State Key Laboratory for Mesoscopic Physics and Department of Physics, Peking University, Beijing , China

Transcription

1 Commun. Theor. Phys. Beijing, China) ) pp c International Academic Publishers Vol. 41, No., February 15, 004 Nonadiabatic Phase Persistent Currents for System of Spin- 1 of Electromagnetic Fields Spin-Orbit Interaction Particles in Presence LI Nian-Bei MA Zhong-Shui State Key Laboratory for Mesoscopic Physics Department of Physics, Peking University, Beijing , China Received May 13, 003) Abstract We present a comprehensive view details of calculations on Aharonov Anan phase for the charged particles in the external electric magnetic fields for a nonadiabatic process. We derive, with consideration of a spin-orbit interaction Zeemann Splitting, the persistent currents as a response to an Aharonov Casher topological interference effect in one-dimensional mesoscopic ring. We also establish a connection to Berry adiabatic phase with deduced dynamical-nature dependence in the nonadiabatic process. The second quantization representation has also been employed in exhibition of persistent currents in the many-body case. PACS numbers: Ej, Bz Key words: Aharonov Casher effect, spin persistent current The geometric phase [1] has attracted wide interest in different areas of modern physics since Berry [] first initiated that there is an observable phase which cannot be removed by any local phase reparametrization of the instantaneous eigenstates of the Hamiltonian when the evolution of the system is adiabatic cyclic in the parameter space. Later, the theory of geometric phase has been generalized in a nonadiabatic sense. By introducing the concept of projective Hilbert space, Aharonov Anan [3] showed that if there is a periodic solution of the time-dependent Schrödinger equation it casts the problem in terms of a phase acquired during a cyclic evolution of this solution. This nonadiabatically geometric phase, referred to as Aharonov Anan AA) phase, can be calculated by using the Lewis invariant method. [4] For a given time-dependent Hamiltonian Ht), it can be supposed that there is a Hermitian invariant It) which satisfies the equation dit) dt = It) t i h 1 [It), Ht)] = 0. If there exists a complete orthonormal set of eigenstates λ, t of It) with eigenvalues λ, i.e., It) λ, t = λ λ, t, we are able to use these eigenstates to diagonalize the operator i / t) h 1 Ht) at each time t then express the exact solutions for the Schrödinger equation i h d ψ = H ψ, dt in terms of instantaneous eigenstates of It), as ψ = expift)) λ, t with phase factor ft) = h 1 t t 0 λ, t i h t H λ, t dt. In connection with a special case of Berry phase, the topological phase of the Aharonov Casher AC) effect, [5] which leads to a topological interference effect of the wave function of a particle with spin, is discovered in the presence of the spin-orbit SO) interactions. The nonzero electric field on the path of the particle produces the AC phase shift. Several authors [6] showed the exact correspondence between AC effect the Aharonov Bohm AB) effect in both nonrelativistic relativistic descriptions. The spin-orbit Berry phases produced by SO interactions were studied in the adiabatic limit [7] as well as its nonadiabatic realization, [8] i.e., AA phase, in the AC effect. [9] The Berry phase has been applied to the mesoscopic systems in studying of various behaviors of electronic transportation. The persistent currents in textured mesoscopic ring were both derived theoretically demonstrated by experiment. [10] As one of its realizations in the inhomogeneous, the persistent spin current due to the Berry phase was discussed in the adiabatic approximation by Loss, Goldbart, Balatsky. [11] More recently, the existence of the same persistent current was studied without