Mechanical analogues of spin Hamiltonians and dynamics

Physics Letters A 378 (2014) 388–392

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Physics Letters A www.elsevier.com/locate/pla

Mechanical analogues of spin Hamiltonians and dynamics Harjeet Kaur a,∗ , Sudhir R. Jain b , Sham S. Malik a a b

Department of Physics, Guru Nanak Dev University, Amritsar 143 005, India Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India

a r t i c l e

i n f o

Article history: Received 11 September 2013 Received in revised form 25 November 2013 Accepted 26 November 2013 Available online 7 December 2013 Communicated by C.R. Doering Keywords: Spin–orbit coupling Spin Hamiltonian Dynamical system

a b s t r a c t Bloch et al. mapped the precession of the spin-half in a magnetic field of variable magnitude and direction to the rotations of a rigid sphere rolling on a curved surface utilizing SU (2)–SO(3) isomorphism. This formalism is extended to study the behaviour of spin–orbit interactions and the mechanical analogy for Rashba–Dresselhauss spin–orbit interaction in two dimensions is presented by making its spin states isomorphic to the rotations of a rigid sphere rolling on a ring. The change in phase of spin is represented by the angle of rotation of sphere after a complete revolution. In order to develop the mechanical analogy for the spin filter, we find that perfect spin filtration of down spin makes the sphere to rotate at some unique angles and the perfect spin filtration of up spin causes the rotations with certain discrete frequencies. © 2013 Elsevier B.V. All rights reserved.

Spin is a purely quantum mechanical concept with no analogue in classical mechanics. Thus, the description of classical aspects of a quantum mechanical phenomenon like spin–orbit interactions becomes a complex issue. Spin–orbit interactions play an important role in understanding atomic and nuclear structure. Littlejohn and Flynn employed multicomponent wavefunctions to WKB quantization of integrable spherical spin–orbit coupled systems [1]. The semiclassical description breaks down at those points (or subspaces) of classical phase space where the spin–orbit interaction locally becomes zero. However, Frisk and Guhr have investigated spin–orbit interactions in non-spherical potentials to study the shell structure in atomic nuclei and resolved the problem of mode conversion in the case of planar orbits [2]. Bolte and Keppeler derived the relativistic trace formula for Dirac equation by following the technique developed by Gutzwiller for Schrödinger’s equation [3]. By a path integral approach, it was Klauder, who gave the formulation for a system with spin in SU (2) spin coherent state representation as an integral over the sphere S 2 [4]. Subsequently, Kuratsuji et al. represented the path integral in SU (2) spin coherent state as an integral over the paths in extended complex plane C¯ 1 [5,6]. But the exact form of SU (2) coherent state path integral representation for transition amplitude, involving the boundary term and appropriate boundary conditions, was developed by Kochetov [7]. Adding to his formalism, Pletyukhov et al. [8] calculated the ingredients of Gutzwiller’s trace formula for the density of states and tested it for a two-dimensional quantum dot with a spin–orbit interaction of Rashba type.

*

We connect the quantum-mechanical phenomenon of spin– orbit interactions to its classical analogue by utilizing SU (2)− SO(3) isomorphism. Bloch et al. discussed the precession of a spin-half in an external magnetic field by mapping SU (2) spin to SO(3) rotations of a rigid sphere rolling on a curved surface [9]. Their formulation is extended and a spinor undergoing spin–orbit interaction is mapped to rigid sphere rolling on a ring. While deriving the formulations for an analogous picture of the eigenstates of the spin Hamiltonian, no approximations are made. Hence, the exact solutions obtained by this formalism provide a classical description of spin–orbit interactions. The dynamics of this model is also very interesting. The trajectories painted on the sphere rolling on the curved surface are actually the instantaneous measure of the magnetic moment associated with a spin-half particle. In Section 1, we present the basic ideas, following [9], and then we allow ourselves a leap of imagination by exploiting the mathematical similarity of B and L as axial vectors. 1. Motivation Consider a rigid sphere rolling along a curve Γ on a plane. An inertial coordinate system called the spatial coordinate system with its origin at the center of the sphere is fixed. The position and the instantaneous velocity with respect to the center of the sphere of a given particle in the body at time t are denoted by X(t ) ˙ (t ). At each instant t, there exists a unique angular velocity ω and X such that, for every particle in the body,

˙ (t ) = ω × X(t ). X Corresponding author. Tel.: +91 946 5335804; fax: +91 183 225819. E-mail address: [email protected] (H. Kaur).

