Eigenstate Fluctuation Theorem in the Short and Long Time Regimes
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Eigenstate Fluctuation Theorem in the Short and Long Time Regimes
Eiki Iyoda , Kazuya Kaneko , Takahiro Sagawa , Department of Physics, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan Department of Applied Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Quantum-Phase Electronics Center (QPEC), The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
The canonical ensemble plays a crucial role in statistical mechanics in and out of equilibrium. Forexample, the standard derivation of the fluctuation theorem relies on the assumption that the initialstate of the heat bath is the canonical ensemble. On the other hand, the recent progress in thefoundation of statistical mechanics has revealed that a thermal equilibrium state is not necessarilydescribed by the canonical ensemble but can be a quantum pure state or even a single energyeigenstate, as formulated by the eigenstate thermalization hypothesis (ETH). Then, a questionraised is how these two pictures, the canonical ensemble and a single energy eigenstate as a thermalequilibrium state, are compatible in the fluctuation theorem. In this paper, we theoretically andnumerically show that the fluctuation theorem holds in both of the long and short-time regimes, evenwhen the initial state of the bath is a single energy eigenstate of a many-body system. Our proof ofthe fluctuation theorem in the long-time regime is based on the ETH, while it was previously shown inthe short-time regime on the basis of the Lieb-Robinson bound and the ETH [Phys. Rev. Lett. ,100601 (2017)]. The proofs for these time regimes are theoretically independent and complementary,implying the fluctuation theorem in the entire time domain. We also perform a systematic numericalsimulation of hard-core bosons by exact diagonalization and verify the fluctuation theorem in both ofthe time regimes by focusing on the finite-size scaling. Our results contribute to the understanding ofthe mechanism that the fluctuation theorem emerges from unitary dynamics of quantum many-bodysystems, and can be tested by experiments with, e.g., ultracold atoms.
I. INTRODUCTION
Conventional statistical mechanics relies on the con-cept of ensembles, such as the microcanonical ensembleand the canonical ensemble. These are characterized asthe maximum entropy states under certain energy con-straints [1] and thus given by statistical mixtures of enor-mous energy eigenstates. The canonical ensemble plays acrucial role even beyond equilibrium situations. For ex-ample, we can prove the second law of thermodynamicsand the fluctuation theorem [2–12] out of equilibrium ofthe system, by relying on the assumption that the initialstate of the heat bath is in the canonical ensemble. Here,the fluctuation theorem is a universal relation that in-corporates the role of fluctuations of the entropy produc-tion, represented as h e − σ i = 1 where σ is the stochasticentropy production. The fluctuation theorem implies thesecond law at the average level, stating that the averageentropy production is nonnegative: h σ i := ∆ S S − βQ ≥ , (1)where ∆ S S is the change of the von Neumann entropy, β is the inverse temperature of the heat bath, and Q is theaverage heat emitted from the bath. We note that thefluctuation theorem reproduces the second law throughthe Jensen inequality h e − σ i ≥ e −h σ i .On the other hand, in recent years, it has been estab-lished that most states, not limited to those ensembles(maximum entropy states), can represent thermal equi-librium states [13–25]. The extreme case opposite to thecanonical ensemble is zero entropy states such as a sin- gle energy eigenstate; The eigenstate thermalization hy-pothesis (ETH) [26–42] states that a single energy eigen-state can represent a thermal equilibrium state. TheETH is a sufficient condition for thermalization in iso-lated quantum many-body systems [26–28] and numeri-cally shown to be valid in various non-integrable quantummany-body systems [28–42]. In recent experiments suchas cold atoms [18–23], trapped ions [24], and supercon-ducting qubits [25], it has been observed that thermalequilibrium states are not