High Landau levels of 2D electrons near the topological transition caused by interplay of spin-orbit and Zeeman energy shifts
HHigh Landau levels of 2D electrons near the topological transition caused by interplayof spin-orbit and Zeeman energy shifts
Rajesh K. Malla and M. E. Raikh
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112
In the presence of spin-orbit coupling two branches of the energy spectrum of 2D electrons getshifted in the momentum space. Application of in-plane magnetic field causes the splitting ofthe branches in energy. When both, spin-orbit coupling and Zeeman splitting are present, thebranches of energy spectrum cross at certain energy. Near this energy, the Landau quantizationbecomes peculiar since semiclassical trajectories, corresponding to individual branches, get coupled.We study this coupling as a function of proximity to the topological transition. Remarkably, thedependence on the proximity is strongly asymmetric reflecting the specifics of the behavior of thetrajectories near the crossing. Equally remarkable, on one side of the transition, the magnitudeof coupling is an oscillating function of this proximity. These oscillations can be interpreted interms of the St¨uckelberg interference. Scaling of characteristic detuning with magnetic length isalso unusual. This unusual behavior cannot be captured by simply linearizing the Fermi contoursnear the crossing point.
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I. INTRODUCTION
It is known for more than 60 years that, in a metal, theperiod of the resistance oscillations with magnetic fieldas well as the period of the oscillations of diamagneticmoment reflect the geometry of its Fermi surface.
Thisrelation originates from the fact that, by virtue of theLandau quantization, the areas of the cross-sections ofthe Fermi surface by the planes perpendicular to mag-netic field are discrete. These areas are encircled in thecourse of semiclassical motion of the electron wave pack-ets in magnetic field and contain half-integer number ofthe flux quanta.In particular situations when energy gaps, correspond-ing to neighboring energy bands, are anomalously small,interband tunneling becomes possible. This tunneling,known as magnetic breakdown, couples the Fermi sur-faces from different bands and modifies the quantizationcondition tocos (cid:32) S + l + S − l ϕ E (cid:33) = T E cos (cid:32) S + l − S − l (cid:33) , (1)where S ± are the areas encircled by the contacting semi-classical trajectories, corresponding to the energy, E , and l is the magnetic length. Parameters T E and ϕ E are, re-spectively, the amplitude and the phase of the couplingcoefficient between the contacting trajectories. The tun-nel probability, |T E | , assumes an appreciable value atenergies when the separation of the Fermi surfaces in themomentum space becomes comparable to l − . Analyticalform of T E was established using the effective mass ap-proximation, within which the band dispersion near thetouching point has the form ε ( k ) = (cid:126) k x m x − (cid:126) k y m y , (2)where m x and m y are the in-plane effective masses (mag- netic field is directed along z ). As E crosses from negativeto positive values, the connectivity of the Fermi surface, ε ( k ) = E , changes. In magnetic field, the tunneling prob-ability between the states k x → −∞ and k x → ∞ reducesto the transmission through the “inverted parabola” po-tential, − (cid:126) m x m y ) / (cid:104) ( x − k y l ) l (cid:105) , the result for which, ob-tained in a celebrated paper by Kemble, reads |T E | = 1exp ( − πµ E ) + 1 , (3)where the parameter µ E is proportional to energy and isgiven by µ E = (cid:126) ( m x m y ) / El .Quantization condition