Quantum charge pumping and electric polarization in Anderson insulators Chyh-Hong Chern∗ Institute for Solid State Physics, University of Tokyo, Kashiwanoha, 5-1-5, Kashiwa 277-8581, Japan Shigeki Onoda and Shuichi Murakami CREST, Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Tokyo 113-8656, Japan 7 0 0 Naoto Nagaosa 2 CREST, Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Tokyo 113-8656, Japan, and Correlated Electron Research Center, n National Institute of Advanced Industrial Science and Technology, 1-1-1, Higashi, Tsukuba, Ibaraki 305-8562 a J (Dated: February 6, 2008) 0 We investigate adiabatic charge pumping in disordered system in one dimension with open and 3 closed boundary conditions. In contrast to the Thouless charge pumping, the system has no gap even though all the states are localized, i.e., strong localization. Charge pumping can be achieved ] n by making a loop adiabatically in the two-dimensional parameter space of the Hamiltonian. It is n because there are many δ-function-like fluxes distributed over the parameter space with random - strength,in sharp contrast tothesingle δ-functionin thepurecase. This providesanewand more s efficient way of charge pumpingand polarization. i d . t a INTRODUCTION dependence since we consider the adiabatic change, but m thedifferentpathC andC intheQ~-spacemightleadto 1 2 - The quantum dynamics of the charge in insulators is the different values of ∆P~. This is related to the single- d a rich and non-trivial issue even in the d.c. limit. Of valueness and Chern number of the Bloch wavefunction. n o course, the d.c. conductivity vanishes, but this does not Onodaetal. addressedthisproblemasfollowsbyanalyz- c meanthatchargemotionisfrozenintheinsulators. One ing the one-dimensional two-band models characterized [ example is the ferroelectricity and electric polarization bythethreedimensionalQ~ [6,7]. Thereappearsasingu- 2 in insulators. The classical definition of the electric po- lar line, i.e., string, in the Q~-space corresponding to the v larization P~ = d~r~rρ(~r) ( ρ(~r): charge density) fails trajectory of the monopole (band crossing point) as the 1 for the extended Bloch wave functions since the ~r is un- momentum k moves, which acts as the “current circuit” 8 bounded. This Rdifficulty is avoided by considering in- to produce the “magnetic field” dP via the Biot-Savart 6 dQ~ 1 stead the new definition for the difference of the polar- law. Awayfromthe string,the systemis alwaysgapped, 0 ization [1, 2, 3, 4, 5], andinsulating. Whentheadiabaticchangeoftheparam- 7 eter Q~ makes a loop enclosing the string, the charge is 0 ∆P~ = T dτdP~, (1) pumped during this process,which is quantizedto be an at/ Z0 dτ integer multiple of e since the strength of the “current” m where the change of the polarization between the initial is quantized. This is a realizationofthe quantumcharge and final states are given by the integral of the polariza- pumping first proposed by Thouless [8]. This analogy - d tion current during the adiabatic change of the param- to the magnetostatics says that the polarization is path on emteernstsQ~. O=n(eQc1a,nQu2s,u·a·l·l,yQcnh)oosusechthaesitnhiteiaaltostmaitcedwisitphlatchee- innodteepnecnldoesnetthaes slotrnignga.sTthheisloisopusCual=lyCth1e+ca(−seCb2e)cdauosees :c inversionsymmetrywithoutelectricpolarization,and(1) the change of the parameter Q~ is rather small and we v uniquely determines the polarization of the final state need the huge variation of Q~ to enclose the string, i.e., i X of our interest. Here by using dP~ = dP~ dQi, the µ- gaplessstates. Inotherwords,theferroelectricitycanbe dτ dQi dτ regardedas“afractionofthequantumchargepumping”. ar component of ∆P~ is expressed as Thischargepumpingcanberegardedastherigidshift ∆P = dQ~ · dPµ (2) of the wavefunction due to the change in the external µ ZC dQ~ parameters such as the atomic positions. It is natural when the wavefunction consists of the Bloch states ex- with the path C specified by Q~ = Q~(τ). What is found tended over the whole sample, and then the quantum by Resta [3] and King-Smith and Vanderbuilt [4] is that interference pattern is modified by the external parame- dPµ canbe representedby the Berryphase,whichfits to ters. One can estimate roughly how much the charge is dQ~ the first-principle band calculation. Then the question pumped as below. Let the dimension of the parameters arises, “Is there any path (C) dependence of the polar- be the energy. Then the distance between the physically ization ?”