Chaotic cyclotron and Hall trajectories due to spin-orbit coupling
aa r X i v : . [ n li n . C D ] M a y Chaotic cyclotron and Hall trajectories due to spin-orbit coupling
E.V. Kirichenko, V. A. Stephanovich, and E. Ya. Sherman
2, 3 Institute of Physics, Opole University, Opole, 45-052, Poland Department of Physical Chemistry, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain IKERBASQUE Basque Foundation for Science, Bilbao, Spain (Dated: May 12, 2020)We demonstrate that the synergistic effect of a gauge field, Rashba spin-orbit coupling (SOC),and Zeeman splitting can generate chaotic cyclotron and Hall trajectories of particles. The physicalorigin of the chaotic behavior is that the SOC produces a spin-dependent (so-called anomalous)contribution to the particle velocity and the presence of Zeeman field reduces the number of integralsof motion. By using analytical and numerical arguments, we study the conditions of chaos emergenceand report the dynamics both in the regular and chaotic regimes. We observe the critical dependenceof the dynamic patterns (such as the chaotic regime onset) on small variations in the initial conditionsand problem parameters, that is the SOC and/or Zeeman constants. The transition to chaotic regimeis further verified by the analysis of phase portraits as well as Lyapunov exponents spectrum. Theconsidered chaotic behavior can occur in solid state systems, weakly-relativistic plasmas, and coldatomic gases with synthetic gauge fields and spin-related couplings.
INTRODUCTION
Puzzling properties of chaotic motion in simple clas-sical and quantum systems are among the most intrigu-ing problems in modern physics. Recently observed fea-tures of a quantum chaos in cold gases [1, 2], Rydbergexcitons [3], and polaritons [4] demonstrated that con-densed matter is an excellent testbed for these studies(see, e.g., [5] and [6]). They also posed new questionsabout relation between quantum chaos manifestations inthe spectrum and corresponding classical motion [7–10]in a broad variety of the systems. Therefore, it would beof interest to get insights into physical mechanisms un-derlying chaotic behavior. This is especially true for thesystems with well-defined classical (position and momen-tum) and quantum (e.g., spin) degrees of freedom. Mo-tivated by these results, we study two-dimensional (2D)motion of particles in magnetic fields with spin-orbit cou-pling (SOC) of the Rashba type [11] and Zeeman split-ting. These fields can be either genuine (electrons insolids or plasma [12, 13]) or synthetic for cold atoms indesigned coherent optical potentials [14–17]. As an inter-esting example we mention that the spectra of billiards[18, 19] and excitons [20] with SOC do not provide unam-biguous relation to the classical chaos since their quan-tum chaotic features do not have classical counterpart,although rectangular billiards with spin-orbit couplingdriven by external electric fields clearly demonstrate achaotic behavior [21]. Also, it is worth mentioning thata chaos existing in a host 2D system can have strongnontrivial effect on the spin transport there [22]. Herewe concentrate on the spin-orbit coupling effect on theemergence of chaos in a simple semiclassical system. Forthis purpose we consider the diamagnetic effects in theorbital motion due to the Lorentz force and spin preces-sion owing to the joint action of SOC and Zeeman effect.We have shown that latter combination can lead to a chaotic behavior due to the anomalous spin-dependentcontribution to the particle velocity [23]. This anoma-lous velocity in semiconductors is the core element ofthe phenomenon, much resembling
Zitterbewegung (trem-bling oscillatory motion) of free relativistic electrons, de-scribed by Dirac equation. In semiconductor structuressuch as III-V quantum wells and wires, the
