Consistency in Formulation of Spin Current and Torque Associated with a Variance of Angular Momentum

Wang, Y.; Xia, Ke; Su, Zhixun; Ma, Zhongshui

doi:10.1103/physrevlett.96.066601

Cited by 52 publications

(41 citation statements)

References 29 publications

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“…[85] A spin current produces a spin accumulation near the sample edges, which in turn causes the sample resistance to decrease by a small amount, whereas an external magnetic field can destroy the edge spin polarization and yield a positive magnetoresistance. An alternative path was followed by Wang et al, [46] who started from the Dirac equation and obtained in the weakly relativistic limit a set of Maxwell equations in the presence of spin-orbit interactions. Although the spin current does not appear explicitly, it is contained in the Maxwell equations and the authors demonstrate that the relativistic conservation laws imply that the spin current yields an electrical polarization, which could be detected directly.…”

Section: Future Directionsmentioning

confidence: 99%

See 1 more Smart Citation

Steady-State Spin Densities and Currents

Culcer

2008

Int. J. Mod. Phys. B

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This article reviews steady-state spin densities and spin currents in materials with strong spinorbit interactions. These phenomena are intimately related to spin precession due to spin-orbit coupling which has no equivalent in the steady state of charge distributions. The focus will be initially on effects originating from the band structure. In this case spin densities arise in an electric field because a component of each spin is conserved during precession. Spin currents arise because a component of each spin is continually precessing. These two phenomena are due to independent contributions to the steady-state density matrix, and scattering between the conserved and precessing spin distributions has important consequences for spin dynamics and spin-related effects in general. In the latter part of the article extrinsic effects such as skew scattering and side jump will be discussed, and it will be shown that these effects are also modified considerably by spin precession. Theoretical and experimental progress in all areas will be reviewed.

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Section: Future Directionsmentioning

confidence: 99%

“…[ [38]- [45]]) and the form of the Maxwell equations in systems with band structure spin-orbit coupling. [46,47] Most theoretical studies have focused on metals, common semiconductors and asymmetric quantum wells with band structure spin-orbit interactions, such as Refs. [[16] - [76]].…”

Section: Introductionmentioning

confidence: 99%

Steady-State Spin Densities and Currents

Culcer

2008

Int. J. Mod. Phys. B

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“…(1) is a 2 × 2 matrix potential ,therefore, the scattering phase associated with it should also be a matrix which would make it a non-abelian phase. In fact the nonabelian nature of scattering phases has been at the root of theoretical problem of defining spin currents and its conservation laws both quantum mechanically [62,63,64,65,66,67,68,69,70] as well within the gauge theoretical formulation [71,72,73]. Before we proceed let us re-look at adiabatic criterion semi-quantitatively.…”

Section: Introductionmentioning

confidence: 99%

Quantum non-Abelian hydrodynamics: Anyonic or spin–orbital entangled liquids, nonunitarity of scattering matrix and charge fractionalization

Pareek

2014

Int. J. Mod. Phys. B

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In this article we develop an exact(non-adiabatic,non-perturbative) density matrix scattering theory for a two component quantum liquid which interacts or scatters off from a generic spin-dependent quantum potential.The generic spin dependent quantum potential[eq. (1)] is a matrix potential, hence, adiabaticity criterion is ill-defined. Therefore the full matrix potential should be treated non-adiabatically. We succeed in doing so using the notion of vectorial matrices which allows us to obtain an exact analytical expression for the scattered density matrix ,̺sc [eq.(30)]. We find that the number or charge density in scattered fluid,Tr(̺sc), expressions in eqs. (32) depends on non-trivial quantum interference coefficients, Q αβ 0ijk which arises due to quantum interference between spin-independent and spin-dependent scattering amplitudes and among spin-dependent scattering amplitudes. Further it is shown that Tr(̺sc) can be expressed in a compact form [eq.(39)] where the effect of quantum interference coefficients can be include using a vector Q αβ which allows us to define a vector order parameter Q. Since the number density is obtained using and exact scattered density matrix, therefore, we do not need to prove that Q is non-zero. However for sake of completeness we make detailed mathematical analysis for the conditions under which the vector order parameter Q would be zero or non-zero.We find that in presence of spin-dependent interaction the vector order parameter Q is necessarily non-zero and is related to the commutator and anti-commutator of scattering matrix S with its dagger S † [eq. (78)]. It is further shown that Q = 0, implies four physically equivalent conditions,i.e, spin-orbital entanglement is non-zero, NonAbelian scattering phase ,i.e, matrices, scattering matrix is NonUnitary and the broken time reversal symmetry for scattered density matrix. This also implies that quasi particle excitation are anyonic in nature, hence, charge fractionalization is a natural consequence. This aspect has also been discussed from the perspective of number or charge density conservation, which implies i.e, Tr(̺sc) = Tr(̺in). On the other hand Q = 0 turns out to be a mathematically forced unphysical solution in presence of spin-dependent potential or scattering which is equivalent 1 to Abelian hydrodynamics ,Unitary scattering matrix, absence of spinspace entanglement, and preserved time reversal symmetry.We have formulated the theory using mesoscopic language, specifically, we have considered two terminal systems connected to spin-dependent scattering region, which is equivalent to having two potential wells separated by a generic spin-dependent potential barrier. The formulation using mesoscopic language is practically useful because it leads directly to the measured quantities such as conductance and spin-polarization density in the leads, however, the presented formulation is not limited to the mesoscopic system only, its generality has been stressed at various places in this article.