. It is clear that there is no parametrization realizedsetofparametersandthatofgaplessstates,i.e., 2 string, is the energy gap E . When ∆ be the change of parameter space including the angle α for the twisted G thetheparametersinunitofenergy,theanglesubtended boundarycondition, andthe roleofmagnetic monopoles by this segment in parameter space is roughly estimated in this space. Section V is devoted to the conclusions. as 2π∆/E . Since the 2π winding corresponds to unit G chargeeshiftbyonelattice constanta,i.e.,P =ea,the 0 polarizationisroughlygivenbyP =ea∆/E [6]. There- A MODEL FOR DISORDERED INSULATOR G fore one can enhance the dielectric response by reducing E , or enlarging ∆. One possible method to reduce E The minimal model for ferroelectrics is given by the G G is by introducing disorder, by which even the gapless following ionic dimer model; insulator can be realized. However, the electron wave- L functions are no longer the extended Bloch states in the H = −1 t −(−)iQ (c† c +h.c.) presence of the disorder, and one needs to worry about pure 2 n.n. 2 i+1 i i=1 the localization,i.e, Andersonlocalization. Whenallthe X(cid:0) (cid:1) L states are strongly localized, one can not transmit the + (−)iQ c†c . (3) 1 i i phaseinformationthoughthesample,andhencecannot i=1 X expect the charge pumping either. On the other hand, as can be seen from the above explanation, the charge Here, ci and c†i are the annihilation and creation opera- pumping is closely related to the topological nature of tors of the electron at the site i=1,···,L with L being the wavefunctions in the parameter space, which is ro- the number of sites in the system. For open system at- bustagainstthedisordertosomedegree. Forexample,in tached to leads at the both ends, cL+1 and c†L+1 repre- the two dimensional electron systems under strong mag- sentsthose operatorsinone ofthe leads,while forclosed netic field, there occur discrete extended states protected system, they are understood as c1 and c†1. tn.n. is the by the topology. Namely, the Chern number is carriedby transfer integral. Q1 and Q2 represent the alternations the extended states only, which is not destroyed by the of the local ionic level and the bond dimerization, re- weak disorder. Therefore similar problem arises even for spectively. The spin degree of freedom is omitted for the one-dimensional systems which we study below. Niu simplicity. Then, of our interest is the half-filling case andThouless[9]wasthefirsttostudythestabilityofthe relevanttotheferroelectrics. Althoughthismodelmight charge pumping against the weak disorder by a topolog- lookspecial,itrepresentsthe twoessentialfeatureofthe ical argument. Even though all the states are localized, ferroelectrics, i.e., (i) the two species of the ions charac- the charge pumping was shown to be unchanged as long terizedbythelevelalternationQ1and(ii)relativeshiftof asthegapremainsfinite. However,questionsstillremain theatomicpositionsdescribedbythedimerizationQ2. It such as what is the physical mechanism of the charge correspondsto the ferroelectricityine.g. BaTiO3, where pumping through the localized states and what happens Ti and O are dimerized to produce the polarization[11]. when the gap collapses. It can be also applied to the quasi-one dimensional fer- roelectric materials such as organic charge transfer com- In this paper, we study the effect of the disorder on pounds TTF-CA [12] and (TMTTF) X [13]. the charge pumping and dielectric response in a one- 2 The Hamiltonian H under the periodic boundary dimensionalmodelforinsulators. Combiningthenumeri- pure calsimulationsandanalyticconsiderations,werevealthe condition yields two bands[6] physical picture of the charge pumping by the localized states both in the case of open boundary condition and ε±(k)=± t2n.n.cos2k+Q21+Q22sin2k. (4) periodic/twisted boundary condition. The former one is q Experimentally, the parameters Q~ ≡ (Q ,Q ) can be more relevant to the experimental situation such as the 1 2 controlledbyapplyingtheelectricfieldEalongthepolar- FeRAM(ferroelectricrandomaccessmemory)wherethe izationdirectionandthe pressurep as follows. Electrons leads are attached to the thin film of insulators, while at high and low deinsity sites shift in relatively opposite the latter is more appropriate to see the role of topol- directions, as shown in open arrows of Fig. 1 (b), chang- ogy. We have published a Letter focusing on the open ing the dimerization Q by δQ ∝ eEQ . Simultane- boundary condition [10]. This paper provides the full 2 2 1 ously,withineachdimer,aleveldifferenceQ changesby details of the formulation, calculations, and additional 1 δQ ∝−eEQ asillustratedbyblackarrowsinFig.1(b). results onthe open system, as well as new results on the 1 2 Therefore, the electric field E mainly controls the angle periodic/twisted boundary condition. The plan of this paper follows. In section II, a model θ =arctan(Q /Q ). (5) 2 1 for disordered insulator showing the charge pumping is introduced. Its analysis with the open boundary condi- Applying the pressure, one can increase the hybridiza- tionisgiveninsectionIII,includingthedetaileddescrip- tion, and then reduces the ratio Q/tn.n. with tionofthe resonanttunnelling mechanism. InsectionIV given the analysis of the model in the three