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Semiclassical theory of transport in a random magnetic field F. Evers, A. D. Mirlin,∗ D. G. Polyakov,†,‡ and P. Wölfle Institut fu¨r Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany We present a systematic description of thesemiclassical kinetics of two-dimensional fermions in a smoothly varying inhomogeneous magnetic field B(r). We show that the nature of the transport 9 depends crucially on both, the strength of the random component of B(r) and its mean value B. 9 9 ForvanishingB,thegoverningparameterisα=d/R0,wheredisthecorrelation lengthofdisorder 1 andR0 istheLarmorradiusinthefieldB0,thecharacteristicamplitudeofthefluctuationsofB(r). Whileforα≪1theconventionalDrudetheoryapplies,inthelimitofstrongdisorder(α≫1)most n particles drift adiabatically along closed contours and are localized within the adiabatic approxi- a mation. The conductivity is then determined by the percolation of a special class of trajectories, J the “snake states”. The unbounded snake states percolate by scattering at the saddle points of 8 B(r) where the adiabaticity of their motion breaks down. The external field B is also shown to suppressthestochasticdiffusionbycreatingapercolationnetworkofdriftingcyclotronorbits. This ] l kindofpercolation isdueonlytothe(exponentiallyweak)violation oftheadiabaticityoftherapid l a cyclotron rotation in the field B, leading to an exponentially fast drop of the conductivity at large h B. We argue that in the regime α ≫ 1 the crossover between the snake-state percolation and the - s percolation of the drift orbits with increasing B is very sharp and has the character of a phase e transition (localization ofsnakestates) smearedexponentiallyweaklybynon-adiabaticeffects. The m ac conductivityalso reflectsthedynamicalpropertiesof particles movingon thefractal percolation . network. In particular, we demonstrate that the conductivity has a sharp kink at zero frequency t a andfallsoffexponentiallyathigherfrequencies. Wealsodiscussthenatureofthequantummagne- m tooscillations. We report detailed numerical studies of the transport in the field B(r): the results ofthenumericalsimulationsconfirmtheanalyticalfindings. Theshapeofthemagnetoresistivity at - d α∼1foundnumericallyisingoodagreementwiththeexperimentaldatainthefractionalquantum n Hall regime for thevicinity of half-filling of thelowest Landau level. o c PACS numbers: 71.10.Pm, 73.40.Hm, 73.50.Bk, 05.60.+w [ 1 I. INTRODUCTION in a random field B(r) shows up most distinctly in sys- v tems with smooth inhomogeneities. The case of long- 0 range disorder is most important also experimentally – 7 The transport properties of two-dimensional(2d) par- 0 ticlesmovinginaspatiallyrandommagneticfield(RMF) sincethecompressiblestateinahalf-filledLandaulevelis 1 observed in high-mobility samples. In the latter, a large B(r)orientedperpendicularlytotheplanehaveattracted 0 correlation radius of potential fluctuations, d, is deter- considerable interest in the last few years. This interest 9 mined by a wide “spacer” between the electron gas and 9 is largely motivated by the relevance of the problem to the doped layer containing ionized impurities. Likewise, / the composite-fermion (CF) description [1,2] of a half- t inhomogeneities of the RMF created by the ferromag- a filled Landau level. Within this approach, the electron m netic overlayers in [6,7] appear to be fairly long-ranged. liquidinastrongmagneticfieldismapped–bymeansof The large value of the correlationradius d (as compared - aChern-Simonsgaugetransformation– toa fermiongas d to the interelectrondistance) allowsto describe the elec- subject to a weak effective magnetic field. Precisely at n tron kinetics quasiclassically. o half-filling, the averagevalue of the Chern-Simons gauge c fieldcompensatestheeffectoftheexternalmagneticfield. Itiswellknownthatquantuminterferenceeffectsmay : The RMF appears in this model after taking static dis- cause localization of noninteracting particles in an infi- v i order into account: fluctuations of the local filling factor nite 2d system even for arbitrarily weak disorder. This X induced by the random potential of impurities lead to a has been shown to be the case for charged particles in r localmismatchbetweenthegaugeandexternalmagnetic a random magnetic field [8,9]. Specifically, the random a fields. Anumber ofobservations[3]ofFermi-surfacefea- magnetic field problem belongs to the unitary universal- tures near half-filling give strong experimental support ityclass,withthelocalizationlengthξgrowingextremely to the model of the effective magnetic field. Apart from fastwiththedimensionlessconductanceg =σxx/(e2/h), the composite-particle models involving fictitious fields, 2d electron systems with a real RMF can be directly ξ exp(π2g2). (1) ∝ realized in semiconductor heterostructures by attaching to the latter superconducting [4,5] or ferromagnetic [6,7] These theoretical results are in full agreement with the overlayers. recent extensive numerical study [10]. According to (1), The peculiarity of transport properties of 2d electrons already for g >1.5 the localization length is larger than ∼ 1 any reasonable system size, and the quasiclassical ap- the trajectories are of the form of drifting cyclotron or- proach is fully justified. bits,andthedynamicsisgovernedbyanadiabaticinvari- Let us stress that we consider the situation, in which ant (magnetic flux through one cyclotron orbit). In the the smooth random magnetic field constitutes the