Eq. (1) describes topologi-cal transitions for spinless electrons with scalar wave-functions. An alternative scenario of this transition unfolds in type-II Weyl semimetals predicted recently and realized experimentally, for review see Refs. 13, 14.In these materials, the contacting contours of the Fermisurface belong to electron and hole pockets, see e.g.Ref. 15. The corresponding states are the eigenfunctionsof the matrix Hamiltonian, the simplest version of whichhas the form ˆ H W = ak x σ + (cid:88) i v i k i σ i , (4)where σ is a unit matrix and σ i are the Pauli matrices.Two branches, E ± ( k ) = ak x ± (cid:34)(cid:88) i v i k i (cid:35) / , (5)of the spectrum defined by the Hamiltonian Eq. (4) touchat the point k = 0. The difference between the spectraEq. (2) and Eq. (5) manifests itself in the expression forthe transmission probability. For type-II Weyl semimet-als it takes the form a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r |T W | = exp ( − πµ W ) , (6)where µ W is proportional to the square of minimum sep-aration between the contours and to the square of mag-netic length. The origin of the difference between Eqs.(3) and (6) is that the Hamiltonian Eq. (4) allows theKlein tunneling between the electron and hole states.With linear dispersion Eq. (5), the calculation of thetunnel probability reduces to the Landau-Zener problem.In Ref. 16 it was noted that the topological transi-tion in the geometry of two Fermi contours can be real-ized for purely two-dimensional electrons subject to in-plane magnetic field and in the presence of spin-orbit cou-pling. The origin of crossing of the two branches of thespectrum is the interplay of the Zeeman and spin-orbitsplittings. The eigenfunctions corresponding to the twocrossing branches are spinors. Then it was concluded inRef. 16 that the semiclassical Landau quantization isgoverned by Eq. (1) with tunnel probability given byEq. (6), similarly to the type-II Weyl semimetals.In the present paper we study in detail the evolutionof the 2D Fermi contours in the vicinity of the topolog-ical transition emerging in the presence of Zeeman andspin-orbit couplings. We show that, linearizing of thespectrum in the very vicinity of crossing is insufficientto describe the transition probability. Magnetic field de-pendence of T as well as its dependence on detuning, isgoverned by the curvature of the Fermi contours. II. EVOLUTION OF THE FERMI CONTOURSNEAR THE CROSSING
We start with a 2D Hamiltonianˆ H = (cid:126) k m + α ( k x σ y − k y σ x ) − ∆ σ y , (7)where the first term is a free-electron Hamiltonian, whilethe second and the third terms describe spin-orbit cou-pling and Zeeman splitting in an in-plane magnetic field,respectively.Two branches of the spectrum of the HamiltonianEq. (7) are given by E ± ( k ) = (cid:126) ( k x + k y )2 m ± (cid:2) (∆ − αk x ) + α k y (cid:3) / . (8)The branches cross at the point k x = ∆ α , k y = 0 , (9)which corresponds to the energy E = E = (cid:126) ∆ mα . (10) - - - - - q x q y ( a ) - - - - - q x q y ( b )- - - - - q x q y ( c ) - - - - - q x q y ( d ) FIG. 1: (Color online) Evolution of the Fermi contours, de-fined by Eq. (12), with the ratio, ν , of the spin-orbit andZeeman energy shifts [Eq. (13)]; (a), (b), (c), and (d) corre-spond to ν = 2, ν = 1 . ν = 1, and ν = 0 .