the adiabatic restriction. [1] In the presence of SO interaction, it is shown that SO interaction in one-dimensional disordered ring induced an effective spin-dependent magnetic flux. [13] With consideration of the AC effect in external electric field Balatsky Altshuler [14] studied the persistent current produced by SO interaction via the AC effect. The phase persistence current problem in AC effect were also studied by Taeseung et al. [15] The relationship between the Pachratnam phase U1) SU) The project supported by National Natural Science Foundation of China under Grant Nos , Natural Science Foundation of Guangdong Province of China under Grant No

2 306 LI Nian-Bei MA Zhong-Shui Vol. 41 field theory was also analyzed. [16] Recently, the nonadiabatic geometric phase analysis has been extended to some quantum computation schemes. [17] Despite the fact that the progress for the various analyses has been made, the consequence of the Berry phase the persistent current being observed in multiply connected geometries with co-existence of an SO interaction for the influence of electric fields on spin-particle is not thoroughly recognized. The nonadiabatic phase for the wave function of cyclic evolution of a physically defined system is expressed in terms of the parameter in the instantaneous solutions of the textured system. Therefore, the question arises as to whether the phase i λ, t / t λ, t is totally a geometrical phase which equals to Berry adiabatic phase. In order to clarify the question raised above, we will discuss in this paper the exact phase for the charged particle in a ring system in presence of external electric magnetic fields of SO interaction. We show that AA phase referred to the exact evolution of a cyclic state in a given Hamiltonian is combined with the Berry phase some deviations from adiabaticity as the non-adiabatic corrections to Berry phase. We find that the AA phase can be adjusted by adequately fixing the relative strengths of the components of the electric magnetic fields. In the present paper, we are interested in showing how the AA phase induced by magnetic field SO interaction in a nonadiabatic process contribute to the persistent currents. In particular, we show that not only the component of electric field in the ring plane, but also the component perpendicular to the plane of the ring do contribute themselves through the SO interactions to the AC effect. WΨ L = 1 m = Simultaneously, they are all manifested by themselves as observable effects in the persistent currents. We firstly consider a charged particle of spin 1/ with anomalous magnetic moment µ = κe h/4mc, where κ is gyromagnetic ratio with κ = 1.79 for a proton while κ = 1.91 for a neutron, moving in external electric magnetic fields. In the case of studying the behavior of the particle in weak field, the Dirac equation describing the system can be written in a nonminimal form as [i h / e c A/ ) mc + µ c F µνσ µν] Ψ = 0, 1) where σ µν = 1/)i[γ µ, γ ν ]. In this equation the spindependent term is entirely involved in a Pauli moment term µ/c)f µν σ µν = µ/c)α E/c + σ B) in terms of interaction with electric magnetic fields. By employing the Pauli metric writing Ψ in the form of two-component representation with large small components of Dirac spinors being Ψ L Ψ S respectively, we have two coupled equations, i.e., σ Π + i µ c σ E )σ Π i µ c σ E ) Ψ L µσ BΨ L [ 1 e h Π Π κ ) m mc σ B µ mc σ E Π c [σ Π + iµ/c )σ E]Ψ S = W + µσ B)Ψ L c [σ Π iµ/c )σ E]Ψ L = W µσ B + mc )Ψ S, where Π = P + e/c)a is the generalized momentum operator W = E mc is related to the total energy ɛ. In the non-relativistic limit the case of the physically weak field, W µ/c)σ B mc µ /c 4 )E E Π Π. So we can replace W µσ B+mc by mc in the second equation to obtain Ψ S = mc) 1 σ Π iµ/c )σ E)Ψ L. Substituting it in the first equation, we then have µ h ] mc E iσ E) Ψ L. ) The third term corresponds to spin-orbit interaction with the motional magnetic field B = v E acting on the particle. In the following discussions we would be restrict to the situation of electric magnetic fields oriented at a tilt to the sample with the constant amplitudes. The last term in Eq. ) can be neglected. We are now ready to consider the system that this spin 1/ particle is confined to a one-dimensional ring of radius R. The ring lies in the x y plane with center being in the origin. We adopt a cylindrical coordinate system for description of the motion of the particle. Assuming that there exists an additional flux Φ passing through the axis normalized the ring plane, we write the vector potential which is tangent to the ring as A ϕ = Φ/πR. The Hamiltonian describing the system can be expressed as Ĥ = hω ϕ + Φ ) + hω σ 1 cos ϕ + σ sin ϕ) ϕ + Φ ) + i hω σ 1 sin ϕ σ cos ϕ) + hω ϕ + Φ ) σ 3 + hω m ˆn σ, 3)

3 No. Nonadiabatic Phase Persistent Currents for System of Spin- 1 Particles in 307 where we have adopted the following shorth notations ω = h/mr, ω = µe z /mc R, ω = µe r /mc R, ω m = κ )eb/mc, = hc/e. The magnetic field is oriented at the direction ˆn = sin θ cos ϕ, sin θ sin ϕ, cos θ). In the derivation of Eq. 30), a little modification is made for the requirement of the hermiticity. [18] Attempting to find the exact solution with this Hamiltonian, we diagonalize the spin-dependent part of the Hamiltonian, treating the angle ϕ as a precessing parameter. First we take a semiclassical quantization by separating out the rotator part of the Hamiltonian with the help of a wave function π) 1/ expinϕ). In this way the Hilbert space of the Hamiltonian 3) is constituted by the set of states Ψ = ψϕ) χϕ, σ, t), where χϕ, σ, t) is the wave function for the spinor satisfying the time-dependent Schrödinger equation i h t χ σ = ĥsχ σ, 4) where the internal Hamiltonian ĥst) is ĥ s t) = hω lσ 1 cosωt) + σ sinωt)) + hω l 1 ) σ 3 + hω m σ 1 sin θ cosωt) + σ sin θ cosωt) + σ 3 cos θ) 5) with l = n + 1/) + Φ/ ), which has an explicit time dependence. Using the method outlined by Mizrahi [4] we find that there exists a Hermitian invariant I = σ 1 sin Θ cosωt) + σ sin Θ sinωt) + σ 3 cos Θ with the precessing frequency. Ω = ωl ω. It is found that dit) dt = It) t + i h 1 [It), h s t)] = 0 when tan Θ = n /Ξ n, where n = [ω l + ω m sin θ] Ξ n = [ω ω )l ω m cos θ]. The eigenstates of the Hermitian invariant It) are obtained as χ + = cos Θ sin Θ, χ = sin Θ e iωt cos Θ 6) e iωt with eigenvalues λ = +1 1, respectively. exact solution of the Schrödinger equation 4) is where the phase factor is χ σ t) = e ift) χ σ t), f = π + [π/ωl ω )] Ξ n + n So the the cyclic condition is χ σ T ) = χ σ 0). f is consisted of two parts associated with cyclic evolution π/ω 0 i χ σt)d/dt) χ σ t)dt = π1 σ cos Θ) π/ω h 1 i χ σt)h s t) χ σ t)dt = [π/ωl ω )][ω l 1/) + ω m cos θ) cos Θ + n sin Θ]. 0 The exact solution of the stationary state Schrödinger equation ĤΨ = E Ψ can then be written as Ψ = π) 1/ expinϕ) χ σ with the energy eigenvalues being [ ε = h ω l + 1 ) 1 4 ω σ ] Ξ n + n. 7) We will show now that the above result is a general one. Let us consider a time-dependent Hamiltonian Ĥt) = hω i ϕ Φ ) + hω cos Ωtσ 1 sin Ωtσ ) ϕ Φ ) + hω ϕ + Φ ) σ 3 + hω m ˆn σ, 8) the corresponding Schrödinger equation i h Ψt) = t Ĥt)Ψt), HT ) = H0). There are two linearly independent solutions Ψ + t) = Ψ n+ exp ie n+ t), Ψ t) = Ψ n exp ie n t), where E are given in Eq. 7). The instantaneous value of the energy is H = hωl 1/) + χ σt)h s t) χ σ t) the momentary value of the spin is S σ = 1/)σsin Θ cos Ωt, sin Θ sin Ωt, cos Θ). After one period, the wave function Ψ ± t) acquires a phase Ψ σ T ) = Ψ σ 0) exp ie σ T ) = Ψ σ 0) exp[ iπ1 σ cos Θ) + i h 1 H ± T ] in which H σ T is the dynamical phase π1 σ cos Θ) is the solid angle at the center subtended by the circle traced by spin on a sphere. This is an AA phase. Let us remark that the phase is different from the half of the solid angle enclosed by the trajectory of the magnetic field in the parameter space π1 cos θ). The exact evolution of the proper states of the instantaneous Hamiltonian is not cyclic, in general. However, we choose the

