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arXiv:0905.1596

Post-adiabatic forces and Lagrangians with higher-order derivatives

ARMEN E. ALLAHVERDYAN, Bahar Mehmani

Published May 11, 2009• 1 citation
arXivHTMLPDF

Abstract

We study a slow classical system [particle] coupled to a fast quantum system with discrete energy spectrum. We adiabatically exclude the quantum system and construct an autonomous dynamics for the classical particle in successive orders of the small ratio $ε$ of the characteristic times. It is known that in the order $ε^0$ the particle gets an additional [Born-Oppenheimer] potential, while in the order $\eps^1$ it feels an effective magnetic field related to the Berry phase. In the order $ε^2$ the motion of the classical particle can be reduced to a free [geodesic] motion on a curved Riemannian manifold, with the metric generated by the excluded quantum system. This motion has a number of unusual features, e.g., it combines subspaces of different (Riemannian and pseudo-Riemannian) signature for the metric tensor. In the order $ε^3$ the motion of the classical particle is still described by a Lagrangian, but the latter linearly depends on the particle's acceleration. This implies the existence of a spin tensor [non-orbital angular momentum] for the particle. This spin tensor is related to the momentum via an analogue of the zitterbewegung effect. The Hamiltonian structure of the system is non-trivial and is defined via non-linear Poisson brackets. The linear dependence of the effective classical Lagrangian on higher-order derivatives is seen as well in the higher orders $ε^n$.

DOI
10.48550/arxiv.0905.1596

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Abstract We illustrate the effects of the classical ‘magnetic’ and ‘electric’ geometric forces that enter into the adiabatic description of the slow motion of a heavy system coupled to a light one, beyond the Born–Oppenheimer approximation of simple averaging. When the fast system is a spin S and the slow system is a massive particle whose spatial position R is coupled to S with energy (fast hamiltonian) S · R, the magnetic force is that of a monopole of strength I (= adiabatic invariant S · R/R) centred at R = 0, and the electric force is inverse-cube repulsion with strength S2 – I2. Confining the slow particle to the surface of a sphere eliminates the Born–Oppenheimer and electric forces, and generates motion with precession and nutation exactly equivalent to that of a heavy symmetrical top. In the adiabatic limit the nutation is small and the averaged precession is precisely reproduced by the magnetic force. Alternatively, choosing the exactly conserved total angular momentum to vanish eliminates the Born–Oppenheimer and magnetic forces, and generates as exact orbits a one-parameter family of curly ‘antelope horns’ coiling in from infinity, reversing hand, and receding to infinity. In the adiabatic limit the repulsion of the ‘guiding centre’ of these coils is exactly reproduced by the electric force. A by-product of the ‘antelope horn’ analysis is a determination of the shape of a curve with a given curvature k and torsion T in terms of the evolution of a quantum 2-spinor driven by a planar ‘magnetic field’ with components k and T.
Abstract We study the dynamics of a heavy (slow) classical system coupled, through its position, to a classical or quantal light (fast) system, and derive the first-order velocity-dependent corrections to the lowest adiabatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-order forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force–force correlation function of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the ‘magnetic field’ is the classical limit of the 2-form generating the quantum geometric phase. The symmetric part is deterministic friction, dissipating slow energy into the fast chaos; previously found by Wilkinson, this involves the same correlation function as governs the fluctuations and drift of the adiabatic invariant. In the ‘half-classical’ case where the fast system is quantal with a discrete spectrum of adiabatic states, the only first-order slow force is geometric magnetism; there is no friction. This discordance between classical and quantal fast motion is explained in terms of the clash between the semiclassical and adiabatic limits. A generalization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, the symplectic form in the lowest-order hamiltonian dynamics is modified.
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We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape distortions, magnetization, currents and so on. Expanding their dy-namics near the adiabatic limit we find the emergent Newton’s second law (force is equal to the mass times acceleration) with an extra dissipative term. In systems with broken time reversal symmetry there is an additional Cori-olis type force proportional to the Berry curvature. We give the microscopic definition of the mass tensor relating it to the non-equal time correlation functions in equilibrium or alternatively expressing it through dressing by virtual excitations in the system. In the classical (high-temperature) limit the mass tensor is given by the product of the inverse temperature and the Fubini-Study metric tensor determining the natural distance between the eigenstates of the Hamiltonian. For free particles this result reduces to the conventional definition of mass. This finding shows that any mass, at least in the classical limit, emerges from the distortions of the Hilbert space high-lighting deep connections between any motion (not necessarily in space) and geometry. We illustrate our findings with four simple examples.
The dynamics of observables which are matrices depending on ℏ and taking values in classical phase-space is defined by retaining the terms up to the first order in ℏ of the Moyal bracket. Within this semiclassical approach a first-order Lagrangian involving gauge fields is studied as a constrained Hamiltonian system. This provides a systematic study of spin dynamics in the presence of non-Abelian Berry gauge fields. We applied the method to various types of dynamical spin systems and clarified some persisting discussions. In particular employing the Berry gauge field which generates the Thomas precession, we calculated the force exerted on an electron in the external electric and magnetic fields. Moreover, a simple semiclassical formulation of the spin Hall effect is accomplished.
The adiabatic classical dynamics of certain constrained nonautonomous systems is discussed in terms of the transport of vectors on a surface in parameter space. The classical analogue of Berry's phase is related to the integrated geodesic curvature of a curve on such a surface and the motion of the system analyzed in terms of the relative rotation rates of different vector fields. The classical dependence of the motion on the topology of the parameter space motivates a particular quantum description of a particle confined to a closed space tube.