only adiabatic approximation, the particles thus drift along type of disorder present. This should be contrastedwith the closed magnetic field contours and hence are local- the starting point of Refs. [11], where the main contri- ized. Itisonlytheweaknonadiabatic scatteringbetween bution to the resistivity was assumed to be given by a drifttrajectoriesthatyieldsafiniteconductivity. Thislo- short-range random scalar potential (treated within the calizationeffectissimilartothe formationofa“stochas- relaxationtimeapproximationforthecollisionintegralin ticweb”[14,15]inaslowlyvaryingscalarrandompoten- the Boltzmann equation), while a weak long-range cor- tialinthepresenceofanexternalhomogeneousmagnetic related random magnetic field was considered as a small field. The conductivity due to the weak nonadiabaticity perturbation. falls off exponentially fast with increasing B. To calcu- The purpose of this paper is to examine the transport late the function σ (B) explicitly, one should consider xx in a long-range RMF in detail, with particular emphasis the averaging procedure with care. An essential point is on the conductivity in an external (homogeneous) mag- thatthenonadiabatictransitionsthatdeterminethecon- netic field B = B(r) and/or at finite frequency ω. As ductivity occur in rare fluctuations distributed sparsely h i we will see, the physics of the problem depends crucially along the percolating trajectories. These rare spots are on the values of B and ω, leading to a rich variety of characterizedbyanomalouslysharpchangesoftheRMF. differenttransportregimes. We complementthe analyti- Using the optimum-fluctuation method, we find a Gaus- calanalysisbynumericalsimulations. Theimportanceof sian behavior of the conductivity in the limit of large the latter is due to the fact that in the most interesting B: lnσ = A(α) B/B 2, where the coefficient A(α) xx 0 − part of the parameter space the transport is dominated scales as α4/3. Note that the conductivity in the CF (cid:0) (cid:1) bythephenomenonofpercolation,sothatonlyestimates problem (A 1) falls off sharply beyond a small devia- “by order of magnitude” are available at the analytical ∼ tion fromhalf-filling. It is worthwhileto notice also that level. theexponentialfalloffofσ withincreasingB islimited xx InSec.II westudy thedc conductivityinalong-range by inelastic scattering and by scattering on short-range RMF δB(r) at zero B. The character of the B = 0 static inhomogeneities – these scattering processes allow transport is determined by the parameter α = d/R , 0 an electron to change drift trajectories and thus provide where R = v (mc/eB ) is the Larmor radius in the 0 F 0 a mechanism of percolation competing with the nonadi- field B which is a characteristic amplitude of the fluc- 0 abatic transitions [16,7]. tuations, v the Fermi velocity. At α 1, the classical F ≪ While increasing B drives the system into the adia- dynamicsisofconventionaldiffusivenature,andthecon- batic regime at any α, the manner in which the conduc- ductivity can be calculated in the Born approximation, tivity crosses over into this regime is qualitatively dif- which gives σ (e2nd/mv )/α2, where n is the par- xx F ∼ ferent in the cases of weak and strong disorder. The ticle density. With increasing α, a crossover to the per- situation is especially interesting at α 1, where the colating kinetics takes place, and at α 1 the conduc- ≫ ≫ transportregimescontrolledbythesnakestates(weakB) tivity is determined by a small fraction of classical tra- andbythenonadiabaticdynamicsofthecyclotronorbits jectories–so-called“snakestates”[12,13]. Theextended (strong B) are separatedby a sharptransition accompa- snake states percolate through the strongly disordered nied by an abrupt change of σ . The transition occurs system by winding around the lines of zero B(r) and xx at B B /α2/3, where B becomes larger than the char- yield σxx (e2nd/mvF)/α1/2 , which falls off slowly ∼ 0 ∼ L acteristicamplitude ofthe magnetic fieldatthe nodes of with increasing strength of disorder [here = (lnα)1/4 L the conductingnetwork. Atthe criticalpoint, the perco- is a weakly varying logarithmic factor]. The crossover lation network formed by the extended snake states falls fromthe 1/α2 to 1/α1/2 behaviorofσ atα 1 is con- xx ∼ apart into disconnected clusters, while the nonadiabatic firmed by our numerical simulations. Furthermore, the scatteringonthehigh-B sideofthetransitionisstrongly latter allow us to find the numerical value of σ for the xx suppressed and yields only a slight smearing of the criti- CF problem (for which α 1/√2 lies in the crossover ≃ cal singularity. region). InSec.IIIweconsiderthecaseofstrongB. Thetrans- For α 1, which is the regime relevant to the CF ∼ port through the snake states, examined in the previous problem, we perform a numerical simulation to calcu- section, is a dominant mechanism of the conductivity at late the magnetoresistance ρ (B). The obtained curves xx strongdisorderandzeromeanB. IncreasingB alsoleads are in good agreement with experimental findings: they toasuppressionoftheconventionaldiffusivemotionand show a weak positive magnetoresistanceat low B, cross- a transition to a percolation regime, even if α 1. The ing over to a falloff of ρ with increasing B (in accor- xx ≪ physics of this phenomenon is, however, quite distinct dancewiththeexponentialsuppressionofσ intheadi- xx fromthe snake-statepercolation. Inthe limitoflargeB, abaticregime). Further,weanalyzethequantumoscilla- 2 tions of the resistivity