5, respectively.As ν decreases, the inner contour grows. The Fermi energy ischosen to be E = 1 . E in all panels. Thus, strictly speaking,the separation between the inner and the outer contours isfinite at q y = 0. This separation can be distinguished in (a)and (b), but cannot be distinguished in (c) and (d). To analyze the behavior of the Fermi contours, E ± ( k ) = E , we introduce the dimensionless variables k x = (cid:32) ∆ α (cid:33) q x , k y = (cid:32) ∆ α (cid:33) q y , (11)and rewrite Eq. (8) in the form EE = q x + q y ± ν (cid:2) ( q x − + q y (cid:3) / , (12)where we have introduced a dimensionless parameter ν = 2 mα (cid:126) ∆ , (13)which measures the ratio of the energy shifts due to thespin-orbit and Zeeman couplings.Near the crossing point ( E − E ) (cid:28) E and q y (cid:28) (cid:34) q x − − E − E ) E (4 − ν ) (cid:35) − (cid:32) ν − ν (cid:33) q y = (cid:34) ν ( E − E )(4 − ν ) E (cid:35) . (14)We see that the behavior of the Fermi contours is differentfor ν > ν <
2. For ν > - - - - - - - q x q y ( a ) - - - - - - - q x q y ( b )- - - - - - - q x q y ( c ) - - - - - - - q x q y ( d ) FIG. 2: (Color online) Evolution of the Fermi contours, de-fined by Eq. (12), with energy. Panels (a), (b), (c), and (d)correspond to energies E = 1 . E , E = 1 . E , E = 1 . E ,and E = 2 E , respectively. We chose ν = 1 in all panels. It isseen that in panel (c) the outer contour is vertical at q y = 0in accordance to Eq. (18). ellipse, i.e. there is only one Fermi contour. For ν < E = E , namely, q y = ± (cid:0) − ν (cid:1) / ν ( q x − . (15)Evolution of the Fermi contours with ν is illustrated inFig. 1. It is seen that, as ν decreases below ν = 2, theinner contours grows. The behavior of the outer contouris quadratic near q y = 0 and also at two finite values ± ˜ q y .To find these values, we differentiate Eq. (12) keeping E constant and obtain ∂q x ∂q y = − q y (cid:40) (cid:104) ( q x − + q y (cid:105) / ± ν (cid:41) q x (cid:104) ( q x − + q y (cid:105) / ± ν ( q x − . (16)The sign “ − ” corresponds to the outer branch. At q = ˜ q y the derivative turns to zero, which, together withEq. (12), yields q x = ˜ q x = 12 (cid:16) ν EE (cid:17) , (17)Substituting this value back into Eq. (12), we find ˜ q y = 12 (cid:34)(cid:16) EE + ν ν − (cid:17)(cid:16) − EE − ν ν +1 (cid:17)(cid:35) / . (18)We see that at energy EE = 1 + ν − ν the outer Fermicontour is vertical at points (˜ q x , ± ˜ q y ), as illustrated inFig. 2. At small ν this energy is close to the crossingpoint of the contours. In magnetic field, this peculiar be-havior manifests itself in the coupling between the semi-classical trajectories as we will see in the next Section. III. TUNNELING BETWEEN THESEMICLASSICAL TRAJECTORIES
Incorporating magnetic field in the z -directionamounts to replacing k x by k x − yl . Then the systemof equations for the components of the spinor, (Ψ , i Ψ ),takes the form E Ψ − (cid:126) m (cid:16) k x − yl (cid:17) Ψ + (cid:126) m ∂ Ψ ∂y = α (cid:16) k x − yl (cid:17) Ψ − ∆Ψ + α ∂ Ψ ∂y , (19) E Ψ − (cid:126) m (cid:16) k x − yl (cid:17) Ψ + (cid:126) m ∂ Ψ ∂y = α (cid:16) k x − yl (cid:17) Ψ − ∆Ψ − α ∂ Ψ ∂y . (20)Upon introducing new functionsΦ = Ψ + Ψ , Φ = Ψ − Ψ , (21)and a dimensionless variable u = ∆ α (cid:0) y − k x l (cid:1) (22)the system can be rewritten as EE Φ + ∂ Φ ∂u − (cid:34)(cid:16) α ∆ l (cid:17) (cid:16) u + ∆ (cid:126) ω c (cid:17) − (cid:16) ν ν (cid:17)(cid:35) Φ = ν ∂ Φ ∂u , (23) EE Φ + ∂ Φ ∂u − (cid:34)(cid:16) α ∆ l (cid:17) (cid:16) u − ∆ (cid:126) ω c (cid:17) − (cid:16) ν − ν (cid:17)(cid:35) Φ = − ν ∂ Φ ∂u . (24)Here (cid:126) ω c = (cid:126) ml is the cyclotron energy. Equations (23)and (24) are obtained by adding and subtracting Eqs.