Zitterbewe-gung of electron wavepackets can be experimentally ob-served due to favorable energy and length scales [24, 25].We note also, that
Zitterbewegung - like motion plays animportant role in cold atomic gases [15].The Hall effect, both in quantum and classical real-izations, plays an important role in condensed matterphysics. In a sufficiently strong magnetic field, the tra-jectory of a particle moving in a smooth 2D potential,resembles a closed narrow stripe in the vicinity of anequipotential line. This leads to the Hall effect quantiza-tion as the conductivity is solely due to the edge states.However, this simple picture does not take into accountthe SOC effects, which can strongly modify the motionand, as a result, the entire cyclotron and Hall effect pic-tures.We demonstrate that at certain values of the Lorentzand electric forces and spin-dependent fields in terms ofanomalous velocity and spin precession rate, the classi-cal cyclotron and Hall trajectories become chaotic. Withfurther increase in the Zeeman field, it becomes dominantand the spin dynamics turns regular. As a result, the ef-fects of SOC decrease, and the chaos disappears althoughthe particle trajectory can be strongly different from thatwithout SOC. As it is customary to chaotic systems, herewe observe the critical dependence of the dynamics onboth the initial conditions and problem parameters suchas the SOC and Zeeman constants. Namely, we haveshown, that system can enter and exit a stability domainby small variations of both initial conditions and aboveparameters.To get further insights into emergent chaotic behavior,we analyze the phase portraits and so-called maximalLyapunov exponent (MLE) λ max both in deterministicand chaotic regimes. The MLE, being the largest valueof the Lyapunov exponents spectrum, is often used as amarker of chaotic (if λ max >
0) or a regular ( λ max < λ max is an additional consistency crite-rion of our numerical procedure. NONLINEAR EQUATIONS OF MOTION
We begin with the full Hamiltonian for a particle inan external electromagnetic field characterized by time-independent vector-potential A = ( A x , A y ): H = p x p y α ( p x ˆ σ y − p y ˆ σ x ) + H Z + ϕ ( r ) , (1)where r = ( x, y ) , and p x ≡ − i∂ x − A x , p y ≡ − i∂ y − A y , (2) ∂ x ≡ ∂/∂x, ∂ y ≡ ∂/∂y, α is the SOC constant, ˆ σ i are thePauli matrices, H Z is the Zeeman term, and ϕ ( r ) is thepotential energy in the electric field. Hereafter we use theunits with ~ = m = e = c = 1 and restore the physicalunits when discussing possible experimental implicationsof the results obtained. Without loss of generality wetake the magnetic field B parallel to the z − axis: B = ∇ × A = ∂ x A y − ∂ y A x . (3)The Zeeman contribution reads: H Z = ∆ x σ x + ∆ z σ z , (4)where ∆ x and ∆ z are Zeeman splittings, which can, e.g.,be produced by material magnetization, and will be as-sumed to be B − independent without loss of generality.We derive the equations of motion for observables O by using commutator-based approach [27]˙ O = i [ H, O ] . (5)Using commutation relation for operators a and b as[ a , b ] = a [ a, b ] + [ a, b ] a with O = x and y , we obtain fol-lowing expression for velocity in terms of time-dependentexpectation values: v x = ˙ x = p x + ασ y ; v y = ˙ y = p y − ασ x . (6)The ασ y and − ασ x terms in Eq. (6) correspond to so-called anomalous velocity, which is explicitly dependent on the spin components. This contribution appears dueto SOC presence in the Hamiltonian (1) and, as it will bedemonstrated below, is responsible for the appearance ofchaotic dynamics.Then, applying Eq. (5) for velocity in Eq. (6) and spincomponents, we obtain the equations of motion:˙ v x = ω c v y − ϕ x ( r )+ α ˙ σ y ; ˙ v y = − ω c v x − ϕ y ( r ) − α ˙ σ x , (7)where the cyclotron frequency in our units ω c ≡ B and ϕ x,y ( r ) ≡ ∂ x,y ϕ ( r ). Equations (7) should be augmentedby those for spin evolution, caused by SOC and Zeemanterms in the form˙ σ x = 2 α ( v x − ασ y ) σ z − ∆ z σ y , (8a)˙ σ y = (2 α ( v y + ασ x ) − ∆ x ) σ z + ∆ z σ x , (8b)˙ σ z = − α ( v x σ x + v y σ y ) + ∆ x σ y . (8c)The equations (7) for accelerations and (8) for spinprecession, being determined by the particle velocity,spin components, SOC, and magnetic field, are gauge-invariant since they do not include vector-potential ex-plicitly.Note that these equations are essentially semiclassicaldespite the quantum character of spin operators and sim-ilar to those of Ref. [17]. In the spirit of Ref. [17], theycan be derived by using the Hamiltonian formalism ofclassical mechanics directly from (1). Namely, classicalHamiltonian equations for coordinate r i and momenta p i ( i = x, y ) components dp i dt = − ∂H∂r i , dr i dt = ∂H∂p i , (9)(where H is Hamiltonian function (1)) should be supple-mented by those for expectation values of spin compo-nents (which we denote as σ x,y,z since they are essentiallythe same as those in Eq. (8)) obeying usual constraint σ x + σ y + σ z = 1 . (10)The constraint (10) corresponds to the spin precession inthe total field given by the sum of the spin-orbit andZeeman contributions. Latter equations yield exactlyEqs.(8) (with v x,y being substituted by sums defined inEq. (6)), while former ones are indeed Eqs. (7). It canbe shown that the equations for spin components remainthe same regardless of the derivation approach: eitherfirst commute the Hamiltonian (1) with spin componentsaccording to the rule (5) and then take expectation val-ues or simply act within the classical approach (9) withrespect to constraint (10).The equations (7), (8) clearly demonstrate the un-usual character of the system nonlinearity, consisting oftwo contributions. First one is constituted by the termslike v a σ b ( a, b = x, y, z ) and second one is due to spinproducts σ a σ b . Both these contributions play an impor-tant role in the motion of the particle since the acceler-ations depend on the spin state, and, in turn, the spinevolution depends on velocity.This geometrical constraint (10), additional to the en-ergy conservation, makes the systems with SOC to bequalitatively different from typical quantum and classi-cal chaotic systems [7–10]. In the absence of Zeemanand electric fields, the time evolution of z - compo-nents of the total angular momentum, L z + σ z / L z = xp y − yp x ) is given by ddt (cid:16) L z + σ z (cid:17) = − ω c ddt r . (11)This SOC-independent constraint with L z + σ z / C − ω c r /
2, where constant C is determined by the ini-tial conditions, strongly influences the chaos emergence,making it less probable.Although analytical investigation of the above systemof nonlinear differential equations is not feasible, one canget a certain insight from a qualitative analysis as pre-sented below. Namely, we trace possible chaotic behaviorfor two electric field realizations: zero field and uniformone ϕ ( r ) = Ey , corresponding to cyclotron motion andHall effect in the electric field E = − E y ( y is a unitvector in the y direction), respectively. CHAOTIC CYCLOTRON MOTION
