Table Of Content

Boson-controlled quantum transport A. Alvermann,1 D. M. Edwards,2 and H. Fehske1 1 Institut fu¨r Physik, Ernst-Moritz-Arndt-Universität Greifswald, 17489 Greifswald, Germany 2 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom We study the interplay of collective dynamics and damping in the presence of correlations and bosonicfluctuationswithintheframework ofanewlyproposedmodel,whichcapturestheprincipal 7 transport mechanisms that apply to a variety of physical systems. We establish close connections 0 to the transport of lattice and spin polarons, or the dynamics of a particle coupled to a bath. We 0 analysethemodelbyexactlycalculatingtheopticalconductivity,Drudeweight,spectralfunctions, 2 groundstate dispersion and particle-boson correlation functions for a 1D infinitesystem. n a J The motion of an electron or hole which interacts _tb_ 2 strongly with some background medium is a constantly ω (II) incoherence (III) strong fluctuations 2 0 recurring theme in condensed matter physics. Media which commonly occur are ordered spin backgrounds ] diffusive l as in the t-J model of doped Mott insulators or vi- e - brating lattices, as in the Holstein and quantized Su- transport boson-assisted r 1 t Schrieffer-Heeger (SSH) models for polarons or charge s quasi-free density waves. A moving particle creates local distor- . at tionsofsubstantialenergyinthe medium,e.g. localspin (I) coherence (IV) strong correlations m fluctuations, which may be able to relax. Their relax- 1 t / t - ation rate determines how fast the particle can move. In b f d thissenseparticlemotionisnotfreeatall;theparticleis n continuouslycreatingastringofdistortionsbutcanmove FIG.1: Schematicviewofthefourphysicalregimesdescribed o on“freely”ataspeedwhichgivesthedistortionstime to byHamiltonian (1), and the transport properties (see text). c [ decay. This picture is very general with wide applicabil- 2 ity, e.g. to charge transport in high-Tc superconducting physicsofthe modelisgovernedbytworatios: Therela- and colossal magnetoresistive materials, mesoscopic de- v tivestrengtht /t ofthetwotransportmechanisms,and 2 viceslikequantumwires,andpresumablyevenbiological b f therateofbosonicfluctuationst /ω . Thereinthemodel 9 systems. [1] b 0 5 In this Letter we investigate transport within some also resembles common electron-phonon models like the 0 Holstein or SSH model. Nevertheless it differs in that background medium by means of an effective lattice 1 particlehoppingcreatesabosononlyonthesitethepar- model with a novel form of electron-boson coupling. [2] 6 ticle leaves, and destroys a boson only on the site the 0 The bosons correspondto local fluctuations of the back- particle enters. As a consequence the ‘string’ effect fa- / ground. To motivate the form of the model let us con- at sider a hole in a 2D antiferromagnet. In a classical miliar from the t-J model is captured within the model, m but also more complex features like in the 2D t-J model Neel background motion of the hole creates a string of occur already in a 1D setting. In this contribution we - misaligned spins. This ‘string effects’ strongly restricts d study the model for a single particle in 1D at tempera- propagation.