(19) and (20). Square brackets in Eqs. (23) and (24) canbe viewed as effective potentials for the functions Φ andΦ . These potentials, sketched in Fig. 3 are parabolasshifted horizontally and vertically These potentials crossat u = u c = ν (cid:126) ω c (cid:18) ∆ lα (cid:19) = (cid:18) ∆ lα (cid:19) . (25)The value of potential at u = u c is equal to δ = E − E E . (26)Parameter δ is the dimensionless measure of the proxim-ity to the crossing. Semiclassical quantization procedureis valid when the Landau levels, corresponding to E = E ,are high. Quantitatively, this condition can be expressedas E (cid:126) ω c = 12 (cid:32) ∆ lα (cid:33) (cid:29) . (27)If the above condition is satisfied, derivation of the equa-tion similar to Eq. (1) for the semiclassical energy lev-els can be outlined as follows. In the absence of theright-hand sides in Eqs. (23), (24), the solution of (23)represents a wave, incident from the left, which is fullyreflected at the turning point (see Fig. 3). The condi-tion that the solution decays to the right from the turningpoint defines the conventional phase shift, 2 × π , betweenthe incident and reflected waves. If the presence of theright-hand side in Eq. (24), there are two channels of re-flection: in addition to the reflected-wave solution of Eq.(23), the incident wave can give rise to the solution of (24)propagating to the left, see Fig. 3. If the amplitude ofthe incident wave is 1, then the amplitude of this secondreflected wave should be identified with T E , the couplingcoefficient in the quantization condition Eq. (1). Calcu-lation of T E is our main goal. To achieve this goal, it isconvenient to analyze the system Eqs. (23), (24) in themomentum space.In the vicinity of u = u c the system (23), (24) takesthe form δ Φ − F u Φ + ∂ Φ ∂u = ν ∂ Φ ∂u ,δ Φ − F u Φ + ∂ Φ ∂u = − ν ∂ Φ ∂u , (28)where u = u − u c . The slopes F , F are defined as F = 2 (cid:16) α ∆ l (cid:17) (cid:16) ν (cid:17) , F = 2 (cid:16) α ∆ l (cid:17) (cid:16) − ν (cid:17) . (29)Upon performing the Fourier transformation in Eq. (28),we arrive to the system of coupled first-order differentialequations for the transformed functions Φ and Φ FIG. 3: (Color online) Without the right-hand sides, Eqs.(23) and (24) are decoupled and describe the electron motionin parabolic potentials (blue and red, respectively). The po-tentials cross at u = u c . Without coupling, the incident wave, i , see inset is fully reflected into the wave, r , propagating inthe red parabola. With right-hand sides caused by spin-orbitcoupling, another channel of reflection into the wave r prop-agating in the blue parabola emerges. The correspondingreflection probability should be identified with transmissionprobability |T E | . δ ˜Φ − i F ∂ ˜Φ ∂κ − κ ˜Φ = iνκ ˜Φ ,δ ˜Φ − i F ∂ ˜Φ ∂κ − κ ˜Φ = − iνκ ˜Φ . (30)To analyze this system, it is convenient to “antisym-metrize” it by eliminating the symmetric phase. Thisis achieved by introducing instead of ˜Φ , ˜Φ the newfunctions defined as˜Υ , ( κ ) = ˜Φ , ( κ ) exp (cid:34) − i (cid:18) δκ − κ (cid:19) F + F F F (cid:35) . (31)Then the system Eq. (30) assumes the form i F ∂ ˜Υ ∂κ + F − F F (cid:0) δ − κ (cid:1) ˜Υ = iνκ ˜Υ ,i F ∂ ˜Υ ∂κ − F − F F (cid:0) δ − κ (cid:1) ˜Υ = − iνκ ˜Υ . (32)The product νκ in the right-hand sides describes the cou-pling between the semiclassical trajectories. We will firstassume that the coupling is weak and find the transmis-sion coefficient perturbatively. In the zeroth order weneglect the right-hand side in the first equation, so that˜Υ ( κ ) = exp (cid:34) i F − F F F (cid:18) δκ − κ (cid:19) (cid:35) . (33)Substituting ˜Υ ( κ ) into the second equation and solvingfor ˜Υ ( κ ) we find | ˜Υ ( ∞ ) | = ν F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:90) −∞ dκ κ exp (cid:104) i F − F F F (cid:18) δκ − κ (cid:19) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (34) - - - - - - - q x q y - - - - - - - q x q y FIG. 4: (Color online) Illustration of the asymmetry of theFermi-contours with respect to the sign of detuning, δ . In theleft panel the contours are shown for δ = 0 .
3, while in theright panel for δ = − .
