arXiv:cond-mat/0307619v2 [cond-mat.mes-hall] 31 Aug 2003 esalueuissc htteeeetr hreis charge elementary the that such units e use shall We a ag nainefor invariance gauge has nFg das h flux the also 1d Fig. In h o.I ecttewr nFg1 egtteopen the many get put we can we Fig.1d Finally in wire Fig.1e. the of cut geometry we lead If two dot. the tato srpeetdi i.2.Te“o”cnbe can “dot” ab- The its and 2c. 1b, Fig. an Fig. by in in represented represented presents is is topology straction ring a with rtehih fsm ares h hr aaee is parameter third The ring: the barriers. through some flux of the boundaries, height some the of location or the determine that voltages r Fg1.Tesaeo h igi otoldb some by controlled is geome- ring ring the a of with shape parameters system The model presentation (Fig.1). to a For try have on problem. focus we the we literature, of purpose definition published better the a to give relation its and hog h ytmdrn n yl faproi driving. periodic called a is of cycle This one during system the through uueqatmcrut rqatmgts osbyfor computing. possibly of quantum gates, realization of the quantum purpose in or the issue circuits major quantum a future as regarded is devices G uofruai h onrsoeo R.I losthe allows particular in It and LRT. coefficients, of the response stone The the corner of the calculation (”fields”). is parameters formula classical Kubo dependent time of aimt elwt rvnsses uhssesare systems Such Hamiltonian a systems. driven by with described deal to malism ecncluaetecag ( charge the calculate can we , .Nt htteHamiltonian the that Note 1. = nodrt xli h oiainfrtepeetwork, present the for motivation the explain to order In upn fcag nmssoi 4 n oeua size molecular and [4] mesoscopic in charge of Pumping ierrsos hoy(R)[–]i h edn for- leading the is [1–3] (LRT) theory response Linear unu upn ncoe ytm,aibtctasot an transport, adiabatic systems, closed in pumping Quantum odcac matrix conductance x oko ierrsos hoy h eaint h omnfo betw common distinguish the We to relation clarified. to The is butions magnetism” theory. “geometric response and linear of work h hsclyrlvn nrdet o h aclto of calculation the for ingredients relevant physically the are oe.I h atrcnetw nlz ielatti site th 3 with connection a the analyze on we remark We context latter device. pumping the simplest translat In by model. pumping barrier dissipative) d (non classical adiabatic examples: classical following the discuss we application rxmto.Tedvainfo xc uniaini rela a is quantized quantization which exact in from circumstances deviation define The we proximation. hand other the On a enue ocluaepmigi pnsystems. open in pumping calculate to used been has 1 pumping unu upn ncoe ytm scniee.W explain We considered. is systems closed in pumping Quantum and .INTRODUCTION I. x 3 .Mdlsystem Model A. S x Φ = arxta eed on depends that matrix 2 . hs aaeescnb gate be can parameters These . Q eateto hsc,BnGro nvriy Beer-Sheva University, Ben-Gurion Physics, of Department x nteoehn eosreta daai upn osntha not does pumping adiabatic that observe we hand one the On . ( 3 G φ ≡ srgre saprmtrof parameter a as regarded is ftesse.I eknow we If system. the of ) 7→ H (¯ h/e φ ( Q x 2 + hc stransported is which ) where ) ) φ H π ( nte system Another . x 1 ( t ) x x , ( x t 2 saset a is ) ( 1 t ) and x , oo Cohen Doron 3 ( x (1) t )) 2 . h omnjro feetia niern hscnbe between Using can distinguish We this driving. conversion. engineering AC-DC periodic electrical as a described of of jargon cycle common per the transported ring is the current”) through probability integrated “net say flux. with system ring driven structure a of periodic analysis driven an to a reduces of quasi is treatment the Hamiltonian mathematical therefore the and case translations, latter momentum periodic the unit the In for getting invariant hence Fig.1f. flux), of (no system series in units such taeta ntelmto pngoer h uofor- Kubo demon- the we geometry [10] open the of paper to the limit reduces up mula of the follow limit in a (subtle) that in strate special a category: regard as also second [6–9] We category category. mathemat- first second as the the [5] to category equivalent last previous ically the the of regard end the we at subsection explained was that reason a For h uofruacnb sd hc hudb distin- be Ref.[12]. of should mechanism which non-linear used, the mechanism be from pumping can guished linear formula of Kubo a case the having in for Only conditions mechanism. the a are having what of ify connotation in Hamiltonian the pumping in with for non-linear interest but synonym recent systems, a again periodic a is also which is present Ratchets[12], the There of con- scope the the beyond Paper. are with systems of systems as type These regarded having periodic be of in can notation pumping These of [11]. studies “ratchets” and cation” [6]. 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Thouless the to ted eHmloin hc ersnsthe represents which Hamiltonian, ce e daai n isptv contri- dissipative and adiabatic een popular e upn ehns.W r on oclar- to going are We mechanism. pumping φ htteKb oml otisall contains formula Kubo the that mltoso daai transport adiabatic of rmulations ibtcpmighlsa nap- an as holds pumping diabatic .Casfiaino pumps of Classification B. sacntn fmto.I olw htthe that follows It motion. of constant a is en htntcag o ab etrto better maybe (or charge net that means S damped 40,Israel 84105, S arxfraimwhich formalism matrix mti oml fB¨uttiker Pr´etre of formula -matrix Q h uoformula Kubo the d ihnteframe- the within ) et equantized. be to ve o-aitna dynamics. non-Hamiltonian linear pumping 2
C. Objectives The last example is the main one. In the context of open geometry it is known as “pumping around a resonance” The purpose of this Paper is to explain and demon- [8]. We explain that this is in fact an diabatic transfer strate that the Kubo formula contains all the physically scheme, and we analyze a particular version of this model relevant ingredients for the calculation of the charge (Q) which is represented by a 3 site lattice Hamiltonian. This which is pumped during one cycle of a periodic driving. is definitely the simplest pump circuit possible, and we In the limit of a very slow time variation (smallx ˙), the believe that it can be realized as a molecular size device. emerging picture coincides with the adiabatic picture of It also can be regarded as an approximation for the closed Refs.[5, 13–15]. In this limit the response of the system geometry version of the two delta potential pump [8]. is commonly described as a non-dissipative “geometric magnetism” [15] effect, or as adiabatic transport. A major objective of this Paper is to bridge between E. Outline the adiabatic picture and the more general LRT / Kubo picture, and to explain how dissipation emerges in the In Section 2 we define the main object of the study, quantum mechanical treatment. For one-parameter driv- which is the conductance matrix G of Eq.