[3]Ifhoweverspinscanexchangequantum- n ture T =0. o mechanicallydistortionsofthespinbackgroundcan‘heal c out’ by local spin fluctuations with a rate controlled by We begin with a discussion of the physical regimes v: the exchange parameter J. This way the hole can move shown in Fig. 1. First, for small tb/tf (left side), trans- i coherently with a bandwidth proportionalto J. [4] Here port takes place through unrestricted hopping. There, X we present a simpler spinless model which captures this the model essentially describes motion of a particle cou- r two-fold physics within a generic framework. pledtoabosonicbath,whenanybosonicfluctuationsre- a Specifically, we consider the Hamiltonian ducethemobilityoftheparticle. Forsmalltb/ω0(regime I), the number of bosons is small. The particle propa- H =−t c†c −t c†c (b†+b ) +ω b†b (1) gates almost coherently, and transport resembles that of f j i b j i i j 0 i i hXi,ji hXi,ji Xi a free particle. If tb/ω0 is larger (regime II), the number of bosons increases, and the bosonic timescale is slower for fermionic particles (c†) coupled to bosonic fluctua- than that of unrestricted hopping. Therefore bosonic i tions (b†) of energy ω . H models two transport pro- fluctuations mainly act as random, incoherent scatter- i 0 cesses, one of unrestricted hopping ∝ t , and a second ers,andthe particlelosesitscoherence. Thetransportis f of boson-controlled hopping ∝ t . While for t = 0 the then diffusive, with a short mean free path. In the sec- b b model reduces to that of a free particle, for t 6= 0 the ond limiting case, for large t /t (right side), transport b b f 2 takesmainlyplacethroughboson-controlledhopping,i.e. 100 palararergtecicrtleeba/tmωed0ot(aiornendgirmceoleinessIuIoIm)n,edtthraiennestxphioserttehnoicspeplioimnfgibtepodsroobncyess,ssw.trhoFincohgr - D / Ekin00..0024 D /t [t = 0]bf1100--42 0 scattering off uncorrelated bosonic fluctuations (similar 0 0.05 0.1 0.5 1 2 4 8 to regime II). For small tb/ω0 however (regime IV), the Ekin tf / tb ω0 / tb bosons instantly follow the particle motion and strong D / 0.5 correlations develop, leading to collective particle-boson - 0.4 dynamics. Note that boson-controlled hopping acts in 0.3 twoopposingways: Dependingonhowmanycorrelations free particle 0.2 ω / t = 2.0 betweenthe bosonspersist, it mayeither limit transport 0 b 0.1 ω0 / tb = 1.0 as a resultofscatteringoff randomfluctuations (regimes ω / t = 0.5 0 b II andIII), but mayalsoenhancetransportthroughcor- 00 10 20 30 40 t / t related emission and absorption of bosons (regime IV). f b Forlarget /t andsmallt /ω ,transportinthemodel b f b 0 FIG. 2: (Color online) Drude weight D scaled to the kinetic resembles that of the hole-doped t-J-model. [4, 5] To energy E . The left inset displays the region t & 0 mag- kin f make this connection more explicit, we perform the uni- nified, while the right inset shows D for t = 0 in depen- f tary transformation bi 7→bi−tf/2tb of H: dence on ω0. The dashed curve gives the asymptotic result D≃t6b/(3ω05)+O(t8b/ω07) for ω0→∞. H′ =−t c†c (b†+b )−λ (b†+b )+ω b†b (2) b j i i j i i 0 i i hXi,ji Xi Xi where E = h0|H −ω b†b |0i is the kinetic energy. kin 0 i i i For a free particle (t = 0), the Drude weight is given (here a constant energy shift is dropped, which is pro- b P by D = t , and the sum rule reads −D/E = 0.5, portionaltothenumberoflatticesitesN andguarantees f kin while −D/E ≪0.5 for diffusive transport in the pres- finite results for N → ∞). In (2) the coherent hopping kin ence of strong fluctuations. We can therefore charac- channelis eliminated in favorof a bosonrelaxationterm λ (b†+b ). If λ = ω0tf > 0, i. e. t > 0 in the orig- terize the different transport regimes through the ratio inalimoidel,ithe decay 2otfb bosonic exciftations allows for −D/Ekin (Fig. 2). The curve for ω0/tb = 2.0 shows P that in a wide range of t /t transport is quasi-free with t-J-like quasiparticle (hole) transport. In the ‘classical’ f b −D/E . 