4 308 LI Nian-Bei MA Zhong-Shui Vol. 41 eigenstates χ σ of invariant operator I because their corresponding evolution is cyclic. Due to Θ being a function of Ω, so doing that of the quantum number l, we might get an exact expression for the AA phase in powers of l. π1 cos Θ) = π1 cos θ) + π/ω m )[ω ω ) sin θ ω sin θ cos θ]l +. From this expression we see that only the first term is independent of l. This term is purely geometrical equal to Berry adiabatic phase. All the other terms depending on l, i.e. the so-called nonadiabatic corrections, [19] are dynamical in nature. We turn to the many-body case. To be definite, we consider N charged particles neglect interaction, as well as b-structure effects alternative spinscattering mechanisms. For an isolated system, this situation may be thought of as a gr-canonical ensemble of the system of N particles. Equilibrium is then assured by a fixed value of the associated chemical potential µ. The Hamiltonian for the system is N H1,..., N) = Hϕ j ) µψ j Ψ j). j For the sake of convenience, we would ignore the µψ j Ψ j term in the flowing which should be understood to be related with the particle number operator ˆN keep always in mind that there is a term µ ˆN in the Hamiltonian for many-body case. It is evident that the system possesses conserved quantities Jj z = i + Φ + 1 ϕ j σz. Obviously, ψ = π) 1/ expinϕ j )ξ σ with σ z ξ σ = σξ σ, σ = ±1 is an eigenfunction of Jj z with eigenvalue J = n + 1/)σ + Φ/. To describe the evolution of the spin states of the particles, we have to modify the states of the particles with a cyclic condition that the initial final spin wave functions would differ in a phase factor after the particle turns around the ring once. [8] We therefore take a second quantization representation for the electrons Ψ = Ψ = a expif )ψ a exp if )ψ, along with the algebras {a, a n,σ = δ n,n δ σ,σ {a, a n,σ = {a, a n,σ = 0. Then the Hamiltonian is transformed into the form Ĥ = [ Ω a a + 1 na n,+ e i f n a n+1, n ] + a n+1, e i f n a n,+ ), 9) where Ω = h[ωn + Φ/ ) + 1/)σω n + Φ/ ) + 1/)σω m cos θ] f n = f n+1, f n,+ corresponding to the spin cyclic evolution. This evolution indicates that the ring is threaded by an AB flux a spin-dependent magnetic flux which leads to the AC effect. [8,10] To diagonalize the Hamiltonian 9) we introduce a Bogoliubov transformation a n,+ = U n C n 1) + Vn C n ), a n+1, = V n C n 1) + UnC n ) with a requirement of U n + V n = 1 for a unitary transformation. The corresponding functions U n = U n expi f n /) V n = V n exp i f n /) could be directly read as U n = 1 V n = 1 [ 1 ± [ 1 Ξ ] n, Ξ n + n Ξ ] n, Ξ n + n thus the Hamiltonian which is diagonal in spin space, is given in the following form, Ĥ = E C C, 10) where E is taken the same values as given in Eq. 7). As a result, we see that there exist two branches corresponding to a picture of two kinds of electron gases for the underlying system. We would like to emphasize here that the Hamiltonian 10) has already included a spindependent shift as mentioned in Ref. [14]. In combination with the above discussions, it is obvious that the topological AC effects from SO interaction AA phase enter the energy spectrums. We are also interested in the persistent current caused by an inhomogeneous time-independent electric magnetic fields. We disregard the