and show that, in contrast to the A. Weak disorder conventional Shubnikov-de Haas effect in a short-range random potential, they start to develop only when the We start with the simple case of α 1. In this limit, dimensionless conductivity σxx/(e2/h) drops down to a the CF trajectories are only slightly≪bent on the scale value of order of unity. of d, so that the Born approximation is valid. Accord- In Sec. IV we discuss the transport in the RMF at fi- ingly, for the transport scattering time one gets [2,19] nite frequency ω. We find strong deviations of the ac τ−1 = v−1(eB /mc)2 ∞drf(r) = 2α2v /d, where the conductivity from the Drude behavior, especially in the tr F 0 0 F CFeffectivemassm=h¯k /v isintroduced. TheDrude F F percolation regime, i.e. when α and/or B is large. At R conductivity at zero B, σ =e2nτ /m, then reads xx tr small ω, we find a nonanalytical ( ω ) contribution to ∝ | | theacconductivity,whichisdeterminedbyreturnsofthe e2k d F particle to the same spatial regions after a time ∼ 1/|ω| σxx = h 4α2, α≪1 . (2) (and is analogous to the long-time “tails” found in the Lorentz gas many years ago [17]). In the large-B regime and at higher frequencies, the ac conductivity takes the form σxx(ω) ω 3/7, since ω itself starts to determine B. Dynamics of the snake states ∝ | | the width of the percolating “stochastic web” responsi- ble for the conductivity. By contrast, in the snake-state Let us now turn to the strong-RMF regime, α 1, percolation regime (α 1, B = 0) σ (ω) shows only a xx ≫ ≫ keeping B = 0. The seemingly innocent assumption weakdispersioninthecorrespondingfrequencyrange. At about the chaotic character of the particle dynamics, still largerfrequencies the ac conductivity starts to drop which enabled us to represent the conductivity in the exponentially reflecting the “ballistic” motion ofdrifting form e2nτ /m, is not valid anymore. Most particles are orbits (or snake states) on short scales. These analytical tr now out of play since they are caught in cyclotron or- estimates are in agreement with our numerical simula- bitsdriftingalongtheclosedlinesofconstantB(r)(“van tions. Alfvéndrift”). Intheadiabaticlimit,theirdrifttrajecto- Sec.Vsummarizesourfindings. Theanalyticalresults ries areperiodic and so do not contribute to the conduc- ofSectionsII,IIIwerepartlypresentedintheLetter[18]. tivity. Still,howeverlargeB is,thereareclassicalpaths 0 whicharenotlocalizedandpercolatethroughthesystem by meandering around the lines of zero B(r). The con- II. DC TRANSPORT IN ZERO MEAN ductivity is determined by the particles that move along MAGNETIC FIELD these extended “snake states” [12] (Fig. 1). We begin by formulating the model to be studied. We B>0 considernon-interactingparticlesintheRMFB+δB(r) withmeanB andthecorrelator δB(0)δB(r) =B2f(r), h i 0 B=0 where f(0) = 1. We assume that the function f(r) is θ R characterized by a single spatial scale, which is the cor- s θ relation length of the RMF. In particular, in the CF model with the electron density n equal to the charged impurity density n we have B = (h¯c/e)(k /√2d) and B<0 i 0 F f(r)=(1+r2/4d2)−3/2, where k2 =4πn (note that the F FIG.1. Typesoftrajectories inastrongrandommagnetic electron gas is fully spin polarized near ν =1/2; we dis- field: drifting orbits along non-zero B contours and snake card the spin degree of freedom throughout the paper). states near B = 0 lines. Geometry of a snake state is char- In this section we confine ourselves to the case of zero acterized by the angle θ (0<θ <π) at which the trajectory B. The RMF with zero mean is characterized by two crosses thezero-field contour. Notethat thedirection of mo- length scales: d and the cyclotron radius R0 in the field tion of the snake state with θ < θc (left) is opposite to that B0. Defining the parameter α = d/R0, we can distin- for θ > θc (right), where θc ≃ 131◦. The width Rs of the guishthe weak-RMFregimeα 1,wherethe meanfree bundleof snakestate trajectories is also indicated. ≪ path l R d, and the regime of strong fluctuations 0 ≫ ≫ α 1, where one should expect drastic deviations from Note that there is only one single percolating path on ≫ the Drude picture. We will explore these two limiting the manifold of the B(r) = 0 contours; yet, the conduc- cases analytically. However, since the value of α corre- tivity is nonzero since the snake-state trajectories form sponding to the CF problem lies in the crossoverregion, a bundle of finite width, R d/α1/2 (see Fig. 1). The s ∼ we will turnto numericalsimulations in orderto getσ conductingnetworkismadeupofthosesnakestatesthat xx of the CFs. can cross over from one critical zero-B line to another. 3 Thiscouplingoftwoadjacentpercolatingclustersoccurs precursorofthebifurcationwhichaccompaniesthebreak near the critical saddle points of B(r), which are nodes awayofthe trajectoryfromthe zero-B line atθ =π (see of the transport network. The crucial role of the saddle Fig. 2, top). The functions F(θ) and G(θ) in the whole points is that they break down the adiabaticity of the range of θ are shown in Fig. 3a,b. snake-state dynamics, as we are going to explain below. Everywhere except in small regions near the saddle points, the motion along the rapidly oscillating snake- state trajectories around the zero-B contours conserves the adiabatic invariant (see also Ref. [13]) I =m y˙dy . (3) ⊥ I 0.0 Here y and y˙ are the distance and the velocity in the direction perpendicular to the zero-B line, and the integral is taken over one period of the oscillations. The −200.0 conservation of the quantity I can be established di- ⊥ rectly by considering the evolution of the angle θ(x) the snake-state trajectory forms with the line of zero field (y =0) at position x along this line as a consequence of −400.0 the smoothly varying gradient b(x) = ∂B(x,y)/∂y . y=0 | | The adiabatic invariant is parametrized as I (b,θ)=4mv3/2(2mc/eb)1/2F(θ) , (4) −600.0 ⊥ F where F(θ) is a dimensionless function of order unity which can be found explicitly: −800.0 1 100.0 300.0 500.0 700.0 F(θ)=(1 cosθ) dξ (1 ξ4)+cosθ(1 ξ2)2 . (5) − − − FIG. 2. “Serpentology”. Top: transformation of a snake Z0 p state with large θ into a drifting orbit with decreasing gradi- Note that I⊥(b,θ) may be written also as (e/c)Φ(b,θ), ent of the magnetic field; bottom: reflection of a snake state where Φ is the magnetic flux through the area encircled bya magnetic bottle-neck. by the snake-state trajectory and the zero-B line in one oscillationperiod. WerepresenttheequationdI /dx=0 The remarkable point to notice is that G(θ) changes ⊥ inthe formofa scalingrelationfor the snake-stateangle sign at some θ = θ (which is 131◦). More specifi- c ≃ cally, G(θ) behaves singularly around θ , as (θ θ )−1, c c dθ d − =G(θ) , G−1(θ)=2 lnF(θ) . (6) which corresponds to a maximum in F(θ) at this point. dlnb dθ This behavior of I (θ) means that the velocity of the ⊥ Thisequationexpressestheadiabaticinvarianceinterms snake states vs(θ) (which is the average of x˙ over one of the fact that, given boundary conditions [θ(x0) and period) must change sign at θ = θc, i.e., the snake state b(x0) at some point x0], the angle θ at a point x of the is “reflected” at the point xc defined by the equation trajectoryis completely determined by the gradientb(x) θ(xc)=θc (Fig. 2, bottom). Indeed, as follows from Eq. at the same point. Eq. (5) gives the asymptotic expres- (4), the constancy of I⊥[θ(x)] cannot be maintained on sions for G(θ): bothsidesofthe pointxc. Note alsothat,intermsofthe time evolution of θ, the change in sign of the function θ G(θ) at θ = θ means that the time derivative θ˙ retains G(θ) , θ 0 ; (7) c ≃ 4 → the sign it had before the reflection. In fact, one can 2 1 show, by solving the problem with constant gradient b G(θ) , θ π . (8) ≃−3(π θ)ln 1 → exactly, that − π−θ Equations (6), (7) tell us that θ(x) obeys the scaling v (θ)=v F′(θ) 1+ 12cos2θ , (10) s F 3sinθF(θ)+cosθF′(θ) 1/4 2 θ(x ) b(x ) 1 1 = (9) i.e., v (θ) interpolates between v (0) = v and v (π) = θ(x ) b(x ) s s F s 2 (cid:20) 2 (cid:21) v and vanishes at θ = θ (see Fig. 3c). It is worth F c − in the limit of small harmonic oscillations θ 0. The noting that the period of the oscillations T (θ) increases s → singularity of G(θ) in the opposite limit of θ π is a monotonically with growing θ: → 4 1 3 ∂ FIG.3. ThefunctionsF(θ)andG(θ) determiningtheadi- T (θ)= +cotθ I (b,θ) , (11) s mv2 2 ∂θ ⊥ abaticdynamicsofthesnakestatesaccordingtoEqs.(4)–(6), F (cid:18) (cid:19) and thesnake state velocity v (θ). s i.e., T (θ) is equal to 2π(mc/ebv )1/2 at θ = 0 and di- C. Snake-state percolation s F verges as 4(mc/ebv )1/2ln[1/(π θ)] at θ π. The F − → “wavelength” of the snake states along the direction of The adiabatic nature of the snake-state dynamics propagation obviously reads ∆x = vs Ts, while the am- means that a typical trajectory is “trapped” between | | plitude of the oscillations in the perpendicular direction two return points x and x with θ(x )=θ (Fig. 4a). + − ± c is given by ∆y =2vF(mc/ebvF)1/2sin(θ/2). Within the adiabatic picture, the drift motion in such a trap is periodic in time, as demonstrated in Fig. 4b. Hence,unlessnonadiabaticcorrectionsaretakenintoac- (a) count,thesetrajectoriesdonotcontributeto thedc con- 1.0 ductivity. Thenonadiabaticcorrectionsforatypicaltra- jectorywithaslowlyvaryingθ(x)areexponentiallyweak, sothatthemotionremainsfiniteonanexponentiallylong ) θ time scale. Yet, this does not mean that σ (α) is expo- ( xx F 0.5 nentially suppressed in the limit of large α. The point is that there are rare (but not exponentially rare) places along the zero-B contours where the adiabatic picture fails completely. These are regions where the contours pass near the saddle points of B(r). 0.0 0 1 2 θ π c 0.2 θ (a) 10.0 0.1 (b) 5.0 y 0.0 ) θ 0.0 ( G −0.1 −5.0 −0.2 −5 0 5 −10.0 x 0 1 2 θ π c θ 10 (b) 1.0 (c) 5 0.5 x F v )/ 0.0 θ ( 0 vs −0.5 −5 −1.0 0 1 2 θ π 0 200 400 600 800 1000 c t θ 5 FIG. 4. Snake state in a trap. (a) Real-space trajectory hits the saddle point. For typical trajectories with θ 1 of a particle trapped between two bottle-necks. The scales and ∆x (dR )1/2 at x d, this condition fails∼at 0 of the x and y axes differ by a factor ≈ 25: the figure is x < x ∼d/α1/3. Now, as i∼s clear from Fig. 6, whether “compressed” in the x-direction. The dashed lines show the th∼e pacr∼ticle will be scattered to the left of the saddle contours of the constant magnetic field. (b) Time evolution pointortothe rightisdeterminedbythe initialphaseof of the x coordinate. It is seen that the drift motion in the thesnake-stateoscillations. Thissensitivitytothe phase trap is almost periodic. signals the breakdown of the adiabaticity. B=0 B>0 B<0 2 R mi ρ B n sp B<0 B>0 FIG.6. Scattering of a snake state at a saddle point. The B=0 particlemayturneitherleftorright,dependingontheinitial conditions. FIG.5. Geometry of a saddle-point. We now turn to the case of finite ρ. At large enough ρ, the typical snake-state trajectory does not change Consider a snake state that is incident on a sad- the