0.5. For smaller ω /t , as the number of limit λ → 0 coherent quasiparticle motion is suppressed kin 0 b fluctuations is larger,−D/E is decreased due to scat- as in the t-J (Ising spin) model. Note that the limit kin z tering. The smaller ω /t , the slower −D/E tends to ω → 0 does not immediately lead to a semi-classical 0 b kin 0 itsasymptoticvalues0.5fort /t →∞. Thisshowshow description established for the Holstein- and SSH-model f b the crossover from the coherent regime (I) with quasi- sincetheelectrondoesnotcoupleexclusivelytooscillator coordinates ∝(b +b†). free transport to the incoherent regime (II) with diffu- i i sive transport is controlled by t /ω . For small t /t , Foraquantitativeanalysisoftransportinthedifferent b 0 f b when boson-controlled hopping is the dominating trans- regimes we employ the optical conductivity Reσ(ω) = port process,D increases with decreasing ω (left inset). 2πDδ(ω)+σ (ω), where the regular part is 0 reg This is the regime of boson-assisted transport, where |hn|j|0i|2 transportismediatedbyvacuum-restoringprocesses(see σreg(ω)=π ω [δ(ω−ωn)+δ(ω+ωn)]. (3) below). For the moment, we note, that D at tf =0 sat- n>0 n urates with ω →0 (right inset), as one passes from the X 0 correlationdominated regime (IV) to the regime (III) of Here|nilabels the eigenstatesofthe one-fermionsystem strong, uncorrelated fluctuations. with excitation energy ω =E −E , |0i is the ground- n n 0 We complete our study by means of three quantities: state. The current operator j =j +j is given by f b First, the groundstate dispersion E(k) provides the effective mass 1/m∗ = ∂2E| , which is renormalized for j =it c† c −c†c , (4) ∂k2 k=0 f f i+1 i i i+1 t 6= 0. By Kohn’s formula, D = 1/(2m∗). Second, we b i X use the particle-boson correlation function j =it c† c b†−c†c b −c† c b†+c†c b . b b i+1 i i i i+1 i i−1 i i i i−1 i χ =h0|b†b c†c |0i, (6) i ij i i j j X and third, the one-particle spectral function TheDrudeweightD servesasameasureofthecoherent, free particle like transport, and fulfils the f-sum rule A(k,ω)= |hn|c†|vaci|2δ(ω−ω ), (7) k n ∞ ∞ n X σ(ω)dω =2πD+2 σ (ω)dω =−πE , (5) reg kin where |vaci denotes the particle vacuum. All quantities Z−∞ Z0 3 1 2 t / t = 2 f b 0.6 χij0000....2468 ttff // ttbb == 420 (E(k)-E) / t0b00..24 σωπ() / reg 1 SSbtot ωA(k,)_π_2k0 S 0 0 f π -5 0 5 0 0.25 0.5 0.75 1 0 i-j k / π 0 5 10 -2 0 2 ω / t ω / t 15 b b Stot _k FIG. 4: (Color online) σreg(ω) for ω0/tb = 1.0, tf/tb = 2.0 σωπ() / reg 150 Sf ωA(k,) π_20 (left0).,2and A(k,ω) for ωt /0 t/ =t 0b.05=0.05.0,6tf/tb =0.04 (right). f b t / t = 1.0 π f b S t / t = 5.0 0 b f b 0.04 0 5 10 -10 0 10 ω / tb ω / tb χij0.1 ) / t0b E FIG. 3: (Color online) Top row: χij (left) and E(k)−E0 k)- (right) for ω0/tb = 0.5 and tf/tb = 2,4,20. Bottom row: (E( σreg(ω)(left) fortf/tb =20andA(k,ω)(right)fortf/tb =4. 0 -5 0 5 00 0.25 0.5 0.75 1 To analyze the relative importance of the two transport i-j k / π processes j and j , we show the corresponding contribu- f b 2 tions σf(ω), σb(ω) to σreg(ω) separately (note that generally σreg(ω)6= σf(ω)+σb(ω)). Stot (and similar Sf, Sb) denotes _k theintegrated conductivity Stot(ω)=R0ωσreg(ω′)dω′. ωπ() / eg 1 Stot ωA(k,) π_0 have been calculated with exact numerical techniques σr S 2 b (Lanczos diagonalization and kernel polynomial meth- π S ods[6]). Weemployedabasisconstructionforthemany- 0 f 0 5 10 -2 0 2 4 6 particle Hilbert space that is variational for an infinite ω / t ω / t lattice. [7] This guarantees data of extremely high preci- b b sion, free of finite size effects. FIG.5: (Coloronline)Topleft: χij forω0/tb =2andtf/tb = Letusfirstdiscussthe‘incoherent’or‘diffusive’regime 0.05(circles),1.0(squares),5.0(diamond). Topright: E(k)− (II)(see Fig.3). As expected,the regularpartofthe op- E0forω0/tb =1(solid)andω0/tb =2(dashed),withtf/tb = ticalconductivityisdominatedbyabroadincoherentab- 0(circles),tf/tb =0.01(squares). Bottomrow: σreg(ω)(left) sorption