For comparison with the conventional cyclotron mo-tion, where v x = v sin( ω c t ) , v y = v cos( ω c t ) , (12)we begin with solving the above equations (7) and (8)at ϕ ( r ) ≡ σ z (0) = 1 ,v x (0) = 0 , and v y (0) = v . Qualitatively, the effect ofSOC on the cyclotron motion is expected to be strong if(1) the typical anomalous velocity α [23] is of the orderof the initial velocity v and(2) spin precession rate 2 v α is of the order of ω c sothat the trajectory radius should be of the order of spinprecession length 1 /α .Although for the Rashba coupling without Zeemanfield the chaos does not appear due to the constraint(11), anisotropic SOC [17] can lead to chaos as the latterconstraint is lifted there.Since Zeeman field is essential in this case, we includeit in the form ∆ = (∆ x , , ∆ z ) in our iterative procedure,presenting the velocity as v = u + V , where u is obtainedin the ”frozen spin” approximation with σ ( t ) = (0 , , V is the corresponding correction. Substitution ofthe above iterative expression for the velocity into Eq.(7) generates the following frozen-spin contribution de-termined by the in-plane Zeeman field component ∆ x : (cid:20) u x u y (cid:21) = (cid:18) v − α ∆ x e ω (cid:19) (cid:20) sin e ωt cos e ωt (cid:21) + α ∆ x e ω (cid:20) (cid:21) , (13) where the renormalized frequency e ω = ω c + 2 α . Theequations for the V − term are determined by the out-of-plane ∆ z and read as:˙ V x = 2 v y α ( σ z −
1) + 2 α σ x σ z + α ∆ z σ x , (14a)˙ V y = − v x α ( σ z −
1) + 2 α σ y σ z + α ∆ z σ y , (14b)determine small- t corrections due to Eqs.(8): V x = v α h α ω c + ∆ z − (∆ x − αv ) i t , (15a) V y = α (cid:0) α + ∆ z (cid:1) (2 αv − ∆ x ) t . (15b)Equations (15) demonstrate that to produce chaos, oneneeds Zeeman field component ∆ x of the order of αv .Now, we can show that in strong Zeeman fields thechaos disappears and the motion returns to a regular be-havior. As an example we take realization with ∆ z = 0and ∆ x ≫ αv . For this realization the ”spin part”, i.e.Eqs. (8) acquire the form ˙ σ x = 0, ˙ σ y = − ∆ x σ z , and˙ σ z = ∆ x σ y with the explicit solution σ y = − sin(∆ x t ) , (16)obtained with the above initial condition σ z (0) = 1,which implies ˙ σ y (0) = − ∆ x . Substitution of the solution(16) into the set (7) generates following inhomogeneoussystem of equations for the velocity components˙ v x = ω c v y − α ∆ x cos(∆ x t ) , (17a)˙ v y = − ω c v x . (17b)After solving it by the variation of constants with initialconditions v x (0) = 0, v y (0) = v , we finally arrive at: (cid:20) v x v y (cid:21) = (cid:18) v − α ∆ x ω c ω c − ∆ x (cid:19) (cid:20) sin ω c t cos ω c t (cid:21) (18)+ α ∆ x ω c − ∆ x (cid:20) ∆ x sin (∆ x t ) ω c cos (∆ x t ) (cid:21) , The equation (18) defines double-periodic regular motionwith, in general, a possible resonance between spin pre-cession and cyclotron frequencies.Since the full description of the system of interest re-quires the set ( r , v , σ , ˙ σ ) , with imposed constraints, theonly way to depict it is to use projections of the abovemultidimensional surface onto specific planes as reportedin Figs. 1 and 2, presenting our main results for the realspace and phase trajectories. It is seen that for regularreal trajectories the phase ones are also regular, while inchaotic case, the phase portrait completely reflects thesituation, being also chaotic.To characterize the chaotic trajectories in determin-istic dynamic systems quantitatively, one usually intro-duces the Lyapunov spectrum (see, e.g., [26, 28] andreferences therein), providing a measure of the rate oftime separation of initially (at t = 0) infinitesimally -1 0 1 2-1012 y ( t ) -4 -2 0 2 4-4-2024-1 0 1 2 30123 y ( t ) -6 -4 -2 0 2 4-4-20246-2 -1 0 1 2 3 4 x ( t ) -1012345 y ( t ) -8 -6 -4 -2 0 2 4 x ( t ) -4-202468 ∆ x =0 ∆ x =-0.1 α=0.1α=0.2α=0.3 (a) (b)(c) (d)(f)(e) FIG. 1. Typical cyclotron trajectories for t < for theinitial spin state σ z (0) = 1. Left column (panels (a), (c), (e))corresponds to Zeeman coupling ∆ x = 0, right column (panels(b), (d), (f)) - to ∆ x = − .
1. Upper row (panels (a),(b))corresponds to SOC constant α = 0 .
1, middle row (panels(c),(d)) corresponds to α = 0 . α = 0 .