zero-B line: the condition is that the angle θ(x ), dle point with the impact parameter ρ (Fig. 5). This c with which the trajectory comes to the saddle point, means that the magnetic field at the saddle point is be much smaller than the ratio R /x . Substituting B B ρ/d and the distance R at which the zero- min c Bspco∼ntou0r passes the saddle poinmtinis Rmin ∼ √dρ. At θH(oxwc)ev∼er,(txhce/add)1ia/b4a[tsieceinEvaqr.ia(n9)c]e,iswberogkeetnρat≫theds/aαd5d/6le. the saddle point, there is an intersection of two lines point in a wider range of ρ: the condition for the curva- of constant B(r) = B , while two zero-B lines, along sp ture of the zero-B line to be large on the scale of the which the snake states can propagate, come within the wavelength is R <x , which gives ρ<ρ , where distance 2R fromeachother. Clearly,if R is small min c s min min ∼ ∼ enough, the snake state can change the zero-field con- ρ d/α2/3 . (12) s tour. The angle θ, which characterizes the type of the ∼ snake-statetrajectory,is then also changed,i.e., the adi- Within this range, the angle θ after the scattering, θ , out abaticity will be broken down upon “scattering” on the is typically of order unity even though the particle is in- saddle point. To understand the parameters, consider cidentonthe saddlepointwithasmallθ (moreover,θ out first the case of ρ = 0 (“direct hit”). The snake state depends strongly on the phase of the oscillations of the propagatesthenalongastraightlinewithdecreasinggra- incoming trajectory). dientb(x) B x/d,wherexismeasuredfromthesaddle Now consider how the particles propagate between 0 ∼ point. According to Eq. (9), θ(x) x1/4 decreases when suchnon-adiabaticsaddlepoints. Thesaddlepointswith the particle approaches the saddl∝e point, while, as fol- ρ < ρ are distributed sparsely along the zero-B trajec- s lowsfromEq.(11),thewavelength∆xdivergesasx−1/2. to∼ries with the linear density ρ /d2. Therefore, only s ∼ The adiabatic picture is valid only as long as ∆x(x) is a small fraction of the snake states can escape the adia- much smaller than the scale on which the magnetic field batic trapsontheirwaybetweentwosuchsaddlepoints: changes, i.e., ∆x(x) x near the saddle point, which most trajectories are localized in between. The snake ≪ gives the condition x x , where x is the characteris- state is not trapped if θ(x) < θ everywhere on its tra- c c c ≫ ticwavelengthofthe“last”oscillationbeforetheparticle jectorybetweenthecollisionswiththesaddlepoints. Ac- 6 cording to Eq. (9), this is possible for trajectories with L(θ ) d α−1/3θo−u4t , θout >α−1/12 sufficiently smallθ. Indeed, consider asnakestate which out ∼ ×(cid:26)exp[(α1/12θout)−8] , θout ∼<α−1/12 . has a small angle θ 1 in a typical place with the gra- ∼ ≪ (14) dientb B /d. Typically,itwillbe able totravelalong 0 ∼ distance, by far larger than d, until its angle reaches the Therefore,theaverage“waitingtime”theparticlespends value θ : this will occur in a fluctuation of the magnetic c in the unsuccessful attempts to reach the next saddle field with the anomalously high gradient b B /dθ4. 0 point is ∼ Since the probability p(b) that the gradient at a given point exceeds some value b is determined by the Gaus- t N dθ L(θ )/v (d/v )α2/3/lnα . (15) sstiaantestwaitlilsttiycsp,icpa(lbly) =runexpb(a−llibs2t/icabl2ly),thweedfiisntdantcheatLt(hθe) w ∼ Zθo(cu)t out out F ∼ F (cid:10) (cid:11) obeying the equation L(θ)p(B /dθ4)/d 1, which gives Thus, the total time it takes to get through from one 0 ∼ L(θ) dexp(θ−8). Hence, using L(θ )/d d/ρ α2/3, saddlepointtoanotherisindeeddeterminedbyt t . s s b w ∼ ∼ ∼ ≫ weconcludethatthestateswiththeanglesθ θ ,where s ≪ D. Conductivity in a strong random magnetic field θ (lnα)−1/8 , (13) s ∼ Now let us calculate the conductivity at α 1. As ≫ was mentioned at the very beginning, most trajectories will typically get through to the saddle point. do not contribute to σ since they follow periodic drift We thus conclude that the particles with θ <θ prop- xx s orbits. Next, we turned to consider a special class of the agate between saddle points “ballistically” wit∼h the lon- trajectories – the snake states. However, as we showed gitudinal velocity v v , while others are simply out s F above, most snake states are also localized in the adi- ≃ of play. Now we turn to construct the overall picture of abatic traps and only those with angles smaller than θ s the snake-state propagation. The scattering on a saddle canpropagatealongthelinesofzeroB. Atthispoint,we point is actually a multi-step process. The fact that the have to be concerned about the topology of the zero-B angle θ is typically 1 means that, having collided out contours. Thefirstthingtonoticeisthatallthecontours ∼ with a saddle point once, the particle almost inevitably are closed except one and this one percolating contour returns back to it with a new angle of incidence θ′: in by itself cannot yield a finite conductivity. The conduc- effect, the trajectory “sticks” to the saddle point. How- tivity is nonetheless finite since the snake states in fact ever, after many mappings θ θ θ′, the multi- out form a conducting network of finite width. The nodes of → → ple reflections establish a stochastic distribution of the the network are critical saddle points, where two adja- angle θ characterizing the outgoing trajectory. Also, out centpercolatingcontourscomeclosetoeachother. Note after many attempts the particle will go to the left or that most of the saddle points that the particle hits on to the right with