within 30% for all regimes. As the temperaturedecreases, however, the quantitative accuracy also de-creases: inaccuracy comes into the entropy calculationsthrough the approximation of the average work, so theregimes of greater/lesser accuracy fairly mirror those ofthe work.

Results for the other two driving potentials confirmthese trends and are shown in the appendix.

Similarly to the ‘comb’ potential, the ‘exact+NI’ ap-proximation with the MI and AEF driving potentialsrecover to a good extent the qualitative behaviour of∆Sexact for low and intermediate temperatures. For hightemperatures the qualitative behaviour is recovered onlyfor U <∼ 2. Quantitatively, the areas of worse perfor-mance are related to the areas of worse performance forthe corresponding 〈W exact+NI

ext 〉, however the approxima-tion performs worse for ∆Sexact+NI than 〈W exact+NI

ext 〉for MI, and better for AEF. Overall the approximationimproves its quantitative performance with temperature,as it reproduces well the limits of the entropy variationrange, and especially so at high temperature.

VII. CONCLUSION

We have presented a comprehensive study of the ex-tracted average quantum work and entropy variation in

complex many-body systems subject to a wealth of driv-ing dynamics and dynamical regimes. We have discussedin details the effects of the interplay of the different en-ergy scales governing the systems – driving potentials,many-body interactions, and thermal energy – on 〈W 〉and ∆S, and compared results as many-body correlationsare turned on up to the strongly coupled regime and asthe dynamical regime is continuously changed from sud-den quench to nearly adiabatic.

For all driving potentials and low to intermediate tem-peratures, intermediate to strong Coulomb repulsion de-creases work extraction by making the system less re-sponsive to the external drive. At weak Coulomb corre-lations, more work can be extracted as the system be-comes adiabatic, while with stronger many-body inter-actions the work production becomes independent of theoverall driving time τ much sooner. For the same pa-rameter sets, at high temperatures, work extraction isgreatly reduced both in absolute values and in variationrange.

At low temperatures the entropy variation presents abehaviour somewhat more dependent on the applied driv-ing potential. We observe a general tendency of lower ∆Svalues for weak and strong Coulomb correlations and in-termediate to large τ ’s, but the onset of an adiabatic-like entropy dynamics varies considerably with the driv-ing potential at intermediate coupling strengths. For a

12

2 4 6 8 10τ × J

0

2

4

6

8

10U/J

Entropy production ∆Sexact+NI0.00 5.49 10.99 16.48 21.97 27.46 32.96 38.45 43.94 49.43 54.93

(a) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(b) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(c) T = 20J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J

|∆Sexact+NI −∆Sexact|/∆Sexact0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(d) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(e) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(f) T = 20J/kB

FIG. 7: Upper panels: Exact + non-interacting entropy production versus τ (x-axis) and U (y-axis) for 6-site chainswith comb driving potential. Lower panels: Exact + non-interacting entropy production relative difference for the

same parameters as the upper panels.

zero temperature, open-boundary, finite, homogeneousHubbard chain, intermediate coupling strengths corre-spond to the transition between metallic and insulatingbehaviour (precursor to a Mott insulator transition). Thebehaviour observed at intermediate coupling strengthsmay be a signal of how the different driving potentialsaffect this transition.

Similarly to the extracted average quantum work, en-tropy production decreases with temperature in both ab-solute value and range of variation, as the system be-comes less and less responsive to the applied potential.

The strong effect of the Coulomb interaction on theextracted average quantum work can be also appreci-ated by comparing exact results to the correspondingnon-interacting approximations. Apart from the obviousindependence on U for all regimes, this approximationstrongly overestimates the minimum work that can be ex-tracted from a system, as it cannot reproduce the freezingof the system dynamics due to strong many-body corre-lations. On the other hand, the value of the maximumaverage work extracted is well captured, albeit often it isattributed to the wrong parameter regions.