necessarily canonical ensem-bles but can be pure states.A critical question here is how universally the thermo-dynamic properties appear without canonical ensembles,even in non-equilibrium situations. In particular, it isinteresting to investigate the validity of the fluctuationtheorem: the question focuses on whether it is possible tounderstand how the fluctuation theorem and the secondlaw emerge from quantum mechanics without assumingthe conventional statistical ensembles.A partial understanding of the above question has beenaddressed in recent papers. Specifically, Ref. [43] theo-retically showed the fluctuation theorem (and the secondlaw) in the short-time regime, which is based on the ETHand the Lieb-Robinson bound [44–46]. We note that inthe numerical simulation of Ref. [43], the initial stateof the system was chosen to be a pure state and thefluctuation theorem was apparently broken in the long-time regime. This apparent breakdown originates fromso-called absolute irreversibility [47, 48] induced by thepure initial state, and as will be shown in this paper, thefluctuation theorem still holds in the long-time regimeif absolute irreversibility is properly treated. Moreover,Ref. [49] theoretically showed the second law at the aver-age level in the long-time regime on the basis of the ETH.Also in previous papers [50, 51], it has been numericallysuggested that the fluctuation theorem holds even in thelong-time regime. Given these researches, it is desirableto make a comprehensive understanding of the validityof the fluctuation theorem in the entire time domain.The main result of this paper is to show, analyticallyand numerically, that the fluctuation theorem holds withthe non-canonical bath in both of the short and long timeregimes. Specifically, the initial state of the bath is sup-posed to be a single energy eigenstate sampled from anenergy shell. This establishes that the universal prop-erty of non-equilibrium fluctuations in entropy produc-tion emerges even without the canonical ensemble. Inparticular, since we prove the fluctuation theorem for theextreme case with a single energy eigenstate, the fluctu-ation theorem holds when the initial state of the bath isany mixture of energy eigenstates in the energy shell.We discuss the entire time domain by considering thelong and short-time regimes. We theoretically show thatthe long-time average of the deviation from the fluctua-tion theorem vanishes in the thermodynamic limit of theheat bath. To confirm the theory, we perform numericalsimulations and show that the deviation decreases withthe bath size N . On the other hand, Ref. [43] theoret-ically showed the fluctuation theorem in the short-timeregime. In the present paper, we perform systematic nu-merical calculations to clarify whether the mechanism inthe short-time regime is indeed relevant. As a conse-quence, our numerical results support the validity andthe relevance of the theory developed in Ref. [43].The rest of this paper is organized as follows. In Sec. II,we introduce the setup of this study. In Sec. III, weoverview the main results of this paper without goinginto details. In Sec. IV, we theoretically derive the fluc-tuation theorem in the long-time regime and show ournumerical results to support the theory. In Sec. V, we dis-cuss the fluctuation theorem in the short-time regime andshow the corresponding numerical results. In Sec. VI,we summarize the results and make some remarks. InAppendix A, we introduce the concept of absolute ir-reversibility [47, 48]. In Appendix B, we examine thedetails of the interaction-induced error of the fluctuationtheorem. In Appendix C, we explain the details of theproof of the fluctuation theorem in the long-time regime.In Appendix D, we show the additional calculations forthe interaction-induced error. In Appendix E, we showthe supplementary numerical results. II. SETUP
In this section, we introduce the setup of the study.In Sec. II A, we explain the conventional setup for thefluctuation theorem with the canonical bath and brieflyoverview the fluctuation theorem [2–12]. In Sec. II B, weexplain the setup of the present study with the energy eigenstate bath. In Sec. II C, we explain the setup of ournumerical simulation.