dimensional Q= Q2+Q2. (6) 1 2 q 3 Todiscussthequantumrelaxorbehavior,weintroduce OPEN BOUNDARY CONDITION theon-siterandompotentialv tothe Hamiltoniangiven i by (3); Inthissection,dielectricpropertyofdisorderedinsula- tors connected to the leads is studied theoretically. This L corresponds to the realistic situation in FeRAM devices, H =Hpure+ vic†ici. (7) wherethepolarizationismeasuredbytheintegratedcur- Xi=1 rent flowing in the leads. Especially in nano-scale sam- ples of the FeRAM, the quantum nature of the polariza- The type of the random distribution takes either a uni- tion is expected to play essential roles via novel quan- form distribution or an alloy model, which are shown tum interference as discussed in the introduction [6, 8]. in Fig. 2. In the following, we study effects of on-site Namely,a theory forthe nano-scalecapacitanceis called disorder on dielectric properties in both open and closed for where the quantum coherence is attained all through systems,particularlyfocusingonthetopologicalaspects. the sample. ¿From the viewpoint of the applications, it is requiredto achieve(i) amagnitude ofthe polarization larger than 10 µC/cm2, (ii) a smaller leak current than 0.1-1 µA/cm2, and (iii) a dielectric constant larger than 300 [14]. By introducing disorder, one can also suppress the dissipation by leak current due to the strong local- ization effect [9, 15] in addition to the reduction of the gap and enhanced dielectric response. In fact, a similar disorder-drivenenhanced response has been found in re- laxorferroelectrics[16,17,18]andpinnedchargedensity waves [19], although its mechanism is due to a classical mesoscopicclusterformationanalogoustospinglasssys- tems. In the disordered case with all the states being local- ized,thechargetransferoccursthroughtheresonanttun- nelling. The quantum nature is topologically protected. Crossoverfromthequantizedchargepumpinginbulkin- sulator [8] to that of quantum dot [20] is clarified in the controlparameter plane as a function of sample size and FIG.1: (Coloronline)Ionicdimersystemsandwitchedbythe disorderstrength[21,22,23]. Then,wetheoreticallypro- leads. posetheenhancementofthedielectricresponseofinsula- tors by disorder in nanoscopic/mesoscopic multichannel systems, with the required time for the dissipationless adiabatic charge displacement being estimated. Appli- cation of this phenomenon to memory devices, like fer- roelectric random access memory (FeRAM) [14], is also discussed. Single-channel problem for uniformly distributed random potential We consider an insulating electronic system sand- wiched by leads (electrodes) as shown in Fig. 2(a). For simplicity, we take a one-dimensional (single-channel) model, but the extension to higher dimensional (multi- channel) cases is straightforwardas discussed later. We take the total Hamiltonian H =H +H (8) tot lead FIG.2: (a)Uniformdistributionoftheon-siterandompoten- with H and H being given by (7) and lead tial. (b) Alloy model for theon-siterandom potential, which mimics effects of substitution in alloys. t −∞ ∞ H = − n.n. + c† c +h.c. , (9) lead 2 i+1 i ! Xi=0 Xi=L (cid:16) (cid:17) 4 respectively. This model is schematically shown in oftheregionintheQ~ spacewithalargeT isoftheorder Fig. 1 (b). The Green’s function Gi,i′(ε) for the above oftn.n./LforalargeL. Namelythevortexcorrespondsto model is readily obtained from the recursion formula in the gapless case, where the extended state at the Fermi the form of the continuued fraction [24]. Here, we can energy carries the charge and causes the perfect trans- concentrate on the case where the chemical potential is mittance. Remarkably, this perfect-transmission point located at the zero energy. Q~ = ~0 governs topological properties of the system in We adopt the Landauer-Bu¨ttiker formalism [24, 25], thewholeQ~ planeinanon-localway. Whenweadiabat- wherethesampleisregardedasascatterercharacterized icallychangetheparameterQ~ alongacyclearoundQ~ =~0 by the scattering S-matrix at which the transmittance T vanishes, the charge e is pumped and the polarization changes by ±2ea (a: lat- r t S = . (10) tice constant) according to the Brouwer’s formula (13). t∗ r′ The vortex is almost isotropic at least in the vicinity of (cid:18) (cid:19) its core. Then, a pumped charge due to a small change Here, the reflection coefficients r and r′, and the trans- ∆~ of Q~ is expressed as q ∼(φ/2π)e ∼(|∆~|/E )e, where mission coefficient t (see Fig. 1) can be calculated from G φ represents a change in the polar angle of the vectorQ~. the Green’s functions [24]. The transmittance through Thereforeonecanenhancethepumped/displacedcharge the sample and the reflectance at the both ends are ex- byenlargingtheangleφsubtendedby∆~ aroundthegap- pressed as less point. This quantized charge pumping [8] through T = |t|2, (11) theadiabaticcyclicchangeofQ~ isconsistentwiththere- R = |r|2 =|r′|2 =1−T, (12) sultspreviouslyfoundintheperiodicsystem[6]byusing the Berry-curvature formulation of the electric polariza- respectively. Then,weemploytheBrouwer’sformula[26, tion [1]. 