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.11.046

The translational velocity of the rolling sphere is

(1)

H. Kaur et al. / Physics Letters A 378 (2014) 388–392

Fig. 1. Illustration of directions n (normal vector), t (tangent vector) and u to the curve Γ .

ds

V = tV (t ) = t

dt

,

where s is the arc length of the curve Γ and t is the tangent to the curve Γ [10]. It is well known that the rolling constraint means that the instantaneous velocity at the point of contact is zero i.e.

ω × nR = t

ds

(2)

dt

where R is the radius of the sphere, n is the normal to the surface. Now the expression for ω can be obtained by taking the cross product of expression (2) with n, i.e.

ω=

V (t ) R

n×t=

1 ds R dt

u

(3)

where u = n × t is the tangent normal as described in Fig. 1 and the rolling without instantaneous rotation about the normal, i.e. ω.n = 0 is considered. Rewriting Eq. (1) by using the expression (3) for ω , we obtain

dX

=

ds

u×X R

(4)

.

If we compare this equation with Larmor precession of the magnetic moment in the external magnetic field as it exerts a torque τ on the magnetic moment

τ = −B ×

μ = −B × J, γ

e where γ = 2m is the gyromagnetic ratio, then X = (x, y , z) is identified as magnetic dipole moment and s as time. Eq. (3) describes precession of the angular momentum vector J with frequency

B=−

(u x , u y , u z ) R

= −ω s

of constant magnitude R1 and direction (−u) varying with s. There is an isomorphism between the rolling sphere written in this way with a spin-half precessing in the magnetic field. This can be seen if (using B = −ω s ) (4) is rewritten in the form

d

  x y z

ds



=

0 −B z By

Bz 0 −Bx

−B y Bx 0

  x y z

(5)

,

which is same as the following equations of motion for two complex numbers a and b

i

d

  a b

ds

=−

1



2

Bz Bx + i B y

Bx − i B y −B z

 

a , b

(6)

where (x, y , z) and (a, b) are related via





x = ab∗ + ba∗ ,





y = i ab∗ − ba∗ ,



∗

∗



z = aa − bb , 2

2

2

x + y + z = 1.

389

From Eq. (7), the behaviour of rolling sphere as a function of arc length, which we are taking as equivalent to time, can be determined. The real numbers (x, y , z) represent the coordinates of a point within the sphere at a unit distance from the center in the spatial coordinate system with its origin at the center of the sphere. Thus, the precession of a spin-half in a magnetic field of variable magnitude and direction is mapped to the rotations of a rigid sphere on a curved surface as the (x, y , z) coordinates have already been identified as component of the magnetic moment. Eq. (6) – the equations of motion for the rolling sphere on a curved surface are made equivalent to the Schrödinger’s equation for the spinor χ T = (a, b) in the presence of a magnetic field B because of SU (2)–SO(3) isomorphism,

i

d ds

χ = −B.Sχ

(8)

where h¯ = 1, S = 12 (σx , σ y , σz ) is the spin operator and (σx , σ y , σz ) are Pauli spin matrices [9]. It is well known that the magnetic field is associated with the angular momentum of the particle by the following relation:

1 ∂ V so B=  L, r ∂ r

where L = r × p

is the angular momentum and V so is the potential energy associated with the spin–orbit interactions in the central field. So, Eq. (8) is now written as

i

d ds

χ =−

1 ∂ V so r ∂ r

L.Sχ = −kL.Sχ

where k = r1 ∂∂Vrso  is the strength of spin–orbit interaction and the Hamiltonian of the system undergoing spin–orbit interaction is given as H = −kL.S. Thus, the behaviour of the spin–orbit interactions can be studied by mapping them to the rotations of the sphere rolling on a curved surface. In [9], precessing of spin about a magnetic field is considered whereas we are considering spin interacting with the orbital angular momentum L. Due to fact that mathematics is similar, we succeed, but the thought of extending it thus is non-trivial. Rashba and Dresselhaus spin–orbit interactions are chosen as special case since they play significant role in dephasing of the spin components in spintronic devices. The mechanical analogy of these interaction Hamiltonian is developed by identifying its isomorphism with a rigid sphere rolling on a ring. A device called, spin filter, allows us to choose only one component of the spin. We develop the mechanical analogue for this device and study how the perfect filtration of each component of spin is related to the rotations of the sphere. 2. Rashba and Dresselhaus spin–orbit interaction Hamiltonian In two dimensional III–V semiconductor systems, there are two distinct Hamiltonian terms contributing to spin dephasing – “bulk inversion asymmetry” term and “structure inversion asymmetry” term. These appear only in asymmetric systems. The bulk inversion asymmetry term arises from Dresselhaus spin-splitting while the structure inversion symmetry arises from Rashba spin splitting [11]. The coupling constant in the case of Rashba spin–orbit interaction Hamiltonian is proportional to the external magnetic field but in case of Dresselhaus interaction Hamiltonian, the coupling constant is proportional to the crystal field. The Hamiltonian for the spin–orbit interaction of Rashba type is given as:

H = k[σ y p x − σx p y ] = k

and (7)

(9)







σ .(ˆz × p) = 2k (ˆz × p).S



where k is the spin–orbit coupling strength. Hence, Schrödinger equation for the spinor ψ T = (a, b), taking s as time, is given as:

390

H. Kaur et al. / Physics Letters A 378 (2014) 388–392

Fig. 3. Geometrical interpretation of cone AOC.

˜ =− H

1



2

α

2kp

2kp

−α

 .

The eigenfrequencies (i.e. the frequencies of rotation in rotating frame) of this Hamiltonian come out to be: Fig. 2. Illustration of rigid sphere rolling of radius R on a ring of radius r.

i

λ=

 ∂ ψ = −2k −(ˆz × p).S ψ = −kL.Sψ. ∂s

(10)

Comparing (10) with (9), we identify V so as the potential energy under central field approximation and the angular momentum as

L = −2(ˆz × p), causing spin-half precession and varying in the x– y plane as kL = (2kp cos α s, 2kp sin α s). This angular momentum vector corresponds to u rotating with the same frequency in the same plane, and the rolling problem becomes that of the sphere of radius 1 rolling counterclockwise on a ring of radius r = α1 as R = 2kp shown in Fig. 2. Similarly, the Hamiltonian for the spin–orbit interactions of Dresselhaus type is given as:

H = k[σx p x − σ y p y ] = −k









σ .(ˆz × K) = −2k (ˆz × K).S ,

where K is another vector whose magnitude is the same as p and its components K x = p y and K y = p x . Schrödinger’s equation for the spinor ψ in case of Dresselhaus spin–orbit interactions is given as:

i

On comparing above equation with (9), we identify angular momentum as

L = 2(ˆz × K) causing spin-half precession and is varying in the x– y plane as kL = (2kp cos α s, 2kp sin α s) = (−ωsx , ωsy ), i.e. the angular frequency is rotating with frequency α in a plane. Now, we will proceed further with the development of mechanical analogy of spin–orbit interactions for Rashba type only. Schrödinger’s equation for this is given as:





∂ 1 0 2k exp(−i α s) ψ. i ψ =− 0 ∂s 2 2k exp(i α s)

(11)

  ∂ ˜ 1 α 2kp ˜ ˜ ˜ ψ = Hψ =− i ψ ∂s 2 2kp −α

 exp( i α2 s ) ψ˜ = 0

and

0 exp(− i α2 s )



rR

(13)

.