A. Fluctuation theorem for the canonical bath
As a preliminary, we consider the conventional setup ofthe fluctuation theorem. The total system is composedof the system S and the heat bath B. The Hamiltonianof the total system is written as H := H S + H I + H B , (2)where H S , H B are the Hamiltonian of system S and bathB respectively, and H I ( = 0) describes the interaction be-tween S and B. The initial state of the total system SBis assumed to be a product state: ρ (0) := ρ S (0) ⊗ ρ B (0) . (3)The time evolution of the total system is given by ρ ( t ) = U ρ S (0) ⊗ ρ B (0) U † , where U ( t ) := e − iHt/ ~ is theunitary time evolution operator. We write the reduceddensity operator of system S and bath B at time t as ρ S ( t ) := tr B [ ρ ( t )] , ρ B ( t ) := tr S [ ρ ( t )], respectively.In this setup, we explain the concept of the stochasticentropy production σ [7]. To introduce it, let us consideran operator σ ( t ) := − ln ρ S ( t ) + βH B . The first termon the right-hand side is the informational contribution,whose average is the von Neumann entropy of the sys-tem S S ( t ) := − tr S [ ρ S ( t ) ln ρ S ( t )]. The second term is thethermal contribution that gives the heat term.We consider the projection measurement of σ ( t ) attime 0 and t . Let the measurement outcomes be σ i at t = 0 and σ f at t . We then define the stochasticentropy production σ as σ := σ f − σ i , whose average h σ i = ∆ S S − βQ is the average entropy production of thetotal system. Here, Q is the heat emitted from bath Bto system S, defined as Q := − tr B [ H B ( ρ B ( t ) − ρ B (0))] . For the conventional fluctuation theorem, the initialstate of bath B is assumed to be the canonical en-semble ρ B (0) = ρ canB , where ρ canB := e − βH B /Z B with Z B := tr B [ e − βH B ]. The fluctuation theorem states that h e − σ i = 1 . (4)It is straightforward to show [7] that h e − σ i can be rewrit-ten as h e − σ i = tr[ e − βH B U e βH B ρ canB U † ρ canS ( t )] . (5)Substituting ρ canB = e − βH B /Z B , we obtain h e − σ i = tr (cid:20) e − βH B U e βH B e − βH B Z B U † ρ canS ( t ) (cid:21) (6)= tr (cid:20) e − βH B Z B U U † ρ canS ( t ) (cid:21) (7)= tr [ ρ canS ( t ) ⊗ ρ canB ] (8)= 1 . (9) FIG. 1. The setup of our study. The total system consistsof the system S and the heat bath B. The system S interactswith a local (bounded) region of B, and the interaction in Bis also local. The bath B is defined on a d -dimensional lattice.We denote the number of sites of B by N and the dimensionof the Hilbert space of B by D B . We also consider an energyshell of B, whose Hilbert-space dimension is D . We note that if absolute irreversibility [47, 48] occurs,the fluctuation theorem is modified (see details in Ap-pendix A). Absolute irreversibility is an apparent viola-tion of the fluctuation theorem that occurs when ρ S (0)does not have the full rank (e.g., ρ S (0) is a pure state).The modification term appears in the form of h e − σ i = 1 − λ ( t ) . (10)See Appendix A for details. We note that absolute irre-versibility has been observed experimentally [53]. B. Fluctuation theorem for the energy eigenstatebath
In this subsection, we introduce the setup of the fluctu-ation theorem for the energy eigenstate bath, which willbe shown as the main result of the present paper.The total system is composed of system S and bathB in the same manner as the setup for the conventionalfluctuation theorem, while in the present setup, bath B isdefined on a lattice as shown in Fig. 1. We assume thatthe Hamiltonian of bath B is local and system S interactswith a local region of B by the interaction H I . Let N bethe number of sites in bath B, and d = 1 , , , · · · be thespatial dimension of it. The size of system S and thesupport of H I are fixed and do not depend on N . Werefer to the dimensions of the Hilbert space of S, B andthe total system SB as D S , D B and D SB , respectively. Wedenote an eigenstate of H with eigenenergy E a as | E a i .In the same manner, we denote eigenstates of H S , H B as | E S i i , | E B α i , respectively. We denote a matrix element of an operator with respect to the eigenstates of H as( · · · ) ab := h E a | · · · | E b i . We also define an operator ofsystem S as q ij S := | E S i ih