effect of the electromagnetic field piercing the body of the ring, focus attention on the magnetic flux threading the ring. Consider the current density j µ = Ψγ µ Ψ. The current can be written as j = Ψ L σψ S + Ψ S σψ L. At the beginning of the discussions in this paper, we have derived that Ψ S = m) 1 σ Π+iµ/c)σ E)Ψ L. With the help of the identities σσ A) = A iσ A σσ ) = iσ, the current can be calculated as j = i h m [Ψ Ψ Ψ )Ψ] e mc Ψ AΨ h m Ψ σψ) µ mc Ψ σ EΨ. 11) The last term corresponds to the AC flux due to the AC effect. On the ring the charge current can be found to be { [ jϕ e = er ω Ψ ϕ + Φ ) Ψ + i ϕ Φ ) ] Ψ Ψ + ω Ψ σ z Ψ ω Ψ σ 1 cos ϕ + σ sin ϕ)ψ. 1)

5 No. Nonadiabatic Phase Persistent Currents for System of Spin- 1 Particles in 309 The equilibrium expectation values of the persistent charge current in canonical ensemble is defined as follows j ϕ = Z 1 tr[ĵ ϕ exp βĥ)], where Z is the partition function at temperature T = βk B) 1. The cyclicity of the trace ensures the calculation of jϕ e with the help of eigenstates of Ĥ. In the single particle case, we have j ϕ = hr/ Z) je exp βe ) with j e = hωl σ ω ω )Ξ n + ω n. 13) Ξ n + n In the many-body case, the current can be calculated as { [ ĵϕ e = R ω l 1 ) + σω ]a a + ω a n,+ e i f n a n+1, + a n+1, e i f n a n,+ ). 14) By expressing it in terms of quasiparticle operators C, then taking ensemble average jϕ e = Z 1 tr[ĵ ϕ exp βĥ)], we have jϕ e = Z 1 je [1 + expβe )] 1. For the persistent spin currents, the nonzero components of the spin-current along the ring read { [ jϕ σz = R ω Ψ ϕ + Φ ) σ z Ψ + i ϕ + Φ ) ] Ψ σ z Ψ + ω Ψ Ψ + ω Ψ σ 1 cos ϕ + σ sin ϕ)σ 3 Ψ, j σr ϕ { [ = R ω Ψ σ 1 cos ϕ + σ sin ϕ) ϕ + Φ ) Ψ + i ϕ + Φ ) ] Ψ σ 1 cos ϕ + σ sin ϕ)ψ + ω Ψ σ 3 σ 1 cos ϕ + σ sin ϕ)ψ ω Ψ Ψ. 15) The ensemble average can be calculated out in the forms with jϕ σz,σr = Z 1 j σz,σr exp βe ) j σz = Rωσl cos Θ Rω ω ), j σr = Rωσl sin Θ Rω. 16) The corresponding currents for many-body case can be calculated from the expressions ĵϕ σz = R [ ωσ n + Φ ) ] + ω a Φ a, 17) 0 ĵ σr ϕ = R ω a a + R n la n,+ e i f n a n+1, + a n+1, e i f n a n,+ ). 18) In these expressions we eliminated the terms proportionating to a n,+ e i f n a n+1, a n+1, e i f n a n,+ ), which have zero contribution to the ensemble averages. Finally, we discuss the adiabatic limit. Following the approach of Ref. [7], we can write the effective Zeemann frequencies for a semiclassical rotator as h[ω mr Ω + ω m sin θ] cos Ωt, h[ω mr Ω + ω m sin θ] sin Ωt, h[ω mr Ω + ω m cos θ]). There still exists the Hermitian-invariant operator I with tan Θ = ω mr Ω + ω m sin θ)ω mr Ω Ω/ + ω m cos θ) 1. In the adiabatic limit we recover Θ = θ AA phase as exactly a Berry phase. The calculations of energy spectrums persistent currents are available by taking tan θ = ω /ω ω ). It is also interesting to notice the relations among the persistent currents, j e = σej σz cos θ + j σr sin θ) j σz sin θ + j σr cos θ = 0. The spin is polarized along the magnetic field. As we have noticed from Eq. 11) that the AC flux due to the AC effect produced by SO interaction would excite the persistent currents even in the absence of external magnetic flux in the situation of neutral particle. The persistent charged spin currents are evidently expressed as a sum of two parts which correspond to spin independence AB) spin dependence AC) respectively. Besides the in plane component of electric field, its component perpendicular to the plane of the ring does result in persistent current. As a result, we conclude that the nonvanishing perpendicular component of electric field not only makes the AC effect to produce nonadiabatic AA phase as mentioned in Ref. [8], but also leads to persistent current excitations. Acknowledgments We are grateful to Prof. SU Zhao-Bin for helpful discussions.