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“…Generally speaking, the spin-orbit coupling arises as relativistic quantum effect from the Dirac equation, and describes the interaction of the electron spin, momentum and electromagnetic field. In the system with spin-orbit coupling, the semiclassical equations of electron were studied recently, and some novel effects have been found [6][7][8][9][10][11][12][13][14][15]. However, in the systems with spin-orbit coupling, there remain some questions to be answered.…”

mentioning

confidence: 99%

“…m = ψ †m ψ, and P is defined by P = ψ †P ψ = ψ † ( π 2mc ×m)ψ, it correlates with the electrical polarization induced by the spin-orbit coupling [10]. B ′ = B − (π/2mc) × E presents the total magnetic field which an electron experiences in its local coordinate frame.…”

mentioning

confidence: 99%

Topological force and torque in spin-orbit coupling systems

Huang

et al. 2007

Europhys. Lett.

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The topological force and torque are investigated in the systems with spin-orbit coupling. It is demonstrated that the topological force and torque appears as a pure relativistic quantum effect in an electromagnetic field. The origin of both topological force and torque is the Zitterbewegung effect. Considering nonlinear behaviors of spin-orbit coupling, we address possible novel phenomena driven by the topological forces.PACS numbers: 03.65. Pm, 72.10.Bg, 71.70.Ej Recently, the spin-orbit coupling has become an interesting topic due to the spin Hall effect [1][2][3][4][5]. It provides an efficient route to generate and control quantum spin state electrically. Generally speaking, the spin-orbit coupling arises as relativistic quantum effect from the Dirac equation, and describes the interaction of the electron spin, momentum and electromagnetic field. In the system with spin-orbit coupling, the semiclassical equations of electron were studied recently, and some novel effects have been found [6][7][8][9][10][11][12][13][14][15]. However, in the systems with spin-orbit coupling, there remain some questions to be answered. As well known, the non-relativistic approximation of Dirac equation can be obtained from Foldy-Wouthuysen transformation. In the transformation, there exists a gauge potential in momentum space [15,16], so one question is, what is the position operator in semiclassical equations? Since the position operator is the space-time parameter, its definition is significant. If we are uncapable of defining it correctly, we couldn't have the complete understanding for spin-orbit coupling.In this paper, we investigate the gauge field of position operator in momentum space, and its related effects. Based on the dynamic continuity equation, we derive the quantum force and torque which contain two parts. The conventional part has the same form as in the classic electrodynamics, and the topological part originates from the spin-orbit coupling and other terms from relativistic quantum correction. We notice that the topological part has a close relation to the Zitterbewegung effect. For a two-dimensional system in a magnetic field, we propose that the topological force and torque can reveal more complicated phenomena. Topological velocity The Dirac equation of electron with the wave function Ψ = (ϕ, χ)T reads i ∂ ∂t Ψ = HΨ, with the Hamiltonian H = cα · π + βmc 2 + V (x), where π = P − e c A, α and β are the 4 × 4 Dirac matrices, V (x) = eφ is a scalar potential, and A is a vector potential for a magnetic field, B = ∇ × A. m and e are the electron mass and charge, and c is the speed of light. To reveal the spin-orbit coupling in Dirac equation, we perform the Foldy-Wouthuysen transformation Ψ ′ = U (π)Ψ, where U (π) = e iS , it is a unitary transformation [17,18]. Choosing S = −i(βα · π/2mc), and substituting it into the Dirac equation, we obtain the transformed Hamiltonian(1) The scalar potential becomes V (D) with covariant derivative defined by D = i ∂ p + A, with the pure gauge potential. By defining the c...

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Consistency in Formulation of Spin Current and Torque Associated with a Variance of Angular Momentum

Cited by 52 publications

References 29 publications

Steady-State Spin Densities and Currents

Steady-State Spin Densities and Currents

Quantum non-Abelian hydrodynamics: Anyonic or spin–orbital entangled liquids, nonunitarity of scattering matrix and charge fractionalization

Topological force and torque in spin-orbit coupling systems

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