(5). The con- ing a unifying picture that bridges between the quan- ductance matrix can be written as the sum of a sym- tum mechanical adiabatic picture and LRT has been pre- metric (η) and an anti-symmetric (B) matrices, which sented in [16–19]. A previous attempt [20] had ended in are later identified as the dissipative and the adiabatic some confusion regarding the identification of the linear contributions respectively. response regime, while [15] had avoided the analysis of In the first part of the paper (Sections 2-8) we analyze the mechanism that leads to dissipation in the quantum the adiabatic equation (Section 3), and illuminate the mechanical case. distinction between its zero order solution (Section 4), The presented (Kubo based) formulation of the pump- its stationary first order solution (Section 5), and its non- ing problem has few advantages: It is not restricted to stationary solution (Section 6). The outcome of the anal- the adiabatic regime; It allows a clear distinction between ysis in Section 5 is Eq.(26) for the conductance matrix dissipative and adiabatic contributions to the pumping; G. This expression is purely adiabatic, and does not give The classical limit is manifest in the formulation; It gives any dissipation. In order to get dissipation we have to a level by level understanding of the pumping process; It look for a non-stationary solution. allows the consideration of any type of occupation (not The standard textbook derivation of the Kubo formula necessarily Fermi occupation); It allows future incorpo- (Eq.(29)) for the conductance matrix G implicitly as- ration of external environmental influences such as that sumes a non-stationary solution. We show how to get of noise; It regards the voltage over the pump as elec- from it Eq.(30) for η and Eq.(31) for B. The latter is tro motive force, rather than adopting the conceptually shown to be identical with the adiabatic result (Eq.(26)). more complicated view [9] of having a chemical potential In Section 7 we further simplify the expression for η lead- difference. ing to the fluctuation-dissipation relation (Eq.(33)). Of particular interest is the possibility to realize a The disadvantages of the standard textbook derivation pumping cycle that transfers exactly one unit of charge of Kubo formula make it is essential to introduce a dif- per cycle. In open systems [7, 8] this “quantization” holds ferent route toward Eq.(33) for η. This route, which is only approximately, and it has been argued [7] that the discusssed in Section 8, explicitly distinguishes the dis- deviation from exact quantization is due to the dissipa- sipative effect from the adiabatic effect, and allows to tive effect. Furthermore it has been claimed [7] that exact determine the conditions for the validity of either the adi- quantization would hold in the strict adiabatic limit, if abatic picture or LRT. In particular it is explained that the system were closed. In this Paper we would like to LRT is based, as strange as it sounds, on perturbation show that the correct picture is quite different. We shall theory to infinite order. demonstrate that the deviation from exact quantization In Section 9 we clarify the general scheme of the pump- is in fact of adiabatic nature. This deviation is related ing calculation (Eq.(39)). Section 10 and Section 11 give to the so-called “Thouless conductance” of the device. two simple classical examples. In Section 12 we turn to discuss quantum pumping, where the cycle is around a chain of degeneracies. The general discussion is fol- D. Examples lowed by presentation of the double barrier model (Sec- tion 13). In order to get a quantitative estimate for the We give several examples for the application of the pumped charge we consider a 3 site lattice Hamiltonian Kubo formula to the calculation of the pumped charge Q: (Section 14). The summary (Section 15) gives some larger perspec- classical dissipative pumping • tive on the subject, pointing out the relation to the S- classical adiabatic pumping (by translation) matrix formalism, and to the Born-Oppenheimer picture. • quantum pumping in the double barrier model In the appendices we give some more details regarding • the derivations, so as to have a self-contained presenta- 3 tion. matrix-vector multiplication. It reduces to the more fa- miliar cross-product in case that we consider 3 param- eters. The dissipation, which is defined as the rate in II. THE CONDUCTANCE MATRIX which energy is absorbed into the system, is given by
˙ d kj Consider the Hamiltonian (x(t)), where x(t) is = = F x˙ = η x˙ kx˙ j (7) H W dt hHi −h i · a set of time dependent parameters (”fields”). For Xkj presentation, as well as for practical reasons, we as- Only the symmetric part contributes to the the dissi- sume later a set of three time dependent parameters pation. The contribution of the antisymmetric part is x (t) = (x1(t), x2(t), x3(t)). We define generalized forces identically zero. in the conventional way as ∂ F k = H (2) III. THE ADIABATIC EQUATION −∂xk
1 Note that if x1 is the location of a wall element, then F is The adiabatic equation is conventionally obtained from the force in the Newtonian sense. If x2 is an electric field, the Schrodinger equation by expanding the wavefunction 2 then F is the polarization. If x3 is the magnetic field or in the x-dependent adiabatic basis: 3 the flux through a ring, then F is the magnetization or d i ψ = (x(t)) ψ (8) the current through the ring. dt| i −¯hH | i In linear response theory (LRT) the response of the ψ = a (t) n(x(t)) (9) system is described by a causal response kernel, namely | i n | i n ∞ X k kj ′ ′ ′ dan i i j F = α (t t ) x (t )dt (3) = Enan + x˙ j A am (10) h it − j dt −¯h ¯h nm j −∞ m j X Z X X where following [13] we define where αkj (τ) = 0 for τ < 0. The Fourier transform kj kj of α (τ) is the generalized susceptibility χ (ω). The j ∂ conductance matrix is defined as: Anm(x)= i¯h n(x) m(x) (11) ∂xj kj ∞ kj Im[χ (ω)] kj Differentiation by parts of ∂j n(x) m(x) = 0 leads to G = lim = α (τ)τdτ (4) j h | i ω→0 ω 0 the conclusion that Anm is a hermitian matrix. Note Z that the effect of gauge transformation is
Consequently, as explained further in Appendix B, the x −i Λn( ) response in the “DC limit” (ω 0) can be written as n(x) e h¯ n(x) (12) → | i 7→ Λn−Λm | i j i h¯ j k kj Anm e Anm + (∂j Λn)δnm (13) F = G x˙ j (5) 7→ h i − j j j Note that the diagonal elements An Ann are real, and X j j ≡ transform as An An + ∂j Λn. As an example for the applicability of this formula note 7→ Associated with An(x) is the gauge invariant 2-form, the following standard examples for one parameter driv- which is defined as: ing: Let x = wall or piston displacement, thenx ˙ = wall ij j i or piston velocity, G = friction coefficient, and F = Gx˙ Bn = ∂iAn ∂j An (14) − − is the friction force. Another standard example is x = = 2¯hIm ∂in ∂j n (15) magnetic flux, x˙ = electro motive force, G = electrical − h | i − 2 i j conductance, and hence F = Gx˙ is Ohm law. = Im AnmAmn (16) −¯h It is convenient to write the− conductance matrix as m kj kj kj kj jk X G η + B , where η = η is the symmetric part This can be written in an abstract notation as B = A. of the≡ conductance matrix, while Bkj = Bjk is the ∇∧ − Using standard manipulations, namely via differentiation antisymmetric part. In case of having three parameters by parts of ∂j n(x) m(x) = 0, we get for n = m the we can arrange the elements of the antisymmetric part as expressions: h |H| i 6 a vector B~ = (B23,B31,B12). Consequently Eq.(5) can i¯h ∂ i¯hF j be written in abstract notation as Aj (x)= n H m nm (17) nm E E ∂x ≡−E E m n j m n F = η x˙ B x˙ (6) − − and hence h i − · − ∧ i j where the dot product should be interpreted as matrix- ij Im FnmFmn vector multiplication, which involves summation over the Bn = 2¯h 2 (18) (Em En) index j. The wedge-product also can be regarded as a mX(6=n) − 4