3. Parameter ν is chosen to be 0 . |T E | , in the rightpanel is much smaller than in the left panel. The meaning of | ˜Υ ( ∞ ) | is the power transmission co-efficient, |T E | , analogously to Eq. (3).It is easy to see that only the imaginary part of theexponent contributes to the integral. Then the integralreduces to the derivative of the Airy function, Ai ( z ). Us-ing the expressions for F , F we rewrite the final resultin the form |T E | = 4 π (cid:16) ν (cid:17) / (cid:18) ∆ lα (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ai (cid:48) (cid:34) − δ (cid:16) ν (cid:17) / (cid:18) ∆ lα (cid:19) / (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (35)Our prime observation is that the coupling is an asym-metric function of the detuning, δ . This, actually, reflectsthe asymmetry of the Fermi-contours’ arrangement withrespect to the sign of δ . The situation is illustrated inFig. 4, where the Fermi contours are plotted for δ = 0 . δ = − .
3. At negative δ the transmission probabilityfalls off with | δ | as exp (cid:104) − | δ | / ν (cid:0) ∆ lα (cid:1) (cid:105) . Note that, bycontrast to Eqs. (3) and (6), characteristic δ scales withmagnetic field as l − / , instead of l − and l − , respec-tively.It is seen from Eq. (35) that at positive δ the transmis-sion coefficient oscillates with δ . Unfortunately, Eq. (35)obtained perturbatively, is not applicable in this domain.This is because it predicts that |T E | exceeds 1 at largepositive δ . For this reason, in the next Section we turnto numerics. IV. NUMERICAL RESULTS
For numerical calculations it is convenient to performa rescaling, κ = Gz , in the system Eq. (32), where theparameter G is equal to G = (cid:18) ν (cid:19) / (cid:16) α ∆ l (cid:17) = (cid:126) ω c ∆ . (36) Then the system assumes the form i ˜Υ (cid:48) + 12 (cid:18) δG − Gz (cid:19) ˜Υ = iz ˜Υ ,i ˜Υ (cid:48) − (cid:18) δG − Gz (cid:19) ˜Υ = − iz ˜Υ . (37)We see that, effectively, the transmission coefficient de-pends only on two parameters, detuning δ and the di-mensionless magnetic field, G . For numerical purposesit is convenient to get rid of the fast oscillations of ˜Υ and ˜Υ by introducing new variables ρ , = ˜Υ , exp (cid:20) ∓ i (cid:18) δG z − G z (cid:19)(cid:21) . (38)With these new variables the oscillating functions appearin the coupling of ρ and ρ , namely i ∂ρ ∂z = iz exp (cid:20) − i (cid:18) δG z − G z (cid:19)(cid:21) ρ ,i ∂ρ ∂z = − iz exp (cid:20) i (cid:18) δG z − G z (cid:19)(cid:21) ρ . (39)In terms of parameter G , the result Eq. (35) reads |T E | = | ρ ( ∞ ) | = 4 π G / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ai (cid:48) (cid:18) − δG / (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (40)In our numerical calculations we first analyzed the be-havior of | ρ , | with z . In general these quantities ex-hibit oscillations on the background of a smooth envelop.There is way to approximately isolate this envelop. Todo so, we integrate the second equation of the systemEq. (20) using the condition ρ ( −∞ ) = 0 and substi-tute the expression for ρ ( z ) into the first equation. Thisyields the following closed integral-differential equationfor ρ ( z ) ∂ρ ∂z = z z (cid:90) −∞ dz (cid:48) z (cid:48) ρ ( z (cid:48) ) exp (cid:40) i (cid:20) δG ( z − z (cid:48) ) − G (cid:0) z − z (cid:48) (cid:1)(cid:21) (cid:41) . (41)The procedure of extracting the envelop from this equa-tion is developed in Ref. 18. Employing this procedureyields | ρ ( z ) | = (cid:0) δG − Gz (cid:1) z + (cid:0) δG − Gz (cid:1) . (42)In Fig. 5 we plot | ρ ( z ) | for two values of detuning δ = 3 and δ = − G = 1. Apparently the smoothpart of δ = − δ = − z . This interference, having the -10 -5 0 5 10 z | ( z ) | (a) -10 -5 0 5 10 z | ( z ) | (b) FIG. 5: (Color online) Numerical solution, | ρ ( z ) | , of thesystem Eq. (20) is plotted for two values of detuning δ = 3(a) and δ = − G = 1. The black line in (a) andthe green line in (b) arethe envelops plotted from Eq. (42).We see that the agreement with theory is much better in (b),since Eq. (42) neglects interference. -10 -5 0 5 10 z | ( z ) | (a) -10 -5 0 5 10 z | ( z ) | (b) FIG. 6: (Color online) Numerical solution, | ρ ( z ) | , of the sys-tem Eq. (20) is plotted for two values of detuning δ = 0 . δ = − . G = 0 .