Including many-body interactions exactly for complexsystem is often an hopeless task. In this paper we haveproposed a relatively simple approximation which relieson been able to provide an accurate approximation forthe system’s initial state, while interactions are com-

pletely neglected in the dynamics.We found this approximation to behave surprisingly

well. Including interactions just through the initial staterecovers the qualitative (and in great part quantitative)behaviour at low and intermediate temperatures for allregimes, driving potentials and chain lengths considered.The greatest improvement is seen at low temperatures,where, for example, with the ‘comb’ driving potentialthe ‘exact+NI’ approximation reproduces the exact workwithin 10-20% up to very strong interactions (U ≈ 9J)for most regimes. At all temperatures, the ‘exact + NI’approximation reproduces fairly well the range of varia-tion for the value of the average extracted quantum workand in particular it significantly improves the value of itsminimum over the NI estimates.

We demonstrate analytically that, independent of sys-tem Hamiltonian and driving potential, including inter-actions in just the initial state is sufficient to recover atlow to intermediate temperatures the characteristic de-pendency on U and τ in the adiabatic region, which wascompletely missed by the non-interacting approximation.This is important as this is the region in which the mostwork can be produced, and the dependency on U is verystrong.

We also demonstrate analytically that this approxi-mation recovers the characteristic U -dependence of theextracted work in the small-τ parameter region for all

13

Hamiltonians and driving potentials.Fully non-interacting estimates of the entropy vari-

ation strongly under-perform, both qualitatively andquantitatively. However at high temperatures the ex-act entropy variation range becomes very small and itsquantitative non-interacting estimate is in the right ball-park due to the decreased influence of many-body inter-actions: as a result, at high temperatures even the non-interacting approximation gives reasonable quantitative(but not qualitative) results.

When we extend the ‘exact+NI’ approximation to theentropy variation, we find qualitative improvements sim-ilar to the work extraction, with the behaviour for lowand intermediate temperatures largely recovered. Quan-titatively this approximation significantly improves overthe non-interacting approximation, albeit not as strik-ingly as for the average quantum work. Overall the ‘ex-act+NI’ approximation improves its quantitative perfor-mance with temperature, as it captures well the entropyvariation range, including at high temperature. As anexample, at high temperature and MI driving potential,this approximation would reproduce exact results within10% for all parameter range, to be contrasted with theperformance by the non interacting approximation whichgives such an accuracy only for U <∼ 3.

Our results show that, even when taking a very crudeapproximation for the evolution operator, starting froman accurate initial state is sufficient for greatly improvingthe estimate of thermodynamical quantities such as theaverage quantum work and the corresponding entropyvariation for the wide range of driving potentials tested,all temperatures, and the great part of interaction anddynamical regimes.

ACKNOWLEDGMENTS

AHS thanks EPSRC for financial support. KZthanks the Schlumberger Foundation for financial sup-port through the Faculty for the Future program. IDAacknowledges support from the Conselho Nacional deDesenvolvimento Cientfico e Tecnologico (CNPq, Grant:PVE Processo: 401414/2014-0) and from the Royal So-ciety (Grant no. NA140436). We thank M. Herrera andR. M. Serra for useful discussions in the early stages ofthis work.

Appendix A: Six-site ‘exact+NI’ approximationresults for MI and AEF driving potentials:

extracted average quantum work

To demonstrate how well the the ‘exact+NI’ approx-imation fares with other driving potentials, we presenthere the related MI and AEF driving potential resultsfor 6 site chains and all three temperatures.

Figure 8 shows results for the MI driving potential:this particular dynamic is captured very well by the ‘ex-

act+NI’ approximation for all temperatures and regimes.As with all other driving potentials considered, the low-est temperature is where quantitatively the approxima-tion struggles the most, though its accuracy is, in mostregions, within 10% of the exact work, otherwise it is20%. At higher temperatures the accuracy remains al-ways within 10%. We note that this approximation re-covers to high accuracy and at all temperatures the over-all range of extractable work (compare colour-scale rangeabove panels (a) to (c) in figure 8 with the correspond-ing ranges in panels (a), (d), and (g) of figure 2). Thisapproximation also recovers most of the work qualitativebehaviour, at least at low and intermediate temperature.At high temperature, although the trends of the exactand approximate results are qualitatively different forτ >∼ 2/J and U >∼ J , the exact amount of work extractedchanges very little with τ and U (panel (g) of figure 2),so, as the quantitative range of the approximated workis close to the exact one, the results in figure 8(c) showhigh quantitative accuracy.