continuum, and −D/E is small. This resem- and A(k,ω) (right) for ω0/tb =2, tf/tb =0.1. kin bles the situation for a large Holstein (lattice) polaron, where the role of bosonic fluctuations is taken by optical phonons. [8] χ shows that bosonic fluctuations are energy, similar to e.g. a double well at certain resonant ij rather weakly correlated. Of course, they form a cloud energies. surrounding the particle, but are not further correlated. From the diffusive regime, we can evolve in two direc- The spectralfunctionA(k,ω)supports this picture. The tions. First, if we increase t /t ( regime (III)), the b f spectral weight is distributed along the ‘free’ dispersion contribution of boson-controlled hopping to the conduc- −2t cosk, like for a weakly bound particle-boson exci- tivity begins to dominate (see Fig. 4, left panel). Still, f tation. Around k = 0 and k = ±π, the overdamped transport is mainly diffusive (cf. Fig. 2). If we further characterofparticle motion is very prominent. Compar- increase t /t while keeping t /ω large (inside regime b f b 0 ing with E(k), we see that the quasi-particle weight is (III)), strong but uncorrelated bosonic fluctuations de- negligible away from k = 0, and a well-defined quasi- velop. Asaresult,thespectralfunctionA(k,ω)becomes particle band does not exist. Somewhat surprisingly, for fullyincoherent(rightpanel),and−D/E issmall. Ap- kin k = ±π/2 almost all weight resides in a single coher- parently, the large number of bosonic fluctuations pre- ent peak at ω = 0. A particle injected with k = ±π/2 ventsstrongcorrelations(cf. χ inFig.3). Inthesecond ij therefore propagates almost unaffected by bosonic fluc- direction, for large t /t and small t /ω (regime (IV)), b f b 0 tuations. In a sense, the system is transparent at this the number of fluctuations is reduced. Then, strong cor- 4 x ⊚y · · → ⋆ ⊚y · → ⋆ ⋆ ⊚x → ⋆ ⊚⋆ ⋆ → ⊚y ⋆ ⋆ → · ⊚y ⋆ → · · ⊚ remark that bosonic fluctuations act in two competing ways: While they limit transport by unrestricted hop- (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ↑ ↓ ⊚ ↓ ↓ ⊚ ↓ ↑ ↓ ↑ ⊚ ↑ ↑ ⊚ ping if strong but weakly correlated, they assist trans- → → → → → → (cid:12)⊚ ↑+ (cid:12)↑ ↑+ (cid:12)↑ ↑+ (cid:12)↑ ⊚+ (cid:12)⊚ ↑+ (cid:12)↓ ↑+ (cid:12)↓ ↑+ portbyboson-controlledhoppinginthe regimeofstrong (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) correlations. Equallyimportant,thesameboson-assisted (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) FIG. 6: Lowest order vacuum-restoring process (upper row), mechanismwhichallowsfortransportislimitedbyitself, which is a one-dimensional representation of the ‘Trugman because strong correlations cannot persist in the pres- path’ [5] in a Néel-ordered spin background (lower row). In ence of many fluctuations: The Drude weight D for steps 1–3, three bosons are excited, which are consumed in tf=0 t =0 increases with decreasing ω as we discussed, but steps 4–6. Note how the correlated many-particle vacuum of f 0 finally saturates. The large number of bosonic fluctua- thespin model is translated tothe bosonic vacuum. tions, which become increasingly uncorrelated for small ω , interfere with the correlatedbosonexcitations in the 0 relations evolve (see χ in Fig. 5). The conductivity vacuum-restoring transport processes, restricting their ij is now entirely given by the contribution from boson- efficiency. Note that these fundamental physical mecha- controlled hopping, but does not show the absorption nisms are not enforced in the model, but emerge natu- continuum we found for diffusive transport (note that, rallyfromthecompetitionbetweenuncorrelatedscatter- although −D/E must be small for large t /t , it is ing and correlation-inducedtransport. kin b f muchlargerthanforlargertb/ω0). Instead,bothσreg(ω) To conclude,the modelthat weproposeprovidesa re- and A(k,ω) consist of a few, well separated peaks. This duced but realisticdescriptionof fundamentalaspects of indicates, that here the model shows collective particle- transport in the presence of bosonic fluctuations. Top- bosondynamics,i.e. awell-definedquasi-particleexists, ics of currentinterest become accessible within the com- likeaspin/magneticpolaroninthet-J-model.