3. Here ω c = 0 .
1, ∆ z = 0 , and the initial velocities v x (0) = 0 , v y (0) = 0 . . The absence of chaos at ∆ x = 0 (leftcolumn) is due to the constraint (11). close trajectories. For this purpose we first introducevector Q ( t ) = [ r ( t ) , v ( t ) , σ ( t )]. In this case, the tra-jectory separation δ Q ( t ) characterizes how close are twotrajectories at arbitrary time instant t , given that at t = 0 they were almost the same, i.e. that the quantity | δ Q | ≡ | δ Q (0) | →
0. Formally, at small separations, theentire set of our dynamical equations can be linearized toyield the sets of Q [ j ]0 and λ j , with | δ Q [ j ] ( t ) | = e λ j t | δ Q [ j ]0 | ,where λ j is the corresponding Lyapunov exponent. Asthe sets of initial conditions may be different (see Figs.1, 2), the initial separation vectors δ Q [ j ]0 have differentdirections. This generates the entire spectrum of Lya-punov exponents, which in our case comprises 7 elementswith j = 1 , . . . ,
7. The most important characteristic ofthe spectrum is the Maximal Lyapunov exponent λ max [26, 28], which determines if system is chaotic ( λ max > λ max < -1 0 1 2 3 -0.100.1 v y ( t ) -2 0 2 4-0.200.20.4 v y ( t ) -4 -2 0 2 4 6-0.4-0.200.20.4 y ( t ) -0.200.2 v y ( t ) -4 -2 0 2 4 6 8 10 y ( t ) -0.4-0.200.20.4 ∆ x =0 α=0.1 (a) α=0.2α=0.3 (c)(e) (b)(d)(f) ∆ x =-0.1 FIG. 2. Phase portraits corresponding to the ( x, y ) trajecto-ries in Fig.1 for the same set of parameters and time intervals. y ( t ) -1 0 1 2 30123-1 0 1 2 3 4 x ( t )-101234 y ( t ) -4 -2 0 2 4 x ( t )-4-20246 (a) (b)(c) (d) ∆ x =0 α=0.2 ∆ x =-0.02 ∆ x =-0.01 ∆ x =-0.03 FIG. 3. Strong dependence of the trajectories on the in-plane Zeeman field ∆ x . The Figure illustrates transition fromhigh density of the regular trajectories to their chaotizationat small variation in the system parameters, as typical forchaotic systems. of MLE, which we used in our calculations, reads [28] λ max = lim t →∞ lim | δ Q |→ t ln | δ Q ( t ) || δ Q | . (19)When the limit (19) is positive, the trajectories show ex-treme sensitivity to the initial conditions and the systembecomes chaotic. Note that the limit t → ∞ is taken innumerical procedure approximately and this makes theproblem of λ max calculation to take quite long time, es- -2 0 2 4 6 8 x ( t ) -4-20246 y ( t ) -2 0 2 4 6 8 x ( t ) -4-20246(a) (b) v y (0)=0.34 ∆ x =-0.1 v y (0)=0.36 FIG. 4. Transition from chaotic to regular high-density cy-clotron trajectories at small variation in the initial veloc-ity v y (0). Parameters are the same as those in Fig. 1(d)(∆ x = − .
1, ∆ z = 0, α = 0 . ω c = 0 . σ z ( t ) time t -101 σ z ( t ) ∆ x =0 ∆ x =0 ∆ x =-0.1 ∆ x =-0.1 α=0.3α=0.20.5 -0.5-0.5 (a)(b) FIG. 5. Typical spin behavior for the cyclotron motion, shownin Fig. 1. SOC constants are reported in the panels and thelines are marked with the values of ∆ x . Regular behavior of σ z ( t ) for ∆ x = 0 clearly corresponds to the quasiperiodic tra-jectories in Fig. 1. Fast spin oscillations correspond to remoteparts of the trajectories, while slow oscillations correspond tothe nearby parts. pecially in the chaotic regime. To calculate the Lyapunovspectrum for our problem, we used the algorithm of Ref.[29] (see also Ref. [30]) for implementation with Wol-fram Mathematica software. Thus, we obtained, for Fig.1(a) λ ( a )max = − . λ ( b )max = 0 . λ ( c )max = − . λ ( d )max = 0 . λ ( e )max = − . λ ( f )max = 0 . α , does not generate chaos. To produce it, aZeeman field is necessary. This is reported in the rightcolumns of these Figures, where the chaotic trajectoriesare due to the interplay between the Zeeman and SOCfields.To confirm the emergence of the chaos, we show othertwo peculiar features of the chaotic behavior such asstrong dependence of the trajectories on the system pa-rameters and initial conditions. Figure 3 shows thedependence on the in-plane magnetic field while Fig.4demonstrates the dependence on the initial velocity. Itis seen from Fig. 3, that while at Zeeman splitting∆ x = − .