E S j | .In the present paper, we consider the thermodynamiclimit that means the large-bath limit ( N → ∞ ) withoutchanging system S. To describe asymptotic behaviors ofthis limit, we use the following asymptotic notations: f ( D ) = Θ( g ( D )) ⇔ < lim D →∞ (cid:12)(cid:12)(cid:12)(cid:12) f ( D ) g ( D ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , (11) f ( D ) = O ( g ( D )) ⇔ lim D →∞ (cid:12)(cid:12)(cid:12)(cid:12) f ( D ) g ( D ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , (12) f ( D ) = o ( g ( D )) ⇔ lim D →∞ (cid:12)(cid:12)(cid:12)(cid:12) f ( D ) g ( D ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (13)We define the operator norm k X k as the largest singularvalue of X and the trace norm k X k as the sum of thesingular values of X . For the sake of simplicity, we use A ≃ B for operators A and B to represent that k A − B k = o (1) or k A − B k = o (1) depending on the context.The initial state of SB is assumed to be a productstate as in (3). We also assume that the initial state of Sis diagonal with respect to H S , because it simplifies thediscussion of the time dependence in Sec. V.Now, a crucial assumption is that the initial state ofbath B is a single energy eigenstate, which is sampledfrom the energy shell [ E − ∆ E, E ]. The energy E isgiven by E := tr B [ ρ canB H B ] with the canonical ensembleat inverse temperature β . The width ∆ E can be chosenas Θ(1) ≤ ∆ E ≤ Θ( N ), which gives the normal thermo-dynamic scaling D = exp( sN ) with the entropy density s [52].We next summarize the assumptions regarding theETH, as summarized in Table I. For the proof of thefluctuation theorem in the long-time regime (Sec. IV),we assume that H satisfies the ETH for all observablesof system S and for all the eigenstates in the energy shell(the “strong” ETH). Here, we can introduce the energyshell of H , which is determined by the energy and energywidth of the initial state (3) (see details in Appendix C).Specifically, we assume the (strong) diagonal and off-diagonal ETH [17, 26–28] for all observables of system S,written as O S , stating that the following relations holdfor all the eigenstates in the energy shell: h E a | O S | E a i = h O S i MC + O ( D ′− / ) , (14) |h E a | O S | E b i| = O ( D ′− / ) ( a = b ) , (15)where h·i MC is the microcanonical average of the energyshell of H . Since the size of system S is Θ(1), the dimen-sion of the energy shell D ′ ( < D SB ) satisfies D ′ = Θ( D ).In Sec. V, we discuss the fluctuation theorem in theshort-time regime, which has been theoretically provedin Ref. [43]. Here, we assume the diagonal ETH of H B : h E B a | L B | E B a i = h L B i MC , B + O ( D − / ) , (16)where h·i MC , B is the microcanonical average of the en-ergy shell [ E − ∆ E, E ], D ( < D B ) is the dimension of TABLE I. The assumptions regarding the ETH.Time region (section) Long time region (Sec. IV) Short time region (Sec. V)Which ETH The diagonal and off-diagonal ETH of H The diagonal ETH of H B For which eigenstates All the eigenstates Only for the given initial state of B the Hilbert space of the energy shell of H B , and L B isa quasi-local operator defined in Sec. V. We note thatEq. (16) is assumed only for the given initial state | E B a i of B.We remark on the validity of the ETH. It has beennumerically shown that the strong ETH holds in vari-ous non-integrable quantum many-body systems [28–42],while does not hold in the presence of quantum many-body scar [54] and in integrable systems [31]. On theother hand, the ETH for a given single energy eigen-state can be valid even in integrable systems [29, 43].Specifically, it has been shown that almost all energyeigenstates in the energy shell satisfy the ETH in in-tegrable systems [43, 55, 56] and quantum many-bodyscars [54]. However, even the ETH for almost all energyeigenstates does not hold in many-body localized (MBL)systems [57, 58].Meanwhile, in the following sections, we sometimesconsider the following condition for the interactionHamiltonian: [ H S + H B , H I ] = 0 , (17)which simplifies the derivation of the fluctuation theo-rem. Under this condition, the sum of the energies ofsystem S and bath B does not change, and we can rewritethe heat Q by the energy change in system S. The con-dition (17) is satisfied in the Jaynes-Cummings model atthe resonant condition [59]. Although the condition isnot necessarily satisfied in general quantum many-bodysystems, it can be approximately satisfied under the ro-tating wave approximation in the long time regime [59].We note that the interaction itself (i.e., k H I k ) is not nec-essarily small for this condition (17) to hold. When thecondition (17) is not satisfied, an error term induced bythe interaction can appear in the fluctuation theoremwith the energy eigenstate bath. C. Hamiltonian for numerical simulation