[4,9]Asa prehensive framework established. Despite its seemingly particularfeature of the correlatedtransportmechanism simple form, the model covers very different physical whichdominatesfortb/tf ≫1,thequasi-particledisper- regimes, identified e.g. by the Drude weight. For these sionE(k)developsak →k+πsymmetryfortb/tf →∞. regimes,we establish links to important many-body sys- At tf =0 the model therefore shows an electronic topo- tems like lattice or spin polarons, and could thereby logical transition, for which the hole doped t-J-model demonstrate the relevance of the model. Based on these provides a specific example. results, the model will certainly prove useful for further The correlated transport mechanism for tb/tf ≫ 1 is studies, e.g. on the possibility of pairing for two parti- best understood in the limit tf = 0. Then, the particles, or on topological phase transitions for finite den- cle can only move by creating bosonic fluctuations, i.e. sities. The whole phase diagram for finite density and transportisfullyboson-assisted. MovingbyLsitescosts temperature is open for investigation. energy Lω , which is just the string effect and tends 0 We acknowledge helpful discussions with G. Wellein to localize the particle. However, there exist processes and financial support by DFG through SFB 652. that propagate the particle but restore the boson vacuum in their course. The lowest order process of this kind comprises 6 steps, promoting the particle by 2 sites (Fig. 6). By this and similar higher order processes the particle is itinerant even at t =0, with a finite, though [1] D.M.Newnsand C.C.Tsuei, cond-mat/0606223 (2006); f small, Drude weight. Since in any hop the boson num- L. G. L. Wegener and P. B. Littlewood, Phys. Rev. B ber changes by one, any vacuum-restoring process prop- 66, 224402 (2002); S.Komineas, G.Kalosakas, and A.R. Bishop, Phys.Rev.E 65, 061905 (2002). agates the particle by an even number of sites. This [2] D. M. Edwards, Physica B 378-380, 133 (2006). immediately explains, why E(k) for tf =0 has period π. [3] W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 1324 The weight of the lowest order process shown in Fig. 6 (1970). scales as t6b/ω05 (cf. Fig. 2). Accordingly, boson-assisted [4] C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39, transport dominates for large (t /ω )5(t /t ). In this 6880 (1989). b 0 b f regime, the mobility of the particle increases if ω de- [5] S. A.Trugman, Phys. Rev.B 37, 1597 (1988). 0 creases, as vacuum-restoring processes become energeti- [6] A.Weiße,G.Wellein,A.Alvermann,andH.Fehske,Rev. Mod. Phys.78, 275 (2006). cally more favorable. This explains the opposite depen- [7] J. Bonˇca, S. A. Trugman, and I. Batistić, Phys. Rev. B denceofDonω0 fortf/tb ≪1andtf/tb &1. Intheplot 60, 1633 (1999). oftheDrudeweightthetransitionfromuncorrelated,dif- [8] B. Bäuml, G. Wellein, and H. Fehske, Phys. Rev. B 58, fusive to correlated, boson-assisted transport therefore 3663 (1998). leads to a crossing of the curves at about t /t ∼ 0.05 [9] J. R. Schrieffer, X.-G. Wen, and S.-C. Zhang, Phys. Rev. f b (see Fig. 2, left inset). This substantiates our initial Lett. 60, 944 (1988).