02, the system trajectory is still regular with λ max = − . x = − .
03 thesystem is already chaotic with λ max = 0 . x (for instance at ∆ x >
0) as well as of∆ y . Fig. 4 reports the same instability with respectto v y (0): at a very small variation 0 . < v y (0) < . λ max = 0 . v y (0) = 0 .
34 (Fig.4(a)) and λ max = − . v y (0) = 0 .
36 (Fig.4(b)). It can be shown that the systemis also sensitive to small variations in v x (0) as well as toall other possible combinations of initial conditions.To understand the spin evolution behind the regu-lar and chaotic trajectories, we present in Fig. 5 thetime dependence σ z ( t ) for four realizations of trajecto-ries shown in Fig. 1. As one can see in the Figure, inthe absence of the Zeeman coupling, spin shows relativelysmall deviations from its initial value, corresponding tothe above frozen spin approximation. The spin behav-ior in the absence of the Zeeman coupling is consistentwith the regular quasiperiodic trajectories in Fig. 1. In-deed, for quasiperiodic trajectories the integral of veloc-ity during one ”period” is small. This smallness leadsto a minute variation in the spin component σ z and,in turn, to regular trajectory, making the pattern con-sistent. At relatively large Zeeman splittings, the spindynamics becomes chaotic, producing chaotic ( y ( t ) , x ( t ))trajectories. Nonzero Zeeman coupling ∆ x enhances thespin rotation, and, therefore, even if the particle displace-ment during one quasiperiod is small, spin precession isessential for the orbital motion. In this case, the dynam-ics of σ z strongly modifies not only the effective cyclotronfrequency ω c + 2 α σ z but also the α - dependent termsin the equations of motion. The spin-orbit coupling hereserves as a mediator between Zeeman-induced rotationand enhanced trajectory chaotization. On the contrary,∆ z suppresses the spin rotation and stabilizes the trajec-tory against the chaos. CHAOTIC HALL EFFECT
To compare the following results of approach with theconventional Hall effect in a uniform electric field E ≪ B , we present the corresponding velocity as: (cid:20) v x v y (cid:21) = u H (cid:20) − cos ( ω c t + φ H )sin ( ω c t + φ H ) (cid:21) + v H (cid:20) (cid:21) , (20)where u H = (cid:0) v + v H (cid:1) / and φ H = arctan( v /v H ).Here v H = − E/B is the conventional Hall velocity inthe given geometry with
E > . For the initial condi-tions v x (0) = 0 , v y (0) = v , chosen here without loss ofgenerality, the Hall velocity in Eq. (20) has the form v x = v H (1 − cos( ω c t )) , v y = v H sin( ω c t ) . (21)Note that here the effect of SOC is stronger since themean value of velocity during one cyclotron period is notsmall. Moreover, the constraint (11) is lifted here, mak-ing the system prone to chaos even at ∆ x = 0. -80 -60 -40 -20 0 x ( t ) -4-2024 y ( t ) time t -101 σ z ( t ) (a)(b) FIG. 6. Typical Hall trajectories (a) and spin evolution (b)for σ z (0) = − α = 0 .
1. Here ω c = 0 .
1, ∆ z = 0, ∆ x =0 . E = 0 .