In this subsection, we explain the setup of our numer-ical simulation discussed in Section III, IV, and V.We perform numerical calculations of hard-core bosonswith nearest-neighbor repulsion using numerically exactdiagonalization. System S is a single site, and bath B ison a two-dimensional lattice. The Hamiltonian is given by H S := ωn , (18) H I := − γ ′ X h ,j i ( c † c j + c † j c ) , (19) H B := ω N X i =1 n i − γ X h i,j i ( c † i c j + c † j c i ) + g X h i,j i n i n j , (20)where ω is the onsite potential, − γ is the hopping inbath B, − γ ′ is the hopping between system S and bathB, and g > h i, j i means the sum over thenearest-neighbor sites. Hard core bosons cannot existsimultaneously on a single site, and their annihilation(creation) operator c i ( c † i ) satisfies the following commu-tation relations [ c i , c j ] = [ c † i , c † j ] = [ c i , c † j ] = 0 for i = j , { c i , c i } = { c † i , c † i } = 0, and { c i , c † i } = 1. The occupa-tion number operator is defined as n i := c † i c i . The siteof i = 0 corresponds to system S. We set bath B as atwo-dimensional lattice with the open boundary condi-tion, and the bath size N is written as N = L x × L y .The operator of the particle number in bath B is writtenas n B := P Ni =1 n i . The Hamiltonian H is non-integrablewhen g = 0 , γ = 0 and γ ′ = 0.We write the initial state of system S as ρ S (0) := p | ih | + (1 − p ) | ih | , (21)where | n S i is the eigenstate of n and satisfies n | n S i = n S | n S i ( n S = 1 , ρ S (0) to be a mixed state by setting p = 0 .
99 in our numerical simulation. See Appendices Aand C 3 for the case with absolute irreversibility. Wedefine the change of the particle number in S as δn S := tr S [ n ( ρ S ( t ) − ρ S (0))] . (22)The initial state of bath B is an energy eigenstate of H B with a particle number N P , which samples from theenergy shell [ E − ∆ E, E ]. We write the width of theenergy shell as ∆ E = N δ E with δ E be a positive con-stant and set δ E = 0 .
02 in our numerical simulation.We perform the calculation of the fluctuation theoremfor each energy eigenstate in the energy shell, and in-vestigate the dependence of the error of the fluctuationtheorem on the bath size and the initial state of bath B.We define the inverse temperature of | E B α i as β α satis-fying E B α = tr B [ ρ canB ( β α ) H B ], where we explicitly writethe β -dependence of the canonical ensemble. The onsitepotential ω is determined such that the canonical ex-pectation of the particle number of the bath equals N P : FIG. 2. The time dependence of h e − σ i . The right inset showsthe time dependence of the error of the fluctuation theorem |h e − σ i − | , whose initial rise is proportionate to t . The leftinset is a schematic of the total system used in numericalcalculations. Parameters: p = 0 . , g = 0 . γ , γ ′ = 0 . γ (pur-ple), γ (green), 4 γ (blue). The initial state of the bath is theenergy eigenstates of H B , whose energy is maximum in theenergy shell at β = 0 .
1. The onsite potential ω is determinedby tr B [ n B ρ B (0)] = N P . tr B [ n B ρ B (0)] = N P . We set ω as above, because the ETHis satisfied only within each particle number sector and N P should be close to the canonical expectation number.To investigate the bath size dependence, we set L x = 3, L y = 3 , , N P = N/ III. OVERVIEW OF THE RESULTS