Results for 〈W exact+NI〉 and the AEF driving poten-tial are shown in figure 9. Once again the approximationaccuracy improves with increasing temperature (panels(d) to (f)), while the qualitative trend is well reproducedfor low and intermediate temperatures, but missed athigh temperatures for intermediate-to-large τ and U val-ues (compare panels (a) to (c) of figure 9 to panels (c),(f), and (i) of figure 2, respectively). For AEF – and atdifference with ‘comb’ and MI driving potentials – theCoulomb repulsion U for U >∼ 4 remains a dominant en-ergy scale at any temperature: in fact even at t = τ thedriving potential difference between nearby sites remainsat most of 4J (see figure 1). This results in a quan-titatively poor approximation for U >∼ 4 even at hightemperature.

Appendix B: Six-site ‘exact+NI’ approximationresults for MI and AEF driving potentials: entropy

variation

Estimates for the entropy variation in the ‘exact+NI’approximation for MI and AEF driving potentials andsix-site chains are shown in figures 10 and 11, respec-tively.

Similarly to the ‘comb’ potential, the ‘exact+NI’ ap-proximation with the MI and AEF driving potentials re-cover to a good extent the qualitatively the behaviour of∆Sexact for low and intermediate temperatures. For hightemperatures the qualitative behaviour is recovered onlyfor U <∼ 2.

Quantitatively, the areas of worse performance are re-lated to the areas of worse performance for the corre-sponding 〈W exact+NI

ext 〉, however the approximation per-forms worse for the entropy than for the extracted workfor MI, and better for AEF. Overall the approximationimproves its quantitative performance with temperature,as it reproduces well the limits of the entropy variation

14

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


−19.18−17.79−16.40−15.01−13.62−12.22−10.83−9.44 −8.05 −6.66 −5.27

(a) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


−19.37−18.86−18.36−17.86−17.36−16.85−16.35−15.85−15.35−14.85−14.34

(b) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


−19.74−19.69−19.64−19.59−19.55−19.50−19.45−19.40−19.35−19.31−19.26

(c) T = 20J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(d) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(e) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(f) T = 20J/kB .

FIG. 8: Panels (a) to (c): Extracted work in the ‘exact + NI’ approximation, for 6-site chains undergoing dynamicsdriven by the MI potential. The lighter the colour shade, the more work is extracted, compatible with the respectivework ranges indicated above each panel. The regimes go from non-interacting to strongly coupled along the y-axis,and from sudden quench closer to adiabatic along the x-axis. Temperature increases from left to right panel, as

indicated.Panels (d) to (f): Relative error for 〈W exact+NI〉 with respect to the exact results for the same parameters as the

upper panels. The darker the colour, the more accurate the approximation is in that regime.

range, and especially so at high temperature.

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15

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


0.48 3.00 5.51 8.02 10.54 13.05 15.57 18.08 20.59 23.11 25.62

(a) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.60 1.46 2.33 3.19 4.06 4.92 5.78 6.65 7.51 8.38 9.24

(b) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


0.41 0.49 0.57 0.65 0.73 0.81 0.89 0.97 1.05 1.13 1.21

(c) T = 20J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(d) T = 0.2J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(e) T = 2.5J/kB .

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J



0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(f) T = 20J/kB .