As the sphere rolls over the ring, the point of contact C moves over the circular rim of the cone AOC and it also paints a circle of diameter AC (see Fig. 2) over the sphere. AC can be found by considering Fig. 3. The right angled  O  CO is similar to right angled  O  MC which implies 

O  OC =  O  CM

r . cos  O  OC = cos  O  CM = √ r2 + R 2 Also, right angled  O  MC is congruent to right angled  O  MA. Thus,

2r R AC = 2MC = 2R cos  O  CM = √ . r2 + R 2 Geometrically, the inverse of the radius of the circle painted on the sphere rolling on the ring represents the eigenfrequencies of the rotation. After a complete revolution over the ring, the angle 2π r of rotation δ for the spin is given as AC /2 , from which we obtain

δ=

2π λ

α

2 r

= 2π 1 +

(14)

.

R

If R  r then δ = 2π Rr which corresponds to rolling of a sphere on a line of length equal to the perimeter of the circle. ˜ is independent of s and the time-dependent solutions of the H ˜ differential equation (12) can be obtained by exponentiation of H. The solutions obtained are exact and no approximations are employed:



α ˜ s) = exp is ψ(

= cos

−α

2kp

2



2kp

λs



2

= exp i σ .m

 ˜ 0) ψ(

+ i σ .m sin λs 2





λs 2

 ˜ 0) ψ(

˜ 0) ψ(

(r ,0, R ) where m =  2 2 , a unit vector representing the axis of a finite r +R

(12)

ψ = exp

2



A transformation to a rotating frame with angular frequency α about z-direction is applied and (11) for the spinor ψ˜ = (˜a, b˜ ) is rewritten as:

r2 + R 2

+α =

4k2 p 2

 ∂ ψ = −2k (ˆz × K).S ψ = −kL .Sψ. ∂s

where

√



i σz α s 2

 ψ

rotation by an angle λ2s . The rotational frequencies of the rolling

˜ The problem are twice as the eigenfrequencies of Hamiltonian H. factor of half rises naturally in mapping to a spin problem as described in (6). These eigenfunctions are in a frame rotating with angular velocity α about the z-direction. By a unitary transformation, and substituting the eigenvalue λ, the eigenfunction ψ is obtained:

H. Kaur et al. / Physics Letters A 378 (2014) 388–392

 ψ(s) =

exp(− i α2 s )



0 exp( i α2 s )

0

exp i σ .m

λs



 αs λs ˜ ψ(0). exp i σ .m = exp −i σz 2



2

˜ 0) ψ(

Considering positive eigenvalue only (the case when spin is aligned with the angular momentum) gives ψβ e −i α s = ψα . If ψα is taken as unity, then ψβ = e i α s . Normalising, we obtain

2







where



λs

A = cos and



−i α s

a(s) e 2 ( Aa(0) + Bb(0)) = iα s b (s) e 2 ( Ba(0) + A ∗ b(0))

2



R

+i√

r2 + R 2

r

λs

sin

λs

(15)

2

x(s) =

1−

2α

2



sin2 λ2s λ2

+

2α 2kp sin2

λ2



y (s) =

2

1−

2α sin

2

cos α s + α



2



2

λ2

z(s) = 2 × 2kp

2α



+

1−

sin2 λ2s λ2



sin λs

λ

sin λs

λ 

x(0) − 2kp



λ

 

(19)

.





1 a(0) = . 1 b(0)

2

 ψ(s) =



1 a(s) =√ b (s) 2

1−



e e

 ( A + B) . (B + A∗)

−i α s 2

iα s 2

 2α 2 sin2 λ2s



sin α s z(0),

y (s) =

1−

 z(s) = 2 × 2kp

λ2

cos α s + α

 2α 2 sin2 λ2s λ2

α

sin2 λ2s λ2

sin α s − α

sin λs

λ sin λs

λ

 sin α s ,

 cos α s , (20)

.