In this section, we give an overview of the main resultsof this paper: the fluctuation theorem holds in both thelong and short-time regimes (as defined below) in the set-ting of the previous section. The proof of the fluctuationtheorem in the long-time regime is based on the ETH of H (Sec. IV). Also, the fluctuation theorem in the short-time regime has been shown on the basis of the ETHof H B and the Lieb-Robinson bound [43] (Sec. V). Theproofs of the fluctuation theorem in these time regimesare theoretically independent of each other and play com-plementary roles. Our systematic numerical calculationsabout the bath size dependence support the theories inthe both of long and short time regimes.First of all, we investigate whether the fluctuation the-orem (4) holds or not by numerically investigating thereal-time dynamics. Figure 2 shows the time dependenceof h e − σ i , which implies that h e − σ i ≃ |h e − σ i − | . The error increases until the relaxation timeof the system S ( t ∼
1) and then saturates. Qualitativelythe same behavior is seen with other interacting param-eters (see Appendix E), and Fig. 3 is a schematic of thetypical time dependence of |h e − σ i − | . FIG. 3. Sketch of a typical time dependence of |h e − σ i − | ,which initially rises in proportionate to t and relaxes to thelong-time average in t & τ relax . We refer to the time regimeafter the relaxation as the long-time regime. We also call thesufficiently shorter time regime than the Lieb-Robinson time τ LR (24) as the short-time regime ( t ≪ τ LR ). We consider the fluctuation theorem in the long-timeregime. In this paper, we prove that h e − σ i , the long-time average of h e − σ i , nearly equals 1 in the large-bathlimit. Note that we denote the long-time average of anyquantity O ( t ) as O ( t ) := lim T →∞ T Z T O ( t ) dt. (23)In the special case that the condition (17) is satisfied,we can prove that |h e − σ i − | = o (1) holds, where theright-hand side represents the asymptotic behavior withrespect to the bath size N . If Eq. (17) is not strictlysatisfied, the interaction-induced error δG I defined inSec. IV A can generally appear in the fluctuation theoremwith the energy eigenstate bath. However, we expect thatthe interaction-induced error does not grow significantly.If the initial state of bath B satisfies the ETH, we canexpect that the interaction-induced error is sufficientlysmaller than 1 (see Sec. IV and Appendix D), which issupported by our numerical simulation (see Sec. IV D).In the absence of localizations [57, 58] or persistent os-cillations [54], we expect that h e − σ i relaxes to the long-time average h e − σ i after the finite relaxation time τ relax .We refer to the regime of t satisfying τ relax . t as thelong-time regime. We expect that the relaxation timesatisfies τ relax = Θ(1) with respect to the bath size N from the following reason. The relaxation of h e − σ i canbe associated with observables of system S, because un-der the condition (17), the heat Q is replaced by theenergy change of system S. Under physically reasonableconditions, the relaxation time of observables of systemS is Θ(1) when the initial state of bath B is the micro-canonical ensemble [60]. Besides, Ref. [61] showed thatthe relaxation time is independent of the size of the totalsystem if the Hamiltonian is random. Numerical resultsconsistent with the random Hamiltonians have been ob-tained with realistic models [62, 63]. Given these obser- FIG. 4. Division of bath B used in the discussion of theshort-time regime in Sec. V. We divide bath B into the nearpart (B ) and the far part (B ). The size of B depends on N as Θ( N µd ) (0 < µ < / (2 d )). System S interacts with apart of B . From the Lieb-Robinson bound, we introduce theLieb-Robinson time τ LR = Θ( N µ ), at which the informationof system S reaches B . We note that the velocity of theinformation propagation does not depend on N . vations, we expect τ relax = Θ(1) in our setup, which isconsistent with our own numerics as well.We next consider the short-time regime. To definethe short-time regime in line with Ref. [43], we dividebath B into B and B , where B is near S and B isfar from S (see Fig. 4). We refer to the boundary be-tween B and B as ∂ B, and the size of B is set tobe Θ( N µd ) (0 < µ < / (2 d )). Then, the short-timeregime is defined as t ≪ τ LR . Here, τ LR is the Lieb-Robinson (LR) time introduced by the Lieb-Robinsonbound [44–46], which, in the present case, represents atime scale that the information of S reaches B . Specif-ically, under