FIG. 9: Panels (a) to (c): Extracted work in the ‘exact + NI’ approximation, for 6-site chains undergoing dynamicsdriven by the AEF potential. The lighter the colour shade, the more work is extracted, compatible with the

respective work ranges indicated above each panel. The regimes go from non-interacting to strongly coupled alongthe y-axis, and from sudden quench closer to adiabatic along the x-axis. Temperature increases from left to right

panel, as indicated.Panels (d) to (f): Relative error for 〈W exact+NI〉 with respect to the exact results for the same parameters as the

upper panels. The darker the colour, the more accurate the approximation is in that regime.

production in quantum many-body systems. Phys. Rev.B, 93:224305, 2016.

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Cambridge University Press, 2005. ISBN 9781139441582.[22] J. P. Coe, V. V. França, and I. D’Amico. Hubbard

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[25] J. P. Coe, V. V. França, and I. D’Amico. Feasibility of ap-proximating spatial and local entanglement in long-rangeinteracting systems using the extended Hubbard model.EPL (Europhysics Letters), 93(1):10001, jan 2011. doi:10.1209/0295-5075/93/10001.

[26] Peter T. Brown, Debayan Mitra, Elmer Guardado-Sanchez, Reza Nourafkan, Alexis Reymbaut, Charles-David Hébert, Simon Bergeron, A.-M. S. Tremblay, JureKokalj, David A. Huse, Peter Schauß, and Waseem S.Bakr. Bad metallic transport in a cold atom fermi-

http://dx.doi.org/10.1103/PhysRevA.74.042325




http://dx.doi.org/10.1007/s13538-018-0592-6

http://dx.doi.org/10.1007/s13538-018-0592-6



http://dx.doi.org/10.1103/PhysRevB.83.161301

http://dx.doi.org/10.1103/PhysRevB.83.161301

http://dx.doi.org/10.1103/PhysRevLett.114.080402


http://dx.doi.org/10.1209/0295-5075/93/10001

http://dx.doi.org/10.1209/0295-5075/93/10001

16

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(a) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(b) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(c) T = 20J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(d) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(e) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(f) T = 20J/kB

FIG. 10: Upper panels: Estimate of the entropy production in the exact + NI approximation versus τ (x-axis) andU (y-axis), for 6 site chains and MI driving potential. Temperatures as indicated. Lower panels: relative differencebetween exact and exact + NI entropy production for MI driving potential. Parameters as for the upper panels.

hubbard system. Science, 363(6425):379–382, 2019. doi:10.1126/science.aat4134.

[27] Matthew A. Nichols, Lawrence W. Cheuk, Melih Okan,Thomas R. Hartke, Enrique Mendez, T. Senthil, EhsanKhatami, Hao Zhang, and Martin W. Zwierlein. Spintransport in a mott insulator of ultracold fermions. Sci-ence, 363(6425):383–387, 2019. ISSN 0036-8075. doi:10.1126/science.aat4387.

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[31] Note1. We note that for the 2 site chain the only relevantdynamics becomes the ‘Applied Electric Field’ one.

[32] Note2. See [16] and [17] for 2-site chain examples, andespecially [16] for 2-site non-interacting results.

[33] Note3. For the ‘exact + NI’ approximation, the resultsof calculating < W > using (2) or (6) differ for thet = 0 terms. The t = 0 term via (2) can be writtenas Σi,nE

NI,0i p0n |an,i|2, while the corresponding term via

(6) is Σnp0nE

NI,0n . The t = τ terms give instead the same

results.

http://dx.doi.org/10.1126/science.aat4134






17

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(a) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(b) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(c) T = 20J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(d) T = 0.2J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10U/J


(e) T = 2.5J/kB

2 4 6 8 10τ × J

0

2

4

6

8

10

U/J


(f) T = 20J/kB

FIG. 11: Upper panels: Estimate of the entropy production in the exact + NI approximation versus τ (x-axis) andU (y-axis), for 6 site chains and AEF driving potential. Temperatures as indicated. Lower panels: relative differencebetween exact and exact + NI entropy production for AEF driving potential. Parameters as for the upper panels.

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