For the case when spin is anti-aligned with the angular momentum vector, (x(s), y (s), z(s)) coordinates are



λ sin λs



(a(0), b(0)) = ( √1 , √1 ), the corresponding (x(0), y (0), 2 2 z(0)) = (1, 0, 0). Substituting these values in (15) and (16), we have (a(s), b(s)):



cos α s x(0)

2kp sin λs

−i α s 2 iα s 2

state, so

x(s) =

sin α s y (0)

sin α s +

e



2kp sin λs

−e

With

 

(18)

.

The (x(s), y (s), z(s)) coordinates for the spin aligned with the angular momentum vector are

sin α s x(0)

λ

sin α s − α

 2α 2 sin2 λ2s λ2

λ



cos α s y (0)

λs 

 2α 2kp sin2 λ2s 2

λ

sin λs

cos α s −

+ cos λs cos α s + α +

sin λs

λs 

λ2

1 2

1

+ − cos λs sin α s + α 



2

χ ↑ (0) = √





χ ↑ (0) = ψ(0), i.e. the system is initially quantized in

Consider m=

.

−i α s

e 2 iα s e 2

Similarly, considering the negative eigenvalue only, we get the eigenspinor for the case when spin is anti-aligned with the angular momentum vector, i.e.

χ ↓ (s) = √

˜ depends on s, then the matrices H ˜ at different s do not If H commute. It is difficult to have the solutions as time ordered exponentials are involved. The corresponding (x(s), y (s), z(s)) coordinates, which we have identified as the magnetic moment components, are obtained by putting the value of a(s) and b(s) in (7), entailing



2

1





B = i√ sin 2 r2 + R 2



1

χ ↑ (s) = √

In the component form, ψ(s) is found to be

ψ(s) =

391



 cos α s z(0),

y (0)

z(0).

(16)

x(s) = −

1−

 2α 2 sin2 λ2s

λ2 

2 2α sin2 y (s) = − 1 − λ2

z(s) = −2 × 2kp

α

sin2 λ2s λ2

λs 2

cos α s + α

 sin α s − α

sin λs

λ sin λs

λ

.

 sin α s ,

 cos α s , (21)

These equations describe the mechanical analogue of the Rashba– Dresselhaus spin–orbit interaction. Mapping the spinor to a sphere rolling on the ring and then following the trajectories painted on the sphere as a function of arc length s leads us to obtain the analogous time-dependent picture of the spin-half interacting with an orbital angular momentum.

Interestingly, the coordinates when spin is aligned in the direction of angular momentum vector come out to be exactly negative of coordinates in the case when spin is aligned against the direction of the angular momentum.

3. Special case for the spin-aligned and anti-aligned – with the angular momentum

Hatano et al. [12] utilized the non-abelian gauge theory for spin–orbit interactions to set the parameters such that they manage to create a phase factor of opposite signs for both the up and down spins. The interference is completely constructive for one component and completely destructive for the other component of spin-half. Thus, they are able to filter out only one component spin and proposed the construction of a device called spin filter. Another spin-filter configuration has been presented in [13]. We can think of spin filtration by finding the probability for having the system in one particular state. Denoted by P the probability of finding the system in χ ↓ state which is given by:

Consider any direction uB such that when uB .σ op  arbitrary erates on

χ=

ψα , it gives the eigenvalues ±1, i.e. we restrict ψβ

ourselves only in the directions parallel and antiparallel with the angular momentum vector. So,

 (uB .σ )

      0 e −i α s ψα ψα ψα = iα s =± . ψβ e 0 ψβ ψβ

(17)

4. Mechanical analogue of a perfect spin filter

392

H. Kaur et al. / Physics Letters A 378 (2014) 388–392



P =

2



χ ↓ (s)∗ ψ(s)

 1  −i α s =  −e 2 2

e

5. Conclusions

 e −i2α s ( A + B ) 2  iα s  e 2 (B + A∗) 

iα s 2

  1 2 ∗   =  −A + A  . 2

Substituting the value of A, we get probability as:

P=

R2 r2 + R 2

sin2

λs 2

= α2

sin2 λ2s

λ2

.