IV. THE STRICTLY ADIABATIC SOLUTION, we ignore the zero order contribution, and go on with the AND THE BERRY PHASE first order contribution:
Wmn ∂ We define the perturbation matrix as F k = n H m + CC h i − En Em ∂xk mX(6=n) − W = x˙ Aj for n = m (19) nm − j nm 6 j = i Ak Aj + CC x˙ X nm mn j j m ! and W j = 0 for n = m. Then the adiabatic equation X X nm = Bkj x˙ (25) can be re-written as follows: − n j j X dan i i = (En xA˙ n)an Wnmam (20) stationary dt −¯h − − ¯h For a general preparation, either pure or m X mixed, one obtains Eq.(5) with If we neglect the perturbation W , then we get the strict Gkj = f(E ) Bkj (26) adiabatic solution: n n n X i t ′ ′ x(t) − En(x(t ))dt − An(x)·dx h¯ x where f(E ) are weighting factors, with the normal- e 0 (0) n(x(t)) (21) n ization f(E ) = 1. For a pure state preparation R R | i n n f(En) distinguishes only one state n, while for canoni- Due to An(x), we have the so called geometric phase. cal preparationP f(E ) exp( E /T ), where T is the This can be gauged away unless we consider a closed n ∝ − n cycle. For a closed cycle, the gauge invariant phase temperature. For a many-body system of non-interacting Fermions f(E ) can be re-interpreted as the Fermi occu- (1/¯h) A d~x is called Berry phase. n pation function, so that f(E ) is the total number of With the· above zero-order solution we can obtain the n n particles. followingH result: Thus we see that theP assumption of a stationary first- order solution leads to a non-dissipative (antisymmet- k ∂ F = n(x) H n(x) (22) ric) conductance matrix This is know as either Adiabatic h i −∂x k Transport [5, 14] or “Geometric Magnetism” [13]. In the
∂ later sections we shall discuss the limitations of the above = n(x) (x) n(x) (23) −∂xk h | H | i result. In case of the standard examples that were mentioned previously this corresponds to conservative force or to VI. THE NON-STATIONARY SOLUTION: THE persistent current. From now on we ignore this trivial KUBO FORMULA contribution to F k , and look for the a first order con- tribution. h i The Kubo formula is an expression for the linear re- sponse of a driven system that goes beyond the stationary V. THE STATIONARY ADIABATIC adiabatic solution of the previous section. The Kubo for- SOLUTION: ADIABATIC TRANSPORT OR mula has many type of derivations. One possibility is to “GEOMETRIC MAGNETISM” use the same procedure as in Section 5 starting with ψ(t) = e−iEnt n For linear driving (unlike the case of a cycle) the An(x) | i | i t ′ i(En−Em)t ′ −iEmt field can be gauged away. Assuming further that the adi- + iWmn e dt e m abatic equation can be treated as parameter independent − 0 | i m(6=n) Z (that means disregarding the parametric dependence of X En and W on x) one realizes that Eq.(20) possesses sta- For completeness we also give in Appendix A a simple tionary solutions. To first order these are: version of the standard derivation, which is based on a conventional fixed-basis first-order-treatment of the per- W ψ = n + mn m (24) turbation. The disadvantages are: | i | i En Em | i mX(6=n) − The standard derivation does not illuminate the un- • derlaying physical mechanisms of the response. Note that in a fixed-basis representation the above sta- The stationary adiabatic limit is not manifest. tionary solution is in fact time-dependent. Hence the no- • tations n(x(t)) , m(x(t)) and ψ(t) are possibly more The fluctuation-dissipation relation is vague. appropriate.| i | i | i • k The validity conditions of the derivation are not With the above solution we can write F as a sum • clear: no identification of the regimes. of zero order and first order contributions.h Fromi now on 5
For now we go on with the conventional approach, but assuming some weak coupling to an environment, or by in a later section we refer to the more illuminating ap- taking the limit of infinite volume. But we are dealing proach of [20] and [16–19]. Then we clarify what is the with a strictly isolated finite system, and therefore the regime (range ofx ˙) were the Kubo formula can be trusted meaning of Γ requires serious consideration. We post- [18, 19], and what is the sub-regime where the response pone the discussion of this issue to the next section. can be described as non-dissipative adiabatic transport. If the smoothed version of K˜ ij (ω) should be used in In order to express the Kubo formula one introduces Eq.(30), then it is possible to obtain ηij from power the following definition: spectrum C˜ij (ω) of the fluctuations. This is called the Fluctuation-Dissipation relation. The spectral function i Kij (τ)= [F i(τ), F j (0)] (27) C˜ij (ω) is defined as the Fourier transform of the sym- ¯hh i metrized correlation function
We use the common interaction picture notation 1 k iHt k −iHt Cij (τ)= (F i(τ)F j (0) + F j (0)F i(τ)) (32) F (τ)=e F e , where = (x) with x = const. h 2 i The expectation value assumesH thatH the system is pre- stationary We use again the interaction picture, as in the definition pared in a state (see previous section). It is ij also implicitly assumed that the result is not sensitive of K (τ). Also this function has a well defined classical to the exact value of x. Note that Kij(τ) has a well de- limit. fined classical limit. Its Fourier transform will be denoted There are several versions for the Fluctuation- K˜ ij (ω). Dissipation relation. The microcanonical version [15] has The expectation value F k is related to the driving been derived using classical considerations, leading to x(t) by the causal responseh kerneli αij (t t′). The Kubo 1 1 d expression for this response kernel, as− derived in Ap- ηij = g(E) C˜ij (ω 0) (33) |E 2 g(E) dE E → pendix A, is h i αij (τ) = Θ(τ) Kij (τ) (28) In Appendix C we introduce its quantum mechanical derivation. The subscript emphasizes that we assume a where the step function Θ(τ) cares for the upper cutoff of microcanonical state with energy E, and g(E) is the den- the integration in Eq.(3). The Fourier transform of αij (τ) sity of states. The traditional version of the Fluctuation- is the generalized susceptibility χij (ω). The conductance Dissipation relation assumes a canonical state. It can be matrix is defined as: obtained by canonical averaging over the microcanonical version leading to: ij ∞ ij Im[χ (ω)] ij G = lim = K (τ)τdτ (29) 1 ω→0 ω 0 ηij = C˜ij (ω 0) (34) Z |T 2T T → This can be split into symmetric and anti-symmetric components (see derivation in Appendix C) as follows: VIII. THE VALIDITY OF LINEAR RESPONSE ˜ ij ij 1 ij ji 1 Im[K (ω)] THEORY AND BEYOND η = 2 (G +G ) = lim (30) 2 ω→0 ω ∞ ˜ ij ij 1 ij ji dω Re[K (ω)] The standard derivation of the dissipative part of the B = 2 (G G ) = 2 (31) Kubo formula, leading to the Fluctuation-Dissipation re- − − −∞ 2π ω Z lation Eq.(33), is not very illuminating physically. More The antisymmetric part is identified [15] (Appendix C) troubling is the realization that one cannot tell from the as corresponding to the stationary solution Eq.(26) of the standard derivation what are the conditions for its va- adiabatic equation. lidity. An alternate, physically appealing derivation [16– 19], is based on the observation that energy absorption is related to having diffusion in energy space [20]. In VII. THE EMERGENCE OF DISSIPATIVE Appendix E we outline the main ingredients of this ap- RESPONSE, AND THE proach. FLUCTUATION-DISSIPATION RELATION It should be clear that the diffusion picture of Ap- pendix E holds only in case of chaotic systems. If this The Kubo formula for the symmetric part of the con- diffusion picture does not hold, then also the Kubo for- ductance matrix (ηij ) can be further simplified. If we mula for ηij does not hold! Driven one-dimensional sys- take Eq.