01 and the initial velocities v x (0) = v y (0) = 0. The numerically obtained trajectories and chaos devel-opment in the Hall regime are reported in Figs. 6 and 7.It is seen, that at a relatively weak SOC α = 0 . α = 0 . σ z ( t ), we need to compare the renormalized by SOC cy-clotron frequency ω c +2 α σ z to the electric field E . When | ω c + 2 α σ z | ≫ E , the system is close to a conventionalHall effect. Otherwise, it is out of this regime, and theparticle acceleration is determined primarily by the elec-tric field. This occurs if α > ω c / t c satisfyingthe condition σ z ( t c ) = − ω c / α . The time τ the particlespends out of the classical Hall regime is of the order of τ ∼ E/α | ˙ σ z ( t c ) | if ˙ σ z ( t c ) = 0 or τ ∼ p E/ | ¨ σ z ( t c ) | /α if˙ σ z ( t c ) = 0. Accordingly, the velocity at this time intervalhas an increment δv y ∼ − Eτ . corresponding to elonga-tion of trajectories along the y - axis in Fig. 7. Notethat all above discussed regularities of chaotic behavior(such as sensitivity to initial conditions and/or problemparameters) take place for chaotic Hall effect as well. POSSIBLE EXPERIMENTAL IMPLICATIONS
Now we are in a position to discuss system param-eters required for observation signatures of the chaoticcyclotron motion and Hall effect for semiconductors andcold atoms. Note that the effects of SOC on the reg-ular cyclotron trajectories in semiconductors have beenexperimentally observed and theoretically studied in Ref.[31]. The role of the anomalous spin-dependent velocityin the ac conductivity of 2D electron gas has been stud-ied experimentally and theoretically in Ref. [32]. Fullquantum mechanical analysis of the electronic wave pack-ets motion in magnetic field has been performed in Ref.[33]. While
Zitterbewegung- like effect has been clearlyrevealed and studied in details, no chaotic behavior ap-peared. The first reason is that the calculations havebeen made for the sets of parameters far away from thechaotic domains. The second reason is that the consider-ation in the paper [33] is explicitly quantum mechanicalwith time-dependent expectation values being calculatedwith the help of corresponding wave functions. Althoughthe relation between quantum [7–10] and classical chaosin spin-orbit coupling systems is very puzzling, such for-malism, which does not deal with explicit time-dependentdifferential equations, would not, most probably, revealfeatures of the classical chaos. It is not excluded, how-ever, that the approaches similar to [33] may reveal somequantum chaotic features such as the energy levels repul-sion, leading to non-Poissonian spectral statistics.Interesting features of the quantum Hall effect in thepresence of SOC have been observed experimentally inRef. [34] and studied theoretically in Refs. [35–38]. Itturns out also, that SOC term in the velocity is criti-cally important for the spin Hall effect [39, 40] and low-temperature transport [41]. -80 -60 -40 -20 0 x ( t ) -8-6-4-2024 y ( t ) time t -101 σ z ( t ) (a)(b) FIG. 7. Typical Hall trajectories (a) and spin evolution (b)for σ z (0) = − α = 0 .
3. Here ω c = 0 .
1, ∆ z = 0, ∆ x =0 . E = 0 .
01 and the initial velocities v x (0) = v y (0) = 0. Now we restore the physical units. To have a strongeffect of spin-orbit coupling in the emerging chaotic be-havior, we need to compare the cyclotron frequency ω c = eB/mc with that corresponding to the shift of theconduction band bottom due to SOC, ω so = mα / ~ .As an example, we take the parameters for GaAs with m = 0 . m ( m is a free electron mass) and typical α = 10 − meVcm . This α corresponds to the anoma-lous velocity α/ ~ ≈ . × cm/s, and 2 ω so ≈ . × s − . Since for B = 0 . ω c ≈ . × s − , we conclude that for chaos emergence, oneneeds either relatively weak magnetic fields or strongerSOC, which occur in In x Ga − x As or InSb 2D structures,albeit having smaller electron effective masses. Takinginto account that at this field, ω c and ω so are of the sameorder of magnitude, we also conclude that electron ve-locity v ≥ α/ ~ and ∆ x / ~ ∼ s − is sufficient to getstrong effects of spin precession and chaos formation. Inthe Hall regime, the condition of fast precession has theform αm | v H | / ~ ω c ∼
1, dependent on the electric field strengths. For the above values of B and α , this con-dition is satisfied at v H ∼ cm/s. The situation issimilar for cold atoms with synthetic SOC [15, 17]. Herethe SOC energy, the typical kinetic energy, and the Zee-man term are of the same order of magnitude [16, 42].Therefore, in the presence of a gauge field producing asynthetic Lorentz force, the cold atoms motion is proneto chaos [17]. CONCLUSIONS