7. The black line in (a) andthe green line in (b) are the envelops plotted from Eq. (42).We see that | ρ ( z ) | approaches finite values at large z . Thisapproach is accompanied by huge oscillations which compli-cate the determination of the transmission coefficient. same origin as St¨ukelberg oscillations takes place at pos-itive δ . Since δ in Fig. 5 was chosen to be big, thevalues of ρ ( z ) approach zero at large z . To capture thefinite transmission, we chose the parameters G = 0 . δ = ± .
5, and plotted | ρ ( z ) | in Fig. 6. The agreementwith Eq. (42) is worse in Fig. 6 since, for chosen param-eters, the regime of transmission is less “semiclassical”.We also see that approaching of | ρ ( z ) | to finite valuesat large z is accompanied by huge oscillations. These os-cillations introduce an uncertainty in the value | ρ ( ∞ ) | due to necessary averaging. This uncertainty manifestsitself as wiggles in the dependencies of | ρ ( ∞ ) | on G and δ to which we now turn. G | () | G ρ ( ∞ ) FIG. 7: (Color online) (Blue curve) Transmision coefficientobtained numerically is plotted versus dimensionless magneticfield, G , directly at topological transition δ = 0. (Black curve)Theoretical prediction for transmision coefficient is plottedfrom Eq. (43). In the inset we plot the prediction,sin ( G ),based on heuristic argument given in Section V. For zero detuning, the theoretical prediction for thetransmission coefficient is | ρ ( ∞ ) | = 2 . G / , (43)as follows from Eq. (40). In Fig. 7 we plot this G -dependence together with | ρ ( ∞ ) | ( G ) obtained numeri-cally. We observe the agreement with theory at large G ,where the theory is applicable. Concerning the theoreti-cally relevant small- G domain, numerical errors did notallow us to establish the G -dependence at real small G .It can be concluded that the averaged over strong oscil-lations transmission coefficient approaches at small G and has a maximum near G = 1. We discuss the theo-retical prediction for | ρ ( ∞ ) | ( G ) at small G in the nextSection.Finally, we studied numerically the dependence of thetransmision coefficient on detuning, δ . The result isshown in Fig. 8 for the value of G = 1 .
5. We see that fornegative δ , the numerics agrees quite well with the theo-retical prediction Eq. (40). For large positive δ , Eq. (40),strictly speaking, does not apply, but qualitative agree-ment is apparent. Oscillatory behavior of the trans-mission coefficient is the consequence of the St¨uckleberginterference of virtual Landau-Zener transitions takingplace at z = ± δ / G . V. DISCUSSION ( i ) It is instructive to compare our analysis to Ref.16, where it was concluded that the probability |T E | isgiven by the Landau-Zener formula. Let us trace howthis formula might emerge from our system Eq. (30) -5 0 5 1000.20.40.60.81 | () | FIG. 8: (Color online) Transmision coefficient (blue curve)obtained numerically is plotted versus the detuning, δ , fordimensionless magnetic field G = 1 .