equal probability. This randomization its way between the critical ones only connect up small of θ and of the direction of motion is clearly seen in out closed loops and so do not create a connected network. the numerical simulation in [13]. Once the particle picks Thishappensonlyatthecriticalsaddlepoints,wherethe up the angle θ θ /α1/12, it will move ballistically out s snakestatescancrossoverfromonecriticalzero-Blineof ∼ until it reaches the next saddle point. Here, the factor length L α14/9d to another. We use here the results α−1/12 is related to the fact that the angle θ(x) will in- s ∼ of the percolation theory (for a review see, e.g., [16]): crease x1/4 onthe scale of d, so that the particle must L d(d/ρ )ν+1, where ν =4/3 is the critical exponent ∝ s s have θout which is (xc/d)1/4 times smaller than θs. At tha∼tcontrolsthesizeofthecriticalclusterξ d(d/ρ )ν, s s the new saddle point the whole process will repeat it- ∼ so that the ratio of the length of the trajectory and the self. Since the saddle points are separated by the large distancefromthestartingpointL /ξ d/ρ . Thechar- s s s distance d2/ρ , the average time it takes the parti- ∼ s acteristicdistancebetweenthe nodes, i.e.,the sizeofthe ∼ cle to move to the next saddle point is determined by elementarycellξ ,isthendα8/9. Onlengthscaleslonger s the ballistic propagation between them, which requires than ξ , the particle dynamics can be viewed as fully the time t d2/ρ v (d/v )α2/3, not by the mul- s b s F F stochastic. We estimate the macroscopicdiffusion coeffi- ∼ ∼ tiple attempts to “break away” with large θout, which cientasD ν D ,whereν L R θ2/ξ2 isthe fraction end in returns to the starting point. Indeed, assuming ∼ s s s ∼ s s s s of particles residing in the delocalized snake-states and the full randomization of θout, we estimate the number D ξ2 v /L is their diffusion coefficient. Note that of such attempts, until the particle picks up the angle ν sc∼ontsai×ns Fa facstor θ2 – since the density of the snake θout < θo(cu)t = α−1/12(lnα)−1/8 necessary to reach the stsates is determined insthe phase space parametrized by next∼saddle-point, as N 1/θ(c). According to what is both the angle θ and real-space coordinate: accordingly, ∼ out said above, the initial condition θ allows the particle onefactorθ comesfromthecalculationofthefractionof out s to advance the distance the plane covered by the conducting snake states, while 7 theotherdescribestheirfractionintheθ space. Wethus have D v R θ2 and, correspondingly [20], ∼ F s s e2 k d 101 σ F , ln1/4α, α>1 . (16) xx ∼ h α1/2 L∼ ∼ h] L 2/ e It is worth noting that the percolation enhances the d [ F conductivity: by comparison with the Born approxima- /kx 100 tion [Eq. (2)], the conductivity is α3/2/ times larger σx ∼ L (thoughthelocalizationeffectsarestrongandnaivelyone mighthaveexpectedthe opposite). Letusalsonotethat σ given by Eq. (16) is larger by a factor of α1/2/ xx ∼ L 10−1 thanthatobtainedforα 1in[21]byusingan“eikonal ≫ 10−1 100 101 approach”. Thefaultin[21]isnotwiththequasiclassical α approximationitself,butwiththemethodofdisorderav- eraging, which neglects the localization of particles and FIG.7. dc conductivity at B =0, as a function of the pa- the percolating character of the transport through the rameter α. The dashed and the full lines correspond to the snake states. theoretical asymptotics (2) and (16), respectively. Statistical We now turn to the numerical simulation. To calcu- errors do not exceed thesymbol size. latethe conductivitytensorcomponentsσ weevaluate µν numerically the classical current response function, III. DC TRANSPORT IN NON-ZERO MEAN ∞ MAGNETIC FIELD σ =e2ρ dt v (0)v (t) , (17) µν F µ ν h i Z0 We now consider the conductivity at finite B. Let us first discuss the case of α 1, when the conductivity where ρ =m/2π¯h2 is the density of states and the av- σ(B) can be parameterized∼as a function of the single F erage is taken over the disorder realizations and starting variable B/B . As shown in the previous section, at 0 pointsofthetrajectory. Typically,evaluationofthecon- smallB/B weareatthecrossoverbetweentheuncorre- 0 ductivities involved averagingover 103 104 trajecto- lated diffusion and the snake-state percolation. Now, at ∼ ÷ ries. Thenumericalresultsforσ inFig.7fullyconfirm B B the particle dynamics is a slow van Alfvén drift xx 0 ≫ theanalyticalfindingsabove. Forsmallα,theresultsare of the cyclotronorbits along the lines of constant δB(r). ingoodagreementwiththeBornapproximationformula, It follows that the conductivity is determined by a per- Eq.(2),whileatα 1acrossovertotheα−1/2 behavior, colation network of trajectories close to the δB(r) = 0 ∼ Eq. (16), takes place. At α = 1/√2 (the value relevant lines. From the point of view of topology of the net- to the CF problem at n = n and in the absence of im- work, the situation is thus similar to that at zero B and i purity correlations) we find σ 1.0(e2/h)k d, which α 1. Whatiscruciallydifferent,however,isthe mech- xx F ≃ ≫ is a factor of 2 larger than the Born approximation anism of the percolation. Specifically, at large B there ∼ value. Thisimprovesthe agreementwiththe experimen- is no stochastic mixing at the nodes of the percolation tally found CF conductivity (defined as the inverse of network: unlike the snake states at B = 0, the rapidly the measured resistivity at ν = 1/2), though the typi- rotatingcyclotronorbitspassunharmedthroughthecrit- cal experimental values of σ are still larger than the ical saddle points of