(30) literally, then ηij = 0 due to the simple fact tems are the obvious example for the failure of linear re- that we have finite spacing between energy levels (see sponse theory (LRT). As in the case of the kicked rotator [21] for a statistical point of view). But if we assume (standard map) [22] there is a complicated route to chaos that the energy levels have some finite width Γ, then the and stochasticity: By increasing the driving amplitude smoothed version of K˜ ij (ω) should be considered. In the phase space structure is changed. If the amplitude common textbooks the introduction of Γ is “justified” by is smaller than a threshold value, then the diffusion is 6 blocked by Kolmogorov-Arnold-Moser curves, and con- mation of diagrams to infinite order, leading to an effec- sequently there is not dissipation. Therefore the Kubo tive broadening of the energy levels. By iterating FOPT, formula is not applicable in such cases. neglecting interference terms, we get a Markovian ap- The following discussion of dissipative response as- proximation for the energy spreading process. This leads sumes that we deal with a quantized chaotic system. We to the diffusion equation of Appendix E. This diffusion would like to discuss the reason and the consequences of can be regarded as arising from Fermi-golden-rule tran- having an energy scale Γ. In the standard derivation the sitions between energy levels. A simple ad-hoc way to assumption of having level broadening as if comes out of determine the energy level broadening is to introduce Γ the blue. As we already noted it is customary in text- as a lower cutoff in the energy distribution which is im- books to argue that either a continuum limit, or some plied by Eq.(24): small coupling to an environment is essential in order to 2 provide Γ. But this is of course just a way to avoid con- 2 ¯hxF˙ mn n ψ = | 4 | 4 (37) frontation with the physical problem of having a driven |h | i| (En Em) + (Γ) − isolated finite mesoscopic system. In fact the energy scale This constitutes a generalization of the well known proce- Γ is related to the rate (x ˙) of the driving: dure used by Wigner in order to obtain the local density 2/3 of states [16, 17]. However, in the present context we ¯hσ Γ = x˙ ∆ (35) do not get a Lorentzian. The width parameter Γ is de- ∆2 | | × termined self consistently from normalization, leading to Eq.(35) [disregarding numerical prefactor]. where for simplicity of presentation we assume one pa- We can summarize the above reasoning by saying that rameter driving. We use ∆ to denote the mean level there is a perturbative regime that includes an adiabatic spacing, and σ is the root mean square value of the ma- (FOPT) sub-regime. Outside of the adiabatic sub-regime trix element F between neighboring levels. In order nm we need all orders of perturbation theory leading to to derive the above expression for Γ we have used the Fermi-golden-rule transitions, diffusion in energy space, result of [19] (Sec.17) for the “core width” at the break- and hence dissipation. Thus the dissipative part of Kubo time t = t of perturbation theory. The purpose of the prt formula emerges only in the regime Γ > ∆, which is just present section is to give an optional ”pedestrian deriva- the opposite of the adiabaticity condition. The next ob- tion” for Γ, and to discuss the physical consequences. vious step is to determine the boundary of the perturba- Looking at the first order solution Eq.(24) of Section 5 tive regime. Following [18, 19] we argue that the required one realizes that it makes sense provided W ∆. mn condition is Γ ∆ . The bandwidth ∆ ¯h is defined This leads to the adiabaticity condition | | ≪ b b as the energy width≪ E E were the matrix∝ elements | n − m| ∆2 Fnm are not vanishingly small. If the condition Γ ∆b is x˙ (36) violated we find ourselves in the non-perturbative≪ regime | | ≪ ¯hσ where the Kubo formula cannot be trusted [18, 19]. If this condition is not satisfied one should go beyond We still have to illuminate why we can get in the per- first order perturbation theory (FOPT), in a sense to be turbative regime a dissipative linear response in spite of explained below. Note that this adiabaticity condition the breakdown of FOPT. The reason is having a separa- can be written as Γ ∆. tion of scales (∆ Γ ∆b). The non-perturbative The adiabaticity condition≪ Eq.(36) can be explained mixing on the small≪ energy≪ scale Γ does not affect in a more illuminating way as follows: Let us assume the rate of first-order transitions between distant lev- that we prepare the system at time t = 0 at the level els (Γ En Em ∆b). Therefore Fermi golden rule n . Using time dependent FOPT we find out that a picture≪ applies | − to the| ≪ description of the coarse grained stationary-like| i solution is reached after the Heisenberg energy spreading, and we get linear response. time tH = 2π¯h/∆. This is of course a valid description The existence of the adiabatic regime is obviously a provided we do not have by then a breakdown of FOPT. quantum mechanical effect. If we take the formal limit The condition for this is easily found to bext ˙ δx , ¯h 0 the adiabaticity condition Γ ∆ breaks down. H ≪ c → ≪ where δxc = ∆/σ. This leads again to the adiabaticity In fact the proper classical limit is non-perturbative, be- condition Eq.(36). cause also the weaker condition Γ ∆b does not sur- Another assumption in the derivation of Section 5 was vive the ¯h 0 limit. For further≪ details see [16–19]. → that we can ignore the parametric dependence of En and In the non-perturbative regime the quantum mechani- W on x. The adiabaticity condition t δx /x˙ mani- cal derivation of Kubo formula is not valid. Indeed we H ≪ c festly justify such an assumption: We should think of tH have demonstrated [18] the failure of Kubo formula in as the transient time for getting a stationary-like state, case of random-matrix models. But if the system has a and we should regard δxc as the parametric correlation classical limit, then Kubo formula still holds in the non- scale. perturbative regime due to semiclassical (rather than As strange as it sounds, in order to have dissipation, quantum-mechanical) reasons. it is essential to have a breakdown of FOPT. In the lan- The discussion of dissipation assumes a generic situa- guage of perturbation theory this implies a required sum- tion such that the Schrodinger equation does not have 7 a stationary solution. This means that driven one- tribution to Q, due to a persistent current, does not ex- dimensional systems are automatically excluded. An- ist. Furthermore, from the reciprocity relations (see Ap- other non-generic possibility is to consider a special driv- pendix B) it follows that G31 = G13, and G32 = G23, ing scheme, such as translation, rotation or dilation [23]. which should be contrasted with−G12 = G21. This means− In such case the time dependent Hamiltonian (x(t)) that a pumping cycle in the Φ = 0 plane is purely adia- possesses a stationary solution (provided the “velocity”H batic: there is no dissipative contribution to Q. Only the x˙ is kept constant). Consequently we do not have a dis- B~ field (second term in Eq.