5. Theoretical predic-tion for transmission coefficient (black curve) is plotted fromEq. (40). Small wiggles in the numerical curves are theartifact of the averaging procedure. For negative detuningsthe agreement with theory is good. For positive detunings,theory predicts oscillations with magnitude bigger than 1,while in numerical curve these oscillations, manifesting theSt¨uckelberg interference, slowly decrease with δ . In the semiclassical limit the solutions of the system are˜Υ , ˜Υ are proportional to exp [ iσ ( κ )], where the deriva-tive of the action, σ ( κ ), is given by σ (cid:48) ( κ ) = ± F − F F F (cid:34) (cid:0) δ − κ (cid:1) + 4 ν F F ( F − F ) κ (cid:35) / . (44)Using Eq. (29), we specify the combinations in the squarebrackets and in the prefactor4 ν F F ( F − F ) = (4 − ν ) , F − F F F = (cid:18) ∆ lα (cid:19) ν − ν . (45)It seems that for small δ (cid:28) κ can bedropped from ( δ − κ ) . Indeed, if this term is dropped,the expression in the square brackets turns to zero at κ = ± iκ − , where κ − ≈ δ (4 − ν ) / ≈ δ . (46)Since κ − = δ is much smaller than δ , dropping κ isjustified. Once δ is dropped, the expression for σ (cid:48) ( κ )assumes the standard Landau-Zener form with transitionprobability given by: exp (cid:20) − πνδ (cid:0) ∆ lα (cid:1) (cid:21) . This is theresult obtained in Ref. 16.In our opinion, the flaw of this approach is that, inaddition κ = ± iκ − , Eq. (44) turns to zero at κ = ± iκ + , where κ + ≈ (cid:0) − ν (cid:1) / ≈
2. The point κ + originatesfrom the second derivatives, ∂ Φ ∂u , ∂ Φ ∂u , in Eq. (28)which accounts for the curvature of the spectrum ne-glected in Ref. 16. The value κ + is much bigger than δ and depends on detuning only weakly. This suggeststhat T E is the result of a “two-stage” process: one involv-ing big momentum transfer ∼ κ + and another involvingsmall momentum transfer, ∼ κ − . The resulting T E isa strongly oscillating function of detuning and magneticfield. In fact, similar situation, i.e. numerous complexzeros in σ (cid:48) , was encountered in Refs. 19–24.( ii ) Overall, we were not able to capture the mostrelevant domain where both δ and G are small neitheranalytically nor numerically. This is due to strongly os-cillating character of | ρ ( z ) | . The physical origin of thiscomplication is that simple linearizng the Fermi contoursnear the crossing is insufficient for finding the transitionprobability. The amplitudes ρ ( z ) and ρ ( z ) keep “talk-ing” to each other outside the domain where lineariza-tion applies. Below we present a heuristic account ofthe behavior of the transmission coefficient at zero de-tuning. Conventionally, the transmission coefficient inthe Landau-Zener problem can be found upon setting κ in the expression for σ (cid:48) ( κ ) to be purely imaginary andintegrating between two turning points. This procedureis applicable when the resulting action is big, so that thetransmission is small. If we adopt this procedure in Eq.(44) after setting δ = 0, we would realize that, unlikeLandau-Zener transition the action is imaginary and isequal to iσ = /G (cid:90) − /G dz (cid:20) z − G z (cid:21) / = 83 G . (47)We see that at small G the magnitude of action is big andthat the transmission coefficient oscillates with G insteadof being exponentially small. We cannot judge aboutthe prefactor, except that in Landau-Zener transition theprefactor is 1. This leads to the prediction | ρ ( ∞ ) | =sin ( G ). This prediction is plotted in the inset of Fig. 7.A maximum at G = 1 . G ∼
1. If theabove heuristic argument applies, then the δ -dependenceof the transmission at small G should be weak.( iii ) The result Eq. (35) can be derived directly fromthe system Eq. (28) without transforming to the mo-mentum space. The zeroth-order solution of the firstequation is Ai (cid:20) F / ( δ F − u ) (cid:21) . Thus, the right-handside in the second equation is the derivative of the Airyfunction. Forced solution of the second equation con-tains the overlap of this right-hand side with the freesolution of the second equation, which is Φ ( u ) = Ai (cid:20) F / ( δ F − u ) (cid:21) . Then the result Eq. (35) followsfrom the identity ∞ (cid:90) −∞ dxAi [ λ ( x − a )] Ai (cid:48) [ µ ( x − b )]= πλ ( λ − µ ) / Ai (cid:48) (cid:34) λµ ( a − b )( λ − µ ) / (cid:35) , (48)which can be easily verified using the integral represen- tation of the Airy function. VI. ACKNOWLEDGEMENTS
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