27], which has been successfully applied to the charge Now we turn on the disorder in order to reduce the pumping in metallic quantum dot systems [20]: the gap EG. In the Q~ plane, the region of |Q~| <∼ v be- charge ∆q pumped from the left to the right during an comes gapless, while the Anderson localization seriously adiabaticchangeofparametersQ~ alongapathC isgiven affects the transport properties [15, 29]. In one dimen- by [26] sion,theeffectispronouncedandallthe statesarelocal- ized [30, 31, 32]. On the other hand, the total vorticity dQ~ ∆q =e ·Im r∗∇~Qr+t∇~Qt∗ . (13) Nv is anintegertopologicalnumber, andis robustunder 2π ZC (cid:16) (cid:17) a continuous change of parameters including the disor- der strength. Therefore, even with disorder, N remains Therefore,evenintheperfectlyreflectingcaset=0,|r|= v unity. Namely, there should exist at least one vortex 1, charge can be pumped by controlling the phase ϕ of r =eiϕ [28]. (r =0)intheQ~ plane;thusperfecttransmittance|t|=1 occurs even though all the states are strongly localized. Let us start with the pure case (v =0). Without the i This is a remarkableconsequencefromthe quantumand random potential v , the bulk system has an energy gap i topological property of the system, and is in sharp con- trast to usual classical tunnelling through disordered in- E =2 Q2+Min{t2 ,Q2}. (14) G0 1 n.n. 2 sulators. q This gap closes at Q~ =~0, where the metallic conduction InFigs.3(a2)and(b2),weshowournumericalresults occurs. This yields a vortex center of the reflective co- of R and ϕ for L=101with a uniform randomdistribu- efficients r and r′. By means of the continuued-fraction tion vi ∈ [−v,v] for the disorder strength v = 0.25tn.n.. expansion of the Green’s function, an analytic form of r ThevortexcenteroftheperfecttransmittanceT =1and in the thermodynamic limit L→∞ is obtained as R=0shiftsintheQ~ space. Besides,ananisotropydevel- opsintheshapeoftheregionofrelativelyhightransmit- r =−Q21+tn.n.Q2+Q t2n.n.+Q21+2itn.n.Q1 (15) tanceT. WefurthercalculateRandϕforlargersystems, Q2+t Q +Q t2 +Q2−2it Q whichareshownin(a3)and(b3)forL=201andin(a4) 1 n.n. 2 p n.n. 1 n.n. 1 and (b4) for L = 401. There, it is evident that the vor- p where Q ≡ Q2+Q2. The panels (a1) and (b1) of tex core, which is almost isotropic without the disorder, 1 2 Fig. 3 show the reflectivity R = |r|2 for L = 10001 and rapidly evolves into the highly anisotropic one. This is p the phase ϕ of r for L → ∞ without randomness. Here associatedwithanincreaseoftheratioL/ξbyincreasing thephasewindsby2πaroundQ~ =(0,0),forminga“vor- thesystemsize. Figure4showsthelocaldensityofstates tex” at which r = 0 and |t| = 1. In the thermodynamic Di of site i(=1,···,L), which has been calculated when limitL→∞,R=1andT =0holdexceptatthevortex Q~ is located at the vortex center Q~ corresponding to c center. For finite-size systems, the transmittance T be- each case of L=101, 201 and 401 for v/t=0.25. When haves as e−L/ξ0 with ξ = t /E . Therefore the size L is equal to 101 or smaller, the state at this energy is 0 n.n. G0 5 (a) (b) 2 FIG. 3: (Color) Reflectance R = |r| , and the phase ϕ = arg r. (a1), (b1): clean bulk system. (a2), (b2): disordered systemoffinitesizeL=401withtherandompotentialofthe strength v/tn.n =0.5. Thecolor codegiven in(a1)/(b1) also applies to (a2,3)/(b2,3). In the white region in (a1,2,3), R is −4 theunity within theaccuracy of 10 . extended over the sample. However, with increasing L, the state is almostlocalized in the middle ofthe sample. The spatial extension of the wave function, i.e., the lo- calization length ξ, can be explicitly evaluated from the (c) second moment of D as the inverse participation ratio; i FIG. 4: Local probability Di of the state at the vortex for ξIPR =( Di)2/ Di2. (16) (a) L = 101, (b) L = 201, and (c) L = 401 in the presence i i of uniformly distributed random potential vi ∈ [−v,v] with X X v=0.25tn.n.. We obtain ξ =51.4, 80.8, 76.4 and 84.7 for L=101, IPR 201, 401 and 501, respectively around the vortex center, indicating that ξIPR almost saturates about 80 sites for the neighboring leads [26]. Since the system is metal- v/t=0.25. lic, the pumped charge is not quantized in this case. In this anisotropic “wing”, ϕ changes rapidly. The Here, the main source of the pumping is a finite trans- width QW of the anisotropic wing decays exponentially mittancebutnotthephaseofreflectioncoefficient. With as exp(−L/ξ) with increasing L, as shown in Fig. 5. increasingthe systemsize,itcrossesoverto the opposite In the rest of this subsection, we comment on the re- regime with vanishing transmittance studied in this pa- lation of the charge displacement discussed above to the per. In the present theory, the charge pumping comes charge pumping through a metallic dot [20] weakly con- from the vortex of the reflection coefficient. Therefore, nected with two leads. In the metallic dot systems, a it is topologically protected against disorder. This topo- charge is pumped from one lead to the other by chang- logicalconstraint guarantees the applicability of the res- ing the heights of potentialbarriersbetweenthe dot and onance tunnelling [33], only near the vortex. 