δB(r) without crossing over to the xx one we obtain by a factor of 2 3. This remaining adjacent cell. In the high-B limit, the mixing occurs on ∼ − discrepancy might be attributed to correlations in the the links of the network and is only due to the weak distribution of the charged donors [22], which reduces scattering between the drift trajectories. the effective strength of the random potential and thus In order to calculate the conductivity at B B , we 0 ≫ reducesα. Theresistivitydatainzeroexternalmagnetic should integrate out the fast cyclotron rotation, taking field(ascontrastedtozeroeffectivemagneticfieldacting care not to lose the effect of the nonadiabatic mixing. onCFs)indeedindicatethatthemodelofstatisticallyin- Specifically, we have to go beyond the standard sepa- dependent impurity positions overestimates the amount ration of the fast and slow degrees of freedom, known of disorder [23,22]. It is also worth noting here, in view as the drift approximation. The parameter that governs of the controversy about the effective mass of the CFs this separationis δ/d, where δ is a characteristic shift of [2,24–26], that in the RMF model σ at zero B [Eqs. the guiding center after one cyclotron revolution. The xx (2),(16)] does not depend on m (neglecting the correc- drift approximation is represented as a power series in tions [27] related to the interaction between the CFs). δ/d 1. In our problem at α 1, this parameter is ≪ ∼ 8 the ratio B /B. Therefore, if B B , the adiabatic de- Thisformulaexpressesthenonadiabaticshiftintermsof 0 0 ≫ scription is good on microscopic scales. The key point, the asymptotics of the Fourier transform of the smooth however, is that the conductivity is strictly zero at the function δB(x) – thus demonstrating explicitly the ex- level of the drift approximation – since the drift orbits ponential smallness of ∆ρ. It shows that the parameter are periodic in the thermodynamic limit. The effects which governs the exponential falloff of ∆ρ is d/δ 1, ≫ thatbreakthe adiabaticinvarianceandleadto the tran- where δ = πǫR , while the ratio d/R may be arbitrary. c c sitionsbetweenthe drift orbitsareexponentially weakat Notethatthepre-exponentialfactorhappenstooscillate δ/d 1. wildly as ǫ 0. These oscillations are geometric reso- ≪ → nances due to the commensurability of two length scales R and δ. Remarkably, the series of the resonances is c A. Single-impurity scattering defined by the properties of the unperturbed solution (“self-commensurability”) and not by the shape of the Theproblemofthescatteringbetweenthedrifttrajec- scatterer. This means that the oscillations are damped tories in the static RMF [18], as well as a similar prob- with increasing strength of the perturbation [29] – since lem for a randomscalarpotential, consideredrecently in the resonance condition cannot be met simultaneously [14], is a particular example of the broad class of prob- everywhere on a strongly perturbed trajectory. Another lems dealing with non-conservation of an adiabatic in- peculiarfeatureofthenonadiabaticshiftisitssensitivity variant. Despite the general interest of this problem, to the phase ϕ of the cyclotron rotation of the incident 0 any systematic expansion capable of giving the scatter- electron (∆ρ cosϕ ). 0 ∝ ing rate beyond the exponential accuracy, has proven to In the CF problem, a charged impurity located at be a tough exercise. To consider a transparent example, a distance d from the plane occupied by the elec- weformulatedandsolvedparametricallyexactlyasingle- tron gas creates the axially symmetric perturbation scatteringproblem[29]. Specifically,weintroduceaweak δB(r) = δB d3(r2 + d2)−3/2. Because of the branch 0 homogeneousgradientǫofthebackgroundmagneticfield points at r = id in this expression, the integrand andconsidertheinteractionwithan“impurity”modeled ± in Eq. (19) will contain the exponentially small factor by a spatially localized perturbation δB(r) of size d, so exp 2ωc [y (t) ρ ]2+d2 . The lengthy general re- that the total field −ǫvF 0 − i sult(cid:16)reduceps to (cid:17) B(r)=B[1+ǫ(y/R )]+δB(r) , (18) c δB √R d where Rc is the cyclotron radius in the field B. The ∆ρ 8πd 0 c (22) guiding center coordinate y averaged over the cyclotron ≃ B δ orbit, ρ = y , plays the role of an impact parame- 2 π 2πd h ic cosϕ0cos exp ter. The particle entering the system at x = with × ǫ − 4 − δ −∞ (cid:18) (cid:19) (cid:18) (cid:19) y =ρ willleaveitatx= alongthetrajectorywith h ic i ∞ y = ρ +∆ρ, where ∆ρ is the nonadiabatic shift we h ic i at ρi = 0 in the limit d Rc. Equation (22) reflects areinterestedin. Inthissingle-impurityscatteringprob- ≫ the features of the non-adiabatic shift discussed above: lem,theshiftisaperfectlywell-definedquantity. Tofirst the exponential smallness, the oscillations with chang- order in δB, the exact solution is given by ing ǫ, and the oscillatory dependence on the phase ϕ . 