(39)) is relevant to the calcu- sipation effect. In Section 11 we discuss the simplest lation of the pumped charge, and its vertical component example of pumping by translation, where the station- B12 vanishes due to the time reversal symmetry. ary adiabatic solution of Section 5 is in fact exact, and The absence of dissipative contribution for a cycle in no dissipation arises. the Φ = 0 plane, does not imply that dissipation is not an issue. The symmetric part of the conductance ma- trix ηij is in general non-zero, leading to an energy ab- IX. APPLICATION TO PUMPING sorption rate which is proportional tox ˙ 2. This implies that the energy absorption per cycle is proportional to So far we have discussed the response for driving in a x˙ . Therefore we are able to minimize the dissipation very general way. From now on we focus on a system with effect| | by making the pumping cycle very slow. Further- a ring geometry as described in the Introduction, and more, if we get into the quantum-mechanical adiabatic illustrated in Fig. 1. The shape of the ring is controlled regime, then ηij becomes extremely small, and then we by some parameters x1 and x2, and x3 is the magnetic can neglect the dissipation effect as long as quantum- flux. The generalized force F 3 which is conjugate to the mechanical adiabaticity can be trusted. flux is the current. The time integral over the current is Whenever the dissipation effect cannot be neglected, the transported charge: one should specify whether or how a stationary operation is achieved. In case of pumping in open system the sta- Q = F 3 dt (38) tionary operation is implicitly guaranteed by having equi- h i librated reservoirs, where the extra energy is dissipated I to infinity. In case of pumping in closed system the issue In fact a less misleading terminology is to talk about of stationary operation is more subtle: In the adiabatic “probability current” and ”integrated probability cur- regime, to the extend that adiabaticity can be trusted, rent”. From a purely mathematical point of view it is we have a stationary solution to the transport problem, not important whether the transported particle has an as defined in Section 5. But outside of the adiabatic electrical charge. regime we have diffusion in energy space (Appendix E) Disregarding a possible persistent current contribu- leading to a slow energy absorption (dissipation). Thus tion, the expression for the pumped charge is: a driven system is heated up gradually (though possibly very slowly). Strictly speaking a stationary operation Q = G dx + B dx (39) is not achieved, unless the system is in (weak) thermal − · ∧ I I k=3 contact with some large bath. Another way to reach a If we neglect the first term, which is associated with the stationary operation, that does not involve an external dissipation effect, and average the second (”adiabatic”) bath, is by having an effectively bounded phase space. term over the flux, then we get This is the case with the mixed phase space example which is discussed in Ref.[12]. There the stochastic-like 1 motion takes place in a bounded chaotic region in phase Q = B d~x d~x (40) |adiabatic −2π¯h · ∧ space. ZZ The integration should be taken over a cylinder of ver- tical height 2π¯h, and whose basis is determined by the X. CLASSICAL DISSIPATIVE PUMPING projection of the pumping cycle onto the (x1, x2) plane. We already pointed out that the Berry phase Before we discuss the quantum mechanical pumping, (1/¯h) A d~x is gauge invariant. Therefore from Stokes n · it is instructive to bring simple examples for classical law it follows that (1/¯h) B d~x d~x is independent of pumping. In the following we consider one particle (r) H · ∧ the surface, and therefore (1/¯h) B d~x d~x with closed in a two dimensional ring as in Fig.1a. RR surface should be 2π integer. Integrating◦ · ∧ over a cylin- The first example is for classical dissipative pumping. der, as in Eq.(40), is× effectivelyRR like integrating over a The conductance G = G33 can be calculated for this closed surface (because of the 2π periodicity in the ver- system [24] leading to a mesoscopic variation of the Drude tical direction). This means that the flux averaged Q of formula. The current is given by Ohm law I = G Φ,˙ Eq.(40) has to be an integer. where Φ˙ is the electro-motive-force. − × The common interest is in pumping cycles in the Φ = 0 Consider− now the following pumping cycle: Change plane. This means that the zero order conservative con- the flux from Φ1 to Φ2, hence pumping charge 8
Q = G(1) (Φ2 Φ1). Change the conductance from XII. QUANTUM PUMPING G(1)− to G(2)× by− modifying the shape of the ring. Change the flux from Φ2 back to Φ1, hence pump- We turn now to the quantum mechanical case. Con- ing charge Q(2) = G(2) (Φ Φ ). Consequently the − × 1 − 2 sider an adiabatic cycle that involves a particular energy net pumping is level n. This level is assumed to have a degeneracy point at (x(0), x(0), Φ(0)). It follows that in fact there is a ver- Q = (G(2) G(1)) (Φ Φ ) (41) 1 2 − × 2 − 1 tical chain of degeneracy points:
Thus we have used the dissipative part of the conduc- (0) (0) chain = (x , x , Φ(0) +2π¯h integer) (46) tance matrix (first term in Eq.(39)) in order to pump 1 2 × charge. In the quantum mechanical version of this exam- These degeneracy points are important for the geo- ple extra care should be taken with respect to the zero metrical understanding of the B~ field, as implied by order contribution of the persistent current. Eq.(18). Every degeneracy point is like a monopole charge. The total flux that emerges from each monopole must be 2π¯h integer for a reason that was explained af- XI. CLASSICAL ADIABATIC PUMPING ter Eq.(40).× Thus the monopoles are quantized in units ofh/ ¯ 2. The B~ field which is created (so to say) by a ver- The second example is for classical adiabatic pumping. tical chain of monopoles may have a different near field The idea is to trap the particle inside the ring by a poten- and far field behavior, which we discuss below. tial well, and then to make a translation of the trap along The far field region exists if the chains are well isolated. a circle. The result of such a cycle is evidently Q = 1. Later we explain that “far” means gT 1, where gT is We would like to see how this trivial result emerges form the Thouless conductance. The far field≪ is obtained by the Kubo formula. regarding the chain as a smooth line. This leads qualita- Let (r, p) be the canonical coordinate of the particle tively to the same field as in Eq.(45). Consequently, for in the ring, while (x1, x2) are the center coordinate of a a “large radius” pumping cycle in the Φ = 0 plane, we trapping potential. The Hamiltonian is: get Q 1. In the following we are interested in the de- viation| |≈ from exact quantization: If φ(0) = 0 we expect to 1 1 have Q 1, while if φ(0) = π we expect Q 1. Only (r, p; x(t)) = p2 + p Φ(t) m ⊥ k 2 2 for the| φ|≥averaged Q of Eq.(40) we get exact| quantization|≤ . H 2 " − 2π x1 + x2 !# The deviation from Q 1 is extremely large if we + Utrap(r1 x1(t), r2 x2(t)) (42) | | ≈ (0) − − p consider a tight pumping cycle around a φ = 0 degen- eracy. After linear transformation of the shape param- where p and p are the components of the momentum k ⊥ eters, the energy splitting ∆ = E E of the energy along the ring and in the perpendicular (transverse) di- n m level n from its neighboring (nearly degenerated)− level m rections. The pumping is done simply by cycling the po- can be written as sition of the trap. The translation of the trap is assumed to be along an inside circle of radius R, ∆ = ((x x(0))2 + (x x(0))2 + c2(φ φ(0))2)1/2 (47) 1− 1 2− 2 − x(t) = (R cos(Ωt), R sin(Ωt), Φ=const) (43) where c is a constant. The monopole field is accordingly