6 -2 10 -3 10 Q -4 10 -5 10 0 100 200 300 400 FIG. 6: Potential profile for the effective model of resonance L tunnelling. Heights of the two potential peaks are character- ized byS1 and S2. FIG.5: TheminimumwidthδQforRdeterminedatR=0.3 as a function of L. the potential peaks are much higher than E . An effec- F tive wave number k(x) ≡ k2−V(x) becomes imagi- nary inside the potential peaks, and the particle tunnels p Resonance tunnelling throughthepeaks. Oneoftheeigenstatesattheleftlead (x < x ) is given by Ψ (x) ∼ exp(ik (x−x )), where 1 1 1 1 In order to understand transport properties of such k1 = k(x1). The time-reversal symmetry of the system one-dimensionaldisorderedsystems, it is sometimes use- requires that the complex conjugate Ψ1∗(x) is also an fulto consideraneffectivemodelwiththe potentialhav- eigenstate. Atthe potentialvalleyaroundx=x2 wecan ing high double peaks. In this model, well-defined local- also consider eigenstates with Ψ2(x) ∼ exp(ik2(x−x2)) ized eigenstates exist between the two potential peaks, and Ψ2∗(x), where k2 =k(x2). Thus general eigenstates and any transport between two ends of the system oc- around x = x1 and around x = x2 are given as Ψ(x) = curs via the localized states through tunnelling. While A1Ψ1(x)+B1Ψ1∗(x) and Ψ(x)= A2Ψ2(x)+B2Ψ2∗(x), suchtunnellinghasexponentiallysmallprobability,itoc- respectively. Next we introduce the transfer matrix Θ1 curs when the Fermi energy of the leads is equal to one from x2 to x1 of the eigenenergies of the localized states, namely when A A 1 =Θ 2 , (17) the resonance takes place. This picture of transport in B 1 B 1 2 disordered systems is called “resonance tunnelling” [33]. (cid:18) (cid:19) (cid:18) (cid:19) where Θ is a 2 × 2 matrix. Time-reversal symmetry Here we describe the resonance-tunnelling theory, fol- 1 requires the transfer matrix Θ to be of the form lowing Ref. 33, and will see whether it fits with the present model with disorder. We define an effective 1 mS¯hc2ohkdr2e¨o/ld2imonfgies“rrteehqseounapatainorctniecΨlet′u′(enxnn)ee+rlgl(iynk,g2a”−ndVas(txhd)e)eΨsecff(rexicb)tei=dve0bpwyohtteehnree- Θ1 =(cid:18)kk21(cid:19)2 (cid:18)scionshh((SS11))ee−iαiβ11 csoisnhh((SS11))ee−iβiα11 (cid:19). (18) In a semiclassical theory we have tial V(x) has two peaks as shown in Fig. 6. It is not trivialwhetherthenumericalresultsofourmodel(3)fits x′1 x2 well with this resonance-tunnelling picture. The present α1 =− k(x)dx− k(x)dx, (19) modelhasmanyvalleyandpeaks,duetoon-siterandom- Zx1 Zx′1′ ness, and is not similar with Fig. 6. Nevertheless, when π x′1 x2 β = − k(x)dx+ k(x)dx. (20) thelocalizationlengthξ ismuchshorterthanthe system 1 2 Zx1 Zx′1′ size, it is indeed the case as we see by fitting our numer- S characterizesthe heightofthe potential,asthetrans- ical results well by the picture of resonance tunnelling. 1 missioncoefficientt isgivenbyt =eiαsechS . Because The basic reason for the applicability of resonance tun- 1 1 1 we assume that the peak is high, which means S ≫ 1. nelling is that the topology guarantees an existence of a 1 The transmission probability |t |2 is then approximated perfect-transmittance point, i.e., the resonanttunnelling 1 as |t |2 ∼4e−2S1 ≪1. inthetwo-dimensionalparameterspace(Q ,Q ),inspite 1 1 2 Similarly,weintroducethe transfermatrixforthe sec- of the complexity of the potential shape in the model. ond peak as First let us consider an open system attached to two ideal leads which are connected to reservoirs. The Fermi A A 2 =Θ 3 , (21) energyE istunedatone’sdisposal,andweassumethat B 2 B F 2 3 (cid:18) (cid:19) (cid:18) (cid:19) 7 and we have times) the amount of charge pumped into the system through the left and the right ends, respectively. Thus, 1 Θ = k3 2 cosh(S2)eiα2 sinh(S2)eiβ2 . (22) by going aroundthe point P, one unit chargeis pumped 2 (cid:18)k2(cid:19) (cid:18)sinh(S2)e−iβ2 cosh(S2)e−iα2 (cid:19) from the left lead to the right. For illustration, let us consider a clockwise cycle around the point P in Fig. 7 The phases α and β can be written similarly to α 2 2 1 (a1)(a2). This pumping from left to right is analogous and β , where S ≫ 1. We assume that S , α , and β 1 2 i i i to the “bicycle pump” [28]. The two potential peaks (i=1,2)aresmoothfunctions ofthe parametersQ ,Q 1 2 correspond to two gates to control the pumping. If the and E . A transfer matrix Θ for the entire system is F systemcrossesthelineω =(2n+1)πattheS <S side, 1 2 given by r undergoes 2π phase changes. It means that tunneling 1 occurs