6 310 LI Nian-Bei MA Zhong-Shui Vol. 41 References [1] Geometric Phase in Physics, eds. A. Shapere F. Wilczek, World Scientific, Singapore 1989). [] M.V. Berry, Proc. R. Soc. London A ) 45. [3] Y. Aharonov J. Anan, Phys. Rev. Lett ) [4] S.S. Mizrahi, Phys. Lett. A ) 456; M.H. Engineer G. Ghosh, J. Phys. A1 1988) L95; A. Bhattacharjee T. Sen, Phys. Rev. A ) 4388; J. Anan, Phys. Lett. A ) 01. [5] Y. Aharonov A. Casher, Phys. Rev. Lett ) 319; J. Anan, Phys. Rev. Lett ) 1660; Phys. Lett. B ) 347. [6] C.R. Hagen, Phys. Rev. Lett ) 347; X.G. He B.H.J. Mckellar, Phys. Lett. B ) 50. [7] A.G. Aronov Y.B. Lya-Geller, Phys. Rev. Lett ) 343. [8] T.Z. Qian Z.B. Su, Phys. Rev. Lett ) 311. [9] C.-M. Ryu, Phys. Rev. Lett ) 968. [10] L.P. Levy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett ) 04; V. Chreskher, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallager, A. Kleinsasser, Phys. Rev. Lett ) 3578; D. Maily, C. Chapelier, A. Benoit, Phys. Rev. Lett ) 00. [11] D. Loss, P.M. Goldbart, A.V. Balatsky, Phys. Rev. Lett ) 1655; D. Loss P.M. Goldbart, Phys. Rev. B45 199) [1] Y.C. Zhou, H.Z. Li, S.L. Zhu, Phys. Lett. A ) 74; X.C. Gao T.Z. Qian, Phys, Rev. B ) 718. [13] H. Mathur A.D. Stone, Phys. Rev. B ) 10957; Phys. Rev. Lett ) 984. [14] A.V. Balatsky B.L. Altshuler, Phys. Rev. Lett ) [15] Taeseung Choi, Sam Young Cho, Chang-Mo Ryu, Phys. Rev. B ) 485. [16] Shi-Liang Zhu Z.D. Wang, Phys. Rev. Lett ) 1076; Z.D. Wang Shi-Liang Zhu, Phys. Rev. B ) [17] Xin-Qi Li, Li-Xiang Cen, Guo-Xiang Huang, Lei Ma, Yi-Jing Yan, Phys. Rev. A66 00) 0430; Shi-Liang Zhu Z.D. Wang, Phys. Rev. A66 00) 043; Li- Xiang Cen, Xin-Qi Li, Yi-Jing Yan, Hou-Zhi Zheng, Shun-Jin Wang, Phys. Rev. Lett ) [18] Y.C. Zhou, H.Z. Li, X. Xue, Phys. Rev. B ) [19] D.J. Fernez, L.M. Nieto, M.A. del Olmo, M. Santer, J. Phys. A5 199) 515.