the above choice of the size of B , the LRtime is given by τ LR = Θ( N µ ) . (24)Again in the special case that the condition (17) is sat-isfied, on the basis of the Lieb-Robinson bound and theETH, |h e − σ i − | = o (1) holds in the short-time regime,as shown in Ref. [43]. Even when the condition (17) doesnot hold, we expect that the interaction-induced errorwill not grow significantly if the initial state of bath Bsatisfies the ETH (see Appendix D 2).In the present paper, to confirm the validity of thefluctuation theorem in the short-time regime, we performsystematic numerical calculations of the bath size depen-dence of the error of the fluctuation theorem. Our resultsin Sec. V show that the error decreases as N increases,which supports the theory of the fluctuation theorem andthe expectation about the interaction-induced error.We note that the error of the fluctuation theorem is0 at t = 0, and initially increases in proportion to t asshown in Fig. 3. We theoretically show this time depen-dence in Sec. V A. In Sec. V, assuming that the error ofthe fluctuation theorem initially grows in the form of at with a being a t -independent coefficient, we numericallyconfirm that this coefficient a decreases as the bath sizeincreases, implying a = o (1). We note, as a side remark, that the time dependence ofthe change of the particle number in S (22) also initiallytakes the form of a n t with a n being a t -independentcoefficient. In this case, we observe that the bath sizedependence of a n is just a n = Θ(1). The bath size inde-pendence of a n is contrastive to the error of the fluctu-ation theorem a = o (1), implying that the ETH plays acrucial role only for the latter.The time regimes defined above have an overlap andthus cover the entire time domain as shown in Fig. 3,when the bath size is sufficiently large (see also Sec.V A).Suppose that the time evolution of the error is given as inFig. 3, i.e., the error of the fluctuation theorem initiallyincreases in t and relaxes monotonically to the long-timeaverage after the N -independent relaxation time. Then,the fluctuation theorem in the long-time regime impliesthat in the short-time regime, and vice versa. In general,however, the fluctuation theorem in these time regimesare theoretically shown independently, and they togethershow the fluctuation theorem in the entire time domain.In fact, only from the fluctuation theorem in the long-time regime, we cannot exclude the situation that the er-ror is not monotonic like Fig. 3, but overshoots to a valuelarger than o (1) after a time evolution of t and then re-laxes to the long-time average. This possibility can beexcluded from the fluctuation theorem in the short-timeregime.Before closing this section, we briefly remark on abso-lute irreversibility (see Appendices A and C 3 for details).As mentioned in the introduction, the numerical calcu-lations in Ref. [43] showed that h e − σ i is smaller than 1in the long-time regime, which was argued to be a viola-tion of the fluctuation theorem. As a matter of fact, thisis due to the numerical setup of Ref. [43] that the ini-tial state of system S was a pure state, causing absoluteirreversibility. That is, the reason why the fluctuationtheorem in the long-time regime appeared to be brokenin the numerical calculations of Ref. [43] is that the cor-rection term λ in Eq. (10) was not taken into account,while the numerical calculation itself was correct. In thepresent paper, we prove and numerically confirm that thefluctuation theorem holds in the long-time regime withabsolute irreversibility and the energy eigenstate bath, ifthe correction term λ is taken into account. Finally, weremark that there is an alternative approach to make thefluctuation theorem hold by regularizing the pure initialstate [64]. IV. LONG-TIME REGIME
In this section, we theoretically and numerically showthe fluctuation theorem in the long-time regime. InSec. IV A, we decompose the error of the fluctuation the-orems into several contributions. In Sec. IV B, we discussthe long-time average and the relaxation time to the long-time average. In Sec. IV C, we outline the proof of thefluctuation theorem. In Sec. IV D, we show our numericalresults to confirm our theory.