through the left potential peak due to resonance, Θ=Θ Θ = k3 2 θ11 θ2∗1 , (23) corresponding to the opening of a “left gate” and a unit 1 2 (cid:18)k1(cid:19) (cid:18)θ21 θ1∗1 (cid:19) charge flows in. After that the left gate closes, and the θ =ei(α1+α2) coshS coshS +sinhS sinhS eiω , rightgateopensinturn,whenthesystemcrossestheline 11 1 2 1 2 ω =(2n+1)π attheS <S side,andr′ undergoes−2π (24) 1 2 (cid:0) (cid:1) phase change. The charge is pumped to the right lead θ =ei(α2−β1) sinhS coshS +coshS sinhS eiω , 21 1 2 1 2 after one cycle. Thus the overall motion of the charge is (cid:0) (2(cid:1)5) as shown in Fig. 7 (a3). For the pumped charge to be x′2 quantized, the cyclic process should be sufficiently slow ω =β1−β2−α1−α2 ∼=2 k(x)dx. (26) to be regarded as “adiabatic”. Otherwise the particle Zx′1′ cannot tunnel through the potential barriers. Below let ¿From unitarity, it follows that |θ |2 −|θ |2 = 1. We us make this physicalpicture moreexplicit in relationto 11 21 assumethattheFermienergiesofthetwoleadsareiden- our numerical results. tical, which means k3 = k1. For a plane wave incident In our numerical results in Fig. 3(b2)-(b4), the phase- from the left lead, let r and t denote the reflection and windingpointP correspondstotheperfecttransmission. transmission coefficients, respectively. Similarly, for a WhenwegofromFig.3(b2)to(b4),L/ξincreases. Inthe plane wave from the right lead, we define r′ and t′ as resonance-tunnelling picture, it means that S increases i well. We then obtain and that the “resonance-tunnelling wing” narrows, as is seen by comparing Fig. 7(a1) and (b1). It is exactly r =θ /θ , t=t′ =1/θ , r′ =−θ∗ /θ . (27) 21 11 11 21 11 observed in our numerical results. It implies the unitarity |r|2 = |r′|2 = 1−|t|2. It also For large L/ξ, the transmissionprobability |t|2 is typ- satisfies t = t′ and rt∗ +r′∗t = 0, as is required from ically ∼ e−2(S1+S2) ∼ e−2L/ξ. Because we have S1 ∼ S2 time-reversalsymmetry. near the resonance, we estimate S1 ∼ S2 ∼ (L/2ξ). In We now apply this framework to fit our numerical re- that case, the change of the phases of r and r′ occurs sults. Because r is written as abruptly at around |ω−(2n+1)π| ∼ e−L/ξ. Along the line ω = (2n+1)π, |r| is given by |r| = tanh(S −S ), 1 2 tanhS +tanhS eiω r =eiθ 1 2 , (28) whichbecomesappreciableonlywhen|S1−S2|∼1. This 1+tanhS1tanhS2eiω is found only in a small region ∆/tn.n. ∼ ξ/L. When L/ξ ≫ 1, because the tunnelling rate Q is exponen- a condition for a total transmission, r=0, is given by W tially small (Q ∼t e−L/ξ), charge pumping requires W n.n. S1 =S2, (29) an exponentially long time τ ∼ (h¯/tn.n.)eL/ξ. This is x′2 a time scale which gives a criterion for adiabatic charge ω =2 k(x)dx=(2n+1)π, (30) pumping in this system. If one changes the parameters Zx′1′ fasterthanthistimescale,thepumpedchargeisreduced withanexponentialfactore−ω/EG asanonadiabaticcor- where n is an integer. The latter condition is equiva- rection,whereω isafrequencyofthechangetheexternal lent to the Bohr quantization condition, that the state parameters [34]. localizedaroundx beaneigenstatewithitseigenenergy 2 equal to E . In other words the total transmission oc- Generally,thereoccursno perfect transmittancepoint F curs at resonance. For fixed E , the two conditions (29) by tuning only two parameters since the effective two F and (30) define isolated points in the Q -Q plane. Let barrier model for the resonanttunnelling applies only to 1 2 P denote oneofsuchpoints oftotaltransmission: r =0. a limited region of the random system and not through One caneasilysee thataroundthe point P,the phaseof a whole sample. In the present case it is guaranteed by r and that of r′ wind by ±2π, as is schematically shown the topologicalconstraintthatthe phase ofr (r′)should in Fig. 7(a1) (a2). As is seen from the Brouwer formula wind by (−)2π for a large cycle far from Q~ = 0, well (13), the phase windings of r and r′ correspond to (2π withinthegappedregion[9]. When|Q~|islargerthanthe 8 Multi-channel problem in alloy model for on-site random potential In the rest of this section, we consider a stronger ran- domness by employing the alloy model (Fig. 2 (b)) with v = s (v+δv ) with s = ±1 being a random sign and i i i i δv /v ∈ [−0.025,0.025] a uniform random distribution, i instead of the uniformly distributed random potential (Fig. 2 (a)). Let us start with the single-channel case. Then, for a fixed disorder strength v, the shape of the vortex core ofthe reflectioncoefficient rapidlyevolvesfromisotropic to anisotropic with increasing the system size L, as in the case of the uniformly distributed random potential discussedinSec . This tendency is generic andalsoreal- izedforafixedsystemsizeLwithincreasingthedisorder strengthv. InFigs.8(a)and(b),weshowthereflectance R and the phase ϕ in the Q~ space for v/t = 1.0 and n.n. a small sample size L = 25. Here, the region in the Q~ plane where the