0 ∞ δB[r (t)] These results were confirmed by numerical simulations 0 ∆ρ=γI , I = dt y˙ (t) . (19) 0 in Ref. [29]. Note that Eq. (22) implies that the drift B Z−∞ trajectory is only slightly perturbed by δB(r). Here r (t) is the unperturbed trajectory for δB =0, 0 t γ(ǫ)=ω y˙ (t) dt′x˙ (t′)/y˙2(t′) , (20) c 0 0 0 (cid:28) Z0 (cid:29)c B. Optimum fluctuation ω =eB/mc, and the angular brackets denote averaging c overonecyclotronperiod. Inthelimitǫ 0theconstant → γ 1. Wecanfurthersimplifythe modelbyassuming In the transport problem one has to average over an →− that δB is a function of x only. In this case, the integral ensemble of impurities. What is crucial for the averag- in Eq. (19) is evaluated at ǫ 1 by the saddle-point ing process is that the drift velocity is itself determined ≪ method to give by the fluctuations of the impurity field. The non-linear problem gets therefore much more involved as compared 2v ǫ 2 π ∆ρ= F cos cosϕ (21) to the single-scattering model above, but the principal 0 − B rπ (cid:18)ǫ − 4(cid:19) featuresofthenonadiabaticscatteringremainunchanged ∞ ǫ and the main message can be simply stated: Because dtcosω t δB v t R . × c 2 F − c of the exponentially strong dependence of the shift on Z−∞ (cid:16) (cid:17) 9 the parameters of the single scatterer, the conductivity ∆ρ(v ) exp( dω /v ). Oneseesthat∆ρ(v )growsex- d c d d ∝ − is determined by rare places with an anomalously high ponentiallywithincreasingv . Now,thelineardensityof d rateofnonadiabatictransitions[14,18]. Accordingly,one the fluctuations with largev along the percolating path d can neglect correlations between consecutive transitions isoforderp(v ),wheretheGaussianprobabilitythatthe d from one drift orbit to another and each nonadiabatic drift velocity at a given point is larger than v reads d shift can be consideredindependently. Since the nonadiabatic scattering rate increases as the drift motion gets p(vd)=exp(−vd2/2 vd2x ); faster, the effective scatterers,sparsely distributed along 2 2 the percolatingtrajectories,arecharacterizedbyanoma- vd2x = 136vF2 Rdc(cid:10) (cid:11)BB0 . (26) louslysharpchangesofthe RMF.Theproblemnowis to (cid:18) (cid:19) (cid:18) (cid:19) (cid:10) (cid:11) find the density and the parameters of these scatterers. Averaging [∆ρ(v )]2 with p(v ), we thus get d d The nonadiabatic shift ∆ρ=∆ρ +i∆ρ (in complex x y notation) after one scattering reads d2ω2 1/3 [∆ρ(v )]2 exp( S ) ; S =3 c . d ∝ − min min 2 v2 ∆ρ=vF dteiωct+iϕ0A(t) , (23) (cid:10) (cid:11) (cid:18) h dxi(cid:19) Z The “optimum” drift velocity which determines this av- wherethesmoothfunctionA(t)variesslowlyonthescale erage is vd0 = vd2x 1/2(4d2ωc2/ vd2x )1/6. As is clear, of ωc−1 and is given by the following average taken over vd0 ≫ vd2x 1/2 (cid:10)at B(cid:11)≫ B0. Th(cid:10)e op(cid:11)timum fluctuations one cyclotron period: yield the Gaussian behavior of the scattering rate: (cid:10) (cid:11) A(t)= eiχ(t) , χ(t)= e tdt′δB[r(t′)] . (24) Smin =c(B/B0)2 , B/B0 ≫1 , (27) D Ec mcZ0 with the coefficient c = 181/3 2.62 (here we assumed The integral which determines the random phase χ(t) α = 1/√2). This simple deriv≃ation of the exponential should be done on the exact trajectory r(t). Note that dependence of [∆ρ(v )]2 captures the essentialphysics d Eq. (23) gives the nonadiabatic shift both along and and yields a correct parametric estimate for S ; how- min (cid:10) (cid:11) across the drift trajectory. Since only the latter is of ever, it is not exact in that v (s) in the optimum fluc- d interest, one should project the result of the integration tuation is not, in fact, constant on the scale of d, and (23) onto the axis perpendicular to the direction of the for this reason it does not give the correct value of the drift of the outgoing particle. numericalcoefficient c in (27). To obtain the asymptoti- Since the nonadiabatic mixing is determined by the cally exact numerical coefficient in S , we have to use min short wavelength Fourier harmonics of the perturbation the optimum-fluctuation method in the whole configura- [Eq. (23)], it is the analytical properties of the function tionalspace. Theoptimumconfigurationischaracterized A(t) and, therefore, of the correlator δB(0)δB(r) that by the function v (s) and the shape of the drift trajec- h i d are important. In the CF problem, this correlator has tory. WewritethephasefactoreiωctinEq. (23)aseiϕ(s), branch points as a function of r at r = 2id. However, where we introduce the s dependent phase ± in order to calculate the scattering probability, which is given by Eq. (23), one has to find the singularities in s ds′ ϕ(s)=ω , (28) δB[r(t)] as a function of time t and average the result. c v (s′) Z0 d For a given perturbation δB(r) this purely mechanical and notice that the exponent S is determined by the problem of finding the Fourier asymptotics of the inte- min gral along the path r(t) may be quite complex, but we phase ϕ(s) pickedup at the singularpoint of the pertur- bationδB[r(t)]regardedasafunctionofthelongitudinal can circumvent the difficulties by performing the con- coordinate s. As can be verified by varying the shape of figurational averaging first. As was already mentioned, the trajectory, the minimum “action” S is acquired the effective scatterers are characterizedby anomalously min along a straight path and the quantity to be calculated large fluctuations of the drift velocity is therefore v (s)=v (R /2B) B(s) , (25) d F c |∇ | id ds S = ln exp iω , (29) min c where s is the coordinate along the path. To see this, − * Z−id vd(s)!+ one can use the exact solution of the single-scattering problem considered above. Let us first assume, for the where the integralshouldbe done along the straightline purpose of illustration, that the large v (s) does not connecting the points s = id and s = id in the com- d − changeappreciablyonthe scaleofd. Equation(22)then plex plane of the variable s [30]. This average deter- tells us that a single impurity located on the trajectory mines,withexponentialaccuracy,thediffusioncoefficient whichpassesthroughthefluctuationwithlargev yields D exp( S ), which is defined as d ⊥ min ∝ − 10