A. Definition of the error
In this subsection, we decompose the error of the fluc-tuation theorem h e − σ i− δG S in Eq. (31)and δG I in Eq. (33). Here, δG S vanishes in the thermo-dynamic limit, while the interaction-induced error δG I might remain. This decomposition of the error will alsobe used when we discuss the short-time regime in Sec. V.We denote h e − σ i when the initial state of bath B is ρ B (0) (resp. ρ canB ) by G (resp. G can ). We define G and G can as G := tr[ e − βH B U e βH B ρ B (0) U † ρ S ( t )] , (25) G can := tr[ e − βH B U e βH B ρ canB U † ρ canS ( t )] = 1 , (26)where ρ canS ( t ) is the reduced density operator of systemS defined as ρ canS ( t ) := tr B [ U ρ S (0) ⊗ ρ canB U † ].In the case where Eq. (17) holds, H B in G and G can can be replaced by − H S , and correspondingly we define G S := tr[ e βH S U e − βH S ρ B (0) U † ρ S ( t )] , (27) G canS := tr[ e βH S U e − βH S ρ canB U † ρ canS ( t )] . (28)Under Eq. (17), G = G S and G can = G canS hold, and theerror of the fluctuation theorem is written as G − G can = G S − G canS . (29)With the above argument, we define the error of thefluctuation theorem in the general case as G − G can = δG S + δG I , (30)where we define δG S := G S − G canS , (31) δG I := G − G can − δG S (32)= G − G S + G canS − G can (33)= δG (1)I + δG (2)I , (34) δG (1)I := G − G S , (35) δG (2)I := G canS − G can . (36)As argued above, δG I = 0 holds under Eq. (17). Wealso note that δG I = 0 holds when ρ B (0) = ρ canB , evenif Eq. (17) does not hold. Thus, we expect that | δG I | issmall when the initial state of bath B satisfies the ETH.See also Appendix B for the form of δG I . B. Long-time average
In this subsection, we consider the long-time averageof δG S = G S − G canS and discuss the relaxation times of δG S and δG I . To analyze the long-time average of G S , we first write G S as follows: G S := tr[ e βH S U e − βH S ρ B (0) U † tr B [ U ρ S (0) ⊗ ρ B (0) U † ]] . (37)We note that Eq. (37) contains four unitary operators.Then, under no degeneracy and the nonresonance con-dition (i.e., E a − E b + E c − E d = 0 holds only when thepair of indexes ( a, c ) equals ( b, d )), the long-time averageof G S can be written as G S = G S1 + G S2 . Here, G S1 and G S2 are defined as G S1 := Z S X a,c tr S [ e βH S tr B [ π a ρ canS ⊗ ρ B (0) π a ]tr B [ π c ρ S (0) ⊗ ρ B (0) π c ]] , (38) G S2 := Z S X a,ba = b tr S [ e βH S tr B [ π a ρ canS ⊗ ρ B (0) π b ]tr B [ π b ρ S (0) ⊗ ρ B (0) π a ]] , (39)where π a := | E a ih E a | , ρ canS := e − βH S /Z S , and Z S :=tr S [ e − βH S ]. We refer to G S1 as the diagonal term be-cause it contains the diagonal ensemble [65], which isthe long-time average of the density operator. We alsorefer to G S2 as the off-diagonal term, because G S2 con-tains the off-diagonal matrix elements with respect toenergy eigenstates of H . In a similar manner, we write G canS = G canS1 + G canS2 .In this paper, we prove | δG S | = o (1) . (40)From the foregoing discussion, the left-hand side is di-vided as δG S = ( G S1 − G canS1 ) + ( G S2 − G canS2 ). In the nextsubsection, we will show that | G S1 − G canS1 | = o (1) , (41) | G S2 | = o (1) , (42) | G canS2 | = o (1) , (43)which imply Eq. (40).We now discuss the relaxation time of δG S . FromEq. (37), we can write G S as G S = X i,j,k,l e β ( E S i − E S j ) p S k h ψ k ( t ) | q il S | ψ k ( t ) ih φ j ( t ) | q li S | φ j ( t ) i , (44) | ψ k (0) i := | p S k i ⊗ | E Bini i , (45) | φ j (0) i := | E S j i ⊗ | E Bini i , (46)where the spectral decomposition of ρ S (0) is ρ S (0) = P k p S k | p S k ih p S k | . As shown in Eq. (44), G S is written asa combination of the expectation values of q il S . Besides,the same rewrite is possible for G canS . In the total systemSB, if the initial state of bath B is the microcanonicalensemble, the relaxation time of an observable of sys-tem S to its long-time average is independent of the bathsize under some conditions on the matrix elements of theoperator and the initial state of SB [60]. Those condi-tions are expected to be satisfied if the amplitudes of thematrix elements obey the Gaussian distribution, whichis numerically confirmed in various quantum many-bodysystems satisfying the off-diagonal ETH [30, 33]. Then,we expect that the relaxation time of any system operatordoes not depend on the bath size also in our setup assum-ing the off-diagonal ETH. Since G S and G canS are writtenas a combination of the expectation values of operatorsin S, the relaxation time of δG S is also independent ofthe bath size.We also discuss the relaxation time of δG I to δG I .Since the interaction H I is local, we expect that the re-laxation time to be Θ(1) by the same argument as above. C. Outline of the proof