gapcollapsesandvortices distribute ex- pandsbecauseofthestrongerdisorderandthusthemuch shorterlocalizationlengthξ ∼3or4. Thetransmittance T is typically obtained as ∼ 10−9, which is practically negligible except at the vortex core. FIG. 7: Schematic contour mapping of the phases of the re- flection coefficients r and r′ in the Q1-Q2 plane. (a1)(a2) correspondtosmallS1,S2,and(b1)(b2)tolargeS1,S2. The ′ phases of r and r wind by 2π during the cycle around the “vortex” P. When we go along the arrows indicated in the ′ figures(a1)(a2),thephaseofr(r)increaseby2π(−2π). (a3) illustrates the motion of the charge in the cycle. Note that the2π phasechangeofr (r′)isassociated with pumpingone unitchargeintothesystemthroughtheleft(right)endofthe system. energy scale of the disorder, a finite gap (∼ |Q~|) opens. The wing will then become as wide as QW ∼ |Q~|. Cor- FIG.8: (a)Reflectanceand(b)thephaseϕofrindisordered respondingly, the typical time scale τ ∼ ¯h/QW ∼ ¯h/|Q~| systemwithL=25andv/tn.n. =1.0forthealloymodelwith becomes smaller, and the adiabaticity condition is easily theon-site random potential given byFig. 2. satisfied. Insuchcase,however,thedielectricresponseis not enhanced. The localization length becomes as long Nowwe considera systemthatconsists ofmanychan- as the system size, and the pumping is accomplished nels,eachofwhichisdescribedbythepresentdisordered through extended states, not by resonance tunnelling. alloymodelbutwithadifferentprofileofrandompoten- Remarkably, this charge pumping in the gapped region tials. Such configuration can be realized in thin films is governed by the vortex which is located deep in the of ferroelectrics. Then, we can design the pattern of disordered regime (ξ ≪ L). In other words, the charge the phase ϕ in the Q~ plane by tuning the disorder. For pumping in the gapped regime (ξ ∼L) is smoothly con- the alloy model with the random on-site potentials, vor- nectedto thatviaresonancetunnelling inthe disordered tices are mostly located around Q = ±v/2 with the 1 regime(ξ ≪L). Wenote thatS andS areregardedas “wing” almost along the Q axis. Therefore, we can en- 1 2 2 effective parameters,although the realpotential is much hance the dielectric response, if we can tune the disor- more complicated than the two-barrierstructure. der strength and choose a sample where the Q~ point of 9 the system is located inside the “wing”. In particular, PERIODIC/TWISTED BOUNDARY CONDITION when the transmittance T is negligibly small, the en- hancement factor of the dielectric response is given by The adiabatic charge pumping in the presence of the (∂ϕ/∂θ)disorder/(∂ϕ/∂θ)pure from (13), since the electric substrate disorder was considered by Niu et al.[9]. They field is proportional to θ = arctan(Q1/Q2). We calcu- showedthat the adiabatic charge transport is still quan- late this enhancement factor for this multichannel sys- tized as long as the excitation gap between the high- tem. Increasing the number N of channels. Figures 9(a) estoccupiedstate (HOMO) andlowestunoccupiedstate and (b) representthe Q~ dependence of the enhancement (LUMO) does not vanish due to the substrate disorder factor in the case of v/tn.n. = 1.6 and 2.4, respectively, and the many-body interaction in the thermodynamic for L = 25 with N = 102. The main structure in this limit. In the case of open boundary condition, it is easy map is almost saturated up to N = 102. These results to imagine the meaning of the charge pumping, namely reveal that around Q1 ∼ ±v/2, the dielectric response the charge transport from one end to the other. In the is significantly enhanced by a factor 30 ∼ 40 compared caseofperiodicboundarycondition,thechargepumping with the pure case. Even for the thin film with a square after one-cycle means that the electronic wavefunction shape of a linear dimension larger than 50˚A, which cor- shiftsbytheminimumnumberoflatticesitessothatthe responds to N =102, the disorder-induced enhancement final wavefunction is the same as the initial one. A car- of the charge transfer rate should be robust. Then, the toon picture of this process is given by Fig. 10. We note applied electric field necessary for switching the polar- that in the absence of the disorder, the adiabatic charge ization is reduced by this enhancement factor. If we re- transport has been related to the field-theoretical model quire the response time τ ∼ eL/ξ/tn.n. of the order of of the one-dimensional chiral anomaly[37]. 