Two-dimensional materials and structures with spinor-bit coupling can exhibit a wealth of unexpected effects,both of fundamental physical interest and important fortheir possible electronic and spintronics applications [12].In the present paper, using analytical and numericalarguments in the semiclassical approximation, we havedemonstrated that joint effect of the Lorentz force, Zee-man splitting, and spin-orbit coupling in 2D systems gen-erates chaotic trajectories of a particle moving in thiscombination of the fields. A typical chaotic trajectorycan be described as a highly entangled path with highsensitivity to the small variations of initial conditionsand/or system parameters. To describe this chaos math-ematically, we utilize the phase portraits of the systemunder consideration as well as the spectrum of its Lya-punov exponents. The main role is played here by theMLE - the maximal exponent in the spectrum, providinga consistency check for our numerical approach. Namely,for chaotic trajectories MLE is positive, while for regularones it is negative. In our case, the reason for the chaoslies in the fact that the system loses integrability since itpossesses only two integrals of motion for its phase space.Dynamically, this effect is clearly seen in the equations ofmotion including the anomalous spin-dependent velocityterm.The specific physical mechanism behind the chaotiza-tion is the emergence of the spin-dependent term causedby the Rashba coupling in the effective Lorentz force re-lated to the particle’s velocity and the
Zitterbewegung effect. In other words, the spin rotation in the Zeemanand Rashba fields is chaotically transformed into time-dependent anomalous (renormalized by spin degrees offreedom) velocity. In this respect, our dependences, re-ported in Figs. 5 - 7 can be considered as chaotic Zit-terbewegung . This interesting phenomenon needs fur-ther studies. Therefore, the Zeeman field plays criticalrole since it can either trigger chaotization or suppressit, stabilizing the regular trajectories. As we have dis-cussed in this paper, the considered effects are commonfor 2D semiconductor structures, weakly relativistic elec-trons and cold atoms with synthetic gauge, spin-orbit,and Zeeman couplings. The appearance of chaos in theHall regime in smooth random potentials and dynamicsof two-component wavepackets in the domains of spin-orbit and Zeeman couplings suitable for the chaos emer-gence are of interest and will be studied separately. Inaddition, generalization of the proposed approach for thespin-orbit coupled Bose-Einstein condensates [43, 44] andcold atomic gases [45] with the effective (pseudo)spin s = 1 , demonstrating a more classical behavior than s = 1 / , can reveal possibly chaos-related properties ofthese systems.E.K. and V.S. acknowledge support of the NarodoweCentrum Nauki in Poland as research Project No. DEC-2017/27/B/ST3/02881. E.S. acknowledges support ofthe Spanish Ministry of Science and the European Re-gional Development Fund through PGC2018-101355-B-I00 (MCIU/AEI/FEDER,UE), and the Basque Govern-ment through Grant No. IT986-16. [1] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn,C. Makrides, A. Petrov and S. Kotochigova, Nature ,475 (2014).[2] T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D.Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ofling,Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R.G. Dall, and E. A. Ostrovskaya, Nature , 554 (2015).[3] M. Aßmann, J. Thewes, D. Fr¨ohlich, and M. Bayer, Nat.Mater. , 741 (2016).[4] E. A. Ostrovskaya and F. Nori, Nat. Mater. , 702(2016).[5] F. Schweiner, J. Main, and G. Wunner, Phys. Rev. E ,062205 (2017).[6] F. Schweiner, J. Main, and G. Wunner, Phys. Rev. Lett. , 046401 (2017).[7] M. C. Gutzwiller, Chaos in Classical and Quantum Me-chanics (Springer-Verlag, New York, 1990).[8] L. E. Reichl, The Transition to Chaos. ConservativeClassical Systems and Quantum Manifestations, 2nd ed.(Springer-Verlag, New York, 2004).[9] F. Haake, Quantum Signatures of Chaos, 3rd ed.(Springer-Verlag, Berlin/Heidelberg, 2010).[10] H.-J. St¨ockmann, Quantum Chaos: An Introduction(Cambridge University Press, Cambridge, U.K., 1999).[11] Yu. A. Bychkov and E. I. Rashba, JETP Lett. , 78(1984)[12] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004).[13] Spin Physics in Semiconductors
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