In this problem the stationary solution of Section 5 is (0) (0) (0) c (x1 x , x2 x , x3 x ) an exact solution. Namely B~ = − 1 − 2 − 3 (48) ±2 (0) 2 (0) 2 c 2 (0) 2 3/2 ((x1 x1 ) + (x2 x2 ) + ( h¯ ) (x3 x3 ) ) mx r − − − ψ(t) =ei ˙ · n(x(t)) (44) | i | i where the prefactor is determined by the requirement where n(x) ψ(n)(r x) are the eigenfunctions of a of having a single (¯h/2) monopole charge. Assuming a particle| in thei 7→ trap. Eq.(44)− is just Galilei transformation pumping cycle of radius R in the Φ = 0 plane we get from the moving (trap) frame to the Laboratory frame. from the second term of Eq.(39) It is a-priori clear that in this problem the pumped charge per cycle is Q = 1, irrespective of Φ. Therefore Q = B dx = π√g (49) − ∧ ∓ T the B~ field must be I 3 (x , x , 0) where B~ = 1 2 (45) −2π(x2 + x2) 1 ∂2∆ c2 1 2 g = = (50) T ∆ ∂φ2 R2 This can be verified by calculation via Eq.(18). The sin- gularity along the x3 axis is not of quantum mechanical is a practical definition for the Thouless conductance in origin: It is not due to degeneracies, but rather due to this context. It is used here simply as a measure for the the diverging current operator (∂ /∂x 1/ x2 + x2). sensitivity of an energy level to the magnetic flux Φ. H 3 ∝ 1 2 p 9
What we want to do in the next sections is to interpo- participate in the pumping cycle carries Q = (1) net O late between the near field result, which is Q = (√gT ), charge: It takes an electron from the left side, and after and the far field result, which is Q = (1). ForO this pur- the bias reversal it emits it into the right side. Thus pose it is convenient to consider a particularO model that the pumping process in this model can be regarded as a can be solved exactly. particular example [14] of an adiabatic transfer scheme: The electrons are adiabatically transfered from state to state, one by one, as in “musical chair game”. XIII. THE DOUBLE BARRIER MODEL For a single occupied level the net Q is the sum of charge transfer events that take place in four avoided A simple example for quantum pumping is the double crossings (two avoided crossings in case of the lowest barrier model. An open geometry version of this model level). For many particle occupation the total Q is the has been analyzed in [8] using the S matrix approach. sum over the net Qs which are carried by individual lev- The analogous closed geometry version is obtained by els. For a dense zero temperature Fermi occupation the considering a one-dimensional ring with two delta bar- summation over all the net Qs is a telescopic sum, leav- riers. As we are going to explain below, the pumping ing non-canceling contributions only from the first and process in this model can be regarded as a particular ex- the last adiabatic crossings. The latter involve the last ample of an adiabatic transfer scheme: The electrons are occupied level at the Fermi energy. adiabatically transfered from state to state one by one as in “musical chair game”. The two delta barriers version of the double barrier XIV. THE THREE SITE LATTICE model is illustrated in Fig.2. The length of the ring is HAMILTONIAN L, with periodic boundary conditions on (L/2)
1 1 2π E = u +2√ cos θ + n (53) XV. SUMMARY AND DISCUSSION n 3 Q 3 3 We have shown how the Kubo formalism can be used where n =0, 1. The corresponding eigenstates are: ± in order to derive both classical and quantum mechanical iφ results for the pumped charge Q in a closed system. In c2e + c1En 1 2 this formulation the distinction between dissipative and n(x) 1 En (54) | i 7→ √S −iφ− non-dissipative contributions is manifest. c1e + c2En ! Within the framework of the Kubo formalism (disre- where S is the normalization, namely garding non-linear corrections) we have made a distinc- tion between the following levels of treatment: S = (1 E2 )2 + (c +c E )2 + (c +c E )2 (55) − n 1 2 n 2 1 n Strict adiabaticity • (outcome of zero order treatment) For the calculation of the pumped charge in the next Adiabatic transport paragraph it is useful to notice that for E = 1 the • normalization is S = 2(c c )2, while for E =± 0 the (outcome of stationary first order treatment) 1 ± 2 normalization is S 1. Dissipation ≈ After some algebra we find that the first component of • (the result of first order transitions) the B~ field in the Φ = 0 plane is In the adiabatic regime one can assume a stationary so- ∂ ∂ lution to the adiabatic equation, which implies no dis- B1 = 2 n(x) n(x) (56) − ℑ ∂u ∂φ sipation effect. This leads to the picture of adiabatic transport, where the Berry phase is the outcome of a 1 ∂S 2 2 zero order treatment, while the “geometric magnetism” = (c1 c2) 2 (57) − − S ∂u of Eq.(26) is the outcome of a first order treatment of the Which is illustrated in Fig.3. From here it follow that if inter-level couplings. we keep constant bias, and change only x2 = u, then the In some very special cases (translations, rotations and pumped charge is: dilations) this assumption (of having a stationary solu- tion) is in fact exact, but in generic circumstance this 1 final assumption is an approximation. Outside of the adia- Q = B1dx = (c2 c2) (58) − 2 − 1 − 2 S batic regime the stationary solution cannot be trusted. Z initial Assuming quantized chaotic dynamics one argues
For a planar (Φ = 0) pumping cycle around the negative that Fermi-golden-rule transitions between levels lead to vertical chain the main contribution to Q comes from the (slow) diffusion in energy (Eq.(E1)). This leads to the emergence of the dissipative part in the Kubo formula. two crossings of the x2 1 line. Hence we get ≈− We have obtained an expression (Eq.(35)) for the en- 2/3 c1 + c2 ergy scale Γ x˙ that controls the dissipative ef- Q = = 1+2gT (59) ∝ | | c1 c2 fect. We have explained that the dissipative contribu- − p tion to the Kubo formula is valid only in the regime where the Thouless conductance in this context refers to ∆ < Γ ∆b. Otherwise the dynamics is either of adia- the avoided crossing, and is defined as batic nature≪ (Γ ∆) or non-perturbative (Γ > ∆). ≪ In order to calculate the pumped charge Q we have 2 1 ∂ ∆ 2c1c2 to perform a closed line integral over the conductance gT = (60) ≡ ∆ ∂φ2 (c c )2 (Eq.(39)). This may have in general both adiabatic and φ=0 1 2 −