10−9 s, we obtain eL/ξ < 106 with the assumption of t ∼ 1015 s−1. This also assures a negligibly small n.n. transmittance T ∼ e−L/ξ ∼ 10−6, and thus the small leak current and low dissipation. FIG. 10: A cartoon picture of the charge transport on the one-dimensional ring. Formalism We consider a one-dimensional charge-density-wave FIG. 9: Relative dielectric response in the presence of the system with disorder, as described in (3) and (7). In disordercompared with thepurecase. Thedisorderstrength is v/tn.n. = 0.8 for (a) and 1.2 for (b). Averages are taken contrasttotheopensystemwithtwoleadsasconsidered over102 randomdisorderconfigurations. Insidewhitebands, in Section III, we deal with a closed system in a ring thereoccurs a gradual sign changein thedielectric response. geometry. Instead of a periodic boundary condition as is usually employed for the ring, we introduce a twisted boundary condition; we impose that the phase gained Possible experimental realization of this quantum- by running around the ring once is eiα. This twisted mechanical disorder-induced enhancement of the dielec- boundary condition is realized by applying a magnetic tricresponse,namelythequantumrelaxor,hasalsobeen flux through the one-dimensional ring. proposed [10] for thin films of solid solution systems like The reason for introducing the twisted boundary con- Pb(Fe Nb )O and Pb(Sc Nb )O prepared with dition is to reveal the topological nature of the charge 0.5 0.5 3 0.5 0.5 3 an adequate slow-anneal process [35, 36]. transport in the system. In the pure case, the lattice 10 momentum k is a good quantum number, and the topo- where L =Na is the circumference of the ring, and a is logical nature is manifest in the three-dimensional space the lattice spacing, and (Q ,Q ,k) as monopoles and antimonopoles. However, 1 2 k is no longera good quantumnumber in the disordered ∂H = (−1)jc†c (38) case, and instead the flux α plays a similar role to k. ∂Q1 j j j Therefore, instead of (Q ,Q ,k), we will be interested X 1 2 ∂H 1 in the adiabatic process such that the system is changed = (−1)j(eiα/Nc†c +e−iα/Nc† c ()3.9) ∂Q 2 j j+1 j+1 j due to the slow variationsof the parameters (Q1,Q2,α), 2 Xj which spans a 3-dimensional parameter space with the property α ∈ [0,2π]. The Hamiltonian with the flux α Wenotethatthechangeofthepolarization,δP,isingen- can be rewritten as eral dependent on the path in the parameter (Q~) space. The|Ψ iin(36)and(37)isthemany-bodygroundstate, 0 N H(α)= − tn.n. (eiα/Nc†c +h.c.) and |Ψmi denote the excited states. E0 and Em are the 2 j j+1 energyforthegroundandtheexcitedstates,respectively. j=1 X In the pure case when the chemical potential lies in the N + Q (−1)jc†c gap, the system is a band insulator. The magnitude of 1 j j the energy gap is given by the magnitude of the CDW j=1 X order parameter (Q ,Q ). In the presence of the disor- 1 2 N Q der, the gap closes when the magnitude of the disorder + 2 (−1)j(eiα/Nc†c +h.c.)+V(,31) 2 j j+1 potential becomes the order of O( Q2+Q2). The sys- 1 2 j=1 X tem remains insulating because each state is localized, p where c ≡ c and V = v c†c is the uniformly i.e., Anderson localization. N+1 1 j j j j distributed disorder potential. Although we have defined (35), (37) and (36) only for In the tight binding modelP, the polarization operator α=0andα=π,wehenceforthextendtheseformulaeto P~ has the following form general α, which makes the topological properties of the chargepumpingmanifest. Wenotethatexceptforα=0 P~ = R~jc†jcj (32) and π, δP(α) does not mean a change of polarization. j With this extension to arbitrary α, let us succinctly ab- X breviate (Q ,Q ,α) as Q~ = (Q ,Q ,Q ) and define the where R~ is the position at site j and c†c is the elec- 1 2 1 2 3 j j j gauge potential as trondensityoperator. The currentis definedasthe time derivative of the polarization operator given by ∂ A~ = ihΨ | |Ψ i. (40) ∂P~ 1 0 ∂Q~ 0 J~= = [P~,H] (33) ∂t i Thisgaugefieldissodefinedthatthecorrespondingfield In the pure case the current is given by ∂H, whereas in strength F~ = ∇×A~ has the components given in (37) ∂k thepresentcasewithdisorder,thecurrentoperatorgiven and (36). Furthermore, when the parameters (Q ,Q ) 1 2 by ∂H has the following form are changed along a cycle, the pumped charge can be ∂α written as J =ie−iα/N (t+Q (−1)j)c† c +h.c. (34) 2 j+1 j Xj δP(α)= dQ~ ×αˆ·F~(Q~) (41) Whenthe flux αis equalto0 orπ,the systemistime- IS reversal symmetric, and there is no persistent current. where S is a loop onthe Q -Q plane. Using the Stoke’s 1 2 In such cases, we can consider a change of the electric theorem, (41) becomes polarization by an adiabatic change of parameters Q 1 and Q . In the linear response theory, the change of the 2 δP(α)= d2QD(Q~) (42) electric polarization is given by Z δP =F ∆Q −F ∆Q , (35) 1 2 2 1 where D(Q~) = ∂ F +∂ F defined as the distribution 1 1 2 2 with function of the polarization. This pumped charge is not necessarily an integer. F =−i (hΨ0|J|ΨmihΨm|∂∂QH2|Ψ0i −c.c.), (36) Nevertheless, in a strongly disordered system it be- 1 L (E −E )2 m 0 comes an integer, with anexponentially small correction m6=0 X of the order of e−L/ξ. To see this, we note that the F = i (hΨ0|J|ΨmihΨm|∂∂QH1|Ψ0i −c.c.), (37) wavefunctions in a strongly localized case are almost in- 2 L (E −E )2 m 0 tact with the twisted boundary condition α within an m6=0 X