(22)

Of course, probability takes values only between zero and one. For perfect spin filtration of the down spin, (22) should be equal to one. Thus,

λ

α

= ± sin

λs 2

 .

(23)

Also, we find that the probability of non-adiabatic transitions depends on the angle of rotation δ of the rolling sphere which is given as δ = 2π αλ . Putting this expression in (23), we get the angle of rotation of the sphere as

δ = ±2π sin

λs 2

 ,

(24)

i.e., perfect spin filtration of down spin is mechanically evolved with the rotations of the sphere rolling on ring at some unique angles given by (24). For perfect spin filtration of the up spin, Eq. (22) should be equal to zero. Thus,

sin

λs 2

 =0

or λs = 2nπ , n = 0, 1, 2, 3 . . . . So, perfect spin filtration of the up spin fixes the eigenfrequencies of rotation of the sphere rolling on the ring as 2nsπ . This is how the perfect spin filtration of each component of the spin affects the motion of the sphere rolling on a ring.

Spin-half experiencing spin–orbit interaction is studied by making it isomorphic to the rotations of a rigid sphere rolling on a curved surface. Because of SU(2)–SO(3) isomorphism, a sphere rolling on a ring is the mechanical analogue for Rashba, Dresselhaus Hamiltonians. The trajectories painted on the sphere presents the analogous picture of the eigenfunctions. These interactions are responsible for spin-dephasing in III–V semiconductors and play a crucial role in the filtration of only one component of the spin-half. Perfect spin filtration of down-component of spin-half is mechanically evolved with the rotation of sphere rolling on a ring at some unique angles whereas the perfect filtration of upcomponent makes the sphere to rotate with certain discrete angular frequencies. This is quite non-trivial and unexpected. As briefly explained in the Introduction, there is a lot of interest in developing the semiclassical understanding of Hamiltonians with spin–orbit interaction. We believe that development of classical ideas for such Hamiltonians will be helpful in this pursuit. Acknowledgements One of us, H.K. is thankful to the Council of Scientific and Industrial Research, India for supporting this work, and, B.A.R.C. for the hospitality provided during her visit. The discussions with Shashi C.L. Srivastava are gratefully acknowledged. S.S.M. acknowledges financial support from D.A.E., India. References [1] R.G. Littlejohn, W.G. Flynn, Phys. Rev. A 44 (1991) 5239; R.G. Littlejohn, W.G. Flynn, Phys. Rev. A 45 (1992) 7697. [2] H. Frisk, T. Guhr, Ann. Phys. 221 (1993) 229. [3] J. Bolte, S. Keppeler, Phys. Rev. Lett. 81 (1998) 1987. [4] J.R. Klauder, Phys. Rev. D, Part. Fields 19 (1979) 2349. [5] H. Kuratsuji, T. Suzuki, J. Math. Phys. 21 (1980) 472. [6] H. Kuratsuji, Y. Mizobuchi, J. Math. Phys. 22 (1981) 757. [7] E.A. Kochetov, J. Math. Phys. 36 (1995) 4667. [8] M. Pletyukhov, Ch. Amann, M. Mehta, M. Brack, Phys. Rev. Lett. 89 (2005) 116601. [9] A.G. Rojo, A.M. Bloch, Am. J. Phys. 78 (2010) 1014. [10] D.D. Holm, T. Schmah, C. Stoica, D.C.P. Ellis, Geometry Mechanics and Symmetry (From Finite to Infinite Dimensions), Oxford University Press, 2009. ` J. Fabian, S. Das Sharma, Rev. Mod. Phys. 76 (2004) 323. [11] I. Žutic, [12] N. Hatano, R. Shirasaki, H. Nakamura, Phys. Rev. A 75 (2007) 032107. [13] B. Basu, D. Chowdhury, Ann. Phys. 335 (2013) 47.