11 dissipative contributions. For the common pumping cy- use the Born-Oppenheimer basis x, n(x) = x n(x) . cle in the Φ = 0 plane, only the adiabatic contribution Then the Hamiltonian can be written| asi | i⊗| i exists. This follows from the reciprocity relations (Sec- 1 tion 9). Still we have emphasized (without any contradic- = (p Aj (x))2 + δ E (x) Htotal 2M j − nm nm n tion) that in the same circumstances a dissipation effect j typically accompanies the pumping process. X The quantum adiabatic contribution to the pumping where the interaction term is consistent with Eq.(19). is determined by a line integral over a B~ field which is Thus it is evident that the theory of driven systems is created by monopoles. The monopoles, which are related a special limit of this problem, which is obtained if we to the degeneracies of the Hamiltonian, are located along treat the xj as classical variables. vertical chains in x space (Eq.(46)). The 3 site model provides the simplest example for such vertical chains: By calculating the B~ field which is created (so to say) by Acknowledgments: It is my pleasure to thank these chains, we were able to determine the charge which Yshai Avishai (Ben-Gurion University), Yosi Avron is pumped during a cycle (e.g. Eq.(59)). (Technion), Thomas Dittrich (Colombia), Shmuel Fish- The (monopoles of the) vertical chains have near field man (Technion), Tsampikos Kottos (Gottingen), and regions (Eq.(48)). If the chains are well isolated in x Holger Schantz (Gottingen) for useful discussions. This space, then there are also far field regions. The far field research was supported by the Israel Science Foundation regions are defined as those where the Thouless conduc- (grant No.11/02), and by a grant from the GIF, the German-Israeli Foundation for Scientific Research and tance is very small (gT 1). Pumping cycles that are contained in the far field≪ region of a given chain lead to an Development. approximately quantized pumping Q = integer + (gT ). It is important to realize that the existence of farO field regions in x space is associated with having a low di- mensional system far away from the classical limit. In a quantized chaotic system it is unlikely to have gT 1 along a pumping cycle. As we take the ¯h 0 limit≪ the vertical chains become very dense, and the→ far field re- gions disappear. In the subtle limiting case of open geometry we expect to get agreement with the S-matrix formula of B¨uttiker Pr´etre and Thomas (BPT) [6]. Using the notations of the present Paper the BPT formula for the current that comes out of (say) the right lead can be written as: e ∂S G3j = trace P S† (62) 2πi ∂x j where P is the projector on the right lead channels. For G33 the above reduces to the Landauer formula. The details regarding the relation between the Kubo formula and the BPT formula will be published in a separate paper [10]. Here we just note that the derivation is based on a generalization of the Fisher-Lee approach [3, 25, 26]. Finally it is important to remember that the theory of driven systems is the corner stone for the analysis of interaction between “slow” and “fast” degrees of freedom. Assume that that the xj are in fact dynamical variable, and that the conjugate momenta are pj . The standard textbook example is the study of diatomic molecules. In such case xj are the locations of the nuclei. The total Hamiltonian is assumed to be of the general form 1 = p2 + (x) (63) Htotal 2M j H j X where is the Hamiltonian of the ”fast” degrees of free- dom (inH the context of molecular physics these are the electrons). Rather than using the standard basis, one can 12
APPENDIX A: THE KUBO FORMULA: APPENDIX B: REMARKS REGARDING THE STANDARD DERIVATION GENERALIZED SUSCEPTIBILITY
In this Appendix we present an elementary textbook- In this appendix we would like to further illuminate style derivation of the Kubo formula. For notational sim- the relation between the generalized susceptibility and plicity we write the Hamiltonian as = H0 f(t)V. It the conductance matrix. The generalized susceptibility is assumed that the system, in theH absence− of driving, χkj (ω) is the Fourier transform of the causal response kj is prepared in a stationary state ρ0. In the presence of kernel α (τ). Therefore it is an analytic function in the driving we look for a first order solution ρ(t)= ρ0 +˜ρ(t). upper half of the complex ω plan, whose real and imag- The equation forρ ˜(t) is: inary parts are related by Hilbert transforms (Kramers- Kronig relations): ∂ρ˜(t) H V i[ 0, ρ˜(t)] + if(t)[ ,ρ0] (A1) ∞ kj ′ ′ ∂t ≈− kj Im[χ (ω )] dω χ (ω) Re[χkj (ω)] = (B1) 0 ≡ ω′ ω π This equation can be re-written as Z−∞ − kj ∂ The imaginary part of χ (ω) is the sine transforms of U −1 U U −1VU kj ( 0(t) ρ˜(t) 0(t)) if(t)[ 0(t) 0(t),ρ0] α (τ), and therefore it is proportional to ω for small ∂t ≈ frequencies. Consequently it is convenient to write the where U0(t) is the evolution operator which is generated Fourier transformed version of Eq.(3) as by H0. The solution of the latter equation is k kj kj [ F ]ω = χ (ω)[xj ]ω µ (ω)[x ˙ j ]ω (B2) t h i 0 − ′ ′ ′ j ρ˜(t) i[V( (t t )),ρ0] f(t )dt (A2) X ≈ − − Z where the dissipation coefficient is defined as where we use the usual definition of the “interaction pic- ture” operator V(τ)= U (τ)−1VU (τ). Im[χkj (ω)] ∞ sin(ωτ) 0 0 µkj (ω)= = αkj (τ) dτ (B3) Consider now the time dependence of the expectation ω ω Z0 value F t = trace(Fρ(t)) of an observable. Disregarding the zeroh i order contribution, the first order expression is In this paper we ignore the first term in Eq.(B2) which signify the non-dissipative in-phase response. Rather t ′ ′ ′ we put the emphasis on the “DC limit” (ω 0) F t itrace (F[V( (t t )),ρ0]) f(t )dt → h i ≈ − − of the second term. Thus the conductance matrix kj kj Z t G = µ (ω 0) is just a synonym for the term “dis- = α(t t′) f(t′)dt′ sipation coefficient”.→ However, “conductance” is a bet- − Z ter (less misleading) terminology: it does not have the where the response kernel α(τ) is defined for τ > 0 as (wrong) connotation of being specifically associated with dissipation, and consequently it is less confusing to say
α(τ) = i trace (F[V( τ),ρ0]) that it contains a (non-dissipative) adiabatic component. − For systems where time reversal symmetry is broken = i trace ([F, V( τ)]ρ ) − 0 due to the presence of a magnetic field , the response = i [F, V( τ)] B h − i kernel, and consequently the generalized susceptibility = i [F(τ), V] (A3) and the conductance matrix satisfies the Onsager reci- h i procity relations We have used above the cyclic property of the trace oper- U U−1 ij ji ation; the stationarity 0ρ0 0 = ρ0 of the unperturbed α (τ, ) = [ ] α (τ, ) (B4) F U −1FU −B ± B state; and the definition (τ)= 0(τ) 0(τ). χij (ω, ) = [ ] χji(ω, ) (B5) −B ± B Gij ( ) = [ ] Gji( ) (B6) −B ± B where the plus (minus) applies if the signs of F i and F j transform (not) in the same way under time reversal. These reciprocity relations follow from the Kubo formula (Eq.(28), using Kij ( τ, ) = [ ]Kij(τ, ), together with the trivial identity− −BKij ( τ,− ±) = KBji(τ, ). In Section. 9 we discuss the implications− B of− the reciprocityB relations in the context of pumping. 13
APPENDIX C: EXPRESSIONS FOR B AND G
The functions Cij (τ) and Kij (τ) are the expectation values of hermitian operators. Therefore they are real functions. It follows that the real part of their Fourier transform is a symmetric function with respect to ω, while the imaginary part of their Fourier transform is anti symmetric with respect to ω. By definition they satisfy Cij (τ)= Cji( τ) and Kij (τ)= Kji( τ). It is convenient to regard them as the real and imaginary parts of one complex function− Φij (τ). Namely, − − ¯h Φij (τ) = F i(τ)F j (0) = Cij (τ) i Kij (τ) (C1) h i − 2 1 Cij (τ) = Φij (τ)+Φji( τ) (C2) 2 − i Kij (τ) = Φij (τ) Φji( τ) (C3) ¯h − − It is possible to express the decomposition Gij = ηij + Bij in terms of K˜ ij (ω). Using the definition Eq.(29) we get:
∞ ∞ ˜ ij ˜ ij ij ij Re[K (ω)] dω 1 Im[K (ω)] G = K (τ)τdτ = 2 + (C4) 0 − −∞ ω 2π 2 ω Z Z " #ω=0 The first term is antisymmetric with respect to its indexes, and is identified as Bij . The second term is symmetric with respect to its indexes, and is identified as ηij . The last step in the above derivation involves the following identity that hold for any real function f(τ)
∞ ∞ dω ∞ ∞ dω 1 f(τ)τdτ = f˜(ω) e−iωτ τdτ = f˜(ω) + iπδ′(ω) = 2π 2π −ω2 Z0 Z−∞ Z0 Z−∞ ∞ ˜ ∞ ˜ ˜ dω Re[f(ω)] ˜ ′ Re[f(ω)] dω 1 Im[f(ω)] = 2 πIm[f(ω)]δ (ω) = 2 + (C5) −∞ 2π − ω − − −∞ ω 2π 2 ω Z ! Z " #ω=0 Note that Im[f˜(ω)] is the sine transform of f(τ), and therefore it is proportional to ω is the limit of small frequencies. ij i It is of practical value to re-derive Eq.(C4) by writing Φ (τ) using the energies En and the matrix elements Fnm. Then we can get from it straightforwardly (using the definitions) all the other expressions. Namely:
E E Φij (τ) = f(E ) F i F j exp i m− n t (C6) n nm mn − ¯h n m X X E E Φ˜ ij (ω) = f(E ) F i F j 2πδ ω m− n (C7) n nm mn − ¯h n m X X F i F j F j F i χij (ω) = f(E ) − nm mn + nm mn (C8) n ¯hω (E E )+i0 ¯hω+(E E )+i0 n,m m n m n X − − − ηij = 2π¯h f(E ) Re F i F j δ′(E E ) (C9) − n nm mn m − n n m(6=n) X X Im F i F j Bij = 2¯h f(E ) nm mn (C10) n (E E )2 n m n X mX(6=n) − One observes that the expression for Bij coincides with the adiabatic transport result Eq.(26). Alternatively this identification can be obtained by expressing the sum in Eq.(C10) as an integral, getting form it the first term in Eq.(C4):
2 ∞ Im[Φ˜ ij (ω)] dω 2 ∞ Im[C˜ij (ω)] h¯ Re[K˜ ij (ω)] dω ∞ Re[K˜ ij (ω)] dω Bij = = − 2 = (C11) ¯h ω2 2π ¯h ω2 2π − ω2 2π Z−∞ Z−∞ Z−∞ 14
APPENDIX D: EXPRESSING K˜ (ω) USING C˜(ω)
We can use the following manipulation in order to relate K˜ ij (ω) to C˜ij (ω),
˜ ij ˜ ij K (ω) = f(En) Kn (ω) (D1) n X i = 2π f(E )(F i F j δ(ω + ω ) F j F i δ(ω ω )) ¯h n nm mn nm − nm mn − nm nm X i = 2π f(E )( F i F j δ(ω + ω )+ F j F i δ(ω ω )) ¯h m − nm mn nm nm mn − nm nm X i f(E ) f(E ) = 2π n − m (F i F j δ(ω + ω ) F j F i δ(ω ω )) ¯h 2 nm mn nm − nm mn − nm nm X f(E ) f(E ) = iωπ n − m (F i F j δ(ω + ω )+ F j F i δ(ω ω )) − E E nm mn nm nm mn − nm nm n m X − = iω f ′(E ) Cij (ω) − n n n X where we use the notation ωnm = (En Em)/¯h. The third line differs from the second line by permutation of the dummy summation indexes, while the fourth− line is the sum of the second and the third lines divided by 2. In the last equality we assume small ω. If the levels are very dense, then we can replace the summation by integration, leading to the relation:
g(E)dE f(E) K˜ ij(ω) = iω g(E)dE f ′(E) C˜ij (ω) (D2) E − E Z Z ˜ ij ˜ij where KE (ω) and CE (ω) are microcanonically smoothed functions. Since this equality hold for any smoothed f(E), it follows that the following relation holds (in the limit ω 0): → 1 d K˜ ij (ω) = iω g(E)Cij (ω) (D3) E g(E) dE E h i If we do not assume small ω, but instead assume canonical state, then a variation on the last steps in Eq.(D1), using the fact that (f(E ) f(E ))/(f(E )+f(E )) = tanh((E E )/(2T )) is an odd function, leads to the relation n − m n m n− m 1 ¯hω K˜ ij (ω) = iω tanh Cij (ω) (D4) T × ¯hω 2T T Upon substitution of the above expressions in the Kubo formula for ηij , one obtains the Fluctuation-Dissipation relation.
APPENDIX E: THE KUBO FORMULA AND THE The energy of the system is = Eρ(E)dE. It follows DIFFUSION IN ENERGY SPACE that the rate of energy absorptionhHi is R d ∞ ∂ ρ(E) = dE g(E) D (E2) The illuminating derivation of Eq.(33) is based on the dt hHi − E ∂E g(E) observation that energy absorption is related to having Z0 diffusion in energy space. Let us assume that the proba- For a microcanonical preparation we get bility distribution ρ(E)= g(E)f(E) of the energy satis- d 1 d fies the following diffusion equation: = [g(E) D ] (E3) dthHi g(E) dE E
∂ρ ∂ ∂ 1 This diffusion-dissipation relation reduces immediately to = g(E)D ρ (E1) ∂t ∂E E ∂E g(E) the fluctuation-dissipation relation if we assume that the 15 diffusion in energy space due to the driving is given by Squaring this expression, and performing microcanonical averaging over initial conditions we obtain: 1 D = C˜ij (ω 0)x ˙ x˙ (E4) E 2 E → i j ij t t X δE2(t) = x˙ x˙ Cij (t′′ t′)dt′dt′′ (E7) i j E − Thus it is clear that a theory for linear response should ij 0 0 X Z Z establish that there is a diffusion process in energy space due to the driving, and that the diffusion coefficient where Cij (t′′ t′) = F i(t′)F j (t′′) is the correlation is given by Eq.(E4). More importantly, this approach E function. For− very shorth times thisi equation implies also allows treating cases where the expression for D is E “ballistic” spreading (δE2 t2) while on intermediate non-perturbative, while the diffusion-dissipation relation time scales it leads to diffusive∝ spreading δE2(t)=2D t, Eq.(E3) still holds! E where A full exposition (and further reference) for this route of derivation can be found in [16–19]. Here we shall 1 ∞ give just the classical derivation of Eq.(E4), which is ex- D = x˙ x˙ Cij (τ)dτ (E8) E 2 i j E tremely simple. We start with the identity ij −∞ X Z d ∂ = H = x˙ F k(t) (E5) dthHi ∂t − k The latter result assumes a short correlation time. This Xk is also the reason that the integration over τ can be ex- tended form to + . Hence we get Eq.(E4). We note Assuming (for presentation purpose) that the ratesx ˙ k that for long−∞ times the∞ systems deviates significantly from are constant numbers, it follows that energy changes are the initial microcanonical preparation. Hence, for long related to the fluctuating F k(t) as follows: times, one should justify the use of the diffusion equa- t tion (E1). This leads to the classical slowness condition k ′ ′ δE = t 0 = x˙ k F (t )dt (E6) which is discussed in Ref.[19]. hHi − hHi − 0 Xk Z
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(a) (b) Flux
Flux Ring wire
dot
(c) (d) (e) wire wire Flux left lead right lead
dot dot dot
(f)
FIG1. Illustration of a ring system (a). The shape of the ring is controlled by some parameters x1 and x2. The flux through the ring is x3 = Φ. A system with equivalent topology, and abstraction of the model are presented in (b) and (c). The “dot” can be represented by an S matrix that depends on x1 and x2. In (d) also the flux x3 is regarded as a parameter of the dot. If we cut the wire in (d) we get the open two lead geometry of (e). If we put many such units in series we get the period system in (f).
En bias E n (x(t))
wire states dot state dot level
position time FIG2. Schematic illustration of quantum pumping in a closed wire-dot system. The net charge via the third level (thick solid line on the right) is vanishingly small: As the dot potential is lowered an electron is taken from the left side (first avoided crossing), and then emitted back to the left side (second avoided crossing). Assuming that the bias is inverted before the dot potential is raised back, only the second level carry a net charge Q = O(1).
1 B1 -2 -1 1 X2 2 -1 -2 2 En
-3 1
-4 -2 -1 1 x2 2 -5 -1 -2 -6 -7 FIG3. The first component of the B~ field for a particle in the middle level of the 3 site lattice model. It is plotted as a function of the dot potential x2 = u. The other parameters are φ = 0, and c1 = 0.1, while c2 = 0.04 for the thick line and c2 = 0.02 for the thin line. In the limit c2 → 0, all the charge that is transfered from the left side into the dot during the first avoided crossing, is emitted back into the left side during the second avoided crossing. Inset: The eigenenergies En(x) for the c2 = 0.04 calculation.